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 |
|---|---|---|---|---|---|---|---|---|---|---|
set1K : pcancel set1 unset1. | Proof. by move=> i; rewrite /unset1 cards1 eqxx pick_set1. Qed. | Lemma | set1K | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"cards1",
"eqxx",
"pick_set1",
"unset1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
omap_unset1K A : #|A| = 1 -> omap set1 (unset1 A) = Some A. | Proof. by move=> /eqP/cards1P[i ->]; rewrite set1K. Qed. | Lemma | omap_unset1K | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"cards1P",
"set1K",
"unset1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unset10 : unset1 set0 = None. | Proof. by rewrite /unset1 cards0. Qed. | Lemma | unset10 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"cards0",
"set0",
"unset1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unset1N1 A : #|A| != 1 -> unset1 A = None. | Proof. by move=> AN1; rewrite /unset1 ifN. Qed. | Lemma | unset1N1 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"unset1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unset1K : ocancel unset1 set1. | Proof.
move=> A; rewrite /unset1.
by case: ifPn => // /cards1P[i ->]/=; rewrite pick_set1.
Qed. | Lemma | unset1K | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"cards1P",
"pick_set1",
"unset1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setXnS (I : finType) (T : I -> finType) (A B : forall i, {set T i}) :
(forall i, A i \subset B i) -> setXn A \subset setXn B. | Proof.
move=> sAB; apply/subsetP => x /setXnP xA.
by apply/setXnP => i; apply/subsetP: (xA i).
Qed. | Lemma | setXnS | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"setXn",
"setXnP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_setXn (I : finType) (T : I -> finType) (A B : forall i, {set T i}) :
(forall i, A i = B i) -> setXn A = setXn B. | Proof.
by move=> eqAB; apply/eqP; rewrite eqEsubset !setXnS// => j; rewrite eqAB.
Qed. | Lemma | eq_setXn | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"eqEsubset",
"setXn",
"setXnS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_set0 F : \big[op/x]_(i in set0) F i = x. | Proof. by apply: big_pred0 => i; rewrite inE. Qed. | Lemma | big_set0 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"big_pred0",
"inE",
"set0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_set1E j F : \big[op/x]_(i in [set j]) F i = op (F j) x. | Proof. by rewrite -big_pred1_eq_id; apply: eq_bigl => i; apply: in_set1. Qed. | Lemma | big_set1E | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"big_pred1_eq_id",
"eq_bigl",
"in_set1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_set (A : pred I) F :
\big[op/x]_(i in [set i | A i]) (F i) = \big[op/x]_(i in A) (F i). | Proof. by apply: eq_bigl => i; rewrite inE. Qed. | Lemma | big_set | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"eq_bigl",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(le_refl : reflexive le) (le_incr : forall x y, le x (op x y)). | Hypotheses | le_refl | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"le"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
subset_le_big_cond (I : finType) (A A' P P' : {pred I}) (F : I -> R) :
[set i in A | P i] \subset [set i in A' | P' i] ->
le (\big[op/x]_(i in A | P i) F i) (\big[op/x]_(i in A' | P' i) F i). | Proof.
by move=> /subsetP AP; apply: sub_le_big => // i; have /[!inE] := AP i.
Qed. | Lemma | subset_le_big_cond | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"A'",
"apply",
"inE",
"le",
"sub_le_big",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_imset_idem [I J : finType] (h : I -> J) (A : pred I) F :
idempotent_op op ->
\big[op/x]_(j in h @: A) F j = \big[op/x]_(i in A) F (h i). | Proof.
rewrite -!big_image => op_idem; rewrite -big_undup// -[RHS]big_undup//.
apply/perm_big/perm_undup => j; apply/imageP.
have [mem_j | /imageP mem_j] := boolP (j \in [seq h j | j in A]).
- by exists j => //; apply/imsetP; apply: imageP mem_j.
- by case=> k /imsetP [i j_in_A ->] eq_i; apply: mem_j; exists i.
