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 |
|---|---|---|---|---|---|---|---|---|---|---|
big_image I (J : finType) (h : J -> I) (A : pred J) F :
\big[op/x]_(i <- [seq h j | j in A]) F i = \big[op/x]_(j in A) F (h j). | Proof. by rewrite big_map big_enum. Qed. | Lemma | big_image | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_enum",
"big_map",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cardD1x (I : finType) (A : pred I) j :
A j -> #|SimplPred A| = 1 + #|[pred i | A i & i != j]|. | Proof.
move=> Aj; rewrite (cardD1 j) [j \in A]Aj; congr (_ + _).
by apply: eq_card => i; rewrite inE /= andbC.
Qed. | Lemma | cardD1x | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"cardD1",
"eq_card",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
reindex_omap (I J : finType) (h : J -> I) h' (P : pred I) F :
(forall i, P i -> omap h (h' i) = some i) ->
\big[op/x]_(i | P i) F i =
\big[op/x]_(j | P (h j) && (h' (h j) == some j)) F (h j). | Proof.
move=> h'K; have [n lePn] := ubnP #|P|; elim: n => // n IHn in P h'K lePn *.
case: (pickP P) => [i Pi | P0]; last first.
by rewrite !big_pred0 // => j; rewrite P0.
have := h'K i Pi; case h'i_eq : (h' i) => [/= j|//] [hj_eq].
rewrite (bigD1 i Pi) (bigD1 j) hj_eq ?Pi ?h'i_eq ?eqxx //=; congr (op : _ -> _).
rewri... | Lemma | reindex_omap | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"P0",
"apply",
"bigD1",
"big_pred0",
"cardD1x",
"eq_bigl",
"eqxx",
"last",
"pickP",
"ubnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
reindex_onto (I J : finType) (h : J -> I) h' (P : pred I) F :
(forall i, P i -> h (h' i) = i) ->
\big[op/x]_(i | P i) F i =
\big[op/x]_(j | P (h j) && (h' (h j) == j)) F (h j). | Proof.
by move=> h'K; rewrite (reindex_omap h (some \o h'))//= => i Pi; rewrite h'K.
Qed. | Lemma | reindex_onto | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"reindex_omap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
reindex (I J : finType) (h : J -> I) (P : pred I) F :
{on [pred i | P i], bijective h} ->
\big[op/x]_(i | P i) F i = \big[op/x]_(j | P (h j)) F (h j). | Proof.
case=> h' hK h'K; rewrite (reindex_onto h h' h'K).
by apply: eq_bigl => j /[!inE]; case Pi: (P _); rewrite //= hK ?eqxx.
Qed. | Lemma | reindex | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"eq_bigl",
"eqxx",
"inE",
"on",
"reindex_onto"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
reindex_inj (I : finType) (h : I -> I) (P : pred I) F :
injective h -> \big[op/x]_(i | P i) F i = \big[op/x]_(j | P (h j)) F (h j). | Proof. by move=> injh; apply: reindex (onW_bij _ (injF_bij injh)). Qed. | Lemma | reindex_inj | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"injF_bij",
"reindex"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigD1_ord n j (P : pred 'I_n) F :
P j -> \big[op/x]_(i < n | P i) F i
= op (F j) (\big[op/x]_(i < n.-1 | P (lift j i)) F (lift j i)). | Proof.
move=> Pj; rewrite (bigD1 j Pj) (reindex_omap (lift j) (unlift j))/=.
by move=> i; case: unliftP => [k ->|->]; rewrite ?eqxx ?andbF.
by under eq_bigl do rewrite liftK eq_sym eqxx neq_lift ?andbT.
