# Library prosa.util.setoid

Setoid Rewriting with boolean inequalities of ssreflect. Solution suggested by Georges Gonthier (ssreflect mailinglist @ 18.12.2016)

From Coq 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).

Lemma leb_eq a b : leb a b (a b).
Proof. move: a b ⇒ [|] [|]; firstorder. Qed.

#[global] Instance : PreOrder leb.
Proof. split ⇒ [[|]|[|][|][|][?][?]]; try exact: Leb. Qed.

#[global] Instance : Proper (leb ==> leb ==> leb) andb.
Proof. move ⇒ [|] [|] [A] c d [B]; exact: Leb. Qed.

#[global] Instance : Proper (leb ==> leb ==> leb) orb.
Proof. move ⇒ [|] [|] [A] [|] d [B]; exact: Leb. Qed.

#[global] Instance : Proper (leb ==> impl) is_true.
Proof. movea b []. exact: implyP. Qed.

#[global] Instance : Proper (le --> le ++> leb) leq.
Proof. moven m /leP ? n' m' /leP ?. apply/leb_eq ⇒ ?. eauto using leq_trans. Qed.

#[global] Instance : Proper (le ==> le ==> le) addn.
Proof. moven m /leP ? n' m' /leP ?. apply/leP. exact: leq_add. Qed.

#[global] Instance : Proper (le ++> le --> le) subn.
Proof. moven m /leP ? n' m' /leP ?. apply/leP. exact: leq_sub. Qed.

#[global] Instance : Proper (le ==> le) S.
Proof. moven m /leP ?. apply/leP. by rewrite ltnS. Qed.

#[global] Instance : RewriteRelation le.
Defined.

Definition leqRW {m n} : m n le m n := leP.

To see the benefits, consider the following example.
(*
Goal
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
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.