phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
in_setC x A : (x \in ~: A) = (x \notin A).	Proof. exact: in_set. Qed.	Lemma	in_setC	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "in_set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setCK : involutive (@setC T).	Proof. by move=> A; apply/setP=> x; rewrite !inE negbK. Qed.	Lemma	setCK	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "setC", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setC_inj : injective (@setC T).	Proof. exact: can_inj setCK. Qed.	Lemma	setC_inj	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "setC", "setCK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsets_disjoint A B : (A \subset B) = [disjoint A & ~: B].	Proof. by rewrite subset_disjoint; apply: eq_disjoint_r => x; rewrite !inE. Qed.	Lemma	subsets_disjoint	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "disjoint", "eq_disjoint_r", "inE", "subset_disjoint" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
disjoints_subset A B : [disjoint A & B] = (A \subset ~: B).	Proof. by rewrite subsets_disjoint setCK. Qed.	Lemma	disjoints_subset	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "disjoint", "setCK", "subsets_disjoint" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
powersetCE A B : (A \in powerset (~: B)) = [disjoint A & B].	Proof. by rewrite inE disjoints_subset. Qed.	Lemma	powersetCE	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "disjoint", "disjoints_subset", "inE", "powerset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setCS A B : (~: A \subset ~: B) = (B \subset A).	Proof. by rewrite !subsets_disjoint setCK disjoint_sym. Qed.	Lemma	setCS	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "disjoint_sym", "setCK", "subsets_disjoint" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setCU A B : ~: (A :\|: B) = ~: A :&: ~: B.	Proof. by apply/setP=> x; rewrite !inE negb_or. Qed.	Lemma	setCU	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setCI A B : ~: (A :&: B) = ~: A :\|: ~: B.	Proof. by apply/setP=> x; rewrite !inE negb_and. Qed.	Lemma	setCI	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setUCr A : A :\|: ~: A = setT.	Proof. by apply/setP=> x; rewrite !inE orbN. Qed.	Lemma	setUCr	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "setP", "setT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setICr A : A :&: ~: A = set0.	Proof. by apply/setP=> x; rewrite !inE andbN. Qed.	Lemma	setICr	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "set0", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setC0 : ~: set0 = [set: T].	Proof. by apply/setP=> x; rewrite !inE. Qed.	Lemma	setC0	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "set0", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setCT : ~: [set: T] = set0.	Proof. by rewrite -setC0 setCK. Qed.	Lemma	setCT	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "set0", "setC0", "setCK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
properC A B : (~: B \proper ~: A) = (A \proper B).	Proof. by rewrite !properE !setCS. Qed.	Lemma	properC	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "proper", "properE", "setCS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setDP A B x : reflect (x \in A /\ x \notin B) (x \in A :\: B).	Proof. by rewrite inE andbC; apply: andP. Qed.	Lemma	setDP	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE" ]	difference	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
in_setD A B x : (x \in A :\: B) = (x \notin B) && (x \in A).	Proof. exact: in_set. Qed.	Lemma	in_setD	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "in_set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setDE A B : A :\: B = A :&: ~: B.	Proof. by apply/setP => x; rewrite !inE andbC. Qed.	Lemma	setDE	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setSD A B C : A \subset B -> A :\: C \subset B :\: C.	Proof. by rewrite !setDE; apply: setSI. Qed.	Lemma	setSD	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "setDE", "setSI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setDS A B C : A \subset B -> C :\: B \subset C :\: A.	Proof. by rewrite !setDE -setCS; apply: setIS. Qed.	Lemma	setDS	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "setCS", "setDE", "setIS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setDSS A 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	setDSS	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "setDS", "setSD", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setD0 A : A :\: set0 = A.	Proof. by apply/setP=> x; rewrite !inE. Qed.	Lemma	setD0	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "set0", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set0D A : set0 :\: A = set0.	Proof. by apply/setP=> x; rewrite !inE andbF. Qed.	Lemma	set0D	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "set0", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setDT A : A :\: setT = set0.	Proof. by apply/setP=> x; rewrite !inE. Qed.	Lemma	setDT	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "set0", "setP", "setT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setTD A : setT :\: A = ~: A.	Proof. by apply/setP=> x; rewrite !inE andbT. Qed.	Lemma	setTD	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "setP", "setT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setDv A : A :\: A = set0.	Proof. by apply/setP=> x; rewrite !inE andNb. Qed.	Lemma	setDv	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "set0", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setCD A B : ~: (A :\: B) = ~: A :\|: B.	Proof. by rewrite !setDE setCI setCK. Qed.	Lemma	setCD	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "setCI", "setCK", "setDE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setID A B : A :&: B :\|: A :\: B = A.	Proof. by rewrite setDE -setIUr setUCr setIT. Qed.	Lemma	setID	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "setDE", "setIT", "setIUr", "setUCr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setDUl A B C : (A :\|: B) :\: C = (A :\: C) :\|: (B :\: C).	Proof. by rewrite !setDE setIUl. Qed.	