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 |
|---|---|---|---|---|---|---|---|---|---|---|
"[ 'set' ~ a ]" | := (~: [set a])
(format "[ 'set' ~ a ]") : set_scope. | Notation | [ 'set' ~ a ] | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A :\: B" | := (setD A B) : set_scope. | Notation | A :\: B | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A :\ a" | := (A :\: [set a]) : set_scope. | Notation | A :\ a | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"P ::&: D" | := (ssetI P D) (at level 48) : set_scope. | Notation | P ::&: D | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"ssetI"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqEsubset A B : (A == B) = (A \subset B) && (B \subset A). | Proof. by apply/eqP/subset_eqP=> /setP. Qed. | Lemma | eqEsubset | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"setP",
"subset_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subEproper A B : (A \subset B) = (A == B) || (A \proper B). | Proof. by rewrite eqEsubset -andb_orr orbN andbT. Qed. | Lemma | subEproper | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"eqEsubset",
"proper"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqVproper A B : A \subset B -> A = B \/ A \proper B. | Proof. by rewrite subEproper => /predU1P. Qed. | Lemma | eqVproper | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"predU1P",
"proper",
"subEproper"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
properEneq A B : (A \proper B) = (A != B) && (A \subset B). | Proof. by rewrite andbC eqEsubset negb_and andb_orr andbN. Qed. | Lemma | properEneq | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"eqEsubset",
"proper"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proper_neq A B : A \proper B -> A != B. | Proof. by rewrite properEneq; case/andP. Qed. | Lemma | proper_neq | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"proper",
"properEneq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqEproper A B : (A == B) = (A \subset B) && ~~ (A \proper B). | Proof. by rewrite negb_and negbK andb_orr andbN eqEsubset. Qed. | Lemma | eqEproper | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"eqEsubset",
"proper"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqEcard A B : (A == B) = (A \subset B) && (#|B| <= #|A|). | Proof.
rewrite eqEsubset; apply: andb_id2l => sAB.
by rewrite (geq_leqif (subset_leqif_card sAB)).
Qed. | Lemma | eqEcard | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"eqEsubset",
"geq_leqif",
"subset_leqif_card"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
properEcard A B : (A \proper B) = (A \subset B) && (#|A| < #|B|). | Proof. by rewrite properEneq ltnNge andbC eqEcard; case: (A \subset B). Qed. | Lemma | properEcard | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"eqEcard",
"ltnNge",
"proper",
"properEneq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subset_leqif_cards A B : A \subset B -> (#|A| <= #|B| ?= iff (A == B)). | Proof. by move=> sAB; rewrite eqEsubset sAB; apply: subset_leqif_card. Qed. | Lemma | subset_leqif_cards | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"eqEsubset",
"subset_leqif_card"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_set0 x : (x \in set0) = false. | Proof. by rewrite in_set. Qed. | Lemma | in_set0 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"in_set",
"set0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub0set A : set0 \subset A. | Proof. by apply/subsetP=> x; rewrite in_set. Qed. | Lemma | sub0set | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"in_set",
"set0",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subset0 A : (A \subset set0) = (A == set0). | Proof. by rewrite eqEsubset sub0set andbT. Qed. | Lemma | subset0 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"eqEsubset",
"set0",
"sub0set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proper0 A : (set0 \proper A) = (A != set0). | Proof. by rewrite properE sub0set subset0. Qed. | Lemma | proper0 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"proper",
"properE",
"set0",
"sub0set",
"subset0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subset_neq0 A B : A \subset B -> A != set0 -> B != set0. | Proof. by rewrite -!proper0 => sAB /proper_sub_trans->. Qed. | Lemma | subset_neq0 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"proper0",
"proper_sub_trans",
"set0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set_0Vmem A : (A = set0) + {x : T | x \in A}. | Proof.
case: (pickP (mem A)) => [x Ax | A0]; [by right; exists x | left].
by apply/setP=> x; rewrite in_set; apply: A0.
