diff --git a/test/bench/Basic.lean.leanInk.expected b/test/bench/Basic.lean.leanInk.expected
index de32329..baad71b 100644
--- a/test/bench/Basic.lean.leanInk.expected
+++ b/test/bench/Basic.lean.leanInk.expected
@@ -45,14 +45,14 @@
     "raw": "induction",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": {"type": "Nat", "name": "a", "_type": "typeinfo"},
     "semanticType": "Name.Variable",
@@ -65,21 +65,21 @@
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": "Keyword",
     "raw": "with",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -132,7 +132,7 @@
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -843,7 +843,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -980,7 +980,7 @@
     "The `simp` tactic uses lemmas and hypotheses to simplify the main goal target or\nnon-dependent hypotheses. It has many variants:\n- `simp` simplifies the main goal target using lemmas tagged with the attribute `[simp]`.\n- `simp [h₁, h₂, ..., hₙ]` simplifies the main goal target using the lemmas tagged\n  with the attribute `[simp]` and the given `hᵢ`'s, where the `hᵢ`'s are expressions.\n  If an `hᵢ` is a defined constant `f`, then the equational lemmas associated with\n  `f` are used. This provides a convenient way to unfold `f`.\n- `simp [*]` simplifies the main goal target using the lemmas tagged with the\n  attribute `[simp]` and all hypotheses.\n- `simp only [h₁, h₂, ..., hₙ]` is like `simp [h₁, h₂, ..., hₙ]` but does not use `[simp]` lemmas.\n- `simp [-id₁, ..., -idₙ]` simplifies the main goal target using the lemmas tagged\n  with the attribute `[simp]`, but removes the ones named `idᵢ`.\n- `simp at h₁ h₂ ... hₙ` simplifies the hypotheses `h₁ : T₁` ... `hₙ : Tₙ`. If\n  the target or another hypothesis depends on `hᵢ`, a new simplified hypothesis\n  `hᵢ` is introduced, but the old one remains in the local context.\n- `simp at *` simplifies all the hypotheses and the target.\n- `simp [*] at *` simplifies target and all (propositional) hypotheses using the\n  other hypotheses.\n",
     "_type": "token"},
    {"typeinfo":
-    {"type": "{α : Type ?u.2394} → [self : BEq α] → α → α → Bool",
+    {"type": "{α : Type ?u.2382} → [self : BEq α] → α → α → Bool",
      "name": "BEq.beq",
      "_type": "typeinfo"},
     "semanticType": null,
@@ -1071,7 +1071,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -1414,7 +1414,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -2134,7 +2134,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -2196,7 +2196,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -2328,7 +2328,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -2428,7 +2428,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -2584,7 +2584,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -2672,7 +2672,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -2724,7 +2724,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -2787,7 +2787,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -2839,7 +2839,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -2902,7 +2902,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -2978,7 +2978,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -3632,7 +3632,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -3730,7 +3730,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -3947,7 +3947,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -4338,7 +4338,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -4729,7 +4729,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -4860,14 +4860,14 @@
     "raw": "induction",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": {"type": "Nat", "name": "n", "_type": "typeinfo"},
     "semanticType": "Name.Variable",
@@ -4880,21 +4880,21 @@
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": "Keyword",
     "raw": "with",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -4989,7 +4989,7 @@
     "raw": "intros",
     "link": null,
     "docstring":
-    "`intros x...` behaves like `intro x...`, but then keeps introducing (anonymous)\nhypotheses until goal is not of a function type.\n",
+    "Introduces zero or more hypotheses, optionally naming them.\n\n- `intros` is equivalent to repeatedly applying `intro`\n  until the goal is not an obvious candidate for `intro`, which is to say\n  that so long as the goal is a `let` or a pi type (e.g. an implication, function, or universal quantifier),\n  the `intros` tactic will introduce an anonymous hypothesis.\n  This tactic does not unfold definitions.\n\n- `intros x y ...` is equivalent to `intro x y ...`,\n  introducing hypotheses for each supplied argument and unfolding definitions as necessary.\n  Each argument can be either an identifier or a `_`.\n  An identifier indicates a name to use for the corresponding introduced hypothesis,\n  and a `_` indicates that the hypotheses should be introduced anonymously.\n\n## Examples\n\nBasic properties:\n```lean\ndef AllEven (f : Nat → Nat) := ∀ n, f n % 2 = 0\n\n-- Introduces the two obvious hypotheses automatically\nexample : ∀ (f : Nat → Nat), AllEven f → AllEven (fun k => f (k + 1)) := by\n  intros\n  /- Tactic state\n     f✝ : Nat → Nat\n     a✝ : AllEven f✝\n     ⊢ AllEven fun k => f✝ (k + 1) -/\n  sorry\n\n-- Introduces exactly two hypotheses, naming only the first\nexample : ∀ (f : Nat → Nat), AllEven f → AllEven (fun k => f (k + 1)) := by\n  intros g _\n  /- Tactic state\n     g : Nat → Nat\n     a✝ : AllEven g\n     ⊢ AllEven fun k => g (k + 1) -/\n  sorry\n\n-- Introduces exactly three hypotheses, which requires unfolding `AllEven`\nexample : ∀ (f : Nat → Nat), AllEven f → AllEven (fun k => f (k + 1)) := by\n  intros f h n\n  /- Tactic state\n     f : Nat → Nat\n     h : AllEven f\n     n : Nat\n     ⊢ (fun k => f (k + 1)) n % 2 = 0 -/\n  apply h\n```\n\nImplications:\n```lean\nexample (p q : Prop) : p → q → p := by\n  intros\n  /- Tactic state\n     a✝¹ : p\n     a✝ : q\n     ⊢ p      -/\n  assumption\n```\n\nLet bindings:\n```lean\nexample : let n := 1; let k := 2; n + k = 3 := by\n  intros\n  /- n✝ : Nat := 1\n     k✝ : Nat := 2\n     ⊢ n✝ + k✝ = 3 -/\n  rfl\n```\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -5056,7 +5056,7 @@
