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
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|---|---|---|---|---|---|---|
orPP: reflect (P \/ Q) (p || q).
Proof. by apply: (iffP orP) => -[/rP ?|/rQ ?]; tauto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orPP | |
implyPP: reflect (P -> Q) (p ==> q).
Proof. by apply: (iffP implyP) => pq /rP /pq /rQ. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | implyPP | |
andTb: left_id true andb. Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andTb | |
andFb: left_zero false andb. Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andFb | |
andbT: right_id true andb. Proof. by case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andbT | |
andbF: right_zero false andb. Proof. by case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andbF | |
andbb: idempotent_op andb. Proof. by case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andbb | |
andbC: commutative andb. Proof. by do 2!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andbC | |
andbA: associative andb. Proof. by do 3!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andbA | |
andbCA: left_commutative andb. Proof. by do 3!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andbCA | |
andbAC: right_commutative andb. Proof. by do 3!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andbAC | |
andbACA: interchange andb andb. Proof. by do 4!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andbACA | |
orTb: forall b, true || b. Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orTb | |
orFb: left_id false orb. Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orFb | |
orbT: forall b, b || true. Proof. by case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orbT | |
orbF: right_id false orb. Proof. by case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orbF | |
orbb: idempotent_op orb. Proof. by case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orbb | |
orbC: commutative orb. Proof. by do 2!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orbC | |
orbA: associative orb. Proof. by do 3!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orbA | |
orbCA: left_commutative orb. Proof. by do 3!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orbCA | |
orbAC: right_commutative orb. Proof. by do 3!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orbAC | |
orbACA: interchange orb orb. Proof. by do 4!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orbACA | |
andbNb : b && ~~ b = false. Proof. by case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andbN | |
andNbb : ~~ b && b = false. Proof. by case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andNb | |
orbNb : b || ~~ b = true. Proof. by case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orbN | |
orNbb : ~~ b || b = true. Proof. by case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orNb | |
andb_orl: left_distributive andb orb. Proof. by do 3!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andb_orl | |
andb_orr: right_distributive andb orb. Proof. by do 3!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andb_orr | |
orb_andl: left_distributive orb andb. Proof. by do 3!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orb_andl | |
orb_andr: right_distributive orb andb. Proof. by do 3!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orb_andr | |
andb_idl(a b : bool) : (b -> a) -> a && b = b.
Proof. by case: a; case: b => // ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andb_idl | |
andb_idr(a b : bool) : (a -> b) -> a && b = a.
Proof. by case: a; case: b => // ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andb_idr | |
andb_id2l(a b c : bool) : (a -> b = c) -> a && b = a && c.
Proof. by case: a; case: b; case: c => // ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andb_id2l | |
andb_id2r(a b c : bool) : (b -> a = c) -> a && b = c && b.
Proof. by case: a; case: b; case: c => // ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andb_id2r | |
orb_idl(a b : bool) : (a -> b) -> a || b = b.
Proof. by case: a; case: b => // ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orb_idl | |
orb_idr(a b : bool) : (b -> a) -> a || b = a.
Proof. by case: a; case: b => // ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orb_idr | |
orb_id2l(a b c : bool) : (~~ a -> b = c) -> a || b = a || c.
Proof. by case: a; case: b; case: c => // ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orb_id2l | |
orb_id2r(a b c : bool) : (~~ b -> a = c) -> a || b = c || b.
Proof. by case: a; case: b; case: c => // ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orb_id2r | |
negb_and(a b : bool) : ~~ (a && b) = ~~ a || ~~ b.
Proof. by case: a; case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | negb_and | |
negb_or(a b : bool) : ~~ (a || b) = ~~ a && ~~ b.
Proof. by case: a; case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | negb_or | |
andbKa b : a && b || a = a. Proof. by case: a; case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andbK | |
andKba b : a || b && a = a. Proof. by case: a; case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andKb | |
orbKa b : (a || b) && a = a. Proof. by case: a; case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orbK | |
orKba b : a && (b || a) = a. Proof. by case: a; case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | orKb | |
implybTb : b ==> true. Proof. by case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | implybT | |
implybFb : (b ==> false) = ~~ b. Proof. by case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | implybF | |
implyFbb : false ==> b. Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | implyFb | |
implyTbb : (true ==> b) = b. Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | implyTb | |
implybbb : b ==> b. Proof. by case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | implybb | |
negb_implya b : ~~ (a ==> b) = a && ~~ b.
Proof. by case: a; case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | negb_imply | |
implybEa b : (a ==> b) = ~~ a || b.
Proof. by case: a; case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | implybE | |
implyNba b : (~~ a ==> b) = a || b.
Proof. by case: a; case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | implyNb | |
implybNa b : (a ==> ~~ b) = (b ==> ~~ a).
Proof. by case: a; case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | implybN | |
implybNNa b : (~~ a ==> ~~ b) = b ==> a.
Proof. by case: a; case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | implybNN | |
implyb_idl(a b : bool) : (~~ a -> b) -> (a ==> b) = b.
Proof. by case: a; case: b => // ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | implyb_idl | |
implyb_idr(a b : bool) : (b -> ~~ a) -> (a ==> b) = ~~ a.
