fact stringlengths 8 1.54k | type stringclasses 19 values | library stringclasses 8 values | imports listlengths 1 10 | filename stringclasses 98 values | symbolic_name stringlengths 1 42 | docstring stringclasses 1 value |
|---|---|---|---|---|---|---|
setTUA : setT :|: A = setT.
Proof. by apply/setP => x; rewrite !inE orTb. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setTU | |
setUTA : A :|: setT = setT.
Proof. by rewrite setUC setTU. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setUT | |
setUidA : A :|: A = A.
Proof. by apply/setP=> x; rewrite inE orbb. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setUid | |
setUUlA B C : A :|: B :|: C = (A :|: C) :|: (B :|: C).
Proof. by rewrite setUA !(setUAC _ C) -(setUA _ C) setUid. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setUUl | |
setUUrA B C : A :|: (B :|: C) = (A :|: B) :|: (A :|: C).
Proof. by rewrite !(setUC A) setUUl. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setUUr | |
setIdPx pA pB : reflect (pA x /\ pB x) (x \in [set y | pA y & pB y]).
Proof. by rewrite !inE; apply: andP. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setIdP | |
setId2Px pA pB pC :
reflect [/\ pA x, pB x & pC x] (x \in [set y | pA y & pB y && pC y]).
Proof. by rewrite !inE; apply: and3P. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setId2P | |
setIdEA pB : [set x in A | pB x] = A :&: [set x | pB x].
Proof. by apply/setP=> x; rewrite !inE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setIdE | |
setIPx A B : reflect (x \in A /\ x \in B) (x \in A :&: B).
Proof. exact: (iffP (@setIdP _ _ _)). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setIP | |
in_setIx A B : (x \in A :&: B) = (x \in A) && (x \in B).
Proof. exact: in_set. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | in_setI | |
setICA B : A :&: B = B :&: A.
Proof. by apply/setP => x; rewrite !inE andbC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setIC | |
setISA B C : A \subset B -> C :&: A \subset C :&: B.
Proof.
move=> sAB; apply/subsetP=> x; rewrite !inE.
by case: (x \in C) => //; apply: (subsetP sAB).
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setIS | |
setSIA B C : A \subset B -> A :&: C \subset B :&: C.
Proof. by move=> sAB; rewrite -!(setIC C) setIS. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setSI | |
setISSA B C D : A \subset C -> B \subset D -> A :&: B \subset C :&: D.
Proof. by move=> /(setSI B) /subset_trans sAC /(setIS C) /sAC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setISS | |
setTIA : setT :&: A = A.
Proof. by apply/setP => x; rewrite !inE andTb. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setTI | |
setITA : A :&: setT = A.
Proof. by rewrite setIC setTI. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setIT | |
set0IA : set0 :&: A = set0.
Proof. by apply/setP => x; rewrite !inE andFb. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set0I | |
setI0A : A :&: set0 = set0.
Proof. by rewrite setIC set0I. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setI0 | |
setIAA B C : A :&: (B :&: C) = A :&: B :&: C.
Proof. by apply/setP=> x; rewrite !inE andbA. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setIA | |
setICAA B C : A :&: (B :&: C) = B :&: (A :&: C).
Proof. by rewrite !setIA (setIC A). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setICA | |
setIACA B C : A :&: B :&: C = A :&: C :&: B.
Proof. by rewrite -!setIA (setIC B). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setIAC | |
setIACAA B C D : (A :&: B) :&: (C :&: D) = (A :&: C) :&: (B :&: D).
Proof. by rewrite -!setIA (setICA B). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setIACA | |
setIidA : A :&: A = A.
Proof. by apply/setP=> x; rewrite inE andbb. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setIid | |
setIIlA B C : A :&: B :&: C = (A :&: C) :&: (B :&: C).
Proof. by rewrite setIA !(setIAC _ C) -(setIA _ C) setIid. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setIIl | |
setIIrA B C : A :&: (B :&: C) = (A :&: B) :&: (A :&: C).
Proof. by rewrite !(setIC A) setIIl. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setIIr | |
setIUrA B C : A :&: (B :|: C) = (A :&: B) :|: (A :&: C).
Proof. by apply/setP=> x; rewrite !inE andb_orr. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setIUr | |
setIUlA B C : (A :|: B) :&: C = (A :&: C) :|: (B :&: C).
Proof. by apply/setP=> x; rewrite !inE andb_orl. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setIUl | |
setUIrA B C : A :|: (B :&: C) = (A :|: B) :&: (A :|: C).
Proof. by apply/setP=> x; rewrite !inE orb_andr. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setUIr | |
setUIlA B C : (A :&: B) :|: C = (A :|: C) :&: (B :|: C).
Proof. by apply/setP=> x; rewrite !inE orb_andl. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setUIl | |
setUKA B : (A :|: B) :&: A = A.
Proof. by apply/setP=> x; rewrite !inE orbK. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setUK | |
setKUA B : A :&: (B :|: A) = A.
Proof. by apply/setP=> x; rewrite !inE orKb. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setKU | |
setIKA B : (A :&: B) :|: A = A.