Qed. | Lemma | big_imset_idem | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"big_image",
"big_undup",
"idempotent_op",
"imageP",
"imsetP",
"perm_big",
"perm_undup",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_set1 a F : \big[op/idx]_(i in [set a]) F i = F a. | Proof. by apply: big_pred1 => i; rewrite !inE. Qed. | Lemma | big_set1 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"big_pred1",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_setID A B F :
\big[aop/idx]_(i in A) F i =
aop (\big[aop/idx]_(i in A :&: B) F i)
(\big[aop/idx]_(i in A :\: B) F i). | Proof.
rewrite (bigID [in B]) setDE.
by congr (aop _ _); apply: eq_bigl => i; rewrite !inE.
Qed. | Lemma | big_setID | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"bigID",
"eq_bigl",
"inE",
"setDE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_setIDcond A B P F :
\big[aop/idx]_(i in A | P i) F i =
aop (\big[aop/idx]_(i in A :&: B | P i) F i)
(\big[aop/idx]_(i in A :\: B | P i) F i). | Proof. by rewrite !big_mkcondr; apply: big_setID. Qed. | Lemma | big_setIDcond | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"big_mkcondr",
"big_setID"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_setD1 a A F : a \in A ->
\big[aop/idx]_(i in A) F i = aop (F a) (\big[aop/idx]_(i in A :\ a) F i). | Proof.
move=> Aa; rewrite (bigD1 a Aa); congr (aop _).
by apply: eq_bigl => x; rewrite !inE andbC.
Qed. | Lemma | big_setD1 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"bigD1",
"eq_bigl",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_setU1 a A F : a \notin A ->
\big[aop/idx]_(i in a |: A) F i = aop (F a) (\big[aop/idx]_(i in A) F i). | Proof. by move=> notAa; rewrite (@big_setD1 a) ?setU11 //= setU1K. Qed. | Lemma | big_setU1 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"big_setD1",
"setU11",
"setU1K"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_subset_idem_cond A B P F :
idempotent_op aop ->
A \subset B ->
aop (\big[aop/idx]_(i in A | P i) F i) (\big[aop/idx]_(i in B | P i) F i)
= \big[aop/idx]_(i in B | P i) F i. | Proof.
by move=> idaop /setIidPr <-; rewrite (big_setIDcond B A) Monoid.mulmA /= idaop.
Qed. | Lemma | big_subset_idem_cond | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"big_setIDcond",
"idempotent_op",
"mulmA",
"setIidPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_subset_idem A B F :
idempotent_op aop ->
A \subset B ->
aop (\big[aop/idx]_(i in A) F i) (\big[aop/idx]_(i in B) F i)
= \big[aop/idx]_(i in B) F i. | Proof. by rewrite -2!big_condT; apply: big_subset_idem_cond. Qed. | Lemma | big_subset_idem | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"big_condT",
"big_subset_idem_cond",
"idempotent_op"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_setU_cond A B P F :
idempotent_op aop ->
\big[aop/idx]_(i in A :|: B | P i) F i
= aop (\big[aop/idx]_(i in A | P i) F i) (\big[aop/idx]_(i in B | P i) F i). | Proof.
move=> idemaop; rewrite (big_setIDcond _ A) setUK setDUl setDv set0U.
rewrite (big_setIDcond B A) Monoid.mulmCA Monoid.mulmA /=.
by rewrite (@big_subset_idem_cond (B :&: A)) // subsetIr.
Qed. | Lemma | big_setU_cond | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"big_setIDcond",
"big_subset_idem_cond",
"idempotent_op",
"mulmA",
"mulmCA",
"set0U",
"setDUl",
"setDv",
"setUK",
"subsetIr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_setU A B F :
idempotent_op aop ->
\big[aop/idx]_(i in A :|: B) F i
= aop (\big[aop/idx]_(i in A) F i) (\big[aop/idx]_(i in B) F i). | Proof. by rewrite -3!big_condT; apply: big_setU_cond. Qed. | Lemma | big_setU | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"big_condT",
"big_setU_cond",
"idempotent_op"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_imset h (A : {pred I}) G : {in A &, injective h} ->
\big[aop/idx]_(j in h @: A) G j = \big[aop/idx]_(i in A) G (h i). | Proof.
move=> injh; pose hA := mem (image h A).
rewrite (eq_bigl hA) => [j|]; first exact/imsetP/imageP.
pose h' := omap (fun u : {j | hA j} => iinv (svalP u)) \o insub.