Proof. by case: a; case: b => // ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | implyb_idr | |
implyb_id2l(a b c : bool) : (a -> b = c) -> (a ==> b) = (a ==> c).
Proof. by case: a; case: b; case: c => // ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | implyb_id2l | |
addFb: left_id false addb. Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | addFb | |
addbF: right_id false addb. Proof. by case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | addbF | |
addbb: self_inverse false addb. Proof. by case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | addbb | |
addbC: commutative addb. Proof. by do 2!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | addbC | |
addbA: associative addb. Proof. by do 3!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | addbA | |
addbCA: left_commutative addb. Proof. by do 3!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | addbCA | |
addbAC: right_commutative addb. Proof. by do 3!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | addbAC | |
addbACA: interchange addb addb. Proof. by do 4!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | addbACA | |
andb_addl: left_distributive andb addb. Proof. by do 3!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andb_addl | |
andb_addr: right_distributive andb addb. Proof. by do 3!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | andb_addr | |
addKb: left_loop id addb. Proof. by do 2!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | addKb | |
addbK: right_loop id addb. Proof. by do 2!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | addbK | |
addIb: left_injective addb. Proof. by do 3!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | addIb | |
addbI: right_injective addb. Proof. by do 3!case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | addbI | |
addTbb : true (+) b = ~~ b. Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | addTb | |
addbTb : b (+) true = ~~ b. Proof. by case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | addbT | |
addbNa b : a (+) ~~ b = ~~ (a (+) b).
Proof. by case: a; case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | addbN | |
addNba b : ~~ a (+) b = ~~ (a (+) b).
Proof. by case: a; case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | addNb | |
addbPa b : reflect (~~ a = b) (a (+) b).
Proof. by case: a; case: b; constructor. Qed.
Arguments addbP {a b}. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | addbP | |
bool_congr:=
match goal with
| |- (?X1 && ?X2 = ?X3) => first
[ symmetry; rewrite -1?(andbC X1) -?(andbCA X1); congr 1 (andb X1); symmetry
| case: (X1); [ rewrite ?andTb ?andbT // | by rewrite ?andbF /= ] ]
| |- (?X1 || ?X2 = ?X3) => first
[ symmetry; rewrite -1?(orbC X1) -?(orbCA X1); congr 1 (orb X1); sym... | Ltac | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | bool_congr | |
eq_bigin bigop).
We also sometimes "instantiate" the predType structure by defining a
coercion to the sort of the predPredType structure, conveniently denoted
{pred T}. This works better for types such as {set T} that have subtypes that
coerce to them, since the same coercion will be inserted by the application
... | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | eq_big | |
predT := T -> bool.
Identity Coercion fun_of_pred : pred >-> Funclass. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | pred | |
subpredT (p1 p2 : pred T) := forall x : T, p1 x -> p2 x. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | subpred | |
predTypeT :=
PredType {pred_sort :> Type; topred : pred_sort -> pred T}. | Structure | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | predType | |
clone_predT U :=
fun pT & @pred_sort T pT -> U =>
fun toP (pT' := @PredType T U toP) & phant_id pT' pT => pT'. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | clone_pred | |
simpl_predT := simpl_fun T bool. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | simpl_pred | |
SimplPred{T} (p : pred T) : simpl_pred T := SimplFun p. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | SimplPred | |
pred0{T} := @SimplPred T xpred0. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | pred0 | |
predT{T} := @SimplPred T xpredT. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | predT | |
predI{T} (p1 p2 : pred T) := SimplPred (xpredI p1 p2). | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | predI | |
predU{T} (p1 p2 : pred T) := SimplPred (xpredU p1 p2). | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | predU | |
predC{T} (p : pred T) := SimplPred (xpredC p). | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | predC | |
predD{T} (p1 p2 : pred T) := SimplPred (xpredD p1 p2). | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | predD | |
preim{aT rT} (f : aT -> rT) (d : pred rT) := SimplPred (xpreim f d). | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | preim | |
coerceT (sp : simpl_pred T) : pred T := fun_of_simpl sp. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | coerce | |
pred_of_simpl: simpl_pred >-> pred.
Canonical simplPredType T := PredType (@pred_of_simpl T). | Coercion | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | pred_of_simpl | |
coerce: forall T, simpl_pred T -> {pred T}. | Parameter | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | coerce | |
PredSortOfSimpl.coerce : simpl_pred >-> pred_sort. | Coercion | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | PredSortOfSimpl | |
predArgType:= Type.
Bind Scope type_scope with predArgType.
Identity Coercion sort_of_predArgType : predArgType >-> Sortclass. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | predArgType | |
pred_of_argType(T : predArgType) : simpl_pred T := predT. | Coercion | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | pred_of_argType | |
relT := T -> pred T.
Identity Coercion fun_of_rel : rel >-> Funclass. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | rel | |
subrelT (r1 r2 : rel T) := forall x y : T, r1 x y -> r2 x y. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | subrel | |
simpl_relT := T -> simpl_pred T. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | simpl_rel |
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