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LogicProgramming bonus: extending an automated-proof example to build a runnable witness finder

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Adam Chlipala 2017-03-18 12:30:50 -04:00
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commit 67ed8a4a63
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LogicProgramming.v
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@ -693,6 +693,35 @@ Proof.
induct e; first_order; eauto.
Qed.
(* Here's a quick tease using a feature that we'll explore fully this time next
* week. Let's use a mysterious construct [sigT] instead of [exists]. *)
Hint Extern 1 (sigT (fun n : nat => _)) => exists 0.
Hint Extern 1 (sigT (fun n : nat => _)) => exists 1.
Hint Extern 1 (sigT (fun n : nat => _)) => eexists (_ + _).
Theorem linear_computable : forall e, sigT (fun k => sigT (fun n =>
forall var, eval var e (k * var + n))).
Proof.
induct e; first_order; eauto.
Defined.
(* Essentially the same proof search ahs completed. This time, though, we ended
* the proof with [Defined], which saves it as _transparent_, so details of the
* "proof" can be consulted from the outside. Actually, this "proof" is more
* like a recursive program that finds [k] and [n], given [e]! Let's add a
* wrapper to make taht idea more concrete. *)
Definition params (e : exp) : nat * nat :=
let '(existT _ k (existT _ n _)) := linear_computable e in
(k, n).
(* Now we can actually _run our proof_ to get normalized versions of particular
* expressions. *)
Compute params (Plus (Const 7) (Plus Var (Plus (Const 8) Var))).
(* With Coq doing so much of the proof-search work for us, we might get
* complacent and consider that any successful [eauto] invocation is as good as
* any other. However, because introduced unification variables may wind up