rewrite (reindex_omap h h') => [j hAj|]; rewrite {}/h'/= ?insubT/= ?f_iinv//.
apply: eq_bigl => i; case: insubP => [u /= -> def_u | nhAhi]; last first... | Lemma | big_imset | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"eq_bigl",
"f_iinv",
"iinv",
"image",
"imageP",
"imsetP",
"insub",
"insubP",
"insubT",
"last",
"mem_iinv",
"reindex_omap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_imset_cond h (A : {pred I}) (P : pred J) G : {in A &, injective h} ->
\big[aop/idx]_(j in h @: A | P j) G j =
\big[aop/idx]_(i in A | P (h i)) G (h i). | Proof. by move=> h_inj; rewrite 2!big_mkcondr big_imset. Qed. | Lemma | big_imset_cond | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"big_imset",
"big_mkcondr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
partition_big_imset h (A : {pred I}) F :
\big[aop/idx]_(i in A) F i =
\big[aop/idx]_(j in h @: A) \big[aop/idx]_(i in A | h i == j) F i. | Proof. by apply: partition_big => i Ai; apply/imsetP; exists i. Qed. | Lemma | partition_big_imset | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"imsetP",
"partition_big"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_cards1 (f : {set I} -> R) :
\big[aop/idx]_(A : {set I} | #|A| == 1) f A
= \big[aop/idx]_(i : I) f [set i]. | Proof.
rewrite (reindex_omap set1 unset1) => [A /cards1P[i ->] /[!set1K]//|].
by apply: eq_bigl => i; rewrite set1K cards1 !eqxx.
Qed. | Lemma | big_cards1 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"cards1",
"cards1P",
"eq_bigl",
"eqxx",
"reindex_omap",
"set1K",
"unset1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigA_distr (R : Type) (zero one : R) (mul : Monoid.mul_law zero)
(add : Monoid.add_law zero mul) (I : finType) (F G : I -> R) :
\big[mul/one]_i add (F i) (G i) =
\big[add/zero]_(J in {set I}) \big[mul/one]_i (if i \in J then F i else G i). | Proof.
under eq_bigr => i _ do rewrite -(big_bool _ (fun b => if b then F i else G i)).
rewrite bigA_distr_bigA.
set f := fun J : {set I} => val J.
transitivity (\big[add/zero]_(f0 in (imset f (mem setT)))
\big[mul/one]_i (if f0 i then F i else G i)).
suff <-: setT = imset f (mem setT) by apply: congr... | Lemma | bigA_distr | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"add",
"add_law",
"apply",
"bigA_distr_bigA",
"big_bool",
"big_imset",
"congr_big",
"eq_bigr",
"imsetP",
"in_setT",
"mul",
"mul_law",
"one",
"setT",
"subTset",
"subsetP",
"val",
"zero"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
imset2_set1l x1 (D2 : {pred aT2}) : f @2: ([set x1], D2) = f x1 @: D2. | Proof.
apply/setP=> y; apply/imset2P/imsetP=> [[x x2 /set1P->]| [x2 Dx2 ->]].
by exists x2.
by exists x1 x2; rewrite ?set11.
Qed. | Lemma | imset2_set1l | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"imset2P",
"imsetP",
"set11",
"set1P",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
imset2_set1r x2 (D1 : {pred aT1}) : f @2: (D1, [set x2]) = f^~ x2 @: D1. | Proof.
apply/setP=> y; apply/imset2P/imsetP=> [[x1 x Dx1 /set1P->]| [x1 Dx1 ->]].
by exists x1.
by exists x1 x2; rewrite ?set11.
Qed. | Lemma | imset2_set1r | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"imset2P",
"imsetP",
"set11",
"set1P",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
imset_card : #|f @: D| = #|image f D|. | Proof. by rewrite [@imset]unlock cardsE. Qed. | Lemma | imset_card | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"cardsE",
"image"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_imset_card : #|f @: D| <= #|D|. | Proof. by rewrite imset_card leq_image_card. Qed. | Lemma | leq_imset_card | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"imset_card",
"leq_image_card"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_in_imset : {in D &, injective f} -> #|f @: D| = #|D|. | Proof. by move=> injf; rewrite imset_card card_in_image. Qed. | Lemma | card_in_imset | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"card_in_image",
"imset_card",
"injf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_imset : injective f -> #|f @: D| = #|D|. | Proof. by move=> injf; rewrite imset_card card_image. Qed. | Lemma | card_imset | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"card_image",
"imset_card",
"injf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
imset_injP : reflect {in D &, injective f} (#|f @: D| == #|D|). | Proof. by rewrite [@imset]unlock cardsE; apply: image_injP. Qed. | Lemma | imset_injP | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"cardsE",
"image_injP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
can2_in_imset_pre :
{in D, cancel f g} -> {on D, cancel g & f} -> f @: D = g @^-1: D. | Proof.
