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restore boolp (rebasing issue)
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@Tragicus
Tragicus committed Jan 8, 2024
1 parent 34c6885 commit d9e1af5
Showing 1 changed file with 22 additions and 31 deletions.
53 changes: 22 additions & 31 deletions classical/boolp.v
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Original file line number Diff line number Diff line change
Expand Up @@ -212,9 +212,6 @@ Qed.
Lemma gen_choiceMixin {T : Type} : Choice.mixin_of T.
Proof. by case: classic. Qed.

Lemma pdegen (P : Prop): P = True \/ P = False.
Proof. by have [p|Np] := pselect P; [left|right]; rewrite propeqE. Qed.

Lemma lem (P : Prop): P \/ ~P.
Proof. by case: (pselect P); tauto. Qed.

Expand Down Expand Up @@ -283,8 +280,7 @@ Proof. by rewrite propeqE; split => -[x [y]]; exists y, x. Qed.
Lemma reflect_eq (P : Prop) (b : bool) : reflect P b -> P = b.
Proof. by rewrite propeqE; exact: rwP. Qed.

Definition asbool (P : Prop) :=
if pselect P then true else false.
Definition asbool (P : Prop) := if pselect P then true else false.

Notation "`[< P >]" := (asbool P) : bool_scope.

Expand Down Expand Up @@ -359,7 +355,8 @@ Definition canonical_of T U (sort : U -> T) := forall (G : T -> Type),
Notation canonical_ sort := (@canonical_of _ _ sort).
Notation canonical T E := (@canonical_of T E id).

Lemma canon T U (sort : U -> T) : (forall x, exists y, sort y = x) -> canonical_ sort.
Lemma canon T U (sort : U -> T) :
(forall x, exists y, sort y = x) -> canonical_ sort.
Proof. by move=> + G Gs x => /(_ x)/cid[x' <-]. Qed.
Arguments canon {T U sort} x.

Expand Down Expand Up @@ -480,36 +477,25 @@ Proof. by case/PropB: P; [left | rewrite orB; right]. Qed.

(* -------------------------------------------------------------------- *)

Lemma notT (P : Prop) : P = False -> ~ P. Proof. by move->. Qed.
Lemma notT (P : Prop) : P = False -> ~ P.
Proof. by move->. Qed.

Lemma contrapT P : ~ ~ P -> P.
Proof.
by move/asboolPn=> nnb; apply/asboolP; apply: contraR nnb => /asboolPn /asboolP.
Qed.
Proof. by case: (PropB P) => //; rewrite not_False. Qed.

Lemma notTE (P : Prop) : (~ P) -> P = False.
Proof. by case: (pdegen P)=> ->. Qed.
Lemma notTE (P : Prop) : (~ P) -> P = False. Proof. by case: (PropB P). Qed.

Lemma notFE (P : Prop) : (~ P) = False -> P.
Proof. move/notT; exact: contrapT. Qed.
Proof. by move/notT; exact: contrapT. Qed.

Lemma notK : involutive not.
Proof.
move=> P; case: (pdegen P)=> ->; last by apply: notTE; intuition.
by rewrite [~ True]notTE //; case: (pdegen (~ False)) => // /notFE.
Qed.
Proof. by case/PropB; rewrite !(not_False,not_True). Qed.

Lemma contra_notP (Q P : Prop) : (~ Q -> P) -> ~ P -> Q.
Proof.
move=> cb /asboolPn nb; apply/asboolP.
by apply: contraR nb => /asboolP /cb /asboolP.
Qed.
Proof. by move: Q P => /PropB[] /PropB[]. Qed.

Lemma contraPP (Q P : Prop) : (~ Q -> ~ P) -> P -> Q.
Proof.
move=> cb /asboolP hb; apply/asboolP.
by apply: contraLR hb => /asboolP /cb /asboolPn.
Qed.
Proof. by move: Q P => /PropB[] /PropB[]//; rewrite not_False not_True. Qed.

Lemma contra_notT b (P : Prop) : (~~ b -> P) -> ~ P -> b.
Proof. by move=> bP; apply: contra_notP => /negP. Qed.
Expand All @@ -526,7 +512,7 @@ Proof. by move=> /contra_notP + /negP => /[apply]. Qed.
Lemma contra_neqP (T : eqType) (x y : T) P : (~ P -> x = y) -> x != y -> P.
Proof. by move=> Pxy; apply: contraNP => /Pxy/eqP. Qed.

Lemma contra_eqP (T : eqType) (x y : T) (Q : Prop) : (~ Q -> x != y) -> x = y -> Q.
Lemma contra_eqP (T : eqType) (x y : T) Q : (~ Q -> x != y) -> x = y -> Q.
Proof. by move=> Qxy /eqP; apply: contraTP. Qed.

Lemma contra_leP {disp1 : unit} {T1 : porderType disp1} [P : Prop] [x y : T1] :
Expand All @@ -544,9 +530,10 @@ by apply: Order.POrderTheory.contra_ltT yx => /asboolPn.
Qed.

Lemma wlog_neg P : (~ P -> P) -> P.
Proof. by move=> ?; case: (pselect P). Qed.
Proof. by case: (PropB P); exact. Qed.

Lemma not_inj : injective not. Proof. exact: can_inj notK. Qed.

Lemma notLR P Q : (P = ~ Q) -> (~ P) = Q. Proof. exact: canLR notK. Qed.

Lemma notRL P Q : (~ P) = Q -> P = ~ Q. Proof. exact: canRL notK. Qed.
Expand Down Expand Up @@ -653,7 +640,7 @@ Lemma implyE (P Q : Prop) : (P -> Q) = (~ P \/ Q).
Proof. by rewrite -[LHS]notE not_implyE propeqE not_andP notE. Qed.

Lemma orC : commutative or.
Proof. by move=> P Q; rewrite propeqE; split; (case=> ?; [right|left]). Qed.
Proof. by move=> /PropB[] /PropB[] => //; rewrite !orB. Qed.

Lemma orA : associative or.
Proof. by move=> P Q R; rewrite propeqE; split=> [|]; tauto. Qed.
Expand All @@ -673,15 +660,19 @@ Proof. by case/PropB: P; rewrite notB orB implyB. Qed.
Lemma orpN P Q : (P \/ ~ Q) = (Q -> P). Proof. by rewrite orC orNp. Qed.

Lemma or3E P Q R : [\/ P, Q | R] = (P \/ Q \/ R).
Proof. by rewrite -(asboolE P) -(asboolE Q) -(asboolE R) (reflect_eq or3P) -2!(reflect_eq orP). Qed.
Proof.
rewrite -(asboolE P) -(asboolE Q) -(asboolE R) (reflect_eq or3P).
by rewrite -2!(reflect_eq orP).
Qed.

Lemma or4E P Q R S : [\/ P, Q, R | S] = (P \/ Q \/ R \/ S).
Proof.
by rewrite -(asboolE P) -(asboolE Q) -(asboolE R) -(asboolE S) (reflect_eq or4P) -3!(reflect_eq orP).
rewrite -(asboolE P) -(asboolE Q) -(asboolE R) -(asboolE S) (reflect_eq or4P).
by rewrite -3!(reflect_eq orP).
Qed.

Lemma andC : commutative and.
Proof. by move=> P Q; rewrite propeqE; split=> [[]|[]]. Qed.
Proof. by move=> /PropB[] /PropB[]; rewrite !andB. Qed.

Lemma andA : associative and.
Proof. by move=> P Q R; rewrite propeqE; split=> [|]; tauto. Qed.
Expand Down

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