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -5434,7 +5434,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -5907,14 +5907,14 @@
     "raw": "induction",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": {"type": "Nat", "name": "m", "_type": "typeinfo"},
     "semanticType": "Name.Variable",
@@ -5927,21 +5927,21 @@
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": "Keyword",
     "raw": "with",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -6002,7 +6002,7 @@
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -6448,7 +6448,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -6620,7 +6620,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -6836,7 +6836,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -6910,7 +6910,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -7035,7 +7035,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -7225,7 +7225,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -7368,14 +7368,14 @@
     "raw": "induction",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
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-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
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     "semanticType": "Name.Variable",
@@ -7388,21 +7388,21 @@
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": "Keyword",
     "raw": "generalizing",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
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     "semanticType": null,
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-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": {"type": "Nat", "name": "m", "_type": "typeinfo"},
     "semanticType": "Name.Variable",
@@ -7415,7 +7415,7 @@
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": {"type": "Nat", "name": "k", "_type": "typeinfo"},
     "semanticType": "Name.Variable",
@@ -7428,21 +7428,21 @@
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": "Keyword",
     "raw": "with",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -7570,7 +7570,7 @@
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -8135,7 +8135,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -8538,7 +8538,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -8761,7 +8761,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -8984,7 +8984,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -9106,7 +9106,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -9394,7 +9394,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -9745,14 +9745,14 @@
     "raw": "induction",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": {"type": "Nat", "name": "m", "_type": "typeinfo"},
     "semanticType": "Name.Variable",
@@ -9765,21 +9765,21 @@
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": "Keyword",
     "raw": "with",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -9839,7 +9839,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -9855,7 +9855,7 @@
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -10028,7 +10028,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -10183,7 +10183,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -10299,7 +10299,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -10426,7 +10426,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -10515,14 +10515,14 @@
     "raw": "induction",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": {"type": "Nat", "name": "m", "_type": "typeinfo"},
     "semanticType": "Name.Variable",
@@ -10535,21 +10535,21 @@
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": "Keyword",
     "raw": "with",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -10657,7 +10657,7 @@
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -10864,7 +10864,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -11386,7 +11386,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -11474,7 +11474,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -11538,7 +11538,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -11725,7 +11725,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12118,7 +12118,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12293,7 +12293,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12471,7 +12471,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12484,7 +12484,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12509,7 +12509,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12522,7 +12522,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12547,7 +12547,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12560,7 +12560,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12658,7 +12658,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12671,7 +12671,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12696,7 +12696,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12709,7 +12709,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12734,7 +12734,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12747,7 +12747,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12845,7 +12845,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12858,7 +12858,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12883,7 +12883,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12896,7 +12896,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12921,7 +12921,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -12934,7 +12934,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -13032,7 +13032,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -13045,7 +13045,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -13070,7 +13070,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -13083,7 +13083,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -13108,7 +13108,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -13121,7 +13121,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -13249,7 +13249,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -13410,7 +13410,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -13611,7 +13611,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -13762,7 +13762,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -13890,7 +13890,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -13980,7 +13980,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -14085,7 +14085,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -14212,7 +14212,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -14350,7 +14350,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -14640,7 +14640,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -14816,7 +14816,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -14956,7 +14956,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -15109,7 +15109,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -15258,7 +15258,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -15407,7 +15407,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -15696,7 +15696,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -16107,7 +16107,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -16322,7 +16322,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -16883,7 +16883,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -17110,7 +17110,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -17288,7 +17288,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -17440,7 +17440,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -17569,7 +17569,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -18142,7 +18142,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -18218,7 +18218,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -18417,7 +18417,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -18985,7 +18985,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -19280,7 +19280,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -19555,7 +19555,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -19568,7 +19568,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -19712,7 +19712,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -19725,7 +19725,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -19931,7 +19931,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -19992,7 +19992,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -20485,7 +20485,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -20546,7 +20546,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -20956,7 +20956,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -21017,7 +21017,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -21249,7 +21249,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -21310,7 +21310,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -21548,7 +21548,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -21663,7 +21663,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -21964,7 +21964,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -22265,7 +22265,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -22960,7 +22960,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -23960,7 +23960,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -24139,7 +24139,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -24240,14 +24240,14 @@
     "raw": "induction",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": {"type": "Nat", "name": "a", "_type": "typeinfo"},
     "semanticType": "Name.Variable",
@@ -24260,21 +24260,21 @@
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": "Keyword",
     "raw": "with",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -24335,7 +24335,7 @@
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -24627,7 +24627,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -24898,7 +24898,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -25547,14 +25547,14 @@
     "raw": "induction",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": " ",
     "link": null,
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-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": {"type": "Nat", "name": "a", "_type": "typeinfo"},
     "semanticType": "Name.Variable",
@@ -25567,21 +25567,21 @@
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": "Keyword",
     "raw": "with",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -25644,7 +25644,7 @@
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -26626,7 +26626,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -27105,7 +27105,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -27246,14 +27246,14 @@
     "raw": "induction",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": {"type": "Nat", "name": "a", "_type": "typeinfo"},
     "semanticType": "Name.Variable",
@@ -27266,21 +27266,21 @@
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": "Keyword",
     "raw": "with",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -27342,7 +27342,7 @@
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -27461,7 +27461,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -28169,7 +28169,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -28308,14 +28308,14 @@
     "raw": "induction",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": {"type": "Nat", "name": "a", "_type": "typeinfo"},
     "semanticType": "Name.Variable",
