feat: omega handles shift operators, and normalises ground term expon…

…entials (#3433)

This is a preliminary to a BitVec frontend for `omega`.

src/Init/Omega/Int.lean

@@ -5,6 +5,8 @@ Authors: Scott Morrison
 -/
 prelude
 import Init.Data.Int.Order
+import Init.Data.Int.DivModLemmas
+import Init.Data.Nat.Lemmas
 
 /-!
 # Lemmas about `Nat` and `Int` needed internally by `omega`.
@@ -43,6 +45,12 @@ theorem ofNat_lt_of_lt {x y : Nat} (h : x < y) : (x : Int) < (y : Int) :=
 theorem ofNat_le_of_le {x y : Nat} (h : x ≤ y) : (x : Int) ≤ (y : Int) :=
   Int.ofNat_le.mpr h
 
+theorem ofNat_shiftLeft_eq {x y : Nat} : (x <<< y : Int) = (x : Int) * (2 ^ y : Nat) := by
+  simp [Nat.shiftLeft_eq]
+
+theorem ofNat_shiftRight_eq_div_pow {x y : Nat} : (x >>> y : Int) = (x : Int) / (2 ^ y : Nat) := by
+  simp [Nat.shiftRight_eq_div_pow]
+
 -- FIXME these are insane:
 theorem lt_of_not_ge {x y : Int} (h : ¬ (x ≤ y)) : y < x := Int.not_le.mp h
 theorem lt_of_not_le {x y : Int} (h : ¬ (x ≤ y)) : y < x := Int.not_le.mp h

src/Lean/Elab/Tactic/Omega/Frontend.lean

@@ -24,6 +24,24 @@ Allow elaboration of `OmegaConfig` arguments to tactics.
 declare_config_elab elabOmegaConfig Lean.Meta.Omega.OmegaConfig
 
 
+
+
+/--
+The current `ToExpr` instance for `Int` is bad,
+so we roll our own here.
+-/
+def mkInt (i : Int) : Expr :=
+  if 0 ≤ i then
+    mkNat i.toNat
+  else
+    mkApp3 (.const ``Neg.neg [0]) (.const ``Int []) (mkNat (-i).toNat)
+      (.const ``Int.instNegInt [])
+where
+  mkNat (n : Nat) : Expr :=
+    let r := mkRawNatLit n
+    mkApp3 (.const ``OfNat.ofNat [0]) (.const ``Int []) r
+        (.app (.const ``instOfNat []) r)
+
 /--
 A partially processed `omega` context.
 
@@ -114,7 +132,7 @@ We also transform the expression as we descend into it:
 -/
 partial def asLinearComboImpl (e : Expr) : OmegaM (LinearCombo × OmegaM Expr × HashSet Expr) := do
   trace[omega] "processing {e}"
-  match e.int? with
+  match groundInt? e with
   | some i =>
     let lc := {const := i}
     return ⟨lc, mkEvalRflProof e lc, ∅⟩
@@ -177,17 +195,20 @@ partial def asLinearComboImpl (e : Expr) : OmegaM (LinearCombo × OmegaM Expr ×
     | some r => pure r
     | none => mkAtomLinearCombo e
   | (``HMod.hMod, #[_, _, _, _, n, k]) =>
-    match natCast? k with
-    | some _ => rewrite e (mkApp2 (.const ``Int.emod_def []) n k)
+    match groundNat? k with
+    | some k' => do
+      let k' := mkInt k'
+      rewrite (← mkAppM ``HMod.hMod #[n, k']) (mkApp2 (.const ``Int.emod_def []) n k')
     | none => mkAtomLinearCombo e
   | (``HDiv.hDiv, #[_, _, _, _, x, z]) =>
-    match intCast? z with
+    match groundInt? z with
     | some 0 => rewrite e (mkApp (.const ``Int.ediv_zero []) x)
-    | some i =>
+    | some i => do
+      let e' ← mkAppM ``HDiv.hDiv #[x, mkInt i]
       if i < 0 then
-        rewrite e (mkApp2 (.const ``Int.ediv_neg []) x (toExpr (-i)))
+        rewrite e' (mkApp2 (.const ``Int.ediv_neg []) x (mkInt (-i)))
       else
-        mkAtomLinearCombo e
+        mkAtomLinearCombo e'
     | _ => mkAtomLinearCombo e
   | (``Min.min, #[_, _, a, b]) =>
     if (← cfg).splitMinMax then
@@ -216,6 +237,9 @@ partial def asLinearComboImpl (e : Expr) : OmegaM (LinearCombo × OmegaM Expr ×
     | (``HMod.hMod, #[_, _, _, _, a, b]) => rewrite e (mkApp2 (.const ``Int.ofNat_emod []) a b)
     | (``HSub.hSub, #[_, _, _, _, mkApp6 (.const ``HSub.hSub _) _ _ _ _ a b, c]) =>
       rewrite e (mkApp3 (.const ``Int.ofNat_sub_sub []) a b c)
+    | (``HPow.hPow, #[_, _, _, _, a, b]) => match groundNat? a, groundNat? b with
+      | some _, some _ => rewrite e (mkApp2 (.const ``Int.ofNat_pow []) a b)
+      | _, _ => mkAtomLinearCombo e
     | (``Prod.fst, #[_, β, p]) => match p with
       | .app (.app (.app (.app (.const ``Prod.mk [0, v]) _) _) x) y =>
         rewrite e (mkApp3 (.const ``Int.ofNat_fst_mk [v]) β x y)
@@ -226,6 +250,10 @@ partial def asLinearComboImpl (e : Expr) : OmegaM (LinearCombo × OmegaM Expr ×
       | _ => mkAtomLinearCombo e
     | (``Min.min, #[_, _, a, b]) => rewrite e (mkApp2 (.const ``Int.ofNat_min []) a b)
     | (``Max.max, #[_, _, a, b]) => rewrite e (mkApp2 (.const ``Int.ofNat_max []) a b)
+    | (``HShiftLeft.hShiftLeft, #[_, _, _, _, a, b]) =>
+      rewrite e (mkApp2 (.const ``Int.ofNat_shiftLeft_eq []) a b)
+    | (``HShiftRight.hShiftRight, #[_, _, _, _, a, b]) =>
+      rewrite e (mkApp2 (.const ``Int.ofNat_shiftRight_eq_div_pow []) a b)
     | (``Int.natAbs, #[n]) =>
       if (← cfg).splitNatAbs then
         rewrite e (mkApp (.const ``Int.ofNat_natAbs []) n)

