fact stringlengths 9 12.1k | type stringclasses 25
values | library stringclasses 6
values | imports listlengths 0 15 | filename stringclasses 105
values | symbolic_name stringlengths 1 50 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
or_asboolP (P Q : Prop) : reflect (P \/ Q) (`[< P >] || `[< Q >]). Proof. apply: (iffP idP); first by case/orP=> /asboolP; [left | right]. by case=> /asboolP-> //=; rewrite orbT. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | or_asboolP | |
or3_asboolP (P Q R : Prop) : reflect [\/ P, Q | R] [|| `[< P >], `[< Q >] | `[< R >]]. Proof. apply: (iffP idP); last by case=> [| |] /asboolP -> //=; rewrite !orbT. by case/orP=> [/asboolP p|/orP[]/asboolP]; [exact:Or31|exact:Or32|exact:Or33]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | or3_asboolP | |
asbool_neg {P : Prop} : `[<~ P>] = ~~ `[<P>]. Proof. by apply/idP/asboolPn=> [/asboolP|/asboolT]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | asbool_neg | |
asbool_or {P Q : Prop} : `[<P \/ Q>] = `[<P>] || `[<Q>]. Proof. exact: (asbool_equiv_eqP (or_asboolP _ _)). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | asbool_or | |
asbool_and {P Q : Prop} : `[<P /\ Q>] = `[<P>] && `[<Q>]. Proof. exact: (asbool_equiv_eqP (and_asboolP _ _)). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | asbool_and | |
imply_asboolP {P Q : Prop} : reflect (P -> Q) (`[<P>] ==> `[<Q>]). Proof. apply: (iffP implyP)=> [PQb /asboolP/PQb/asboolW //|]. by move=> PQ /asboolP/PQ/asboolT. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | imply_asboolP | |
asbool_imply {P Q : Prop} : `[<P -> Q>] = `[<P>] ==> `[<Q>]. Proof. exact: (asbool_equiv_eqP imply_asboolP). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | asbool_imply | |
imply_asboolPn (P Q : Prop) : reflect (P /\ ~ Q) (~~ `[<P -> Q>]). Proof. apply: (iffP idP). by rewrite asbool_imply negb_imply -asbool_neg => /and_asboolP. by move/and_asboolP; rewrite asbool_neg -negb_imply asbool_imply. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | imply_asboolPn | |
forall_asboolP {T : Type} (P : T -> Prop) : reflect (forall x, `[<P x>]) (`[<forall x, P x>]). Proof. apply: (iffP idP); first by move/asboolP=> Px x; apply/asboolP. by move=> Px; apply/asboolP=> x; apply/asboolP. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | forall_asboolP | |
exists_asboolP {T : Type} (P : T -> Prop) : reflect (exists x, `[<P x>]) (`[<exists x, P x>]). Proof. apply: (iffP idP); first by case/asboolP=> x Px; exists x; apply/asboolP. by case=> x bPx; apply/asboolP; exists x; apply/asboolP. Qed. (* -------------------------------------------------------------------- *) | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | exists_asboolP | |
BoolProp : Prop -> Type := | TrueProp : BoolProp True | FalseProp : BoolProp False. | Variant | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | BoolProp | |
PropB P : BoolProp P. Proof. by case: (asboolP P) => [/propT-> | /propF->]; [left | right]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | PropB | |
notB : ((~ True) = False) * ((~ False) = True). Proof. by rewrite /not; split; eqProp. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | notB | |
andB : left_id True and * right_id True and * (left_zero False and * right_zero False and * idempotent_op and). Proof. by do ![split] => /PropB[]; eqProp=> // -[]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | andB | |
orB : left_id False or * right_id False or * (left_zero True or * right_zero True or * idempotent_op or). Proof. do ![split] => /PropB[]; eqProp=> -[] //; by [left | right]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | orB | |
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 | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | implyB | |
decide_or P Q : P \/ Q -> {P} + {Q}. Proof. by case/PropB: P; [left | rewrite orB; right]. Qed. (* -------------------------------------------------------------------- *) | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | decide_or | |
notT (P : Prop) : P = False -> ~ P. Proof. by move->. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | notT | |
contrapT P : ~ ~ P -> P. Proof. by case: (PropB P) => //; rewrite not_False. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | contrapT | |
notTE (P : Prop) : (~ P) -> P = False. Proof. by case: (PropB P). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | notTE | |
notFE (P : Prop) : (~ P) = False -> P. Proof. by move/notT; exact: contrapT. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | notFE | |
notK : involutive not. Proof. by case/PropB; rewrite !(not_False,not_True). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | notK | |