@@ -28328,21 +28328,21 @@
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": "Keyword",
     "raw": "with",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -28405,7 +28405,7 @@
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -29092,7 +29092,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -29418,7 +29418,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -29549,14 +29549,14 @@
     "raw": "induction",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": {"type": "Nat", "name": "k", "_type": "typeinfo"},
     "semanticType": "Name.Variable",
@@ -29569,21 +29569,21 @@
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": "Keyword",
     "raw": "with",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -29647,7 +29647,7 @@
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -29885,7 +29885,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -30283,7 +30283,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -30798,7 +30798,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -31142,7 +31142,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -31203,7 +31203,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -31810,7 +31810,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -32206,7 +32206,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -32519,7 +32519,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -33736,7 +33736,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -33825,14 +33825,14 @@
     "raw": "induction",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": {"type": "Nat", "name": "n", "_type": "typeinfo"},
     "semanticType": "Name.Variable",
@@ -33845,21 +33845,21 @@
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": "Keyword",
     "raw": "with",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -33919,7 +33919,7 @@
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -34130,7 +34130,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -34487,7 +34487,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -34917,7 +34917,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -36096,7 +36096,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -36813,7 +36813,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -37669,7 +37669,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -38627,7 +38627,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -39053,7 +39053,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -39129,7 +39129,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -39193,7 +39193,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -39246,7 +39246,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -39295,7 +39295,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -39359,7 +39359,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -40218,7 +40218,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -40686,7 +40686,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -41024,7 +41024,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -41155,14 +41155,14 @@
     "raw": "induction",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": {"type": "Nat", "name": "k", "_type": "typeinfo"},
     "semanticType": "Name.Variable",
@@ -41175,21 +41175,21 @@
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": "Keyword",
     "raw": "with",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -41253,7 +41253,7 @@
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -41640,7 +41640,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -41783,14 +41783,14 @@
     "raw": "induction",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": " ",
     "link": null,
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-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": {"type": "Nat", "name": "m", "_type": "typeinfo"},
     "semanticType": "Name.Variable",
@@ -41803,21 +41803,21 @@
     "raw": " ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": "Keyword",
     "raw": "with",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -41881,7 +41881,7 @@
     "raw": "\n  ",
     "link": null,
     "docstring":
-    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a theorem whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
+    "Assuming `x` is a variable in the local context with an inductive type,\n`induction x` applies induction on `x` to the main goal,\nproducing one goal for each constructor of the inductive type,\nin which the target is replaced by a general instance of that constructor\nand an inductive hypothesis is added for each recursive argument to the constructor.\nIf the type of an element in the local context depends on `x`,\nthat element is reverted and reintroduced afterward,\nso that the inductive hypothesis incorporates that hypothesis as well.\n\nFor example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,\n`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,\nand one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.\nHere the names `a` and `ih₁` are chosen automatically and are not accessible.\nYou can use `with` to provide the variables names for each constructor.\n- `induction e`, where `e` is an expression instead of a variable,\n  generalizes `e` in the goal, and then performs induction on the resulting variable.\n- `induction e using r` allows the user to specify the principle of induction that should be used.\n  Here `r` should be a term whose result type must be of the form `C t`,\n  where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables\n- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,\n  generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.\n  In other words, the net effect is that each inductive hypothesis is generalized.\n- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`\n  uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.\n",