Qed. | Lemma | bigD1_ord | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"bigD1",
"eq_bigl",
"eq_sym",
"eqxx",
"lift",
"liftK",
"neq_lift",
"reindex_omap",
"unlift",
"unliftP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_enum_val_cond (I : finType) (A : pred I) (P : pred I) F :
\big[op/x]_(x in A | P x) F x =
\big[op/x]_(i < #|A| | P (enum_val i)) F (enum_val i). | Proof.
have [A_eq0|/card_gt0P[x0 x0A]] := posnP #|A|.
rewrite !big_pred0 // => i; first by rewrite card0_eq.
by have: false by move: i => []; rewrite A_eq0.
rewrite (reindex (enum_val : 'I_#|A| -> I)).
by apply: subon_bij (enum_val_bij_in x0A) => y /andP[].
by apply: eq_big => [y|y Py]; rewrite ?enum_valP.
Qed. | Lemma | big_enum_val_cond | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"big_pred0",
"card0_eq",
"card_gt0P",
"enum_val",
"enum_valP",
"enum_val_bij_in",
"eq_big",
"posnP",
"reindex"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_enum_rank_cond (I : finType) (A : pred I) z (zA : z \in A) P F
(h := enum_rank_in zA) :
\big[op/x]_(i < #|A| | P i) F i = \big[op/x]_(s in A | P (h s)) F (h s). | Proof.
rewrite big_enum_val_cond {}/h.
by apply: eq_big => [i|i Pi]; rewrite ?enum_valK_in.
Qed. | Lemma | big_enum_rank_cond | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"big_enum_val_cond",
"enum_rank_in",
"enum_valK_in",
"eq_big"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_nat_rev m n P F :
\big[op/x]_(m <= i < n | P i) F i
= \big[op/x]_(m <= i < n | P (m + n - i.+1)) F (m + n - i.+1). | Proof.
case: (ltnP m n) => ltmn; last by rewrite !big_geq.
rewrite -{3 4}(subnK (ltnW ltmn)) addnA.
do 2!rewrite (big_addn _ _ 0) big_mkord; rewrite (reindex_inj rev_ord_inj)/=.
by apply: eq_big => [i | i _]; rewrite /= -addSn subnDr addnC addnBA.
Qed. | Lemma | big_nat_rev | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"addSn",
"addnA",
"addnBA",
"addnC",
"apply",
"big_addn",
"big_geq",
"big_mkord",
"eq_big",
"last",
"ltnP",
"ltnW",
"reindex_inj",
"rev_ord_inj",
"subnDr",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_rev_mkord m n P F :
\big[op/x]_(m <= k < n | P k) F k
= \big[op/x]_(k < n - m | P (n - k.+1)) F (n - k.+1). | Proof.
rewrite big_nat_rev (big_addn _ _ 0) big_mkord.
by apply: eq_big => [i|i _]; rewrite -addSn addnC subnDr.
Qed. | Lemma | big_rev_mkord | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"addSn",
"addnC",
"apply",
"big_addn",
"big_mkord",
"big_nat_rev",
"eq_big",
"subnDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_mkcond_idem I r (P : pred I) F :
\big[op/x]_(i <- r | P i) F i = \big[op/x]_(i <- r) (if P i then F i else x). | Proof.
elim: r => [|i r]; rewrite ?(big_nil, big_cons)//.
by case: ifPn => Pi ->//; rewrite -[in LHS]big_id_idem // opC.
Qed. | Lemma | big_mkcond_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_cons",
"big_id_idem",
"big_nil",
"opC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_mkcondr_idem I r (P Q : pred I) F :
\big[op/x]_(i <- r | P i && Q i) F i =
\big[op/x]_(i <- r | P i) (if Q i then F i else x). | Proof. by rewrite -big_filter_cond big_mkcond_idem big_filter. Qed. | Lemma | big_mkcondr_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_filter",
"big_filter_cond",
"big_mkcond_idem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_mkcondl_idem I r (P Q : pred I) F :
\big[op/x]_(i <- r | P i && Q i) F i =
\big[op/x]_(i <- r | Q i) (if P i then F i else x). | Proof. by rewrite big_andbC big_mkcondr_idem. Qed. | Lemma | big_mkcondl_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_andbC",
"big_mkcondr_idem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_rmcond_idem I (r : seq I) (P : pred I) F :
(forall i, ~~ P i -> F i = x) ->
\big[op/x]_(i <- r | P i) F i = \big[op/x]_(i <- r) F i. | Proof.
move=> F_eq1; rewrite big_mkcond_idem; apply: eq_bigr => i.
by case: (P i) (F_eq1 i) => // ->.