move=> fK gK; apply/setP=> y; rewrite inE.
by apply/imsetP/idP=> [[x Ax ->] | Agy]; last exists (g y); rewrite ?(fK, gK).
Qed. | Lemma | can2_in_imset_pre | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"fK",
"gK",
"imsetP",
"inE",
"last",
"on",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
can2_imset_pre : cancel f g -> cancel g f -> f @: D = g @^-1: D. | Proof. by move=> fK gK; apply: can2_in_imset_pre; apply: in1W. Qed. | Lemma | can2_imset_pre | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"can2_in_imset_pre",
"fK",
"gK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
on_card_preimset (aT rT : finType) (f : aT -> rT) (R : {pred rT}) :
{on R, bijective f} -> #|f @^-1: R| = #|R|. | Proof.
case=> g fK gK; rewrite -(can2_in_imset_pre gK) // card_in_imset //.
exact: can_in_inj gK.
Qed. | Lemma | on_card_preimset | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"aT",
"can2_in_imset_pre",
"card_in_imset",
"fK",
"gK",
"on"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
can_imset_pre (T : finType) f g (A : {set T}) :
cancel f g -> f @: A = g @^-1: A :> {set T}. | Proof.
move=> fK; apply: can2_imset_pre => // x.
suffices fx: x \in codom f by rewrite -(f_iinv fx) fK.
exact/(subset_cardP (card_codom (can_inj fK)))/subsetP.
Qed. | Lemma | can_imset_pre | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"can2_imset_pre",
"card_codom",
"codom",
"fK",
"f_iinv",
"subsetP",
"subset_cardP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
imset_id (T : finType) (A : {set T}) : [set x | x in A] = A. | Proof. by apply/setP=> x; rewrite (@can_imset_pre _ _ id) ?inE. Qed. | Lemma | imset_id | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"can_imset_pre",
"id",
"inE",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_preimset (T : finType) (f : T -> T) (A : {set T}) :
injective f -> #|f @^-1: A| = #|A|. | Proof.
move=> injf; apply: on_card_preimset; apply: onW_bij.
have ontof: _ \in codom f by apply/(subset_cardP (card_codom injf))/subsetP.
by exists (fun x => iinv (ontof x)) => x; rewrite (f_iinv, iinv_f).
Qed. | Lemma | card_preimset | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"card_codom",
"codom",
"f_iinv",
"iinv",
"iinv_f",
"injf",
"on_card_preimset",
"subsetP",
"subset_cardP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_powerset (T : finType) (A : {set T}) : #|powerset A| = 2 ^ #|A|. | Proof.
rewrite -card_bool -(card_pffun_on false) -(card_imset _ val_inj).
apply: eq_card => f; pose sf := false.-support f; pose D := finset sf.
have sDA: (D \subset A) = (sf \subset A) by apply: eq_subset; apply: in_set.
have eq_sf x : sf x = f x by rewrite /= negb_eqb addbF.
have valD: val D = f by rewrite /D unlock;... | Lemma | card_powerset | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"card_bool",
"card_imset",
"card_pffun_on",
"eq_card",
"eq_subset",
"ffunE",
"ffunP",
"imsetP",
"inE",
"in_set",
"last",
"negb_eqb",
"pffun_onP",
"powerset",
"support",
"val",
"valD",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
imset_comp (f : T' -> U) (g : T -> T') (H : {pred T}) :
(f \o g) @: H = f @: (g @: H). | Proof.
apply/setP/subset_eqP/andP.
split; apply/subsetP=> _ /imsetP[x0 Hx0 ->]; apply/imsetP.
by exists (g x0); first apply: imset_f.
by move/imsetP: Hx0 => [x1 Hx1 ->]; exists x1.