     "_type": "token"}],
   "_type": "sentence"},
  {"messages": [],
@@ -42375,7 +42375,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
diff --git a/test/bugs/GH_15.lean.leanInk.expected b/test/bugs/GH_15.lean.leanInk.expected
index bb09453..d24e574 100644
--- a/test/bugs/GH_15.lean.leanInk.expected
+++ b/test/bugs/GH_15.lean.leanInk.expected
@@ -16,7 +16,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -29,7 +29,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
diff --git a/test/playground/Function.lean.leanInk.expected b/test/playground/Function.lean.leanInk.expected
index 7d9d4b5..4805953 100644
--- a/test/playground/Function.lean.leanInk.expected
+++ b/test/playground/Function.lean.leanInk.expected
@@ -1352,7 +1352,7 @@
     "raw": "id",
     "link": null,
     "docstring":
-    "The identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n",
+    "The identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and Lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -1378,7 +1378,7 @@
     "raw": "id",
     "link": null,
     "docstring":
-    "The identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n",
+    "The identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and Lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -1433,7 +1433,7 @@
     "raw": "id",
     "link": null,
     "docstring":
-    "The identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n",
+    "The identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and Lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -1485,7 +1485,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -1540,7 +1540,7 @@
     "raw": "id",
     "link": null,
     "docstring":
-    "The identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n",
+    "The identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and Lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -2215,7 +2215,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -2434,7 +2434,7 @@
     "raw": "rfl",
     "link": null,
     "docstring":
-    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nlean by `rfl`, because both sides are the same up to definitional equality.\n",
+    "`rfl : a = a` is the unique constructor of the equality type. This is the\nsame as `Eq.refl` except that it takes `a` implicitly instead of explicitly.\n\nThis is a more powerful theorem than it may appear at first, because although\nthe statement of the theorem is `a = a`, Lean will allow anything that is\ndefinitionally equal to that type. So, for instance, `2 + 2 = 4` is proven in\nLean by `rfl`, because both sides are the same up to definitional equality.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -2636,7 +2636,7 @@
     "raw": "id",
     "link": null,
     "docstring":
-    "The identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n",
+    "The identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and Lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -2867,7 +2867,7 @@
     "raw": "id",
     "link": null,
     "docstring":
-    "The identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n",
+    "The identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and Lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
diff --git a/test/playground/Simple.lean.leanInk.expected b/test/playground/Simple.lean.leanInk.expected
index e9884ee..251f81c 100644
--- a/test/playground/Simple.lean.leanInk.expected
+++ b/test/playground/Simple.lean.leanInk.expected
@@ -52,7 +52,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
diff --git a/test/playground/infoTree.lean.leanInk.expected b/test/playground/infoTree.lean.leanInk.expected
index 7170d97..d3d51e8 100644
--- a/test/playground/infoTree.lean.leanInk.expected
+++ b/test/playground/infoTree.lean.leanInk.expected
@@ -70,7 +70,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -83,7 +83,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -204,7 +204,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -217,7 +217,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -230,7 +230,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -292,7 +292,7 @@
     "raw": "id",
     "link": null,
     "docstring":
-    "The identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n",
+    "The identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and Lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -368,7 +368,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -610,7 +610,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -648,7 +648,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -913,7 +913,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -926,7 +926,7 @@
     "raw": "Array",
     "link": null,
     "docstring":
-    "`Array α` is the type of [dynamic arrays](https://en.wikipedia.org/wiki/Dynamic_array)\nwith elements from `α`. This type has special support in the runtime.\n\nAn array has a size and a capacity; the size is `Array.size` but the capacity\nis not observable from lean code. Arrays perform best when unshared; as long\nas they are used \"linearly\" all updates will be performed destructively on the\narray, so it has comparable performance to mutable arrays in imperative\nprogramming languages.\n\nFrom the point of view of proofs `Array α` is just a wrapper around `List α`.\n",
+    "`Array α` is the type of [dynamic arrays](https://en.wikipedia.org/wiki/Dynamic_array)\nwith elements from `α`. This type has special support in the runtime.\n\nAn array has a size and a capacity; the size is `Array.size` but the capacity\nis not observable from Lean code. Arrays perform best when unshared; as long\nas they are used \"linearly\" all updates will be performed destructively on the\narray, so it has comparable performance to mutable arrays in imperative\nprogramming languages.\n\nFrom the point of view of proofs `Array α` is just a wrapper around `List α`.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -939,7 +939,7 @@