Qed. | Lemma | big_rmcond_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"big_mkcond_idem",
"eq_bigr",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_rmcond_in_idem (I : eqType) (r : seq I) (P : pred I) F :
(forall i, i \in r -> ~~ P i -> F i = x) ->
\big[op/x]_(i <- r | P i) F i = \big[op/x]_(i <- r) F i. | Proof.
move=> F_eq1; rewrite big_seq_cond [RHS]big_seq_cond !big_mkcondl_idem.
by rewrite big_rmcond_idem => // i /F_eq1; case: ifP => // _ ->.
Qed. | Lemma | big_rmcond_in_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_mkcondl_idem",
"big_rmcond_idem",
"big_seq_cond",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_cat_idem I r1 r2 (P : pred I) F :
\big[op/x]_(i <- r1 ++ r2 | P i) F i =
op (\big[op/x]_(i <- r1 | P i) F i) (\big[op/x]_(i <- r2 | P i) F i). | Proof.
elim: r1 => [/=|i r1 IHr1]; first by rewrite big_nil opC big_id_idem.
by rewrite /= big_cons IHr1 big_cons; case: (P i); rewrite // opA.
Qed. | Lemma | big_cat_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_cons",
"big_id_idem",
"big_nil",
"opA",
"opC",
"r1",
"r2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_allpairs_dep_idem I1 (I2 : I1 -> Type) J (h : forall i1, I2 i1 -> J)
(r1 : seq I1) (r2 : forall i1, seq (I2 i1)) (F : J -> R) :
\big[op/x]_(i <- [seq h i1 i2 | i1 <- r1, i2 <- r2 i1]) F i =
\big[op/x]_(i1 <- r1) \big[op/x]_(i2 <- r2 i1) F (h i1 i2). | Proof.
elim: r1 => [|i1 r1 IHr1]; first by rewrite !big_nil.
by rewrite big_cat_idem IHr1 big_cons big_map.
Qed. | Lemma | big_allpairs_dep_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_cat_idem",
"big_cons",
"big_map",
"big_nil",
"r1",
"r2",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_allpairs_idem I1 I2 (r1 : seq I1) (r2 : seq I2) F :
\big[op/x]_(i <- [seq (i1, i2) | i1 <- r1, i2 <- r2]) F i =
\big[op/x]_(i1 <- r1) \big[op/x]_(i2 <- r2) F (i1, i2). | Proof. exact: big_allpairs_dep_idem. Qed. | Lemma | big_allpairs_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_allpairs_dep_idem",
"r1",
"r2",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_cat_nat_idem n m p (P : pred nat) F : m <= n -> n <= p ->
\big[op/x]_(m <= i < p | P i) F i =
op (\big[op/x]_(m <= i < n | P i) F i) (\big[op/x]_(n <= i < p | P i) F i). | Proof.
move=> le_mn le_np; rewrite -big_cat_idem -{2}(subnKC le_mn) -iotaD subnDA.
by rewrite subnKC // leq_sub.
Qed. | Lemma | big_cat_nat_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_cat_idem",
"iotaD",
"leq_sub",
"nat",
"subnDA",
"subnKC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_split_idem I r (P : pred I) F1 F2 :
\big[op/x]_(i <- r | P i) op (F1 i) (F2 i) =
op (\big[op/x]_(i <- r | P i) F1 i) (\big[op/x]_(i <- r | P i) F2 i). | Proof.
by elim/big_rec3: _ => [|i x' y _ _ ->]; rewrite ?opxx// opCA -!opA opCA.