Qed. | Lemma | set_0Vmem | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"in_set",
"pickP",
"set0",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set_enum A : [set x | x \in enum A] = A. | Proof. by apply/setP => x; rewrite in_set mem_enum. Qed. | Lemma | set_enum | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"enum",
"in_set",
"mem_enum",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
enum_set0 : enum set0 = [::] :> seq T. | Proof. by rewrite (eq_enum (in_set _)) enum0. Qed. | Lemma | enum_set0 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"enum",
"enum0",
"eq_enum",
"in_set",
"seq",
"set0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subsetT A : A \subset setT. | Proof. by apply/subsetP=> x; rewrite in_set. Qed. | Lemma | subsetT | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"in_set",
"setT",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subsetT_hint mA : subset mA (mem [set: T]). | Proof. by rewrite unlock; apply/pred0P=> x; rewrite !inE in_set. Qed. | Lemma | subsetT_hint | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"inE",
"in_set",
"pred0P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subTset A : (setT \subset A) = (A == setT). | Proof. by rewrite eqEsubset subsetT. Qed. | Lemma | subTset | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"eqEsubset",
"setT",
"subsetT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
properT A : (A \proper setT) = (A != setT). | Proof. by rewrite properEneq subsetT andbT. Qed. | Lemma | properT | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"proper",
"properEneq",
"setT",
"subsetT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set1P x a : reflect (x = a) (x \in [set a]). | Proof. by rewrite set1.unlock in_set; apply: eqP. Qed. | Lemma | set1P | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"in_set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
enum_setT : enum [set: T] = Finite.enum T. | Proof. by rewrite (eq_enum (in_set _)) enumT. Qed. | Lemma | enum_setT | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"enum",
"enumT",
"eq_enum",
"in_set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_set1 x a : (x \in [set a]) = (x == a). | Proof. by rewrite set1.unlock in_set. Qed. | Lemma | in_set1 | 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 | |
inE | := (in_set, in_set1, inE). | Definition | inE | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"in_set",
"in_set1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set11 x : x \in [set x]. | Proof. by rewrite !inE. Qed. | Lemma | set11 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set1_inj : injective (@set1 T). | Proof. by move=> a b eqsab; apply/set1P; rewrite -eqsab set11. Qed. | Lemma | set1_inj | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"set11",
"set1P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
enum_set1 a : enum [set a] = [:: a]. | Proof. by rewrite set1.unlock (eq_enum (in_set _)) enum1. Qed. | Lemma | enum_set1 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"enum",
"enum1",
"eq_enum",
"in_set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setU1P x a B : reflect (x = a \/ x \in B) (x \in a |: B). | Proof. by rewrite !inE; apply: predU1P. Qed. | Lemma | setU1P | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"inE",
"predU1P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_setU1 x a B : (x \in a |: B) = (x == a) || (x \in B). | Proof. by rewrite !inE. Qed. | Lemma | in_setU1 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set_nil : [set:: nil] = @set0 T. | Proof. by rewrite -enum_set0 set_enum. Qed. | Lemma | set_nil | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"enum_set0",
"set0",
"set_enum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set_seq1 a : [set:: [:: a]] = [set a]. | Proof. by rewrite -enum_set1 set_enum. Qed. | Lemma | set_seq1 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"enum_set1",
"set_enum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set_cons a s : [set:: a :: s] = a |: [set:: s]. | Proof. by apply/setP=> x; rewrite !inE. Qed. | Lemma | set_cons | 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 | |
setU11 x B : x \in x |: B. | Proof. by rewrite !inE eqxx. Qed. | Lemma | setU11 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"eqxx",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setU1r x a B : x \in B -> x \in a |: B. | Proof. by move=> Bx; rewrite !inE predU1r. Qed. | Lemma | setU1r | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"inE",
"predU1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set1Ul x A b : x \in A -> x \in A :|: [set b]. | Proof. by move=> Ax; rewrite !inE Ax. Qed. | Lemma | set1Ul | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"inE"
] | associate on the left. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
set1Ur A b : b \in A :|: [set b]. | Proof. by rewrite !inE eqxx orbT. Qed. | Lemma | set1Ur | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"eqxx",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_setC1 x a : (x \in [set~ a]) = (x != a). | Proof. by rewrite !inE. Qed. | Lemma | in_setC1 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setC11 x : (x \in [set~ x]) = false. | Proof. by rewrite !inE eqxx. Qed. | Lemma | setC11 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"eqxx",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setD1P x A b : reflect (x != b /\ x \in A) (x \in A :\ b). | Proof. by rewrite !inE; apply: andP. Qed. | Lemma | setD1P | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_setD1 x A b : (x \in A :\ b) = (x != b) && (x \in A) . | Proof. by rewrite !inE. Qed. | Lemma | in_setD1 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setD11 b A : (b \in A :\ b) = false. | Proof. by rewrite !inE eqxx. Qed. | Lemma | setD11 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"eqxx",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setD1K a A : a \in A -> a |: (A :\ a) = A. | Proof. by move=> Aa; apply/setP=> x /[!inE]; case: eqP => // ->. Qed. | Lemma | setD1K | 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 | |
setU1K a B : a \notin B -> (a |: B) :\ a = B. | Proof.
by move/negPf=> nBa; apply/setP=> x /[!inE]; case: eqP => // ->.