Qed. | Lemma | imset_comp | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"T'",
"apply",
"imsetP",
"imset_f",
"setP",
"split",
"subsetP",
"subset_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcup_ ( i <- r | P ) F" | :=
(\big[@setU _/set0]_(i <- r | P) F%SET) : set_scope. | Notation | \bigcup_ ( i <- r | P ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"set0",
"setU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcup_ ( i <- r ) F" | :=
(\big[@setU _/set0]_(i <- r) F%SET) : set_scope. | Notation | \bigcup_ ( i <- r ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"set0",
"setU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcup_ ( m <= i < n | P ) F" | :=
(\big[@setU _/set0]_(m <= i < n | P%B) F%SET) : set_scope. | Notation | \bigcup_ ( m <= i < n | P ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"set0",
"setU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcup_ ( m <= i < n ) F" | :=
(\big[@setU _/set0]_(m <= i < n) F%SET) : set_scope. | Notation | \bigcup_ ( m <= i < n ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"set0",
"setU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcup_ ( i | P ) F" | :=
(\big[@setU _/set0]_(i | P%B) F%SET) : set_scope. | Notation | \bigcup_ ( i | P ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"set0",
"setU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcup_ i F" | :=
(\big[@setU _/set0]_i F%SET) : set_scope. | Notation | \bigcup_ i F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"set0",
"setU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcup_ ( i : t | P ) F" | :=
(\big[@setU _/set0]_(i : t | P%B) F%SET) (only parsing): set_scope. | Notation | \bigcup_ ( i : t | P ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"set0",
"setU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcup_ ( i : t ) F" | :=
(\big[@setU _/set0]_(i : t) F%SET) (only parsing) : set_scope. | Notation | \bigcup_ ( i : t ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"set0",
"setU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcup_ ( i < n | P ) F" | :=
(\big[@setU _/set0]_(i < n | P%B) F%SET) : set_scope. | Notation | \bigcup_ ( i < n | P ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"set0",
"setU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcup_ ( i < n ) F" | :=
(\big[@setU _/set0]_ (i < n) F%SET) : set_scope. | Notation | \bigcup_ ( i < n ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"set0",
"setU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcup_ ( i 'in' A | P ) F" | :=
(\big[@setU _/set0]_(i in A | P%B) F%SET) : set_scope. | Notation | \bigcup_ ( i 'in' A | P ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"set0",
"setU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcup_ ( i 'in' A ) F" | :=
(\big[@setU _/set0]_(i in A) F%SET) : set_scope. | Notation | \bigcup_ ( i 'in' A ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"set0",
"setU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i <- r | P ) F" | :=
(\big[@setI _/setT]_(i <- r | P%B) F%SET) : set_scope. | Notation | \bigcap_ ( i <- r | P ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setI",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i <- r ) F" | :=
(\big[@setI _/setT]_(i <- r) F%SET) : set_scope. | Notation | \bigcap_ ( i <- r ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setI",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( m <= i < n | P ) F" | :=
(\big[@setI _/setT]_(m <= i < n | P%B) F%SET) : set_scope. | Notation | \bigcap_ ( m <= i < n | P ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setI",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( m <= i < n ) F" | :=
(\big[@setI _/setT]_(m <= i < n) F%SET) : set_scope. | Notation | \bigcap_ ( m <= i < n ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setI",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i | P ) F" | :=
(\big[@setI _/setT]_(i | P%B) F%SET) : set_scope. | Notation | \bigcap_ ( i | P ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setI",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ i F" | :=
(\big[@setI _/setT]_i F%SET) : set_scope. | Notation | \bigcap_ i F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setI",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i : t | P ) F" | :=
(\big[@setI _/setT]_(i : t | P%B) F%SET) (only parsing): set_scope. | Notation | \bigcap_ ( i : t | P ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setI",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i : t ) F" | :=