Qed. | Lemma | big_split_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"F1",
"F2",
"big_rec3",
"opA",
"opCA",
"opxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_id_idem_AC I (r : seq I) P F :
\big[op/x]_(i <- r | P i) op (F i) x = \big[op/x]_(i <- r | P i) F i. | Proof. by rewrite big_split_idem big_const_idem ?big_id_idem. Qed. | Lemma | big_id_idem_AC | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_const_idem",
"big_id_idem",
"big_split_idem",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigID_idem I r (a P : pred I) F :
\big[op/x]_(i <- r | P i) F i =
op (\big[op/x]_(i <- r | P i && a i) F i)
(\big[op/x]_(i <- r | P i && ~~ a i) F i). | Proof.
rewrite -big_id_idem_AC big_mkcond_idem !(big_mkcond_idem _ _ F) -big_split_idem.
by apply: eq_bigr => i; case: ifPn => //=; case: ifPn; rewrite // opC.
Qed. | Lemma | bigID_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"big_id_idem_AC",
"big_mkcond_idem",
"big_split_idem",
"eq_bigr",
"opC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigU_idem (I : finType) (A B : pred I) F :
[disjoint A & B] ->
\big[op/x]_(i in [predU A & B]) F i =
op (\big[op/x]_(i in A) F i) (\big[op/x]_(i in B) F i). | Proof.
move=> dAB; rewrite (bigID_idem (mem A)).
congr (op : _ -> _); apply: eq_bigl => i; first by rewrite orbK.
by have:= pred0P dAB i; rewrite andbC /= !inE; case: (i \in A).
Qed. | Lemma | bigU_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"bigID_idem",
"disjoint",
"eq_bigl",
"inE",
"pred0P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
partition_big_idem I (s : seq I)
(J : finType) (P : pred I) (p : I -> J) (Q : pred J) F :
(forall i, P i -> Q (p i)) ->
\big[op/x]_(i <- s | P i) F i =
\big[op/x]_(j : J | Q j) \big[op/x]_(i <- s | (P i) && (p i == j)) F i. | Proof.
move=> Qp; transitivity (\big[op/x]_(i <- s | P i && Q (p i)) F i).
by apply: eq_bigl => i; case Pi: (P i); rewrite // Qp.
have [n leQn] := ubnP #|Q|; elim: n => // n IHn in Q {Qp} leQn *.
case: (pickP Q) => [j Qj | Q0]; last first.
by rewrite !big_pred0 // => i; rewrite Q0 andbF.
rewrite (bigD1 j) // -IHn; ... | Lemma | partition_big_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"bigD1",
"bigID_idem",
"big_pred0",
"cardD1x",
"eq_bigl",
"last",
"ltnS",
"pickP",
"seq",
"ubnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sig_big_dep_idem (I : finType) (J : I -> finType)
(P : pred I) (Q : forall {i}, pred (J i)) (F : forall {i}, J i -> R) :
\big[op/x]_(i | P i) \big[op/x]_(j : J i | Q j) F j =
\big[op/x]_(p : {i : I & J i} | P (tag p) && Q (tagged p)) F (tagged p). | Proof.
pose s := [seq Tagged J j | i <- index_enum I, j <- index_enum (J i)].
rewrite [LHS]big_mkcond_idem big_mkcondl_idem.
rewrite [RHS]big_mkcond_idem -[RHS](@perm_big _ s); last first.
rewrite big_allpairs_dep_idem/=; apply: eq_bigr => i _.
by rewrite -big_mkcond_idem/=; case: P; rewrite // big1_idem.
