fact stringlengths 6 2.88k | type stringclasses 17
values | library stringclasses 2
values | imports listlengths 0 16 | filename stringclasses 89
values | symbolic_name stringlengths 1 36 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
is_leftA B (u : {A} + {B}) := if u is left _ then true else false. | Coercion | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | is_left | |
is_inleftA B (u : A + {B}) := if u is inleft _ then true else false.
Prenex Implicits isSome is_inl is_left is_inleft. | Coercion | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | is_inleft | |
decidableP := {P} + {~ P}. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | decidable | |
if_spec(not_b : Prop) : bool -> A -> Set :=
| IfSpecTrue of b : if_spec not_b true vT
| IfSpecFalse of not_b : if_spec not_b false vF. | Variant | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | if_spec | |
ifP: if_spec (b = false) b (if b then vT else vF).
Proof. by case def_b: b; constructor. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | ifP | |
ifPn: if_spec (~~ b) b (if b then vT else vF).
Proof. by case def_b: b; constructor; rewrite ?def_b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | ifPn | |
ifT: b -> (if b then vT else vF) = vT. Proof. by move->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | ifT | |
ifF: b = false -> (if b then vT else vF) = vF. Proof. by move->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | ifF | |
ifN: ~~ b -> (if b then vT else vF) = vF. Proof. by move/negbTE->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | ifN | |
if_same: (if b then vT else vT) = vT.
Proof. by case b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | if_same | |
if_neg: (if ~~ b then vT else vF) = if b then vF else vT.
Proof. by case b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | if_neg | |
fun_if: f (if b then vT else vF) = if b then f vT else f vF.
Proof. by case b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | fun_if | |
if_arg(fT fF : A -> B) :
(if b then fT else fF) x = if b then fT x else fF x.
Proof. by case b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | if_arg | |
if_expr:= if b then vT else vF. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | if_expr | |
ifE: (if b then vT else vF) = if_expr. Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | ifE | |
introNTF: (if c then ~ P else P) -> ~~ b = c.
Proof. by case c; case Hb. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | introNTF | |
introTF: (if c then P else ~ P) -> b = c.
Proof. by case c; case Hb. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | introTF | |
elimNTF: ~~ b = c -> if c then ~ P else P.
Proof. by move <-; case Hb. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | elimNTF | |
elimTF: b = c -> if c then P else ~ P.
Proof. by move <-; case Hb. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | elimTF | |
equivPif: (Q -> P) -> (P -> Q) -> if b then Q else ~ Q.
Proof. by case Hb; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | equivPif | |
xorPif: Q \/ P -> ~ (Q /\ P) -> if b then ~ Q else Q.
Proof. by case Hb => [? _ H ? | ? H _]; case: H. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | xorPif | |
introTFn: (if c then ~ P else P) -> b = c.
Proof. by move/(introNTF Hb) <-; case b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | introTFn | |
elimTFn: b = c -> if c then ~ P else P.
Proof. by move <-; apply: (elimNTF Hb); case b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | elimTFn | |
equivPifn: (Q -> P) -> (P -> Q) -> if b then ~ Q else Q.
Proof. by rewrite -if_neg; apply: equivPif. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | equivPifn | |
xorPifn: Q \/ P -> ~ (Q /\ P) -> if b then Q else ~ Q.
Proof. by rewrite -if_neg; apply: xorPif. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | xorPifn | |
introT: P -> b. Proof. exact: introTF true _. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | introT | |
introF: ~ P -> b = false. Proof. exact: introTF false _. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | introF | |
introN: ~ P -> ~~ b. Proof. exact: introNTF true _. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | introN | |
introNf: P -> ~~ b = false. Proof. exact: introNTF false _. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | introNf | |
introTn: ~ P -> b'. Proof. exact: introTFn true _. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | introTn | |
introFn: P -> b' = false. Proof. exact: introTFn false _. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | introFn | |
elimT: b -> P. Proof. exact: elimTF true _. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | elimT | |
elimF: b = false -> ~ P. Proof. exact: elimTF false _. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | elimF | |
elimN: ~~ b -> ~P. Proof. exact: elimNTF true _. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | elimN | |
elimNf: ~~ b = false -> P. Proof. exact: elimNTF false _. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | elimNf | |
elimTn: b' -> ~ P. Proof. exact: elimTFn true _. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | elimTn | |
elimFn: b' = false -> P. Proof. exact: elimTFn false _. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | elimFn | |
introP: (b -> Q) -> (~~ b -> ~ Q) -> reflect Q b.
Proof. by case b; constructor; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | introP | |
iffP: (P -> Q) -> (Q -> P) -> reflect Q b.
Proof. by case: Pb; constructor; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | iffP | |
equivP: (P <-> Q) -> reflect Q b.
