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Batch of changes based on proofreader feedback
author Adam Chlipala <adam@chlipala.net>
date Sun, 11 Nov 2012 18:17:23 -0500
parents 1fd4109f7b31
children f38a3af9dd17
comparison
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477:6769ef9688f2 478:f02b698aadb1
72 | First _ => x 72 | First _ => x
73 | Next _ idx' => get ls' idx' 73 | Next _ idx' => get ls' idx'
74 end 74 end
75 end. 75 end.
76 ]] 76 ]]
77 %\vspace{-.15in}%Now the first [match] case type-checks, and we see that the problem with the [Cons] case is that the pattern-bound variable [idx'] does not have an apparent type compatible with [ls']. In fact, the error message Coq gives for this exact code can be confusing, thanks to an overenthusiastic type inference heuristic. We are told that the [Nil] case body has type [match X with | 0 => A | S _ => unit end] for a unification variable [X], while it is expected to have type [A]. We can see that setting [X] to [0] resolves the conflict, but Coq is not yet smart enough to do this unification automatically. Repeating the function's type in a [return] annotation, used with an [in] annotation, leads us to a more informative error message, saying that [idx'] has type [fin n1] while it is expected to have type [fin n0], where [n0] is bound by the [Cons] pattern and [n1] by the [Next] pattern. As the code is written above, nothing forces these two natural numbers to be equal, though we know intuitively that they must be. 77 %\vspace{-.15in}%Now the first [match] case type-checks, and we see that the problem with the [Cons] case is that the pattern-bound variable [idx'] does not have an apparent type compatible with [ls']. In fact, the error message Coq gives for this exact code can be confusing, thanks to an overenthusiastic type inference heuristic. We are told that the [Nil] case body has type [match X with | O => A | S _ => unit end] for a unification variable [X], while it is expected to have type [A]. We can see that setting [X] to [O] resolves the conflict, but Coq is not yet smart enough to do this unification automatically. Repeating the function's type in a [return] annotation, used with an [in] annotation, leads us to a more informative error message, saying that [idx'] has type [fin n1] while it is expected to have type [fin n0], where [n0] is bound by the [Cons] pattern and [n1] by the [Next] pattern. As the code is written above, nothing forces these two natural numbers to be equal, though we know intuitively that they must be.
78 78
79 We need to use [match] annotations to make the relationship explicit. Unfortunately, the usual trick of postponing argument binding will not help us here. We need to match on both [ls] and [idx]; one or the other must be matched first. To get around this, we apply the convoy pattern that we met last chapter. This application is a little more clever than those we saw before; we use the natural number predecessor function [pred] to express the relationship between the types of these variables. 79 We need to use [match] annotations to make the relationship explicit. Unfortunately, the usual trick of postponing argument binding will not help us here. We need to match on both [ls] and [idx]; one or the other must be matched first. To get around this, we apply the convoy pattern that we met last chapter. This application is a little more clever than those we saw before; we use the natural number predecessor function [pred] to express the relationship between the types of these variables.
80 [[ 80 [[
81 Fixpoint get n (ls : ilist n) : fin n -> A := 81 Fixpoint get n (ls : ilist n) : fin n -> A :=
82 match ls with 82 match ls with
621 crush. 621 crush.
622 Qed. 622 Qed.
623 623
624 Lemma sum_inc' : forall n (f1 f2 : ffin n -> nat), 624 Lemma sum_inc' : forall n (f1 f2 : ffin n -> nat),
625 (forall idx, f1 idx >= f2 idx) 625 (forall idx, f1 idx >= f2 idx)
626 -> rifoldr plus 0 f1 >= rifoldr plus 0 f2. 626 -> rifoldr plus O f1 >= rifoldr plus O f2.
627 Hint Resolve plus_ge. 627 Hint Resolve plus_ge.
628 628
629 induction n; crush. 629 induction n; crush.
630 Qed. 630 Qed.
631 631