Proving simple inequality in Ssreflect

Question

I'm having trouble with some pretty simple proofs in Coq, using the MathComp library for reflection.

Suppose I want to prove this Lemma:

From mathcomp Require Import ssreflect ssrbool ssrnat.

Lemma example m n: n.+1 < m -> n < m.
Proof.
      have predn_ltn_k k: (0 < k.-1) -> (0 < k).
          by case: k.
      rewrite -subn_gt0 subnS => submn_pred_gt0.
      by rewrite -subn_gt0; apply predn_ltn_k.
Qed.

This approach seems a little "unorthodox" to me for such a simple task. Is there a better/simpler way to do this?

Please always include the list of imports when you're posting a code snippet. — gallais, Apr 10 '17 at 23:48

score 5 · Accepted Answer · answered Apr 11 '17 at 00:04

5

Yes, there is a better way. Your lemma is a special case of ltnW : forall m n, m < n -> m <= n:

Lemma example n m : n.+1 < m -> n < m.
Proof. exact: ltnW. Qed.

This works because n < m is actually syntactic sugar for n.+1 <= m.

answered Apr 11 '17 at 00:04

Arthur Azevedo De Amorim

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score 1 · Answer 2 · answered Apr 11 '17 at 00:02

1

I haven't practiced ssreflect a lot so I can't really tell whether this can be golfed down but basically the idea is that n < n.+1 < m:

Require Import mathcomp.ssreflect.ssrnat.
Require Import mathcomp.ssreflect.ssrbool.
Require Import mathcomp.ssreflect.ssreflect.

Lemma example m n: n.+1 < m -> n < m.
Proof.
move => ltSnm; apply: ltn_trans; by [apply ltnSn | apply ltSnm].
Qed.

answered Apr 11 '17 at 00:02

gallais

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Tip: change your three first lines into: `From mathcomp Require Import ssrnat ssrbool ssreflect.` – Zimm i48 Apr 12 '17 at 10:10

Proving simple inequality in Ssreflect

2 Answers2