rewrite u... | Lemma | sig_big_dep_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"allpairsPdep",
"allpairs_uniq_dep",
"apply",
"big1_idem",
"big_allpairs_dep_idem",
"big_mkcond_idem",
"big_mkcondl_idem",
"eq_bigr",
"index_enum",
"index_enum_uniq",
"last",
"mem_index_enum",
"perm_big",
"seq",
"uniq_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_big_dep_idem (I J : finType) (P : pred I) (Q : I -> pred J) F :
\big[op/x]_(i | P i) \big[op/x]_(j | Q i j) F i j =
\big[op/x]_(p | P p.1 && Q p.1 p.2) F p.1 p.2. | Proof.
rewrite sig_big_dep_idem; apply: (reindex (fun x => Tagged (fun=> J) x.2)).
by exists (fun x => (projT1 x, projT2 x)) => -[].
Qed. | Lemma | pair_big_dep_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"reindex",
"sig_big_dep_idem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_big_idem (I J : finType) (P : pred I) (Q : pred J) F :
\big[op/x]_(i | P i) \big[op/x]_(j | Q j) F i j =
\big[op/x]_(p | P p.1 && Q p.2) F p.1 p.2. | Proof. exact: pair_big_dep_idem. Qed. | Lemma | pair_big_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"pair_big_dep_idem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_bigA_idem (I J : finType) (F : I -> J -> R) :
\big[op/x]_i \big[op/x]_j F i j = \big[op/x]_p F p.1 p.2. | Proof. exact: pair_big_dep_idem. Qed. | Lemma | pair_bigA_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"pair_big_dep_idem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exchange_big_dep_idem I J rI rJ (P : pred I) (Q : I -> pred J)
(xQ : pred J) F :
(forall i j, P i -> Q i j -> xQ j) ->
\big[op/x]_(i <- rI | P i) \big[op/x]_(j <- rJ | Q i j) F i j =
\big[op/x]_(j <- rJ | xQ j) \big[op/x]_(i <- rI | P i && Q i j) F i j. | Proof.
move=> PQxQ; pose p u := (u.2, u.1).
under [LHS]eq_bigr do rewrite big_tnth; rewrite [LHS]big_tnth.
under [RHS]eq_bigr do rewrite big_tnth; rewrite [RHS]big_tnth.
rewrite !pair_big_dep_idem (reindex_onto (p _ _) (p _ _)) => [[]|] //=.
apply: eq_big => [] [j i] //=; symmetry; rewrite eqxx andbT andb_idl //.
by ca... | Lemma | exchange_big_dep_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"big_tnth",
"eq_big",
"eq_bigr",
"eqxx",
"pair_big_dep_idem",
"reindex_onto"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exchange_big_idem I J rI rJ (P : pred I) (Q : pred J) F :
\big[op/x]_(i <- rI | P i) \big[op/x]_(j <- rJ | Q j) F i j =
\big[op/x]_(j <- rJ | Q j) \big[op/x]_(i <- rI | P i) F i j. | Proof.
rewrite (exchange_big_dep_idem Q) //.
by under eq_bigr => i Qi do under eq_bigl do rewrite Qi andbT.
Qed. | Lemma | exchange_big_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"eq_bigl",
"eq_bigr",
"exchange_big_dep_idem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exchange_big_dep_nat_idem m1 n1 m2 n2 (P : pred nat) (Q : rel nat)
(xQ : pred nat) F :
(forall i j, m1 <= i < n1 -> m2 <= j < n2 -> P i -> Q i j -> xQ j) ->
\big[op/x]_(m1 <= i < n1 | P i) \big[op/x]_(m2 <= j < n2 | Q i j) F i j =
\big[op/x]_(m2 <= j < n2 | xQ j)
\big[op/x]_(... | Proof.
move=> PQxQ; under eq_bigr do rewrite big_seq_cond.
rewrite big_seq_cond /= (exchange_big_dep_idem xQ) => [i j|].
by rewrite !mem_index_iota => /andP[mn_i Pi] /andP[mn_j /PQxQ->].
