# Library mathcomp.ssreflect.ssrbool

From mathcomp Require Import ssreflect ssrfun.
From Coq Require Export ssrbool.

8.11 addition in Coq but renamed
Notation rel_of_simpl_rel := rel_of_simpl.

Local additions: [in A] == the applicative counterpart of a collective predicate A: [in A] x beta-expands to x \in A. This is similar to mem A, except that mem A x only simplfies to x \in A. > These will become part of the core SSReflect library in later Coq versions. For the sake of backwards compatibility, this file also replicates v8.13-15 additions, including a generalization of the statments of homoRL_in, homoLR_in, homo_mono_in, monoLR_in, monoRL_in, and can_mono_in.

Section ReflectCombinators.

Variables (P Q : Prop) (p q : bool).

Hypothesis rP : reflect P p.
Hypothesis rQ : reflect Q q.

Lemma negPP : reflect (¬ P) (~~ p).

Lemma andPP : reflect (P Q) (p && q).

Lemma orPP : reflect (P Q) (p || q).

Lemma implyPP : reflect (P Q) (p ==> q).

End ReflectCombinators.
Arguments negPP {P p}.
Arguments andPP {P Q p q}.
Arguments orPP {P Q p q}.
Arguments implyPP {P Q p q}.

pred_oapp T D := [pred x | oapp (mem D) false x]

Lemma mono1W_in (aT rT : predArgType) (f : aT rT) (aD : {pred aT})
(aP : pred aT) (rP : pred rT) :
{in aD, {mono f : x / aP x >-> rP x}}
{in aD, {homo f : x / aP x >-> rP x}}.
Arguments mono1W_in [aT rT f aD aP rP].

Set Implicit Arguments.

Lemma all_sig2_cond {I T} (C : pred I) P Q :
T ( x, C x {y : T | P x y & Q x y})
{f : I T | x, C x P x (f x) & x, C x Q x (f x)}.

Lemma can_in_pcan [rT aT : Type] (A : {pred aT}) [f : aT rT] [g : rT aT] :
{in A, cancel f g} {in A, pcancel f (fun y : rTSome (g y))}.

Lemma pcan_in_inj [rT aT : Type] [A : {pred aT}]
[f : aT rT] [g : rT option aT] :
{in A, pcancel f g} {in A &, injective f}.

Lemma in_inj_comp A B C (f : B A) (h : C B) (P : pred B) (Q : pred C) :
{in P &, injective f} {in Q &, injective h} {homo h : x / Q x >-> P x}
{in Q &, injective (f \o h)}.

Lemma can_in_comp [A B C : Type] (D : {pred B}) (D' : {pred C})
[f : B A] [h : C B] [f' : A B] [h' : B C] :
{homo h : x / x \in D' >-> x \in D}
{in D, cancel f f'} {in D', cancel h h'}
{in D', cancel (f \o h) (h' \o f')}.

Lemma pcan_in_comp [A B C : Type] (D : {pred B}) (D' : {pred C})
[f : B A] [h : C B] [f' : A option B] [h' : B option C] :
{homo h : x / x \in D' >-> x \in D}
{in D, pcancel f f'} {in D', pcancel h h'}
{in D', pcancel (f \o h) (obind h' \o f')}.

Definition pred_oapp T (D : {pred T}) : pred (option T) :=
[pred x | oapp (mem D) false x].

Lemma ocan_in_comp [A B C : Type] (D : {pred B}) (D' : {pred C})
[f : B option A] [h : C option B] [f' : A B] [h' : B C] :
{homo h : x / x \in D' >-> x \in pred_oapp D}
{in D, ocancel f f'} {in D', ocancel h h'}
{in D', ocancel (obind f \o h) (h' \o f')}.

Lemma eqbLR (b1 b2 : bool) : b1 = b2 b1 b2.

Lemma eqbRL (b1 b2 : bool) : b1 = b2 b2 b1.

Lemma homo_mono1 [aT rT : Type] [f : aT rT] [g : rT aT]
[aP : pred aT] [rP : pred rT] :
cancel g f
{homo f : x / aP x >-> rP x}
{homo g : x / rP x >-> aP x} {mono g : x / rP x >-> aP x}.

Lemma if_and b1 b2 T (x y : T) :
(if b1 && b2 then x else y) = (if b1 then if b2 then x else y else y).

Lemma if_or b1 b2 T (x y : T) :
(if b1 || b2 then x else y) = (if b1 then x else if b2 then x else y).

Lemma if_implyb b1 b2 T (x y : T) :
(if b1 ==> b2 then x else y) = (if b1 then if b2 then x else y else x).

Lemma if_implybC b1 b2 T (x y : T) :
(if b1 ==> b2 then x else y) = (if b2 then x else if b1 then y else x).

Lemma if_add b1 b2 T (x y : T) :
(if b1 (+) b2 then x else y) = (if b1 then if b2 then y else x else if b2 then x else y).