Proof. by case; apply: iffP. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | equivP | |
sumboolP(decQ : decidable Q) : reflect Q decQ.
Proof. by case: decQ; constructor. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | sumboolP | |
appP: reflect Q b -> P -> Q.
Proof. by move=> Qb; move/introT; case: Qb. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | appP | |
sameP: reflect P c -> b = c.
Proof. by case; [apply: introT | apply: introF]. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | sameP | |
decPcases: if b then P else ~ P. Proof. by case Pb. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | decPcases | |
decP: decidable P. by case: b decPcases; [left | right]. Defined. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | decP | |
rwP: P <-> b. Proof. by split; [apply: introT | apply: elimT]. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | rwP | |
rwP2: reflect Q b -> (P <-> Q).
Proof. by move=> Qb; split=> ?; [apply: appP | apply: elimT; case: Qb]. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | rwP2 | |
alt_spec: bool -> Type :=
| AltTrue of P : alt_spec true
| AltFalse of ~~ b : alt_spec false. | Variant | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | alt_spec | |
altP: alt_spec b.
Proof. by case def_b: b / Pb; constructor; rewrite ?def_b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | altP | |
eqbLR(b1 b2 : bool) : b1 = b2 -> b1 -> b2.
Proof. by move->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | eqbLR | |
eqbRL(b1 b2 : bool) : b1 = b2 -> b2 -> b1.
Proof. by move->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | eqbRL | |
elimT: reflect >-> Funclass.
#[universes(template)] | Coercion | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | elimT | |
impliesP Q := Implies of P -> Q. | Variant | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | implies | |
impliesPP Q : implies P Q -> P -> Q. Proof. by case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | impliesP | |
impliesPn(P Q : Prop) : implies P Q -> ~ Q -> ~ P.
Proof. by case=> iP ? /iP. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | impliesPn | |
impliesP: implies >-> Funclass.
Hint View for move/ impliesPn|2 impliesP|2.
Hint View for apply/ impliesPn|2 impliesP|2. | Coercion | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | impliesP | |
unlesscondition property : Prop :=
forall goal : Prop, (condition -> goal) -> (property -> goal) -> goal. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | unless | |
unlessLC P : implies C (\unless C, P).
Proof. by split=> hC G /(_ hC). Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | unlessL | |
unlessRC P : implies P (\unless C, P).
Proof. by split=> hP G _ /(_ hP). Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | unlessR | |
unless_symC P : implies (\unless C, P) (\unless P, C).
Proof. by split; apply; [apply/unlessR | apply/unlessL]. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | unless_sym | |
unlessP(C P : Prop) : (\unless C, P) <-> C \/ P.
Proof. by split=> [|[/unlessL | /unlessR]]; apply; [left | right]. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | unlessP | |
bind_unlessC P {Q} : implies (\unless C, P) (\unless (\unless C, Q), P).
Proof. by split; apply=> [hC|hP]; [apply/unlessL/unlessL | apply/unlessR]. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | bind_unless | |
unless_contrab C : implies (~~ b -> C) (\unless C, b).
Proof. by split; case: b => [_ | hC]; [apply/unlessR | apply/unlessL/hC]. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | unless_contra | |
classicallyP : Prop := forall b : bool, (P -> b) -> b. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | classically | |
classicP(P : Prop) : classically P <-> ~ ~ P.
Proof.
split=> [cP nP | nnP [] // nP]; last by case nnP; move/nP.
by have: P -> false; [move/nP | move/cP].
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | classicP | |
classicWP : P -> classically P. Proof. by move=> hP _ ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | classicW | |
classic_bindP Q : (P -> classically Q) -> classically P -> classically Q.
Proof. by move=> iPQ cP b /iPQ-/cP. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | classic_bind | |
classic_EMP : classically (decidable P).
Proof.
by case=> // undecP; apply/undecP; right=> notP; apply/notF/undecP; left.
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | classic_EM | |
classic_pickT P : classically ({x : T | P x} + (forall x, ~ P x)).
Proof.
case=> // undecP; apply/undecP; right=> x Px.
by apply/notF/undecP; left; exists x.
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | classic_pick | |
classic_implyP Q : (P -> classically Q) -> classically (P -> Q).
Proof.
move=> iPQ []// notPQ; apply/notPQ=> /iPQ-cQ.
by case: notF; apply: cQ => hQ; apply: notPQ.
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | classic_imply | |
classic_sigWT (P : T -> Prop) :
classically (exists x, P x) <-> classically ({x | P x}).
Proof. by split; apply: classic_bind => -[x Px]; apply/classicW; exists x. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | classic_sigW | |
classic_exT (P : T -> Prop) :
~ (forall x, ~ P x) -> classically (exists x, P x).