Lemma	setDUl	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "setDE", "setIUl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setDUr A B C : A :\: (B :\|: C) = (A :\: B) :&: (A :\: C).	Proof. by rewrite !setDE setCU setIIr. Qed.	Lemma	setDUr	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "setCU", "setDE", "setIIr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setDIl A B C : (A :&: B) :\: C = (A :\: C) :&: (B :\: C).	Proof. by rewrite !setDE setIIl. Qed.	Lemma	setDIl	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "setDE", "setIIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setIDA A B C : A :&: (B :\: C) = (A :&: B) :\: C.	Proof. by rewrite !setDE setIA. Qed.	Lemma	setIDA	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "setDE", "setIA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setIDAC A B C : (A :\: B) :&: C = (A :&: C) :\: B.	Proof. by rewrite !setDE setIAC. Qed.	Lemma	setIDAC	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "setDE", "setIAC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setDIr A B C : A :\: (B :&: C) = (A :\: B) :\|: (A :\: C).	Proof. by rewrite !setDE setCI setIUr. Qed.	Lemma	setDIr	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "setCI", "setDE", "setIUr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setDDl A B C : (A :\: B) :\: C = A :\: (B :\|: C).	Proof. by rewrite !setDE setCU setIA. Qed.	Lemma	setDDl	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "setCU", "setDE", "setIA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setDDr A B C : A :\: (B :\: C) = (A :\: B) :\|: (A :&: C).	Proof. by rewrite !setDE setCI setIUr setCK. Qed.	Lemma	setDDr	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "setCI", "setCK", "setDE", "setIUr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setUD B A C : B \subset A -> C \subset B -> (A :\: B) :\|: (B :\: C) = (A :\: C).	Proof. move=> subBA subCB; apply/setP=> x; rewrite !inE. have /implyP := subsetP subBA x; have /implyP := subsetP subCB x. by do !case: (_ \in _). Qed.	Lemma	setUD	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "setP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setUDl A B: A :\|: B :\: A = A :\|: B.	Proof. by apply/setP=> x; rewrite !inE; do !case: (_ \in _). Qed.	Lemma	setUDl	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
powersetE A B : (A \in powerset B) = (A \subset B).	Proof. by rewrite inE. Qed.	Lemma	powersetE	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "inE", "powerset" ]	powerset	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
powersetS A 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	powersetS	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "powerset", "powersetE", "subsetP", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
powerset0 : powerset set0 = [set set0] :> {set {set T}}.	Proof. by apply/setP=> A; rewrite set1.unlock !inE subset0. Qed.	Lemma	powerset0	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "powerset", "set0", "setP", "subset0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
powersetT : powerset [set: T] = [set: {set T}].	Proof. by apply/setP=> A; rewrite !inE subsetT. Qed.	Lemma	powersetT	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "powerset", "setP", "subsetT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setI_powerset P A : P :&: powerset A = P ::&: A.	Proof. by apply/setP=> B; rewrite !inE. Qed.	Lemma	setI_powerset	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "powerset", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardsE pA : #\|[set x in pA]\| = #\|pA\|.	Proof. exact/eq_card/in_set. Qed.	Lemma	cardsE	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "eq_card", "in_set", "pA" ]	cardinal lemmas for sets	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum1dep_card pA : \sum_(x \| pA x) 1 = #\|[set x \| pA x]\|.	Proof. by rewrite sum1_card cardsE. Qed.	Lemma	sum1dep_card	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "cardsE", "pA", "sum1_card" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum_nat_cond_const pA n : \sum_(x \| pA x) n = #\|[set x \| pA x]\| * n.	Proof. by rewrite sum_nat_const cardsE. Qed.	Lemma	sum_nat_cond_const	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "cardsE", "pA", "sum_nat_const" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cards0 : #\|@set0 T\| = 0.	Proof. by rewrite cardsE card0. Qed.	Lemma	cards0	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "card0", "cardsE", "set0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cards_eq0 A : (#\|A\| == 0) = (A == set0).	Proof. by rewrite (eq_sym A) eqEcard sub0set cards0 leqn0. Qed.	Lemma	cards_eq0	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "cards0", "eqEcard", "eq_sym", "leqn0", "set0", "sub0set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set0Pn A : reflect (exists x, x \in A) (A != set0).	Proof. by rewrite -cards_eq0; apply: existsP. Qed.	Lemma	set0Pn	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "cards_eq0", "existsP", "set0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set0_Nexists A : (A == set0) = ~~ [exists x, x \in A].	Proof. by rewrite -(sameP (set0Pn _) existsP) negbK. Qed.	Lemma	set0_Nexists	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "existsP", "set0", "set0Pn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_gt0 A : (0 < #\|A\|) = (A != set0).	Proof. by rewrite lt0n cards_eq0. Qed.	Lemma	card_gt0	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "cards_eq0", "lt0n", "set0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cards0_eq A : #\|A\| = 0 -> A = set0.	Proof. by move=> A_0; apply/setP=> x; rewrite inE (card0_eq A_0). Qed.	Lemma	cards0_eq	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "card0_eq", "inE", "set0", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cards1 x : #\|[set x]\| = 1.	Proof. by rewrite set1.unlock cardsE card1. Qed.	Lemma	cards1	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "card1", "cardsE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardsUI A B : #\|A :\|: B\| + #\|A :&: B\| = #\|A\| + #\|B\|.	Proof. by rewrite !cardsE cardUI. Qed.	