rewrite 2!(big_seq_cond _ _ _ xQ); apply: eq_bigr => j /andP[-> _] /=.
by rewrite [rhs in _ = rhs]big_seq_cond; apply: eq_bigl => i... | Lemma | exchange_big_dep_nat_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"big_seq_cond",
"eq_bigl",
"eq_bigr",
"exchange_big_dep_idem",
"mem_index_iota",
"nat",
"rel",
"rhs"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exchange_big_nat_idem m1 n1 m2 n2 (P Q : pred nat) F :
\big[op/x]_(m1 <= i < n1 | P i) \big[op/x]_(m2 <= j < n2 | Q j) F i j =
\big[op/x]_(m2 <= j < n2 | Q j) \big[op/x]_(m1 <= i < n1 | P i) F i j. | Proof.
rewrite (exchange_big_dep_nat_idem Q) //.
by under eq_bigr => i Qi do under eq_bigl do rewrite Qi andbT.
Qed. | Lemma | exchange_big_nat_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"eq_bigl",
"eq_bigr",
"exchange_big_dep_nat_idem",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"1" | := idx. | Notation | 1 | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"*%M" | := op. | Notation | *%M | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x * y" | := (op x y). | Notation | x * y | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
foldlE x r : foldl *%M x r = \big[*%M/1]_(y <- x :: r) y. | Proof.
by rewrite -foldrE; elim: r => [|y r IHr]/= in x *; rewrite ?mulm1 ?mulmA ?IHr.
Qed. | Lemma | foldlE | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"foldl",
"foldrE",
"mulm1",
"mulmA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
foldl_idx r : foldl *%M 1 r = \big[*%M/1]_(x <- r) x. | Proof. by rewrite foldlE big_cons mul1m. Qed. | Lemma | foldl_idx | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_cons",
"foldl",
"foldlE",
"mul1m"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_big_idx_seq idx' I r (P : pred I) F :
right_id idx' *%M -> has P r ->
\big[*%M/idx']_(i <- r | P i) F i = \big[*%M/1]_(i <- r | P i) F i. | Proof.
move=> op_idx'; rewrite -!(big_filter _ _ r) has_count -size_filter.
case/lastP: (filter P r) => {r}// r i _.
by rewrite -cats1 !(big_cat_nested, big_cons, big_nil) op_idx' mulm1.
Qed. | Lemma | eq_big_idx_seq | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_cat_nested",
"big_cons",
"big_filter",
"big_nil",
"cats1",
"filter",
"has",
"has_count",
"lastP",
"mulm1",
"size_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_big_idx idx' (I : finType) i0 (P : pred I) F :
P i0 -> right_id idx' *%M ->
\big[*%M/idx']_(i | P i) F i = \big[*%M/1]_(i | P i) F i. | Proof.
by move=> Pi0 op_idx'; apply: eq_big_idx_seq => //; apply/hasP; exists i0.
Qed. | Lemma | eq_big_idx | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"Pi0",
"apply",
"eq_big_idx_seq",
"hasP",
"i0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_change_idx I x r (P : pred I) F :
\big[*%M/x]_(j <- r | P j) F j = (\big[*%M/1]_(j <- r | P j) F j) * x. | Proof.
elim: r => [|i r]; rewrite ?(big_nil, big_cons, mul1m)// => ->.
by case: ifP => // Pi; rewrite mulmA.