     "raw": "Array",
     "link": null,
     "docstring":
-    "`Array α` is the type of [dynamic arrays](https://en.wikipedia.org/wiki/Dynamic_array)\nwith elements from `α`. This type has special support in the runtime.\n\nAn array has a size and a capacity; the size is `Array.size` but the capacity\nis not observable from lean code. Arrays perform best when unshared; as long\nas they are used \"linearly\" all updates will be performed destructively on the\narray, so it has comparable performance to mutable arrays in imperative\nprogramming languages.\n\nFrom the point of view of proofs `Array α` is just a wrapper around `List α`.\n",
+    "`Array α` is the type of [dynamic arrays](https://en.wikipedia.org/wiki/Dynamic_array)\nwith elements from `α`. This type has special support in the runtime.\n\nAn array has a size and a capacity; the size is `Array.size` but the capacity\nis not observable from Lean code. Arrays perform best when unshared; as long\nas they are used \"linearly\" all updates will be performed destructively on the\narray, so it has comparable performance to mutable arrays in imperative\nprogramming languages.\n\nFrom the point of view of proofs `Array α` is just a wrapper around `List α`.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -952,7 +952,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -965,7 +965,7 @@
     "raw": "Array",
     "link": null,
     "docstring":
-    "`Array α` is the type of [dynamic arrays](https://en.wikipedia.org/wiki/Dynamic_array)\nwith elements from `α`. This type has special support in the runtime.\n\nAn array has a size and a capacity; the size is `Array.size` but the capacity\nis not observable from lean code. Arrays perform best when unshared; as long\nas they are used \"linearly\" all updates will be performed destructively on the\narray, so it has comparable performance to mutable arrays in imperative\nprogramming languages.\n\nFrom the point of view of proofs `Array α` is just a wrapper around `List α`.\n",
+    "`Array α` is the type of [dynamic arrays](https://en.wikipedia.org/wiki/Dynamic_array)\nwith elements from `α`. This type has special support in the runtime.\n\nAn array has a size and a capacity; the size is `Array.size` but the capacity\nis not observable from Lean code. Arrays perform best when unshared; as long\nas they are used \"linearly\" all updates will be performed destructively on the\narray, so it has comparable performance to mutable arrays in imperative\nprogramming languages.\n\nFrom the point of view of proofs `Array α` is just a wrapper around `List α`.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -978,7 +978,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -1136,7 +1136,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -1255,7 +1255,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -1281,7 +1281,7 @@
     "raw": "id",
     "link": null,
     "docstring":
-    "The identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n",
+    "The identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and Lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -1295,7 +1295,7 @@
     "raw": "id",
     "link": null,
     "docstring":
-    "The identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n",
+    "The identity function. `id` takes an implicit argument `α : Sort u`\n(a type in any universe), and an argument `a : α`, and returns `a`.\n\nAlthough this may look like a useless function, one application of the identity\nfunction is to explicitly put a type on an expression. If `e` has type `T`,\nand `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and Lean\nknows that this expression has type `T'` rather than `T`. This can make a\ndifference for typeclass inference, since `T` and `T'` may have different\ntypeclass instances on them. `show T' from e` is sugar for an `@id T' e`\nexpression.\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,
@@ -1391,7 +1391,7 @@
     "raw": "Nat",
     "link": null,
     "docstring":
-    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
+    "The type of natural numbers, starting at zero. It is defined as an\ninductive type freely generated by \"zero is a natural number\" and\n\"the successor of a natural number is a natural number\".\n\nYou can prove a theorem `P n` about `n : Nat` by `induction n`, which will\nexpect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming\na proof of `P i`. The same method also works to define functions by recursion\non natural numbers: induction and recursion are two expressions of the same\noperation from Lean's point of view.\n\n```\nopen Nat\nexample (n : Nat) : n < succ n := by\n  induction n with\n  | zero =>\n    show 0 < 1\n    decide\n  | succ i ih => -- ih : i < succ i\n    show succ i < succ (succ i)\n    exact Nat.succ_lt_succ ih\n```\n\nThis type is special-cased by both the kernel and the compiler:\n* The type of expressions contains \"`Nat` literals\" as a primitive constructor,\n  and the kernel knows how to reduce zero/succ expressions to nat literals.\n* If implemented naively, this type would represent a numeral `n` in unary as a\n  linked list with `n` links, which is horribly inefficient. Instead, the\n  runtime itself has a special representation for `Nat` which stores numbers up\n  to 2^63 directly and larger numbers use an arbitrary precision \"bignum\"\n  library (usually [GMP](https://gmplib.org/)).\n",
     "_type": "token"},
    {"typeinfo": null,
     "semanticType": null,