contra_notP (Q P : Prop) : (~ Q -> P) -> ~ P -> Q. Proof. by move: Q P => /PropB[] /PropB[]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | contra_notP | |
contraPP (Q P : Prop) : (~ Q -> ~ P) -> P -> Q. Proof. by move: Q P => /PropB[] /PropB[]//; rewrite not_False not_True. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | contraPP | |
contra_notT b (P : Prop) : (~~ b -> P) -> ~ P -> b. Proof. by move=> bP; apply: contra_notP => /negP. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | contra_notT | |
contraPT (P : Prop) b : (~~ b -> ~ P) -> P -> b. Proof. by move=> /contra_notT; rewrite notK. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | contraPT | |
contraTP b (Q : Prop) : (~ Q -> ~~ b) -> b -> Q. Proof. by move=> QB; apply: contraPP => /QB/negP. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | contraTP | |
contraNP (P : Prop) (b : bool) : (~ P -> b) -> ~~ b -> P. Proof. by move=> /contra_notP + /negP => /[apply]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | contraNP | |
contra_neqP (T : eqType) (x y : T) P : (~ P -> x = y) -> x != y -> P. Proof. by move=> Pxy; apply: contraNP => /Pxy/eqP. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | contra_neqP | |
contra_eqP (T : eqType) (x y : T) Q : (~ Q -> x != y) -> x = y -> Q. Proof. by move=> Qxy /eqP; apply: contraTP. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | contra_eqP | |
contra_leP {disp1} {T1 : porderType disp1} [P : Prop] [x y : T1] : (~ P -> (x < y)%O) -> (y <= x)%O -> P. Proof. move=> Pxy yx; apply/asboolP. by apply: Order.POrderTheory.contra_leT yx => /asboolPn. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | contra_leP | |
contra_ltP {disp1} {T1 : porderType disp1} [P : Prop] [x y : T1] : (~ P -> (x <= y)%O) -> (y < x)%O -> P. Proof. move=> Pxy yx; apply/asboolP. by apply: Order.POrderTheory.contra_ltT yx => /asboolPn. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | contra_ltP | |
wlog_neg P : (~ P -> P) -> P. Proof. by case: (PropB P); exact. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | wlog_neg | |
not_inj : injective not. Proof. exact: can_inj notK. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | not_inj | |
notLR P Q : (P = (~ Q)) -> (~ P) = Q. Proof. exact: canLR notK. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | notLR | |
notRL P Q : (~ P) = Q -> P = (~ Q). Proof. exact: canRL notK. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | notRL | |
iff_notr (P Q : Prop) : (P <-> ~ Q) <-> (~ P <-> Q). Proof. by split=> [/propext ->|/propext <-]; rewrite notK. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | iff_notr | |
iff_not2 (P Q : Prop) : (~ P <-> ~ Q) <-> (P <-> Q). Proof. by split=> [/iff_notr|PQ]; [|apply/iff_notr]; rewrite notK. Qed. (* assia : let's see if we need the simplpred machinery. In any case, we sould first try definitions + appropriate Arguments setting to see whether these can replace the canonical structures mach... | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | iff_not2 | |
predp T := T -> Prop. Identity Coercion fun_of_pred : predp >-> Funclass. | Definition | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | predp | |
relp T := T -> predp T. Identity Coercion fun_of_rel : rel >-> Funclass. | Definition | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | relp | |
xpredp0 := (fun _ => False). | Notation | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | xpredp0 | |
xpredpT := (fun _ => True). | Notation | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | xpredpT | |
xpredpI := (fun (p1 p2 : predp _) x => p1 x /\ p2 x). | Notation | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | xpredpI | |
xpredpU := (fun (p1 p2 : predp _) x => p1 x \/ p2 x). | Notation | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | xpredpU | |
xpredpC := (fun (p : predp _) x => ~ p x). | Notation | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | xpredpC | |
xpredpD := (fun (p1 p2 : predp _) x => ~ p2 x /\ p1 x). | Notation | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | xpredpD | |
xpreimp := (fun f (p : predp _) x => p (f x)). | Notation | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | xpreimp | |
xrelpU := (fun (r1 r2 : relp _) x y => r1 x y \/ r2 x y). | Notation | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | xrelpU | |
pred0p (T : Type) (P : predp T) : bool := `[<P =1 xpredp0>]. Prenex Implicits pred0p. | Definition | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | pred0p | |
pred0pP (T : Type) (P : predp T) : reflect (P =1 xpredp0) (pred0p P). Proof. by apply: (iffP (asboolP _)). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | pred0pP | |