Qed. | Lemma | big_change_idx | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_cons",
"big_nil",
"mul1m",
"mulmA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big1_eq I r (P : pred I) : \big[*%M/1]_(i <- r | P i) 1 = 1. | Proof. by rewrite big1_idem //= mul1m. Qed. | Lemma | big1_eq | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big1_idem",
"mul1m"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big1 I r (P : pred I) F :
(forall i, P i -> F i = 1) -> \big[*%M/1]_(i <- r | P i) F i = 1. | Proof. by move/(eq_bigr _)->; apply: big1_eq. Qed. | Lemma | big1 | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"big1_eq",
"eq_bigr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big1_seq (I : eqType) r (P : pred I) F :
(forall i, P i && (i \in r) -> F i = 1) ->
\big[*%M/1]_(i <- r | P i) F i = 1. | Proof. by move=> eqF1; rewrite big_seq_cond big_andbC big1. Qed. | Lemma | big1_seq | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big1",
"big_andbC",
"big_seq_cond"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_seq1 I (i : I) F : \big[*%M/1]_(j <- [:: i]) F j = F i. | Proof. by rewrite big_seq1_id mulm1. Qed. | Lemma | big_seq1 | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_seq1_id",
"mulm1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_rcons I i r (P : pred I) F :
\big[*%M/1]_(j <- rcons r i | P j) F j =
(\big[*%M/1]_(j <- r | P j) F j) * (if P i then F i else idx). | Proof. by rewrite big_rcons_op big_change_idx mulm1. Qed. | Lemma | big_rcons | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_change_idx",
"big_rcons_op",
"mulm1",
"rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_mkcond I r (P : pred I) F :
\big[*%M/1]_(i <- r | P i) F i =
\big[*%M/1]_(i <- r) (if P i then F i else 1). | Proof. by rewrite unlock; elim: r => //= i r ->; case P; rewrite ?mul1m. Qed. | Lemma | big_mkcond | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"mul1m"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_mkcondr I r (P Q : pred I) F :
\big[*%M/1]_(i <- r | P i && Q i) F i =
\big[*%M/1]_(i <- r | P i) (if Q i then F i else 1). | Proof. by rewrite -big_filter_cond big_mkcond big_filter. Qed. | Lemma | big_mkcondr | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_filter",
"big_filter_cond",
"big_mkcond"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_mkcondl I r (P Q : pred I) F :
\big[*%M/1]_(i <- r | P i && Q i) F i =
\big[*%M/1]_(i <- r | Q i) (if P i then F i else 1). | Proof. by rewrite big_andbC big_mkcondr. Qed. | Lemma | big_mkcondl | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_andbC",
"big_mkcondr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_rmcond I (r : seq I) (P : pred I) F :
(forall i, ~~ P i -> F i = 1) ->
\big[*%M/1]_(i <- r | P i) F i = \big[*%M/1]_(i <- r) F i. | Proof.
move=> F_eq1; rewrite big_mkcond; apply: eq_bigr => i.
by case: (P i) (F_eq1 i) => // ->.
Qed. | Lemma | big_rmcond | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"big_mkcond",
"eq_bigr",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_rmcond_in (I : eqType) (r : seq I) (P : pred I) F :
(forall i, i \in r -> ~~ P i -> F i = 1) ->
\big[*%M/1]_(i <- r | P i) F i = \big[*%M/1]_(i <- r) F i. | Proof.
move=> F_eq1; rewrite big_seq_cond [RHS]big_seq_cond !big_mkcondl big_rmcond//.
by move=> i /F_eq1; case: ifP => // _ ->.
Qed. | Lemma | big_rmcond_in | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_mkcondl",
"big_rmcond",
"big_seq_cond",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_cat I r1 r2 (P : pred I) F :
\big[*%M/1]_(i <- r1 ++ r2 | P i) F i =
\big[*%M/1]_(i <- r1 | P i) F i * \big[*%M/1]_(i <- r2 | P i) F i. | Proof.
rewrite !(big_mkcond _ P) unlock.
by elim: r1 => /= [|i r1 ->]; rewrite (mul1m, mulmA).
