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Use eval! to evaluate functions that use sorry. #133

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Use eval! to evaluate functions that use sorry. #133

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13 changes: 9 additions & 4 deletions induction_and_recursion.md
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Expand Up @@ -796,7 +796,11 @@ The following example is similar: it converts any natural number to a
binary expression, represented as a list of 0's and 1's. We have to
provide evidence that the recursive call is
decreasing, which we do here with a ``sorry``. The ``sorry`` does not
prevent the interpreter from evaluating the function successfully.
prevent the interpreter from evaluating the function successfully, but
by default it will refuse to do so as evaluating a non-terminating
function could cause Lean to crash.
We can use the command ``#eval!`` to override this behavior and evaluate it
anyway.

```lean
def natToBin : Nat → List Nat
Expand All @@ -806,7 +810,7 @@ def natToBin : Nat → List Nat
have : (n + 2) / 2 < n + 2 := sorry
natToBin ((n + 2) / 2) ++ [n % 2]

#eval natToBin 1234567
#eval! natToBin 1234567
```

As a final example, we observe that Ackermann's function can be
Expand Down Expand Up @@ -882,7 +886,8 @@ decreasing_by
· apply Prod.Lex.left; simp_arith
```

We can use `decreasing_by sorry` to instruct Lean to "trust" us that the function terminates.
We can use `decreasing_by sorry` to instruct Lean to "trust" us that the function terminates
and use the command ``#eval!`` to evaluate it.

```lean
def natToBin : Nat → List Nat
Expand All @@ -891,7 +896,7 @@ def natToBin : Nat → List Nat
| n + 2 => natToBin ((n + 2) / 2) ++ [n % 2]
decreasing_by sorry

#eval natToBin 1234567
#eval! natToBin 1234567
```

Recall that using `sorry` is equivalent to using a new axiom, and should be avoided. In the following example, we used the `sorry` to prove `False`.
Expand Down