forallp_asboolPn {T} {P : T -> Prop} : reflect (forall x : T, ~ P x) (~~ `[<exists x : T, P x>]). Proof. apply: (iffP idP)=> [/asboolPn NP x Px|NP]. by apply/NP; exists x. by apply/asboolP=> -[x]; apply/NP. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | forallp_asboolPn | |
existsp_asboolPn {T} {P : T -> Prop} : reflect (exists x : T, ~ P x) (~~ `[<forall x : T, P x>]). Proof. apply: (iffP idP); last by case=> x NPx; apply/asboolPn=> /(_ x). move/asboolPn=> NP; apply/asboolP/negbNE/asboolPn=> h. by apply/NP=> x; apply/asboolP/negbNE/asboolPn=> NPx; apply/h; exists x. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | existsp_asboolPn | |
asbool_forallNb {T : Type} (P : pred T) : `[< forall x : T, ~~ (P x) >] = ~~ `[< exists x : T, P x >]. Proof. apply: (asbool_equiv_eqP forallp_asboolPn); by split=> h x; apply/negP/h. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | asbool_forallNb | |
asbool_existsNb {T : Type} (P : pred T) : `[< exists x : T, ~~ (P x) >] = ~~ `[< forall x : T, P x >]. Proof. apply: (asbool_equiv_eqP existsp_asboolPn); by split=> -[x h]; exists x; apply/negP. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | asbool_existsNb | |
not_implyP (P Q : Prop) : ~ (P -> Q) <-> P /\ ~ Q. Proof. split=> [/asboolP|[p nq pq]]; [|exact/nq/pq]. by rewrite asbool_neg => /imply_asboolPn. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | not_implyP | |
not_andE (P Q : Prop) : (~ (P /\ Q)) = (~ P \/ ~ Q). Proof. eqProp=> [/asboolPn|[|]]; try by apply: contra_not => -[]. by rewrite asbool_and negb_and => /orP[]/asboolPn; [left|right]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | not_andE | |
not_andP (P Q : Prop) : ~ (P /\ Q) <-> ~ P \/ ~ Q. Proof. by rewrite not_andE. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | not_andP | |
not_and3P (P Q R : Prop) : ~ [/\ P, Q & R] <-> [\/ ~ P, ~ Q | ~ R]. Proof. split=> [/and3_asboolP|/or3_asboolP]. by rewrite 2!negb_and -3!asbool_neg => /or3_asboolP. by rewrite 3!asbool_neg -2!negb_and => /and3_asboolP. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | not_and3P | |
not_notP (P : Prop) : ~ ~ P <-> P. Proof. by split => [|p]; [exact: contrapT|exact]. Qed. #[deprecated(since="mathcomp-analysis 1.15.0", note="Renamed to `not_notP`. Warning: a different `notP` is provided by `contra.v`.")] | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | not_notP | |
notP := not_notP (only parsing). | Notation | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | notP | |
not_notE (P : Prop) : (~ ~ P) = P. Proof. by rewrite propeqE not_notP. Qed. #[deprecated(since="mathcomp-analysis 1.15.0", note="Renamed to `not_notE`. Warning: a different `notE` is provided by `contra.v`.")] | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | not_notE | |
notE := not_notE (only parsing). | Notation | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | notE | |
not_orE (P Q : Prop) : (~ (P \/ Q)) = (~ P /\ ~ Q). Proof. by rewrite -[_ /\ _]not_notE not_andE 2!not_notE. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | not_orE | |
not_orP (P Q : Prop) : ~ (P \/ Q) <-> ~ P /\ ~ Q. Proof. by rewrite not_orE. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | not_orP | |
not_implyE (P Q : Prop) : (~ (P -> Q)) = (P /\ ~ Q). Proof. by rewrite propeqE not_implyP. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | not_implyE | |
implyE (P Q : Prop) : (P -> Q) = (~ P \/ Q). Proof. by rewrite -[LHS]not_notE not_implyE propeqE not_andP not_notE. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | implyE | |
orC : commutative or. Proof. by move=> /PropB[] /PropB[] => //; rewrite !orB. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | orC | |
orA : associative or. Proof. by move=> P Q R; rewrite propeqE; split=> [|]; tauto. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | orA | |
orCA : left_commutative or. Proof. by move=> P Q R; rewrite !orA (orC P). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | orCA | |
orAC : right_commutative or. Proof. by move=> P Q R; rewrite -!orA (orC Q). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | orAC | |
orACA : interchange or or. Proof. by move=> P Q R S; rewrite !orA (orAC P). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | orACA | |
orNp P Q : (~ P \/ Q) = (P -> Q). Proof. by case/PropB: P; rewrite notB orB implyB. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | orNp | |
orpN P Q : (P \/ ~ Q) = (Q -> P). Proof. by rewrite orC orNp. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | orpN | |