Proof.
move=> NfNP; apply/classicP => exPF; apply: NfNP => x Px.
by apply: exPF; exists x.
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | classic_ex | |
and3(P1 P2 P3 : Prop) : Prop := And3 of P1 & P2 & P3. | Inductive | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | and3 | |
and4(P1 P2 P3 P4 : Prop) : Prop := And4 of P1 & P2 & P3 & P4. | Inductive | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | and4 | |
and5(P1 P2 P3 P4 P5 : Prop) : Prop :=
And5 of P1 & P2 & P3 & P4 & P5. | Inductive | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | and5 | |
or3(P1 P2 P3 : Prop) : Prop := Or31 of P1 | Or32 of P2 | Or33 of P3. | Inductive | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | or3 | |
or4(P1 P2 P3 P4 : Prop) : Prop :=
Or41 of P1 | Or42 of P2 | Or43 of P3 | Or44 of P4. | Inductive | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | or4 | |
all_and2: implies (forall x, [/\ P1 x & P2 x]) [/\ a P1 & a P2].
Proof. by split=> haveP; split=> x; case: (haveP x). Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | all_and2 | |
all_and3: implies (forall x, [/\ P1 x, P2 x & P3 x])
[/\ a P1, a P2 & a P3].
Proof. by split=> haveP; split=> x; case: (haveP x). Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | all_and3 | |
all_and4: implies (forall x, [/\ P1 x, P2 x, P3 x & P4 x])
[/\ a P1, a P2, a P3 & a P4].
Proof. by split=> haveP; split=> x; case: (haveP x). Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | all_and4 | |
all_and5: implies (forall x, [/\ P1 x, P2 x, P3 x, P4 x & P5 x])
[/\ a P1, a P2, a P3, a P4 & a P5].
Proof. by split=> haveP; split=> x; case: (haveP x). Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | all_and5 | |
pair_andPP Q : P /\ Q <-> P * Q. Proof. by split; case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | pair_andP | |
idP: reflect b1 b1.
Proof. by case b1; constructor. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | idP | |
boolP: alt_spec b1 b1 b1.
Proof. exact: (altP idP). Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | boolP | |
idPn: reflect (~~ b1) (~~ b1).
Proof. by case b1; constructor. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | idPn | |
negP: reflect (~ b1) (~~ b1).
Proof. by case b1; constructor; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | negP | |
negPn: reflect b1 (~~ ~~ b1).
Proof. by case b1; constructor. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | negPn | |
negPf: reflect (b1 = false) (~~ b1).
Proof. by case b1; constructor. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | negPf | |
andP: reflect (b1 /\ b2) (b1 && b2).
Proof. by case b1; case b2; constructor=> //; case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andP | |
and3P: reflect [/\ b1, b2 & b3] [&& b1, b2 & b3].
Proof. by case b1; case b2; case b3; constructor; try by case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | and3P | |
and4P: reflect [/\ b1, b2, b3 & b4] [&& b1, b2, b3 & b4].
Proof. by case b1; case b2; case b3; case b4; constructor; try by case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | and4P | |
and5P: reflect [/\ b1, b2, b3, b4 & b5] [&& b1, b2, b3, b4 & b5].
Proof.
by case b1; case b2; case b3; case b4; case b5; constructor; try by case.
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | and5P | |
orP: reflect (b1 \/ b2) (b1 || b2).
Proof. by case b1; case b2; constructor; auto; case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orP | |
or3P: reflect [\/ b1, b2 | b3] [|| b1, b2 | b3].
Proof.
case b1; first by constructor; constructor 1.
case b2; first by constructor; constructor 2.
case b3; first by constructor; constructor 3.
by constructor; case.
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | or3P | |
or4P: reflect [\/ b1, b2, b3 | b4] [|| b1, b2, b3 | b4].
Proof.
case b1; first by constructor; constructor 1.
case b2; first by constructor; constructor 2.
case b3; first by constructor; constructor 3.
case b4; first by constructor; constructor 4.
by constructor; case.
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | or4P | |
nandP: reflect (~~ b1 \/ ~~ b2) (~~ (b1 && b2)).
Proof. by case b1; case b2; constructor; auto; case; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | nandP | |
norP: reflect (~~ b1 /\ ~~ b2) (~~ (b1 || b2)).
Proof. by case b1; case b2; constructor; auto; case; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | norP | |
implyP: reflect (b1 -> b2) (b1 ==> b2).
Proof. by case b1; case b2; constructor; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | implyP | |
negPP: reflect (~ P) (~~ p).
Proof. by apply:(iffP negP); apply: contra_not => /rP. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | negPP | |
andPP: reflect (P /\ Q) (p && q).
Proof. by apply: (iffP andP) => -[/rP ? /rQ ?]. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andPP |
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