Lemma	cardsUI	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "cardUI", "cardsE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardsU A B : #\|A :\|: B\| = #\|A\| + #\|B\| - #\|A :&: B\|.	Proof. by rewrite -cardsUI addnK. Qed.	Lemma	cardsU	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "addnK", "cardsUI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardsI A B : #\|A :&: B\| = #\|A\| + #\|B\| - #\|A :\|: B\|.	Proof. by rewrite -cardsUI addKn. Qed.	Lemma	cardsI	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "addKn", "cardsUI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardsT : #\|[set: T]\| = #\|T\|.	Proof. by rewrite cardsE. Qed.	Lemma	cardsT	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "cardsE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardsID B A : #\|A :&: B\| + #\|A :\: B\| = #\|A\|.	Proof. by rewrite !cardsE cardID. Qed.	Lemma	cardsID	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "cardID", "cardsE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardsD A B : #\|A :\: B\| = #\|A\| - #\|A :&: B\|.	Proof. by rewrite -(cardsID B A) addKn. Qed.	Lemma	cardsD	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "addKn", "cardsID" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardsC A : #\|A\| + #\|~: A\| = #\|T\|.	Proof. by rewrite cardsE cardC. Qed.	Lemma	cardsC	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "cardC", "cardsE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardsCs A : #\|A\| = #\|T\| - #\|~: A\|.	Proof. by rewrite -(cardsC A) addnK. Qed.	Lemma	cardsCs	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "addnK", "cardsC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardsU1 a A : #\|a \|: A\| = (a \notin A) + #\|A\|.	Proof. by rewrite -cardU1; apply: eq_card=> x; rewrite !inE. Qed.	Lemma	cardsU1	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "cardU1", "eq_card", "inE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cards2 a b : #\|[set a; b]\| = (a != b).+1.	Proof. by rewrite -card2; apply: eq_card=> x; rewrite !inE. Qed.	Lemma	cards2	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "card2", "eq_card", "inE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardsC1 a : #\|[set~ a]\| = #\|T\|.-1.	Proof. by rewrite -(cardC1 a); apply: eq_card=> x; rewrite !inE. Qed.	Lemma	cardsC1	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "cardC1", "eq_card", "inE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardsD1 a A : #\|A\| = (a \in A) + #\|A :\ a\|.	Proof. by rewrite (cardD1 a); congr (_ + _); apply: eq_card => x; rewrite !inE. Qed.	Lemma	cardsD1	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "cardD1", "eq_card", "inE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetIl A B : A :&: B \subset A.	Proof. by apply/subsetP=> x /[!inE] /andP[]. Qed.	Lemma	subsetIl	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "subsetP" ]	other inclusions	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetIr A B : A :&: B \subset B.	Proof. by apply/subsetP=> x /[!inE] /andP[]. Qed.	Lemma	subsetIr	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetUl A B : A \subset A :\|: B.	Proof. by apply/subsetP=> x /[!inE] ->. Qed.	Lemma	subsetUl	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetUr A B : B \subset A :\|: B.	Proof. by apply/subsetP=> x; rewrite inE orbC => ->. Qed.	Lemma	subsetUr	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetU1 x A : A \subset x \|: A.	Proof. exact: subsetUr. Qed.	Lemma	subsetU1	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "subsetUr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetDl A B : A :\: B \subset A.	Proof. by rewrite setDE subsetIl. Qed.	Lemma	subsetDl	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "setDE", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subD1set A x : A :\ x \subset A.	Proof. by rewrite subsetDl. Qed.	Lemma	subD1set	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "subsetDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetDr A B : A :\: B \subset ~: B.	Proof. by rewrite setDE subsetIr. Qed.	Lemma	subsetDr	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "setDE", "subsetIr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub1set A x : ([set x] \subset A) = (x \in A).	Proof. by rewrite -subset_pred1; apply: eq_subset=> y; rewrite !inE. Qed.	Lemma	sub1set	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "eq_subset", "inE", "subset_pred1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cards_eq_spec A : seq T -> {set T} -> nat -> Type	:= \| CardEq (s : seq T) & uniq s : cards_eq_spec A s [set x \| x \in s] (size s).	Variant	cards_eq_spec	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "nat", "seq", "size", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cards_eqP A : cards_eq_spec A (enum A) A #\|A\|.	Proof. by move: (enum A) (cardE A) (set_enum A) (enum_uniq A) => s -> <-; constructor. Qed.	Lemma	cards_eqP	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "cardE", "cards_eq_spec", "enum", "enum_uniq", "set_enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cards1P A : reflect (exists x, A = [set x]) (#\|A\| == 1).	Proof. apply: (iffP idP) => [\|[x ->]]; last by rewrite cards1. by have [[\|x []]// _] := cards_eqP; exists x; apply/setP => y; rewrite !inE. Qed.	Lemma	cards1P	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "cards1", "cards_eqP", "inE", "last", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cards2P A : reflect (exists x y : T, x != y /\ A = [set x; y]) (#\|A\| == 2).	Proof. apply: (iffP idP) => [\|[x] [y] [xy ->]]; last by rewrite cards2 xy. have [[\|x [\|y []]]//=] := cards_eqP; rewrite !inE andbT => neq_xy. by exists x, y; split=> //; apply/setP => z; rewrite !inE. Qed.	Lemma	cards2P	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "cards2", "cards_eqP", "inE", "last", "setP", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset1 A x : (A \subset [set x]) = (A == [set x]) \|\| (A == set0).	Proof. rewrite eqEcard cards1 -cards_eq0 orbC andbC. by case: posnP => // A0; rewrite (cards0_eq A0) sub0set. Qed.	Lemma	subset1	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "cards0_eq", "cards1", "cards_eq0", "eqEcard", "posnP", "set0", "sub0set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