or3E P Q R : [\/ P, Q | R] = (P \/ Q \/ R). Proof. rewrite -(asboolE P) -(asboolE Q) -(asboolE R) (reflect_eq or3P). by rewrite -2!(reflect_eq orP). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | or3E | |
or4E P Q R S : [\/ P, Q, R | S] = (P \/ Q \/ R \/ S). Proof. rewrite -(asboolE P) -(asboolE Q) -(asboolE R) -(asboolE S) (reflect_eq or4P). by rewrite -3!(reflect_eq orP). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | or4E | |
andC : commutative and. Proof. by move=> /PropB[] /PropB[]; rewrite !andB. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | andC | |
andA : associative and. Proof. by move=> P Q R; rewrite propeqE; split=> [|]; tauto. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | andA | |
andCA : left_commutative and. Proof. by move=> P Q R; rewrite !andA (andC P). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | andCA | |
andAC : right_commutative and. Proof. by move=> P Q R; rewrite -!andA (andC Q). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | andAC | |
andACA : interchange and and. Proof. by move=> P Q R S; rewrite !andA (andAC P). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | andACA | |
and3E P Q R : [/\ P, Q & R] = (P /\ Q /\ R). Proof. by eqProp=> [[] | [? []]]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | and3E | |
and4E P Q R S : [/\ P, Q, R & S] = (P /\ Q /\ R /\ S). Proof. by eqProp=> [[] | [? [? []]]]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | and4E | |
and5E P Q R S T : [/\ P, Q, R, S & T] = (P /\ Q /\ R /\ S /\ T). Proof. by eqProp=> [[] | [? [? [? []]]]]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | and5E | |
implyNp P Q : (~ P -> Q : Prop) = (P \/ Q). Proof. by rewrite -orNp notK. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | implyNp | |
implypN (P Q : Prop) : (P -> ~ Q) = (~ (P /\ Q)). Proof. by case/PropB: P; rewrite implyB andB ?notB. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | implypN | |
implyNN P Q : (~ P -> ~ Q) = (Q -> P). Proof. by rewrite implyNp orpN. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | implyNN | |
or_andr : right_distributive or and. Proof. by case/PropB=> Q R; rewrite !orB ?andB. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | or_andr | |
or_andl : left_distributive or and. Proof. by move=> P Q R; rewrite -!(orC R) or_andr. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | or_andl | |
and_orr : right_distributive and or. Proof. by move=> P Q R; apply/not_inj; rewrite !(not_andE, not_orE) or_andr. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | and_orr | |
and_orl : left_distributive and or. Proof. by move=> P Q R; apply/not_inj; rewrite !(not_andE, not_orE) or_andl. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | and_orl | |
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. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | forallNE | |
existsNE {T} (P : T -> Prop) : (exists x, ~ P x) = (~ forall x, P x). Proof. rewrite propeqE; split=> [[x Px] aP //|NaP]. by apply: contrapT; rewrite -forallNE => aP; apply: NaP => x; apply: contrapT. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | existsNE | |
existsNP T (P : T -> Prop) : (exists x, ~ P x) <-> ~ forall x, P x. Proof. by rewrite existsNE. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | existsNP | |
not_existsP T (P : T -> Prop) : (exists x, P x) <-> ~ forall x, ~ P x. Proof. by rewrite forallNE notK. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | not_existsP | |
forallNP T (P : T -> Prop) : (forall x, ~ P x) <-> ~ exists x, P x. Proof. by rewrite forallNE. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | forallNP | |
not_forallP T (P : T -> Prop) : (forall x, P x) <-> ~ exists x, ~ P x. Proof. by rewrite existsNE notK. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | not_forallP | |
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 | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | exists2E | |
exists2P T (P Q : T -> Prop) : (exists2 x, P x & Q x) <-> exists x, P x /\ Q x. Proof. by rewrite exists2E. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | exists2P | |
not_exists2P T (P Q : T -> Prop) : (exists2 x, P x & Q x) <-> ~ forall x, ~ P x \/ ~ Q x. Proof. rewrite exists2P not_existsP. by split; apply: contra_not => PQx x; apply/not_andP; apply: PQx. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | not_exists2P | |
forall2NP T (P Q : T -> Prop) : (forall x, ~ P x \/ ~ Q x) <-> ~ (exists2 x, P x & Q x). Proof. split=> [PQ [t Pt Qt]|PQ t]; first by have [] := PQ t. by rewrite -not_andP => -[Pt Qt]; apply PQ; exists t. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | forall2NP |
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