Qed. | Lemma | setU1K | 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 | |
set2P x a b : reflect (x = a \/ x = b) (x \in [set a; b]). | Proof. by rewrite !inE; apply: pred2P. Qed. | Lemma | set2P | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"inE",
"pred2P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_set2 x a b : (x \in [set a; b]) = (x == a) || (x == b). | Proof. by rewrite !inE. Qed. | Lemma | in_set2 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set21 a b : a \in [set a; b]. | Proof. by rewrite !inE eqxx. Qed. | Lemma | set21 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"eqxx",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set22 a b : b \in [set a; b]. | Proof. by rewrite !inE eqxx orbT. Qed. | Lemma | set22 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"eqxx",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
disjoint_set1l pA x : [disjoint [set x] & pA] = (x \notin pA). | Proof.
apply/pred0P/idP=> [/(_ x)/=|]; first by rewrite inE eqxx /= => ->.
by move=> xNA y; rewrite !inE; case: eqP => //= ->; apply/negbTE.
Qed. | Lemma | disjoint_set1l | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"disjoint",
"eqxx",
"inE",
"pA",
"pred0P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
disjoint_set1r pA x : [disjoint pA & [set x]] = (x \notin pA). | Proof. by rewrite disjoint_sym disjoint_set1l. Qed. | Lemma | disjoint_set1r | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"disjoint",
"disjoint_set1l",
"disjoint_sym",
"pA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setUP x A B : reflect (x \in A \/ x \in B) (x \in A :|: B). | Proof. by rewrite !inE; apply: orP. Qed. | Lemma | setUP | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_setU x A B : (x \in A :|: B) = (x \in A) || (x \in B). | Proof. exact: in_set. Qed. | Lemma | in_setU | 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 | |
setUC A B : A :|: B = B :|: A. | Proof. by apply/setP => x; rewrite !inE orbC. Qed. | Lemma | setUC | 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 | |
setUS A 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 | setUS | 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 | |
setSU A B C : A \subset B -> A :|: C \subset B :|: C. | Proof. by move=> sAB; rewrite -!(setUC C) setUS. Qed. | Lemma | setSU | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setUC",
"setUS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setUSS A B C D : A \subset C -> B \subset D -> A :|: B \subset C :|: D. | Proof. by move=> /(setSU B) /subset_trans sAC /(setUS C)/sAC. Qed. | Lemma | setUSS | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setSU",
"setUS",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set0U A : set0 :|: A = A. | Proof. by apply/setP => x; rewrite !inE orFb. Qed. | Lemma | set0U | 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 | |
setU0 A : A :|: set0 = A. | Proof. by rewrite setUC set0U. Qed. | Lemma | setU0 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"set0",
"set0U",
"setUC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setUA A B C : A :|: (B :|: C) = A :|: B :|: C. | Proof. by apply/setP => x; rewrite !inE orbA. Qed. | Lemma | setUA | 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 | |
setUCA A B C : A :|: (B :|: C) = B :|: (A :|: C). | Proof. by rewrite !setUA (setUC B). Qed. | Lemma | setUCA | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setUA",
"setUC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setUAC A B C : A :|: B :|: C = A :|: C :|: B. | Proof. by rewrite -!setUA (setUC B). Qed. | Lemma | setUAC | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setUA",
"setUC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setUACA A B C D : (A :|: B) :|: (C :|: D) = (A :|: C) :|: (B :|: D). | Proof. by rewrite -!setUA (setUCA B). Qed. | Lemma | setUACA | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setUA",
"setUCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setTU A : setT :|: A = setT. | Proof. by apply/setP => x; rewrite !inE orTb. Qed. | Lemma | setTU | 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 | |
setUT A : A :|: setT = setT. | Proof. by rewrite setUC setTU. Qed. | Lemma | setUT | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setT",
"setTU",
"setUC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setUid A : A :|: A = A. | Proof. by apply/setP=> x; rewrite inE orbb. Qed. | Lemma | setUid | 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 | |
setUUl A B C : A :|: B :|: C = (A :|: C) :|: (B :|: C). | Proof. by rewrite setUA !(setUAC _ C) -(setUA _ C) setUid. Qed. | Lemma | setUUl | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setUA",
"setUAC",
"setUid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setUUr A B C : A :|: (B :|: C) = (A :|: B) :|: (A :|: C). | Proof. by rewrite !(setUC A) setUUl. Qed. | Lemma | setUUr | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setUC",
"setUUl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setIdP x pA pB : reflect (pA x /\ pB x) (x \in [set y | pA y & pB y]). | Proof. by rewrite !inE; apply: andP. Qed. | Lemma | setIdP | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"inE",
"pA"
] | setIdP is a generalisation of setIP that applies to comprehensions. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
setId2P x 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 | setId2P | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"inE",