powerset1 x : powerset [set x] = [set set0; [set x]].	Proof. by apply/setP=> A; rewrite inE subset1 orbC set1.unlock !inE. Qed.	Lemma	powerset1	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "powerset", "set0", "setP", "subset1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setIidPl A B : reflect (A :&: B = A) (A \subset B).	Proof. apply: (iffP subsetP) => [sAB \| <- x /setIP[] //]. by apply/setP=> x /[1!inE]; apply/andb_idr/sAB. Qed.	Lemma	setIidPl	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "setIP", "setP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setIidPr A B : reflect (A :&: B = B) (B \subset A).	Proof. by rewrite setIC; apply: setIidPl. Qed.	Lemma	setIidPr	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "setIC", "setIidPl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardsDS A B : B \subset A -> #\|A :\: B\| = #\|A\| - #\|B\|.	Proof. by rewrite cardsD => /setIidPr->. Qed.	Lemma	cardsDS	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "cardsD", "setIidPr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setUidPl A B : reflect (A :\|: B = A) (B \subset A).	Proof. by rewrite -setCS (sameP setIidPl eqP) -setCU (inj_eq setC_inj); apply: eqP. Qed.	Lemma	setUidPl	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inj_eq", "setCS", "setCU", "setC_inj", "setIidPl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setUidPr A B : reflect (A :\|: B = B) (A \subset B).	Proof. by rewrite setUC; apply: setUidPl. Qed.	Lemma	setUidPr	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "setUC", "setUidPl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setDidPl A B : reflect (A :\: B = A) [disjoint A & B].	Proof. by rewrite setDE disjoints_subset; apply: setIidPl. Qed.	Lemma	setDidPl	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "disjoint", "disjoints_subset", "setDE", "setIidPl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subIset A B C : (B \subset A) \|\| (C \subset A) -> (B :&: C \subset A).	Proof. by case/orP; apply: subset_trans; rewrite (subsetIl, subsetIr). Qed.	Lemma	subIset	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "subsetIl", "subsetIr", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetI A B C : (A \subset B :&: C) = (A \subset B) && (A \subset C).	Proof. rewrite !(sameP setIidPl eqP) setIA; have [-> //\|] := eqVneq (A :&: B) A. by apply: contraNF => /eqP <-; rewrite -setIA -setIIl setIAC. Qed.	Lemma	subsetI	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "eqVneq", "setIA", "setIAC", "setIIl", "setIidPl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetIP A B C : reflect (A \subset B /\ A \subset C) (A \subset B :&: C).	Proof. by rewrite subsetI; apply: andP. Qed.	Lemma	subsetIP	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "subsetI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetIidl A B : (A \subset A :&: B) = (A \subset B).	Proof. by rewrite subsetI subxx. Qed.	Lemma	subsetIidl	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "subsetI", "subxx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetIidr A B : (B \subset A :&: B) = (B \subset A).	Proof. by rewrite setIC subsetIidl. Qed.	Lemma	subsetIidr	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "setIC", "subsetIidl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
powersetI A B : powerset (A :&: B) = powerset A :&: powerset B.	Proof. by apply/setP=> C; rewrite !inE subsetI. Qed.	Lemma	powersetI	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "powerset", "setP", "subsetI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subUset A B C : (B :\|: C \subset A) = (B \subset A) && (C \subset A).	Proof. by rewrite -setCS setCU subsetI !setCS. Qed.	Lemma	subUset	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "setCS", "setCU", "subsetI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetU A B C : (A \subset B) \|\| (A \subset C) -> A \subset B :\|: C.	Proof. by rewrite -!(setCS _ A) setCU; apply: subIset. Qed.	Lemma	subsetU	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "setCS", "setCU", "subIset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subUsetP A B C : reflect (A \subset C /\ B \subset C) (A :\|: B \subset C).	Proof. by rewrite subUset; apply: andP. Qed.	Lemma	subUsetP	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "subUset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetC A B : (A \subset ~: B) = (B \subset ~: A).	Proof. by rewrite -setCS setCK. Qed.	Lemma	subsetC	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "setCK", "setCS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subCset A B : (~: A \subset B) = (~: B \subset A).	Proof. by rewrite -setCS setCK. Qed.	Lemma	subCset	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "setCK", "setCS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetD A B C : (A \subset B :\: C) = (A \subset B) && [disjoint A & C].	Proof. by rewrite setDE subsetI -disjoints_subset. Qed.	Lemma	subsetD	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "disjoint", "disjoints_subset", "setDE", "subsetI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subDset A B C : (A :\: B \subset C) = (A \subset B :\|: C).	Proof. apply/subsetP/subsetP=> sABC x; rewrite !inE. by case Bx: (x \in B) => // Ax; rewrite sABC ?inE ?Bx. by case Bx: (x \in B) => // /sABC; rewrite inE Bx. Qed.	Lemma	subDset	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "inE", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsetDP A B C : reflect (A \subset B /\ [disjoint A & C]) (A \subset B :\: C).	Proof. by rewrite subsetD; apply: andP. Qed.	Lemma	subsetDP	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "apply", "disjoint", "subsetD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setU_eq0 A B : (A :\|: B == set0) = (A == set0) && (B == set0).	Proof. by rewrite -!subset0 subUset. Qed.	Lemma	setU_eq0	boot	boot/finset.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]	[ "set0", "subUset", "subset0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

