From c185c60d78031ee6d796792b46fc2ad9f5cf9467 Mon Sep 17 00:00:00 2001 From: Quentin Vermande Date: Sun, 7 Jan 2024 17:40:49 +0100 Subject: [PATCH] helper lemmas for contra (PR #1119) --- CHANGELOG_UNRELEASED.md | 12 ++++ classical/boolp.v | 129 +++++++++++++++++++++++++++++++++++++--- 2 files changed, 133 insertions(+), 8 deletions(-) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index 496ab1bb9..fe47e5a9f 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -132,6 +132,15 @@ - in `realfun.v`: + lemmas `lime_sup_lim`, `lime_inf_lim` +- in `boolp.v`: + + tactic `eqProp` + + definition `BoolProp` + + lemmas `PropB`, `notB`, `andB`, `orB`, `implyB`, `decide_or`, `not_andE`, + `not_orE`, `orCA`, `orAC`, `orACA`, `orNp`, `orpN`, `or3E`, `or4E`, `andCA`, + `andAC`, `andACA`, `and3E`, `and4E`, `and5E`, `implyNp`, `implypN`, + `implyNN`, `or_andr`, `or_andl`, `and_orr`, `and_orl`, `exists2E`, + `inhabitedE`, `inhabited_witness` + ### Changed - in `normedtype.v`: @@ -169,6 +178,9 @@ - in `sequences.v`: + `limn_esup` now defined from `lime_sup` + `limn_einf` now defined from `limn_esup` + +-in `boolp.v` + - lemmas `orC` and `andC` now use `commutative` ### Renamed diff --git a/classical/boolp.v b/classical/boolp.v index 1bb538d61..fc25b4115 100644 --- a/classical/boolp.v +++ b/classical/boolp.v @@ -90,6 +90,8 @@ Qed. Lemma propext (P Q : Prop) : (P <-> Q) -> (P = Q). Proof. by have [propext _] := extensionality; apply: propext. Qed. +Ltac eqProp := apply: propext; split. + Lemma funext {T U : Type} (f g : T -> U) : (f =1 g) -> f = g. Proof. by case: extensionality=> _; apply. Qed. @@ -448,6 +450,36 @@ apply: (iffP idP); first by case/asboolP=> x Px; exists x; apply/asboolP. by case=> x bPx; apply/asboolP; exists x; apply/asboolP. Qed. +(* -------------------------------------------------------------------- *) + +Variant BoolProp : Prop -> Type := + | TrueProp : BoolProp True + | FalseProp : BoolProp False. + +Lemma PropB P : BoolProp P. +Proof. by case: (asboolP P) => [/propT-> | /propF->]; [left | right]. Qed. + +Lemma notB : ((~ True) = False) * ((~ False) = True). +Proof. by rewrite /not; split; eqProp. Qed. + +Lemma andB : left_id True and * right_id True and + * (left_zero False and * right_zero False and * idempotent and). +Proof. by do ![split] => /PropB[]; eqProp=> // -[]. Qed. + +Lemma orB : left_id False or * right_id False or + * (left_zero True or * right_zero True or * idempotent or). +Proof. do ![split] => /PropB[]; eqProp=> -[] //; by [left | right]. Qed. + +Lemma implyB : let imply (P Q : Prop) := P -> Q in + (imply False =1 fun=> True) * (imply^~ False =1 not) + * (left_id True imply * right_zero True imply * self_inverse True imply). +Proof. by do ![split] => /PropB[]; eqProp=> //; apply. Qed. + +Lemma decide_or P Q : P \/ Q -> {P} + {Q}. +Proof. by case/PropB: P; [left | rewrite orB; right]. Qed. + +(* -------------------------------------------------------------------- *) + Lemma notT (P : Prop) : P = False -> ~ P. Proof. by move->. Qed. Lemma contrapT P : ~ ~ P -> P. @@ -587,12 +619,15 @@ split=> [/asboolP|[p nq pq]]; [|exact/nq/pq]. by rewrite asbool_neg => /imply_asboolPn. Qed. -Lemma not_andP (P Q : Prop) : ~ (P /\ Q) <-> ~ P \/ ~ Q. +Lemma not_andE (P Q : Prop) : (~ (P /\ Q)) = ~ P \/ ~ Q. Proof. -split => [/asboolPn|[|]]; try by apply: contra_not => -[]. +eqProp=> [/asboolPn|[|]]; try by apply: contra_not => -[]. by rewrite asbool_and negb_and => /orP[]/asboolPn; [left|right]. Qed. +Lemma not_andP (P Q : Prop) : ~ (P /\ Q) <-> ~ P \/ ~ Q. +Proof. by rewrite not_andE. Qed. + Lemma not_and3P (P Q R : Prop) : ~ [/\ P, Q & R] <-> [\/ ~ P, ~ Q | ~ R]. Proof. split=> [/and3_asboolP|/or3_asboolP]. @@ -605,8 +640,11 @@ Proof. by split => [|p]; [exact: contrapT|exact]. Qed. Lemma notE (P : Prop) : (~ ~ P) = P. Proof. by rewrite propeqE notP. Qed. +Lemma not_orE (P Q : Prop) : (~ (P \/ Q)) = ~ P /\ ~ Q. +Proof. by rewrite -[_ /\ _]notE not_andE 2!notE. Qed. + Lemma