Library prosa.util.setoid
Setoid Rewriting with boolean inequalities of ssreflect. Solution
suggested by Georges Gonthier (ssreflect mailinglist @ 18.12.2016)
From Stdlib Require Import Basics Setoid Morphisms.
From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat.
This construction allows us to "rewrite" inequalities with ≤.
The following establishes a number of "plumbing facts" that are necessary to
interface leb with the rewrite tactic.
leb expresses an implication.
leb is a preorder.
Relation with leq.
Relation with addn.
Relation with subn.
Relation with the successor constructor.
Establishing le as a rewrite relation.
Wrapper for use in rewrite steps:
To see the benefits, consider the following example.
Another benefit of leqRW is that it allows
to avoid unnecessary eapply leq_trans.
Goal forall a b c d x y, x <= y -> a + (b + (x + c)) + d <= a + (b + (y + c)) + d. Proof. move => a b c d x y LE. (* This could be an unpleasant proof, but we can just [rewrite LE] *) by rewrite (leqRW LE). (* magic here *) Qed.
Goal forall a b x y z, a <= x -> b <= y -> x + y <= z -> a + b <= z. Proof. move => a b x y z LE1 LE2 LE3. rewrite -(leqRW LE3) leq_add //. Qed.