in_setC x A : (x \in ~: A) = (x \notin A).

Proof. exact: in_set. Qed.

Lemma

in_setC

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "in_set" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setCK : involutive (@setC T).

Proof. by move=> A; apply/setP=> x; rewrite !inE negbK. Qed.

Lemma

setCK

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "setC", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setC_inj : injective (@setC T).

Proof. exact: can_inj setCK. Qed.

Lemma

setC_inj

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "setC", "setCK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsets_disjoint A B : (A \subset B) = [disjoint A & ~: B].

Proof. by rewrite subset_disjoint; apply: eq_disjoint_r => x; rewrite !inE. Qed.

Lemma

subsets_disjoint

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "disjoint", "eq_disjoint_r", "inE", "subset_disjoint" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

disjoints_subset A B : [disjoint A & B] = (A \subset ~: B).

Proof. by rewrite subsets_disjoint setCK. Qed.

Lemma

disjoints_subset

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "disjoint", "setCK", "subsets_disjoint" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

powersetCE A B : (A \in powerset (~: B)) = [disjoint A & B].

Proof. by rewrite inE disjoints_subset. Qed.

Lemma

powersetCE

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "disjoint", "disjoints_subset", "inE", "powerset" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setCS A B : (~: A \subset ~: B) = (B \subset A).

Proof. by rewrite !subsets_disjoint setCK disjoint_sym. Qed.

Lemma

setCS

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "disjoint_sym", "setCK", "subsets_disjoint" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setCU A B : ~: (A :|: B) = ~: A :&: ~: B.

Proof. by apply/setP=> x; rewrite !inE negb_or. Qed.

Lemma

setCU

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setCI A B : ~: (A :&: B) = ~: A :|: ~: B.

Proof. by apply/setP=> x; rewrite !inE negb_and. Qed.

Lemma

setCI

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setUCr A : A :|: ~: A = setT.

Proof. by apply/setP=> x; rewrite !inE orbN. Qed.

Lemma

setUCr

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "setP", "setT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setICr A : A :&: ~: A = set0.

Proof. by apply/setP=> x; rewrite !inE andbN. Qed.

Lemma

setICr

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "set0", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setC0 : ~: set0 = [set: T].

Proof. by apply/setP=> x; rewrite !inE. Qed.

Lemma

setC0

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "set0", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setCT : ~: [set: T] = set0.

Proof. by rewrite -setC0 setCK. Qed.

Lemma

setCT

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "set0", "setC0", "setCK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

properC A B : (~: B \proper ~: A) = (A \proper B).

Proof. by rewrite !properE !setCS. Qed.

Lemma

properC

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "proper", "properE", "setCS" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setDP A B x : reflect (x \in A /\ x \notin B) (x \in A :\: B).

Proof. by rewrite inE andbC; apply: andP. Qed.

Lemma

setDP

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE" ]

difference

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

in_setD A B x : (x \in A :\: B) = (x \notin B) && (x \in A).

Proof. exact: in_set. Qed.

Lemma

in_setD

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "in_set" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setDE A B : A :\: B = A :&: ~: B.

Proof. by apply/setP => x; rewrite !inE andbC. Qed.

Lemma

setDE

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setSD A B C : A \subset B -> A :\: C \subset B :\: C.

Proof. by rewrite !setDE; apply: setSI. Qed.

Lemma

setSD

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "setDE", "setSI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setDS A B C : A \subset B -> C :\: B \subset C :\: A.

Proof. by rewrite !setDE -setCS; apply: setIS. Qed.

Lemma

setDS

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "setCS", "setDE", "setIS" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setDSS A 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

setDSS

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "setDS", "setSD", "subset_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setD0 A : A :\: set0 = A.

Proof. by apply/setP=> x; rewrite !inE. Qed.

Lemma

setD0

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "set0", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set0D A : set0 :\: A = set0.

Proof. by apply/setP=> x; rewrite !inE andbF. Qed.

Lemma

set0D

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "set0", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setDT A : A :\: setT = set0.

Proof. by apply/setP=> x; rewrite !inE. Qed.

Lemma

setDT

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "set0", "setP", "setT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setTD A : setT :\: A = ~: A.

Proof. by apply/setP=> x; rewrite !inE andbT. Qed.

Lemma

setTD

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "setP", "setT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setDv A : A :\: A = set0.

Proof. by apply/setP=> x; rewrite !inE andNb. Qed.

Lemma

setDv

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "set0", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setCD A B : ~: (A :\: B) = ~: A :|: B.

Proof. by rewrite !setDE setCI setCK. Qed.

Lemma

setCD

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "setCI", "setCK", "setDE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setID A B : A :&: B :|: A :\: B = A.

Proof. by rewrite setDE -setIUr setUCr setIT. Qed.

Lemma

setID

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "setDE", "setIT", "setIUr", "setUCr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setDUl A B C : (A :|: B) :\: C = (A :\: C) :|: (B :\: C).

Proof. by rewrite !setDE setIUl. Qed.

Lemma

setDUl

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "setDE", "setIUl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setDUr A B C : A :\: (B :|: C) = (A :\: B) :&: (A :\: C).

Proof. by rewrite !setDE setCU setIIr. Qed.