Proof. by apply/setP=> x; rewrite !inE andbK. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setIK | |
setKIA B : A :|: (B :&: A) = A.
Proof. by apply/setP=> x; rewrite !inE andKb. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setKI | |
setCPx A : reflect (~ x \in A) (x \in ~: A).
Proof. by rewrite !inE; apply: negP. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setCP | |
in_setCx A : (x \in ~: A) = (x \notin A).
Proof. exact: in_set. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | in_setC | |
setCK: involutive (@setC T).
Proof. by move=> A; apply/setP=> x; rewrite !inE negbK. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setCK | |
setC_inj: injective (@setC T).
Proof. exact: can_inj setCK. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setC_inj | |
subsets_disjointA B : (A \subset B) = [disjoint A & ~: B].
Proof. by rewrite subset_disjoint; apply: eq_disjoint_r => x; rewrite !inE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | subsets_disjoint | |
disjoints_subsetA B : [disjoint A & B] = (A \subset ~: B).
Proof. by rewrite subsets_disjoint setCK. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | disjoints_subset | |
powersetCEA B : (A \in powerset (~: B)) = [disjoint A & B].
Proof. by rewrite inE disjoints_subset. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | powersetCE | |
setCSA B : (~: A \subset ~: B) = (B \subset A).
Proof. by rewrite !subsets_disjoint setCK disjoint_sym. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setCS | |
setCUA B : ~: (A :|: B) = ~: A :&: ~: B.
Proof. by apply/setP=> x; rewrite !inE negb_or. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setCU | |
setCIA B : ~: (A :&: B) = ~: A :|: ~: B.
Proof. by apply/setP=> x; rewrite !inE negb_and. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setCI | |
setUCrA : A :|: ~: A = setT.
Proof. by apply/setP=> x; rewrite !inE orbN. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setUCr | |
setICrA : A :&: ~: A = set0.
Proof. by apply/setP=> x; rewrite !inE andbN. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setICr | |
setC0: ~: set0 = [set: T].
Proof. by apply/setP=> x; rewrite !inE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setC0 | |
setCT: ~: [set: T] = set0.
Proof. by rewrite -setC0 setCK. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setCT | |
properCA B : (~: B \proper ~: A) = (A \proper B).
Proof. by rewrite !properE !setCS. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | properC | |
setDPA B x : reflect (x \in A /\ x \notin B) (x \in A :\: B).
Proof. by rewrite inE andbC; apply: andP. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setDP | |
in_setDA B x : (x \in A :\: B) = (x \notin B) && (x \in A).
Proof. exact: in_set. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | in_setD | |
setDEA B : A :\: B = A :&: ~: B.
Proof. by apply/setP => x; rewrite !inE andbC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setDE | |
setSDA B C : A \subset B -> A :\: C \subset B :\: C.
Proof. by rewrite !setDE; apply: setSI. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setSD | |
setDSA B C : A \subset B -> C :\: B \subset C :\: A.
Proof. by rewrite !setDE -setCS; apply: setIS. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setDS | |
setDSSA B C D : A \subset C -> D \subset B -> A :\: B \subset C :\: D.
Proof. by move=> /(setSD B) /subset_trans sAC /(setDS C) /sAC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setDSS | |
setD0A : A :\: set0 = A.
Proof. by apply/setP=> x; rewrite !inE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setD0 | |
set0DA : set0 :\: A = set0.
Proof. by apply/setP=> x; rewrite !inE andbF. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set0D | |
setDTA : A :\: setT = set0.
Proof. by apply/setP=> x; rewrite !inE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setDT | |
setTDA : setT :\: A = ~: A.
Proof. by apply/setP=> x; rewrite !inE andbT. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setTD | |
setDvA : A :\: A = set0.
Proof. by apply/setP=> x; rewrite !inE andNb. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setDv | |
setCDA B : ~: (A :\: B) = ~: A :|: B.
Proof. by rewrite !setDE setCI setCK. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setCD | |
setIDA B : A :&: B :|: A :\: B = A.
Proof. by rewrite setDE -setIUr setUCr setIT. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setID | |
setDUlA B C : (A :|: B) :\: C = (A :\: C) :|: (B :\: C).
Proof. by rewrite !setDE setIUl. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setDUl | |
setDUrA B C : A :\: (B :|: C) = (A :\: B) :&: (A :\: C).
Proof. by rewrite !setDE setCU setIIr. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setDUr | |
setDIlA B C : (A :&: B) :\: C = (A :\: C) :&: (B :\: C).
Proof. by rewrite !setDE setIIl. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setDIl | |
setIDAA B C : A :&: (B :\: C) = (A :&: B) :\: C.
Proof. by rewrite !setDE setIA. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setIDA | |
setIDACA B C : (A :\: B) :&: C = (A :&: C) :\: B.
Proof. by rewrite !setDE setIAC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setIDAC | |
setDIrA B C : A :\: (B :&: C) = (A :\: B) :|: (A :\: C).
Proof. by rewrite !setDE setCI setIUr. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setDIr | |
setDDlA B C : (A :\: B) :\: C = A :\: (B :|: C).
