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Proofreading LogicProgramming

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Adam Chlipala 2018-03-13 20:26:17 -04:00
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19
LogicProgramming.v
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@ -380,7 +380,7 @@ Proof.
(* Coq leaves for us two subgoals to prove... [nat]?! We are being asked to
* show that natural numbers exists. Why? Some unification variables of that
* type were left undetermiend, by the end of the proof. Specifically, these
* type were left undetermined, by the end of the proof. Specifically, these
* variables stand for the 2 elements of the list we find. Of course it makes
* sense that the list length follows without knowing the data values. In Coq
* 8.6, the [Unshelve] command brings these goals to the forefront, where we
@ -394,7 +394,7 @@ Proof.
Show Proof.
Abort.
(* Paradoxically, we can make the proof search process easier by constraining
(* Paradoxically, we can make the proof-search process easier by constraining
* the list further, so that proof search naturally locates appropriate data
* elements by unification. The library predicate [Forall] will be helpful. *)
@ -410,9 +410,8 @@ Qed.
Print length_is_2.
(* Let us try one more, fancier example. First, we use a standard higher
* order function to define a function for summing all data elements of a
* list. *)
(* Let us try one more, fancier example. First, we use a standard higher-order
* function to define a function for summing all data elements of a list. *)
Definition sum := fold_right plus O.
@ -693,8 +692,8 @@ 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]. *)
(* Here's a quick tease using a feature that we'll explore fully in a later
* class. 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.
@ -706,11 +705,11 @@ Proof.
induct e; first_order; eauto.
Defined.
(* Essentially the same proof search ahs completed. This time, though, we ended
(* Essentially the same proof search has 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. *)
* wrapper to make that idea more concrete. *)
Definition params (e : exp) : nat * nat :=
let '(existT _ k (existT _ n _)) := linear_computable e in
@ -885,7 +884,7 @@ Hint Extern 1 =>
(** * Rewrite Hints *)
(* Another dimension of extensibility with hints is rewriting with quantified
* equalities. We have used the associated command %[Hint Rewrite] in several
* equalities. We have used the associated command [Hint Rewrite] in several
* examples so far. The [simplify] tactic uses these hints by calling the
* built-in tactic [autorewrite]. Our rewrite hints have taken the form
* [Hint Rewrite lemma], which by default adds them to the default hint database