(\big[@setI _/setT]_(i : t) F%SET) (only parsing) : set_scope. | Notation | \bigcap_ ( i : t ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setI",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i < n | P ) F" | :=
(\big[@setI _/setT]_(i < n | P%B) F%SET) : set_scope. | Notation | \bigcap_ ( i < n | P ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setI",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i < n ) F" | :=
(\big[@setI _/setT]_(i < n) F%SET) : set_scope. | Notation | \bigcap_ ( i < n ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setI",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i 'in' A | P ) F" | :=
(\big[@setI _/setT]_(i in A | P%B) F%SET) : set_scope. | Notation | \bigcap_ ( i 'in' A | P ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setI",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i 'in' A ) F" | :=
(\big[@setI _/setT]_(i in A) F%SET) : set_scope. | Notation | \bigcap_ ( i 'in' A ) F | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setI",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcup_sup j P F : P j -> F j \subset \bigcup_(i | P i) F i. | Proof. by move=> Pj; rewrite (bigD1 j) //= subsetUl. Qed. | Lemma | bigcup_sup | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"bigD1",
"subsetUl"
] | defer the F j pattern (even though it's a Miller pattern!). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
bigcup_max j U P F :
P j -> U \subset F j -> U \subset \bigcup_(i | P i) F i. | Proof. by move=> Pj sUF; apply: subset_trans (bigcup_sup _ Pj). Qed. | Lemma | bigcup_max | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"bigcup_sup",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcupP x P F :
reflect (exists2 i, P i & x \in F i) (x \in \bigcup_(i | P i) F i). | Proof.
apply: (iffP idP) => [|[i Pi]]; last first.
by apply: subsetP x; apply: bigcup_sup.
by elim/big_rec: _ => [|i _ Pi _ /setUP[|//]]; [rewrite inE | exists i].
Qed. | Lemma | bigcupP | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"big_rec",
"bigcup_sup",
"inE",
"last",
"setUP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcupsP U P F :
reflect (forall i, P i -> F i \subset U) (\bigcup_(i | P i) F i \subset U). | Proof.
apply: (iffP idP) => [sFU i Pi| sFU].
by apply: subset_trans sFU; apply: bigcup_sup.
by apply/subsetP=> x /bigcupP[i Pi]; apply: (subsetP (sFU i Pi)).
Qed. | Lemma | bigcupsP | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"bigcupP",
"bigcup_sup",
"subsetP",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcup0P P F :
reflect (forall i, P i -> F i = set0) (\bigcup_(i | P i) F i == set0). | Proof.
rewrite -subset0; apply: (iffP (bigcupsP _ _ _)) => sub0 i /sub0; last by move->.
by rewrite subset0 => /eqP.
Qed. | Lemma | bigcup0P | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"bigcupsP",
"last",
"set0",
"subset0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcup_disjointP U P F :
reflect (forall i : I, P i -> [disjoint U & F i])
[disjoint U & \bigcup_(i | P i) F i]. | Proof.
apply: (iffP idP) => [dUF i Pp|dUF].
by apply: disjointWr dUF; apply: bigcup_sup.
rewrite disjoint_sym disjoint_subset.
by apply/bigcupsP=> i /dUF; rewrite disjoint_sym disjoint_subset.
Qed. | Lemma | bigcup_disjointP | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"bigcup_sup",
"bigcupsP",
"disjoint",
"disjointWr",
"disjoint_subset",
"disjoint_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcup_disjoint U P F :
(forall i, P i -> [disjoint U & F i]) -> [disjoint U & \bigcup_(i | P i) F i]. | Proof. by move/bigcup_disjointP. Qed. | Lemma | bigcup_disjoint | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"bigcup_disjointP",
"disjoint"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcup_setU A B F :
\bigcup_(i in A :|: B) F i =
(\bigcup_(i in A) F i) :|: (\bigcup_ (i in B) F i). | Proof.
apply/setP=> x; apply/bigcupP/setUP=> [[i] | ].
by case/setUP; [left | right]; apply/bigcupP; exists i.
by case=> /bigcupP[i Pi]; exists i; rewrite // inE Pi ?orbT.
Qed. | Lemma | bigcup_setU | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"bigcupP",
"inE",
"setP",
"setUP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcup_seq r F : \bigcup_(i <- r) F i = \bigcup_(i in r) F i. | Proof.
elim: r => [|i r IHr]; first by rewrite big_nil big_pred0.
rewrite big_cons {}IHr; case r_i: (i \in r).
rewrite (setUidPr _) ?bigcup_sup //.
by apply: eq_bigl => j /[!inE]; case: eqP => // ->.
rewrite (bigD1 i (mem_head i r)) /=; congr (_ :|: _).
by apply: eq_bigl => j /=; rewrite andbC; case: eqP => // ->.