Proof. by rewrite !setDE setCU setIA. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setDDl | |
setDDrA B C : A :\: (B :\: C) = (A :\: B) :|: (A :&: C).
Proof. by rewrite !setDE setCI setIUr setCK. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setDDr | |
powersetEA B : (A \in powerset B) = (A \subset B).
Proof. by rewrite inE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | powersetE | |
powersetSA B : (powerset A \subset powerset B) = (A \subset B).
Proof.
apply/subsetP/idP=> [sAB | sAB C /[!inE]/subset_trans->//].
by rewrite -powersetE sAB // inE.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | powersetS | |
powerset0: powerset set0 = [set set0] :> {set {set T}}.
Proof. by apply/setP=> A; rewrite set1.unlock !inE subset0. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | powerset0 | |
powersetT: powerset [set: T] = [set: {set T}].
Proof. by apply/setP=> A; rewrite !inE subsetT. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | powersetT | |
setI_powersetP A : P :&: powerset A = P ::&: A.
Proof. by apply/setP=> B; rewrite !inE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | setI_powerset | |
cardsEpA : #|[set x in pA]| = #|pA|.
Proof. exact/eq_card/in_set. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | cardsE | |
sum1dep_cardpA : \sum_(x | pA x) 1 = #|[set x | pA x]|.
Proof. by rewrite sum1_card cardsE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | sum1dep_card | |
sum_nat_cond_constpA n : \sum_(x | pA x) n = #|[set x | pA x]| * n.
Proof. by rewrite sum_nat_const cardsE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | sum_nat_cond_const | |
cards0: #|@set0 T| = 0.
Proof. by rewrite cardsE card0. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | cards0 | |
cards_eq0A : (#|A| == 0) = (A == set0).
Proof. by rewrite (eq_sym A) eqEcard sub0set cards0 leqn0. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | cards_eq0 | |
set0PnA : reflect (exists x, x \in A) (A != set0).
Proof. by rewrite -cards_eq0; apply: existsP. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set0Pn | |
set0_NexistsA : (A == set0) = ~~ [exists x, x \in A].
Proof. by rewrite -(sameP (set0Pn _) existsP) negbK. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | set0_Nexists | |
card_gt0A : (0 < #|A|) = (A != set0).
Proof. by rewrite lt0n cards_eq0. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | card_gt0 | |
cards0_eqA : #|A| = 0 -> A = set0.
Proof. by move=> A_0; apply/setP=> x; rewrite inE (card0_eq A_0). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | cards0_eq | |
cards1x : #|[set x]| = 1.
Proof. by rewrite set1.unlock cardsE card1. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | cards1 | |
cardsUIA B : #|A :|: B| + #|A :&: B| = #|A| + #|B|.
Proof. by rewrite !cardsE cardUI. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | cardsUI | |
cardsUA B : #|A :|: B| = #|A| + #|B| - #|A :&: B|.
Proof. by rewrite -cardsUI addnK. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | cardsU | |
cardsIA B : #|A :&: B| = #|A| + #|B| - #|A :|: B|.
Proof. by rewrite -cardsUI addKn. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | cardsI | |
cardsT: #|[set: T]| = #|T|.
Proof. by rewrite cardsE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | cardsT | |
cardsIDB A : #|A :&: B| + #|A :\: B| = #|A|.
Proof. by rewrite !cardsE cardID. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | cardsID | |
cardsDA B : #|A :\: B| = #|A| - #|A :&: B|.
Proof. by rewrite -(cardsID B A) addKn. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | cardsD | |
cardsCA : #|A| + #|~: A| = #|T|.
Proof. by rewrite cardsE cardC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | cardsC | |
cardsCsA : #|A| = #|T| - #|~: A|.
Proof. by rewrite -(cardsC A) addnK. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | cardsCs | |
cardsU1a A : #|a |: A| = (a \notin A) + #|A|.
Proof. by rewrite -cardU1; apply: eq_card=> x; rewrite !inE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | cardsU1 | |
cards2a b : #|[set a; b]| = (a != b).+1.
Proof. by rewrite -card2; apply: eq_card=> x; rewrite !inE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | cards2 | |
cardsC1a : #|[set~ a]| = #|T|.-1.
Proof. by rewrite -(cardC1 a); apply: eq_card=> x; rewrite !inE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | cardsC1 | |
cardsD1a A : #|A| = (a \in A) + #|A :\ a|.
Proof.
by rewrite (cardD1 a); congr (_ + _); apply: eq_card => x; rewrite !inE.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | cardsD1 | |
subsetIlA B : A :&: B \subset A.
Proof. by apply/subsetP=> x /[!inE] /andP[]. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | subsetIl | |
subsetIrA B : A :&: B \subset B.
Proof. by apply/subsetP=> x /[!inE] /andP[]. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | subsetIr | |
subsetUlA B : A \subset A :|: B.
Proof. by apply/subsetP=> x /[!inE] ->. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | subsetUl | |
subsetUrA B : B \subset A :|: B.
Proof. by apply/subsetP=> x; rewrite inE orbC => ->. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop"
] | boot/finset.v | subsetUr |
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