Lemma

setDUr

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "setCU", "setDE", "setIIr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setDIl A B C : (A :&: B) :\: C = (A :\: C) :&: (B :\: C).

Proof. by rewrite !setDE setIIl. Qed.

Lemma

setDIl

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "setDE", "setIIl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setIDA A B C : A :&: (B :\: C) = (A :&: B) :\: C.

Proof. by rewrite !setDE setIA. Qed.

Lemma

setIDA

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "setDE", "setIA" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setIDAC A B C : (A :\: B) :&: C = (A :&: C) :\: B.

Proof. by rewrite !setDE setIAC. Qed.

Lemma

setIDAC

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "setDE", "setIAC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setDIr A B C : A :\: (B :&: C) = (A :\: B) :|: (A :\: C).

Proof. by rewrite !setDE setCI setIUr. Qed.

Lemma

setDIr

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "setCI", "setDE", "setIUr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setDDl A B C : (A :\: B) :\: C = A :\: (B :|: C).

Proof. by rewrite !setDE setCU setIA. Qed.

Lemma

setDDl

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "setCU", "setDE", "setIA" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setDDr A B C : A :\: (B :\: C) = (A :\: B) :|: (A :&: C).

Proof. by rewrite !setDE setCI setIUr setCK. Qed.

Lemma

setDDr

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "setCI", "setCK", "setDE", "setIUr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setUD B A C : B \subset A -> C \subset B -> (A :\: B) :|: (B :\: C) = (A :\: C).

Proof. move=> subBA subCB; apply/setP=> x; rewrite !inE. have /implyP := subsetP subBA x; have /implyP := subsetP subCB x. by do !case: (_ \in _). Qed.

Lemma

setUD

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "setP", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setUDl A B: A :|: B :\: A = A :|: B.

Proof. by apply/setP=> x; rewrite !inE; do !case: (_ \in _). Qed.

Lemma

setUDl

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

powersetE A B : (A \in powerset B) = (A \subset B).

Proof. by rewrite inE. Qed.

Lemma

powersetE

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "inE", "powerset" ]

powerset

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

powersetS A 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

powersetS

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "powerset", "powersetE", "subsetP", "subset_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

powerset0 : powerset set0 = [set set0] :> {set {set T}}.

Proof. by apply/setP=> A; rewrite set1.unlock !inE subset0. Qed.

Lemma

powerset0

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "powerset", "set0", "setP", "subset0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

powersetT : powerset [set: T] = [set: {set T}].

Proof. by apply/setP=> A; rewrite !inE subsetT. Qed.

Lemma

powersetT

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "powerset", "setP", "subsetT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setI_powerset P A : P :&: powerset A = P ::&: A.

Proof. by apply/setP=> B; rewrite !inE. Qed.

Lemma

setI_powerset

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "powerset", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardsE pA : #|[set x in pA]| = #|pA|.

Proof. exact/eq_card/in_set. Qed.

Lemma

cardsE

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "eq_card", "in_set", "pA" ]

cardinal lemmas for sets

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum1dep_card pA : \sum_(x | pA x) 1 = #|[set x | pA x]|.

Proof. by rewrite sum1_card cardsE. Qed.

Lemma

sum1dep_card

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "cardsE", "pA", "sum1_card" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum_nat_cond_const pA n : \sum_(x | pA x) n = #|[set x | pA x]| * n.

Proof. by rewrite sum_nat_const cardsE. Qed.

Lemma

sum_nat_cond_const

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "cardsE", "pA", "sum_nat_const" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cards0 : #|@set0 T| = 0.

Proof. by rewrite cardsE card0. Qed.

Lemma

cards0

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "card0", "cardsE", "set0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cards_eq0 A : (#|A| == 0) = (A == set0).

Proof. by rewrite (eq_sym A) eqEcard sub0set cards0 leqn0. Qed.

Lemma

cards_eq0

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "cards0", "eqEcard", "eq_sym", "leqn0", "set0", "sub0set" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set0Pn A : reflect (exists x, x \in A) (A != set0).

Proof. by rewrite -cards_eq0; apply: existsP. Qed.

Lemma

set0Pn

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "cards_eq0", "existsP", "set0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set0_Nexists A : (A == set0) = ~~ [exists x, x \in A].

Proof. by rewrite -(sameP (set0Pn _) existsP) negbK. Qed.

Lemma

set0_Nexists

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "existsP", "set0", "set0Pn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_gt0 A : (0 < #|A|) = (A != set0).

Proof. by rewrite lt0n cards_eq0. Qed.

Lemma

card_gt0

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "cards_eq0", "lt0n", "set0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cards0_eq A : #|A| = 0 -> A = set0.

Proof. by move=> A_0; apply/setP=> x; rewrite inE (card0_eq A_0). Qed.

Lemma

cards0_eq

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "card0_eq", "inE", "set0", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cards1 x : #|[set x]| = 1.

Proof. by rewrite set1.unlock cardsE card1. Qed.

Lemma

cards1

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "card1", "cardsE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardsUI A B : #|A :|: B| + #|A :&: B| = #|A| + #|B|.

Proof. by rewrite !cardsE cardUI. Qed.

Lemma

cardsUI

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "cardUI", "cardsE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardsU A B : #|A :|: B| = #|A| + #|B| - #|A :&: B|.

Proof. by rewrite -cardsUI addnK. Qed.