"pA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setIdE A pB : [set x in A | pB x] = A :&: [set x | pB x]. | Proof. by apply/setP=> x; rewrite !inE. Qed. | Lemma | setIdE | 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 | |
setIP x A B : reflect (x \in A /\ x \in B) (x \in A :&: B). | Proof. exact: (iffP (@setIdP _ _ _)). Qed. | Lemma | setIP | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setIdP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_setI x A B : (x \in A :&: B) = (x \in A) && (x \in B). | Proof. exact: in_set. Qed. | Lemma | in_setI | 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 | |
setIC A B : A :&: B = B :&: A. | Proof. by apply/setP => x; rewrite !inE andbC. Qed. | Lemma | setIC | 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 | |
setIS A 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 | setIS | 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 | |
setSI A B C : A \subset B -> A :&: C \subset B :&: C. | Proof. by move=> sAB; rewrite -!(setIC C) setIS. Qed. | Lemma | setSI | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setIC",
"setIS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setISS A 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 | setISS | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setIS",
"setSI",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setTI A : setT :&: A = A. | Proof. by apply/setP => x; rewrite !inE andTb. Qed. | Lemma | setTI | 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 | |
setIT A : A :&: setT = A. | Proof. by rewrite setIC setTI. Qed. | Lemma | setIT | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setIC",
"setT",
"setTI"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set0I A : set0 :&: A = set0. | Proof. by apply/setP => x; rewrite !inE andFb. Qed. | Lemma | set0I | 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 | |
setI0 A : A :&: set0 = set0. | Proof. by rewrite setIC set0I. Qed. | Lemma | setI0 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"set0",
"set0I",
"setIC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setIA A B C : A :&: (B :&: C) = A :&: B :&: C. | Proof. by apply/setP=> x; rewrite !inE andbA. Qed. | Lemma | setIA | 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 | |
setICA A B C : A :&: (B :&: C) = B :&: (A :&: C). | Proof. by rewrite !setIA (setIC A). Qed. | Lemma | setICA | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setIA",
"setIC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setIAC A B C : A :&: B :&: C = A :&: C :&: B. | Proof. by rewrite -!setIA (setIC B). Qed. | Lemma | setIAC | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setIA",
"setIC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setIACA A B C D : (A :&: B) :&: (C :&: D) = (A :&: C) :&: (B :&: D). | Proof. by rewrite -!setIA (setICA B). Qed. | Lemma | setIACA | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setIA",
"setICA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setIid A : A :&: A = A. | Proof. by apply/setP=> x; rewrite inE andbb. Qed. | Lemma | setIid | 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 | |
setIIl A B C : A :&: B :&: C = (A :&: C) :&: (B :&: C). | Proof. by rewrite setIA !(setIAC _ C) -(setIA _ C) setIid. Qed. | Lemma | setIIl | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setIA",
"setIAC",
"setIid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setIIr A B C : A :&: (B :&: C) = (A :&: B) :&: (A :&: C). | Proof. by rewrite !(setIC A) setIIl. Qed. | Lemma | setIIr | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setIC",
"setIIl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setIUr A B C : A :&: (B :|: C) = (A :&: B) :|: (A :&: C). | Proof. by apply/setP=> x; rewrite !inE andb_orr. Qed. | Lemma | setIUr | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"inE",
"setP"
] | distribute /cancel | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
setIUl A B C : (A :|: B) :&: C = (A :&: C) :|: (B :&: C). | Proof. by apply/setP=> x; rewrite !inE andb_orl. Qed. | Lemma | setIUl | 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 | |
setUIr A B C : A :|: (B :&: C) = (A :|: B) :&: (A :|: C). | Proof. by apply/setP=> x; rewrite !inE orb_andr. Qed. | Lemma | setUIr | 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 | |
setUIl A B C : (A :&: B) :|: C = (A :|: C) :&: (B :|: C). | Proof. by apply/setP=> x; rewrite !inE orb_andl. Qed. | Lemma | setUIl | 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 | |
setUK A B : (A :|: B) :&: A = A. | Proof. by apply/setP=> x; rewrite !inE orbK. Qed. | Lemma | setUK | 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 | |
setKU A B : A :&: (B :|: A) = A. | Proof. by apply/setP=> x; rewrite !inE orKb. Qed. | Lemma | setKU | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"inE",
"orKb",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setIK A B : (A :&: B) :|: A = A. | Proof. by apply/setP=> x; rewrite !inE andbK. Qed. | Lemma | setIK | 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 | |
setKI A B : A :|: (B :&: A) = A. | Proof. by apply/setP=> x; rewrite !inE andKb. Qed. | Lemma | setKI | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"andKb",
"apply",
"inE",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setCP x A : reflect (~ x \in A) (x \in ~: A). | Proof. by rewrite !inE; apply: negP. Qed. | Lemma | setCP | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"inE"
] | complement | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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