Library mathcomp.ssreflect.ssrbool

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

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`. v8.15 addtions

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}.

v8.16 additions 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.

v8.17 additions

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 : rT ⇒ Some (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 ).

Future additions

Notation "[ 'in' A ]" := (in_mem ^~ (mem A))
(at level 0, format "[ 'in' A ]") : function_scope.

Lemma relpre_trans {T' T : Type} {leT : rel T} {f : T' → T} :
transitive leT → transitive (relpre f leT).

The notations below are already defined in Coq.ssr.ssrbool, but we redefine them in different notation scopes (mostly fun_scope -> function_scope, in order to replace the former with the latter).

Notation "[ 'pred' : T | E ]" := (SimplPred (fun _ : T ⇒ E%B)) :
  function_scope.
Notation "[ 'pred' x | E ]" := (SimplPred (fun x ⇒ E%B)) : function_scope.
Notation "[ 'pred' x | E1 & E2 ]" := [pred x | E1 && E2 ] : function_scope.
Notation "[ 'pred' x : T | E ]" :=
  (SimplPred (fun x : T ⇒ E%B)) (only parsing) : function_scope.
Notation "[ 'pred' x : T | E1 & E2 ]" :=
  [pred x : T | E1 && E2 ] (only parsing) : function_scope.

Notation "[ 'rel' x y | E ]" := (SimplRel (fun x y ⇒ E%B))
  (only parsing) : function_scope.
Notation "[ 'rel' x y : T | E ]" :=
  (SimplRel (fun x y : T ⇒ E%B)) (only parsing) : function_scope.

Notation "[ 'mem' A ]" :=
  (pred_of_simpl (simpl_of_mem (mem A))) (only parsing) : function_scope.

Notation "[ 'pred' x 'in' A ]" := [pred x | x \in A] : function_scope.
Notation "[ 'pred' x 'in' A | E ]" := [pred x | x \in A & E] : function_scope.
Notation "[ 'pred' x 'in' A | E1 & E2 ]" :=
  [pred x | x \in A & E1 && E2 ] : function_scope.

Notation "[ 'rel' x y 'in' A & B | E ]" :=
  [rel x y | (x \in A) && (y \in B) && E] : function_scope.
Notation "[ 'rel' x y 'in' A & B ]" :=
  [rel x y | (x \in A) && (y \in B)] : function_scope.
Notation "[ 'rel' x y 'in' A | E ]" := [rel x y in A & A | E] : function_scope.
Notation "[ 'rel' x y 'in' A ]" := [rel x y in A & A] : function_scope.

Both bodies and the scope have been changed for the following notations.

Notation "[ 'predI' A & B ]" := (predI [in A] [in B]) : function_scope.
Notation "[ 'predU' A & B ]" := (predU [in A] [in B]) : function_scope.
Notation "[ 'predD' A & B ]" := (predD [in A] [in B]) : function_scope.
Notation "[ 'predC' A ]" := (predC [in A]) : function_scope.
Notation "[ 'preim' f 'of' A ]" := (preim f [in A]) : function_scope.