Lemma

cardsU

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "addnK", "cardsUI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardsI A B : #|A :&: B| = #|A| + #|B| - #|A :|: B|.

Proof. by rewrite -cardsUI addKn. Qed.

Lemma

cardsI

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "addKn", "cardsUI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardsT : #|[set: T]| = #|T|.

Proof. by rewrite cardsE. Qed.

Lemma

cardsT

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "cardsE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardsID B A : #|A :&: B| + #|A :\: B| = #|A|.

Proof. by rewrite !cardsE cardID. Qed.

Lemma

cardsID

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "cardID", "cardsE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardsD A B : #|A :\: B| = #|A| - #|A :&: B|.

Proof. by rewrite -(cardsID B A) addKn. Qed.

Lemma

cardsD

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "addKn", "cardsID" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardsC A : #|A| + #|~: A| = #|T|.

Proof. by rewrite cardsE cardC. Qed.

Lemma

cardsC

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "cardC", "cardsE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardsCs A : #|A| = #|T| - #|~: A|.

Proof. by rewrite -(cardsC A) addnK. Qed.

Lemma

cardsCs

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "addnK", "cardsC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardsU1 a A : #|a |: A| = (a \notin A) + #|A|.

Proof. by rewrite -cardU1; apply: eq_card=> x; rewrite !inE. Qed.

Lemma

cardsU1

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "cardU1", "eq_card", "inE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cards2 a b : #|[set a; b]| = (a != b).+1.

Proof. by rewrite -card2; apply: eq_card=> x; rewrite !inE. Qed.

Lemma

cards2

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "card2", "eq_card", "inE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardsC1 a : #|[set~ a]| = #|T|.-1.

Proof. by rewrite -(cardC1 a); apply: eq_card=> x; rewrite !inE. Qed.

Lemma

cardsC1

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "cardC1", "eq_card", "inE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardsD1 a A : #|A| = (a \in A) + #|A :\ a|.

Proof. by rewrite (cardD1 a); congr (_ + _); apply: eq_card => x; rewrite !inE. Qed.

Lemma

cardsD1

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "cardD1", "eq_card", "inE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetIl A B : A :&: B \subset A.

Proof. by apply/subsetP=> x /[!inE] /andP[]. Qed.

Lemma

subsetIl

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "subsetP" ]

other inclusions

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetIr A B : A :&: B \subset B.

Proof. by apply/subsetP=> x /[!inE] /andP[]. Qed.

Lemma

subsetIr

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetUl A B : A \subset A :|: B.

Proof. by apply/subsetP=> x /[!inE] ->. Qed.

Lemma

subsetUl

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetUr A B : B \subset A :|: B.

Proof. by apply/subsetP=> x; rewrite inE orbC => ->. Qed.

Lemma

subsetUr

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetU1 x A : A \subset x |: A.

Proof. exact: subsetUr. Qed.

Lemma

subsetU1

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "subsetUr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetDl A B : A :\: B \subset A.

Proof. by rewrite setDE subsetIl. Qed.

Lemma

subsetDl

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "setDE", "subsetIl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subD1set A x : A :\ x \subset A.

Proof. by rewrite subsetDl. Qed.

Lemma

subD1set

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "subsetDl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetDr A B : A :\: B \subset ~: B.

Proof. by rewrite setDE subsetIr. Qed.

Lemma

subsetDr

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "setDE", "subsetIr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub1set A x : ([set x] \subset A) = (x \in A).

Proof. by rewrite -subset_pred1; apply: eq_subset=> y; rewrite !inE. Qed.

Lemma

sub1set

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "eq_subset", "inE", "subset_pred1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cards_eq_spec A : seq T -> {set T} -> nat -> Type

:= | CardEq (s : seq T) & uniq s : cards_eq_spec A s [set x | x \in s] (size s).

Variant

cards_eq_spec

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "nat", "seq", "size", "uniq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cards_eqP A : cards_eq_spec A (enum A) A #|A|.

Proof. by move: (enum A) (cardE A) (set_enum A) (enum_uniq A) => s -> <-; constructor. Qed.

Lemma

cards_eqP

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "cardE", "cards_eq_spec", "enum", "enum_uniq", "set_enum" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cards1P A : reflect (exists x, A = [set x]) (#|A| == 1).

Proof. apply: (iffP idP) => [|[x ->]]; last by rewrite cards1. by have [[|x []]// _] := cards_eqP; exists x; apply/setP => y; rewrite !inE. Qed.

Lemma

cards1P

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "cards1", "cards_eqP", "inE", "last", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cards2P A : reflect (exists x y : T, x != y /\ A = [set x; y]) (#|A| == 2).

Proof. apply: (iffP idP) => [|[x] [y] [xy ->]]; last by rewrite cards2 xy. have [[|x [|y []]]//=] := cards_eqP; rewrite !inE andbT => neq_xy. by exists x, y; split=> //; apply/setP => z; rewrite !inE. Qed.

Lemma

cards2P

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "cards2", "cards_eqP", "inE", "last", "setP", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset1 A x : (A \subset [set x]) = (A == [set x]) || (A == set0).

Proof. rewrite eqEcard cards1 -cards_eq0 orbC andbC. by case: posnP => // A0; rewrite (cards0_eq A0) sub0set. Qed.

Lemma

subset1

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "cards0_eq", "cards1", "cards_eq0", "eqEcard", "posnP", "set0", "sub0set" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

powerset1 x : powerset [set x] = [set set0; [set x]].