not_orP (P Q : Prop) : ~ (P \/ Q) <-> ~ P /\ ~ Q. -Proof. by rewrite -(notP (_ /\ _)) not_andP 2!notE. Qed. +Proof. by rewrite not_orE. Qed. Lemma not_implyE (P Q : Prop) : (~ (P -> Q)) = (P /\ ~ Q). Proof. by rewrite propeqE not_implyP. Qed. @@ -614,18 +652,79 @@ Proof. by rewrite propeqE not_implyP. Qed. Lemma implyE (P Q : Prop) : (P -> Q) = (~ P \/ Q). Proof. by rewrite -[LHS]notE not_implyE propeqE not_andP notE. Qed. -Lemma orC (P Q : Prop) : (P \/ Q) = (Q \/ P). -Proof. by rewrite propeqE; split=> [[]|[]]; [right|left|right|left]. Qed. +Lemma orC : commutative or. +Proof. by move=> P Q; rewrite propeqE; split; (case=> ?; [right|left]). Qed. Lemma orA : associative or. Proof. by move=> P Q R; rewrite propeqE; split=> [|]; tauto. Qed. -Lemma andC (P Q : Prop) : (P /\ Q) = (Q /\ P). -Proof. by rewrite propeqE; split=> [[]|[]]. Qed. +Lemma orCA : left_commutative or. +Proof. by move=> P Q R; rewrite !orA (orC P). Qed. + +Lemma orAC : right_commutative or. +Proof. by move=> P Q R; rewrite -!orA (orC Q). Qed. + +Lemma orACA : interchange or or. +Proof. by move=> P Q R S; rewrite !orA (orAC P). Qed. + +Lemma orNp P Q : (~ P \/ Q) = (P -> Q). +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. + +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). +Qed. + +Lemma andC : commutative and. +Proof. by move=> P Q; rewrite propeqE; split=> [[]|[]]. Qed. Lemma andA : associative and. Proof. by move=> P Q R; rewrite propeqE; split=> [|]; tauto. Qed. +Lemma andCA : left_commutative and. +Proof. by move=> P Q R; rewrite !andA (andC P). Qed. + +Lemma andAC : right_commutative and. +Proof. by move=> P Q R; rewrite -!andA (andC Q). Qed. + +Lemma andACA : interchange and and. +Proof. by move=> P Q R S; rewrite !andA (andAC P). Qed. + +Lemma and3E P Q R : [/\ P, Q & R] = (P /\ Q /\ R). +Proof. by eqProp=> [[] | [? []]]. Qed. + +Lemma and4E P Q R S : [/\ P, Q, R & S] = (P /\ Q /\ R /\ S). +Proof. by eqProp=> [[] | [? [? []]]]. Qed. + +Lemma and5E P Q R S T : [/\ P, Q, R, S & T] = (P /\ Q /\ R /\ S /\ T). +Proof. by eqProp=> [[] | [? [? [? []]]]]. Qed. + +Lemma implyNp P Q : (~ P -> Q : Prop) = (P \/ Q). +Proof. by rewrite -orNp notK. Qed. + +Lemma implypN (P Q : Prop) : (P -> ~ Q) = ~ (P /\ Q). +Proof. by case/PropB: P; rewrite implyB andB ?notB. Qed. + +Lemma implyNN P Q : (~ P -> ~ Q) = (Q -> P). +Proof. by rewrite implyNp orpN. Qed. + +Lemma or_andr : right_distributive or and. +Proof. by case/PropB=> Q R; rewrite !orB ?andB. Qed. + +Lemma or_andl : left_distributive or and. +Proof. by move=> P Q R; rewrite -!(orC R) or_andr. Qed. + +Lemma and_orr : right_distributive and or. +Proof. by move=> P Q R; apply/not_inj; rewrite !(not_andE, not_orE) or_andr. Qed. + +Lemma and_orl : left_distributive and or. +Proof. by move=> P Q R; apply/not_inj; rewrite !(not_andE, not_orE) or_andl. Qed. + Lemma forallNE {T} (P : T -> Prop) : (forall x, ~ P x) = ~ exists x, P x. Proof. by rewrite propeqE; split => [fP [x /fP]//|nexP x Px]; apply: nexP; exists x. @@ -649,9 +748,12 @@ Proof. by rewrite forallNE. Qed. Lemma not_forallP T (P : T -> Prop) : (forall x, P x) <-> ~ exists x, ~ P x. Proof. by rewrite existsNE notK. Qed. +Lemma exists2E A P Q : (exists2 x : A, P x & Q x) = (exists x, P x /\ Q x). +Proof. by eqProp=> -[x]; last case; exists x. Qed. + Lemma exists2P T (P Q : T -> Prop) : (exists2 x, P x & Q x) <-> exists x, P x /\ Q x. -Proof. by split=> [[x ? ?] | [x []]]; exists x. Qed. +Proof. by rewrite exists2E. Qed. Lemma not_exists2P T (P Q : T -> Prop) : (exists2 x, P x & Q x) <-> ~ forall x, ~ P x \/ ~ Q x. @@ -801,3 +903,14 @@ Proof. by apply/funeqP => ?; rewrite iterSr. Qed. Lemma iter0 {T} (f : T -> T) : iter 0 f = id. Proof. by []. Qed. + +Section Inhabited. +Variable (T : Type). + +Lemma inhabitedE: inhabited T = exists x : T, True. +Proof. by eqProp; case. Qed. + +Lemma inhabited_witness: inhabited T -> T. +Proof. by rewrite inhabitedE => /cid[]. Qed. + +End Inhabited.