... | Lemma | bigcup_seq | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"bigD1",
"big_cons",
"big_nil",
"big_pred0",
"bigcup_sup",
"eq_bigl",
"inE",
"mem_head",
"setUidPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcap_inf j P F : P j -> \bigcap_(i | P i) F i \subset F j. | Proof. by move=> Pj; rewrite (bigD1 j) //= subsetIl. Qed. | Lemma | bigcap_inf | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"bigD1",
"subsetIl"
] | Unlike its setU counterpart, this lemma is useable. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
bigcap_min j U P F :
P j -> F j \subset U -> \bigcap_(i | P i) F i \subset U. | Proof. by move=> Pj; apply: subset_trans (bigcap_inf _ Pj). Qed. | Lemma | bigcap_min | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"bigcap_inf",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcapsP U P F :
reflect (forall i, P i -> U \subset F i) (U \subset \bigcap_(i | P i) F i). | Proof.
apply: (iffP idP) => [sUF i Pi | sUF].
by apply: subset_trans sUF _; apply: bigcap_inf.
elim/big_rec: _ => [|i V Pi sUV]; apply/subsetP=> x Ux; rewrite inE //.
by rewrite !(subsetP _ x Ux) ?sUF.
Qed. | Lemma | bigcapsP | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"big_rec",
"bigcap_inf",
"inE",
"subsetP",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcapP x P F :
reflect (forall i, P i -> x \in F i) (x \in \bigcap_(i | P i) F i). | Proof.
rewrite -sub1set.
by apply: (iffP (bigcapsP _ _ _)) => Fx i /Fx; rewrite sub1set.
Qed. | Lemma | bigcapP | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"bigcapsP",
"sub1set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setC_bigcup J r (P : pred J) (F : J -> {set T}) :
~: (\bigcup_(j <- r | P j) F j) = \bigcap_(j <- r | P j) ~: F j. | Proof. by apply: big_morph => [A B|]; rewrite ?setC0 ?setCU. Qed. | Lemma | setC_bigcup | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"big_morph",
"setC0",
"setCU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setC_bigcap J r (P : pred J) (F : J -> {set T}) :
~: (\bigcap_(j <- r | P j) F j) = \bigcup_(j <- r | P j) ~: F j. | Proof. by apply: big_morph => [A B|]; rewrite ?setCT ?setCI. Qed. | Lemma | setC_bigcap | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"big_morph",
"setCI",
"setCT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcap_setU A B F :
(\bigcap_(i in A :|: B) F i) =
(\bigcap_(i in A) F i) :&: (\bigcap_(i in B) F i). | Proof. by apply: setC_inj; rewrite setCI !setC_bigcap bigcup_setU. Qed. | Lemma | bigcap_setU | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"bigcup_setU",
"setCI",
"setC_bigcap",
"setC_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcap_seq r F : \bigcap_(i <- r) F i = \bigcap_(i in r) F i. | Proof. by apply: setC_inj; rewrite !setC_bigcap bigcup_seq. Qed. | Lemma | bigcap_seq | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"bigcup_seq",
"setC_bigcap",
"setC_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
curry_imset2X : f @2: (A1, A2) = uncurry f @: (setX A1 A2). | Proof.
rewrite [@imset]unlock unlock; apply/setP=> x; rewrite !in_set; congr (x \in _).
by apply: eq_image => u //=; rewrite !inE.
Qed. | Lemma | curry_imset2X | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"eq_image",
"inE",
"in_set",
"setP",
"setX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
curry_imset2l : f @2: (D1, D2) = \bigcup_(x1 in D1) f x1 @: D2. | Proof.
apply/setP=> y; apply/imset2P/bigcupP => [[x1 x2 Dx1 Dx2 ->{y}] | [x1 Dx1]].
by exists x1; rewrite // imset_f.
by case/imsetP=> x2 Dx2 ->{y}; exists x1 x2.
