# 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 ≤.
Inductive leb a b := Leb of (a ==> b).
The following establishes a number of "plumbing facts" that are necessary to interface leb with the rewrite tactic.
leb expresses an implication.
Lemma leb_eq a b : leb a b ↔ (a → b).
Proof. move: a b ⇒ [|] [|]; firstorder. Qed.
leb is a preorder.
#[global] Instance : PreOrder leb.
Proof. split ⇒ [[|]|[|][|][|][?][?]]; try exact: Leb. Qed.
Various leb required for the interface:
Relation of leb and andb.
#[global] Instance : Proper (leb ==> leb ==> leb) andb.
Proof. move ⇒ [|] [|] [A] c d [B]; exact: Leb. Qed.
Relation of leb and orb.
#[global] Instance : Proper (leb ==> leb ==> leb) orb.
Proof. move ⇒ [|] [|] [A] [|] d [B]; exact: Leb. Qed.
Relation of leb and is_true.
#[global] Instance : Proper (leb ==> impl) is_true.
Proof. move ⇒ a b []. exact: implyP. Qed.
Relation with leq.
#[global] Instance : Proper (le --> le ++> leb) leq.
Proof. move ⇒ n m /leP ? n' m' /leP ?. apply/leb_eq ⇒ ?. eauto using leq_trans. Qed.
Relation with addn.
#[global] Instance : Proper (le ==> le ==> le) addn.
Proof. move ⇒ n m /leP ? n' m' /leP ?. apply/leP. exact: leq_add. Qed.
Relation with subn.
#[global] Instance : Proper (le ++> le --> le) subn.
Proof. move ⇒ n m /leP ? n' m' /leP ?. apply/leP. exact: leq_sub. Qed.
Relation with the successor constructor.
#[global] Instance : Proper (le ==> le) S.
Proof. move ⇒ n m /leP ?. apply/leP. by rewrite ltnS. Qed.
Establishing le as a rewrite relation.
#[global] Instance : RewriteRelation le.
Defined.
Wrapper for use in rewrite steps:
Definition leqRW {m n} : m ≤ n → le m n := leP.
To see the benefits, consider the following example.

    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.

Another benefit of leqRW is that it allows to avoid unnecessary eapply leq_trans.

    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.