Proof. by apply/setP=> A; rewrite inE subset1 orbC set1.unlock !inE. Qed.

Lemma

powerset1

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "powerset", "set0", "setP", "subset1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setIidPl A B : reflect (A :&: B = A) (A \subset B).

Proof. apply: (iffP subsetP) => [sAB | <- x /setIP[] //]. by apply/setP=> x /[1!inE]; apply/andb_idr/sAB. Qed.

Lemma

setIidPl

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "setIP", "setP", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setIidPr A B : reflect (A :&: B = B) (B \subset A).

Proof. by rewrite setIC; apply: setIidPl. Qed.

Lemma

setIidPr

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "setIC", "setIidPl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardsDS A B : B \subset A -> #|A :\: B| = #|A| - #|B|.

Proof. by rewrite cardsD => /setIidPr->. Qed.

Lemma

cardsDS

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "cardsD", "setIidPr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setUidPl A B : reflect (A :|: B = A) (B \subset A).

Proof. by rewrite -setCS (sameP setIidPl eqP) -setCU (inj_eq setC_inj); apply: eqP. Qed.

Lemma

setUidPl

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inj_eq", "setCS", "setCU", "setC_inj", "setIidPl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setUidPr A B : reflect (A :|: B = B) (A \subset B).

Proof. by rewrite setUC; apply: setUidPl. Qed.

Lemma

setUidPr

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "setUC", "setUidPl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setDidPl A B : reflect (A :\: B = A) [disjoint A & B].

Proof. by rewrite setDE disjoints_subset; apply: setIidPl. Qed.

Lemma

setDidPl

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "disjoint", "disjoints_subset", "setDE", "setIidPl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subIset A B C : (B \subset A) || (C \subset A) -> (B :&: C \subset A).

Proof. by case/orP; apply: subset_trans; rewrite (subsetIl, subsetIr). Qed.

Lemma

subIset

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "subsetIl", "subsetIr", "subset_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetI A B C : (A \subset B :&: C) = (A \subset B) && (A \subset C).

Proof. rewrite !(sameP setIidPl eqP) setIA; have [-> //|] := eqVneq (A :&: B) A. by apply: contraNF => /eqP <-; rewrite -setIA -setIIl setIAC. Qed.

Lemma

subsetI

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "eqVneq", "setIA", "setIAC", "setIIl", "setIidPl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetIP A B C : reflect (A \subset B /\ A \subset C) (A \subset B :&: C).

Proof. by rewrite subsetI; apply: andP. Qed.

Lemma

subsetIP

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "subsetI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetIidl A B : (A \subset A :&: B) = (A \subset B).

Proof. by rewrite subsetI subxx. Qed.

Lemma

subsetIidl

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "subsetI", "subxx" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetIidr A B : (B \subset A :&: B) = (B \subset A).

Proof. by rewrite setIC subsetIidl. Qed.

Lemma

subsetIidr

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "setIC", "subsetIidl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

powersetI A B : powerset (A :&: B) = powerset A :&: powerset B.

Proof. by apply/setP=> C; rewrite !inE subsetI. Qed.

Lemma

powersetI

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "powerset", "setP", "subsetI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subUset A B C : (B :|: C \subset A) = (B \subset A) && (C \subset A).

Proof. by rewrite -setCS setCU subsetI !setCS. Qed.

Lemma

subUset

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "setCS", "setCU", "subsetI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetU A B C : (A \subset B) || (A \subset C) -> A \subset B :|: C.

Proof. by rewrite -!(setCS _ A) setCU; apply: subIset. Qed.

Lemma

subsetU

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "setCS", "setCU", "subIset" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subUsetP A B C : reflect (A \subset C /\ B \subset C) (A :|: B \subset C).

Proof. by rewrite subUset; apply: andP. Qed.

Lemma

subUsetP

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "subUset" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetC A B : (A \subset ~: B) = (B \subset ~: A).

Proof. by rewrite -setCS setCK. Qed.

Lemma

subsetC

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "setCK", "setCS" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subCset A B : (~: A \subset B) = (~: B \subset A).

Proof. by rewrite -setCS setCK. Qed.

Lemma

subCset

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "setCK", "setCS" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetD A B C : (A \subset B :\: C) = (A \subset B) && [disjoint A & C].

Proof. by rewrite setDE subsetI -disjoints_subset. Qed.

Lemma

subsetD

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "disjoint", "disjoints_subset", "setDE", "subsetI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subDset A B C : (A :\: B \subset C) = (A \subset B :|: C).

Proof. apply/subsetP/subsetP=> sABC x; rewrite !inE. by case Bx: (x \in B) => // Ax; rewrite sABC ?inE ?Bx. by case Bx: (x \in B) => // /sABC; rewrite inE Bx. Qed.

Lemma

subDset

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "inE", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsetDP A B C : reflect (A \subset B /\ [disjoint A & C]) (A \subset B :\: C).

Proof. by rewrite subsetD; apply: andP. Qed.

Lemma

subsetDP

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "apply", "disjoint", "subsetD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setU_eq0 A B : (A :|: B == set0) = (A == set0) && (B == set0).

Proof. by rewrite -!subset0 subUset. Qed.

Lemma

setU_eq0

boot

boot/finset.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "Monoid" ]

[ "set0", "subUset", "subset0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d