Qed. | Lemma | curry_imset2l | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"bigcupP",
"imset2P",
"imsetP",
"imset_f",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
curry_imset2r : f @2: (D1, D2) = \bigcup_(x2 in D2) f^~ x2 @: D1. | Proof.
apply/setP=> y; apply/imset2P/bigcupP => [[x1 x2 Dx1 Dx2 ->{y}] | [x2 Dx2]].
by exists x2; rewrite // (imset_f (f^~ x2)).
by case/imsetP=> x1 Dx1 ->{y}; exists x1 x2.
Qed. | Lemma | curry_imset2r | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"bigcupP",
"imset2P",
"imsetP",
"imset_f",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
imset2Ul (A B : {set aT1}) (C : {set aT2}) :
f @2: (A :|: B, C) = f @2: (A, C) :|: f @2: (B, C). | Proof. by rewrite !curry_imset2l bigcup_setU. Qed. | Lemma | imset2Ul | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"bigcup_setU",
"curry_imset2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
imset2Ur (A : {set aT1}) (B C : {set aT2}) :
f @2: (A, B :|: C) = f @2: (A, B) :|: f @2: (A, C). | Proof. by rewrite !curry_imset2r bigcup_setU. Qed. | Lemma | imset2Ur | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"bigcup_setU",
"curry_imset2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cover P | := \bigcup_(B in P) B. | Definition | cover | 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 | |
pblock P x | := odflt set0 (pick [pred B in P | x \in B]). | Definition | pblock | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"pick",
"set0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trivIset P | := \sum_(B in P) #|B| == #|cover P|. | Definition | trivIset | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"cover"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
partition P D | := [&& cover P == D, trivIset P & set0 \notin P]. | Definition | partition | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"cover",
"set0",
"trivIset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_transversal X P D | :=
[&& partition P D, X \subset D & [forall B in P, #|X :&: B| == 1]]. | Definition | is_transversal | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"partition"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
transversal P D | := [set odflt x [pick y in pblock P x] | x in D]. | Definition | transversal | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"pblock",
"pick"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
transversal_repr x0 X B | := odflt x0 [pick x in X :&: B]. | Definition | transversal_repr | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"pick"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_card_setU A B : #|A :|: B| <= #|A| + #|B| ?= iff [disjoint A & B]. | Proof.
rewrite -(addn0 #|_|) -setI_eq0 -cards_eq0 -cardsUI eq_sym.
by rewrite (mono_leqif (leq_add2l _)).
Qed. | Lemma | leq_card_setU | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"addn0",
"cardsUI",
"cards_eq0",
"disjoint",
"eq_sym",
"leq_add2l",
"mono_leqif",
"setI_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_card_cover P : #|cover P| <= \sum_(A in P) #|A| ?= iff trivIset P. | Proof.
split; last exact: eq_sym.
rewrite /cover; elim/big_rec2: _ => [|A n U _ leUn]; first by rewrite cards0.
by rewrite (leq_trans (leq_card_setU A U).1) ?leq_add2l.
Qed. | Lemma | leq_card_cover | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"big_rec2",
"cards0",
"cover",
"eq_sym",
"last",
"leq_add2l",
"leq_card_setU",
"leq_trans",
"split",
"trivIset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
imset_cover (T' : finType) P (f : T -> T') :
[set f x | x in cover P] = \bigcup_(i in P) [set f x | x in i]. | Proof.
apply/setP=> y; apply/imsetP/bigcupP => [|[A AP /imsetP[x xA ->]]].
by move=> [x /bigcupP[A AP xA] ->]; exists A => //; rewrite imset_f.
by exists x => //; apply/bigcupP; exists A.
Qed. | Lemma | imset_cover | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"T'",
"apply",
"bigcupP",
"cover",
"imsetP",
"imset_f",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cover1 A : cover [set A] = A. | Proof. by rewrite /cover big_set1. Qed. | Lemma | cover1 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"big_set1",
"cover"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subset_cover P P' : P \subset P' -> cover P \subset cover P'. | Proof.
move=> /subsetP subP; apply/subsetP=> x /bigcupP [scc /subP].
by move=> scc' x_in; apply/bigcupP; exists scc.
Qed. | Lemma | subset_cover | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"bigcupP",
"cover",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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