src/Lean/Elab/Tactic/Omega/OmegaM.lean

@@ -108,6 +108,45 @@ def intCast? (n : Expr) : Option Int :=
   | (``Nat.cast, #[_, _, n]) => n.nat?
   | _ => n.int?
 
+/--
+If `groundNat? e = some n`, then `e` is definitionally equal to `OfNat.ofNat n`.
+-/
+-- We may want to replace this with an implementation using
+-- the internals of `simp (config := {ground := true})`
+partial def groundNat? (e : Expr) : Option Nat :=
+  match e.getAppFnArgs with
+  | (``Nat.cast, #[_, _, n]) => groundNat? n
+  | (``HAdd.hAdd, #[_, _, _, _, x, y]) => op (· + ·) x y
+  | (``HMul.hMul, #[_, _, _, _, x, y]) => op (· * ·) x y
+  | (``HSub.hSub, #[_, _, _, _, x, y]) => op (· - ·) x y
+  | (``HDiv.hDiv, #[_, _, _, _, x, y]) => op (· / ·) x y
+  | (``HPow.hPow, #[_, _, _, _, x, y]) => op (· ^ ·) x y
+  | _ => e.nat?
+where op (f : Nat → Nat → Nat) (x y : Expr) : Option Nat :=
+  match groundNat? x, groundNat? y with
+    | some x', some y' => some (f x' y')
+    | _, _ => none
+
+/--
+If `groundInt? e = some i`,
+then `e` is definitionally equal to the standard expression for `i`.
+-/
+partial def groundInt? (e : Expr) : Option Int :=
+  match e.getAppFnArgs with
+  | (``Nat.cast, #[_, _, n]) => groundNat? n
+  | (``HAdd.hAdd, #[_, _, _, _, x, y]) => op (· + ·) x y
+  | (``HMul.hMul, #[_, _, _, _, x, y]) => op (· * ·) x y
+  | (``HSub.hSub, #[_, _, _, _, x, y]) => op (· - ·) x y
+  | (``HDiv.hDiv, #[_, _, _, _, x, y]) => op (· / ·) x y
+  | (``HPow.hPow, #[_, _, _, _, x, y]) => match groundInt? x, groundNat? y with
+    | some x', some y' => some (x' ^ y')
+    | _, _ => none
+  | _ => e.int?
+where op (f : Int → Int → Int) (x y : Expr) : Option Int :=
+  match groundNat? x, groundNat? y with
+    | some x', some y' => some (f x' y')
+    | _, _ => none
+
 /-- Construct the term with type hint `(Eq.refl a : a = b)`-/
 def mkEqReflWithExpectedType (a b : Expr) : MetaM Expr := do
   mkExpectedTypeHint (← mkEqRefl a) (← mkEq a b)

tests/lean/run/omega.lean

@@ -381,6 +381,11 @@ example (i : Fin 7) : (i : Nat) < 8 := by omega
 
 example (x y z i : Nat) (hz : z ≤ 1) : x % 2 ^ i + y % 2 ^ i + z < 2 * 2^ i := by omega
 
+/-! ### Ground terms -/
+
+example : 2^7 < 165 := by omega
+example (_ : x % 2^7 < 3) : x % 128 < 5 := by omega
+
 /-! ### BitVec -/
 -- Currently these tests require calling `simp` with many lemmas,
 -- and sometimes adding `toNat_lt` as a hypothesis.
@@ -392,15 +397,16 @@ example (x y : BitVec 8) (hx : x < 16) (hy : y < 16) : x + y < 31 := by
   simp [BitVec.lt_def] at *
   omega
 
-example (x y z : BitVec 8) (hx : x >>> 1 < 16) (hy : y < 16) (hz : z = x + 2 * y) : z ≤ 64 := by
-  simp [BitVec.lt_def, BitVec.le_def, BitVec.toNat_eq, Nat.shiftRight_eq_div_pow, BitVec.toNat_mul] at *
+example (x y z : BitVec 8)
+    (hx : x >>> 1 < 16) (hy : y < 16) (hz : z = x + 2 * y) : z ≤ 64 := by
+  simp [BitVec.lt_def, BitVec.le_def, BitVec.toNat_eq, BitVec.toNat_mul] at *
   omega
 
 example (x : BitVec 8) (hx : (x + 1) <<< 1 = 3) : False := by
-  simp [BitVec.toNat_eq, Nat.shiftLeft_eq] at *
+  simp [BitVec.toNat_eq] at *
   omega
 
 example (x : BitVec 8) (hx : (x + 1) <<< 1 = 4) : x = 1 ∨ x = 129 := by
   have := toNat_lt x
-  simp [BitVec.toNat_eq, Nat.shiftLeft_eq, BitVec.lt_def] at *
+  simp [BitVec.toNat_eq, BitVec.lt_def] at *
   omega