Qed. | Lemma | big_cat | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_mkcond",
"mul1m",
"mulmA",
"r1",
"r2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_allpairs_dep I1 (I2 : I1 -> Type) J (h : forall i1, I2 i1 -> J)
(r1 : seq I1) (r2 : forall i1, seq (I2 i1)) (F : J -> R) :
\big[*%M/1]_(i <- [seq h i1 i2 | i1 <- r1, i2 <- r2 i1]) F i =
\big[*%M/1]_(i1 <- r1) \big[*%M/1]_(i2 <- r2 i1) F (h i1 i2). | Proof.
elim: r1 => [|i1 r1 IHr1]; first by rewrite !big_nil.
by rewrite big_cat IHr1 big_cons big_map.
Qed. | Lemma | big_allpairs_dep | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_cat",
"big_cons",
"big_map",
"big_nil",
"r1",
"r2",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_allpairs I1 I2 (r1 : seq I1) (r2 : seq I2) F :
\big[*%M/1]_(i <- [seq (i1, i2) | i1 <- r1, i2 <- r2]) F i =
\big[*%M/1]_(i1 <- r1) \big[op/idx]_(i2 <- r2) F (i1, i2). | Proof. exact: big_allpairs_dep. Qed. | Lemma | big_allpairs | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_allpairs_dep",
"r1",
"r2",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_big_rev I (r : seq I) P F :
\big[*%M/1]_(i <- rev r | P i) F i =
\big[(fun x y => y * x)/1]_(i <- r | P i) F i. | Proof.
elim: r => [|i r IHr]; rewrite ?big_nil// big_cons rev_cons big_rcons IHr.
by case: (P i); rewrite ?mulm1.
Qed. | Lemma | rev_big_rev | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_cons",
"big_nil",
"big_rcons",
"mulm1",
"rev",
"rev_cons",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_only1 (I : finType) (i : I) (P : pred I) (F : I -> R) : P i ->
(forall j, j != i -> P j -> F j = idx) ->
\big[op/idx]_(j | P j) F j = F i. | Proof.
move=> Pi Fisx; have := index_enum_uniq I.
have : i \in index_enum I by rewrite mem_index_enum.
elim: index_enum => //= j r IHr /[!inE]; case: eqVneq => [<-|nij]//=.
move=> _ /andP[iNr runiq]; rewrite big_cons/= Pi big1_seq ?Monoid.mulm1//.
by move=> {}j /andP[/Fisx + jr] => ->//; apply: contraNneq iNr => <-... | Lemma | big_only1 | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"big1_seq",
"big_cons",
"contraNneq",
"eqVneq",
"eq_sym",
"inE",
"index_enum",
"index_enum_uniq",
"mem_index_enum",
"mul1m",
"mulm1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_pred1_eq (I : finType) (i : I) F : \big[*%M/1]_(j | j == i) F j = F i. | Proof. by rewrite (@big_only1 _ i)// => j /negPf->. Qed. | Lemma | big_pred1_eq | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_only1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_pred1 (I : finType) i (P : pred I) F :
P =1 pred1 i -> \big[*%M/1]_(j | P j) F j = F i. | Proof. by move/(eq_bigl _ _)->; apply: big_pred1_eq. Qed. | Lemma | big_pred1 | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"big_pred1_eq",
"eq_bigl",
"pred1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_ord1 F : \big[op/idx]_(i < 1) F i = F ord0. | Proof. by rewrite big_ord_recl big_ord0 Monoid.mulm1. Qed. | Lemma | big_ord1 | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_ord0",
"big_ord_recl",
"mulm1",
"ord0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_ord1_cond P F :
\big[op/idx]_(i < 1 | P i) F i = if P ord0 then F ord0 else idx. | Proof. by rewrite big_mkcond big_ord1. Qed. | Lemma | big_ord1_cond | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_mkcond",
"big_ord1",
"ord0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_ord1_eq (F : nat -> R) i n :
\big[op/idx]_(j < n | j == i :> nat) F j = if i < n then F i else idx. | Proof.
case: ltnP => [i_lt|i_ge]; first by rewrite (big_pred1_eq (Ordinal _)).
by rewrite big_pred0// => j; apply: contra_leqF i_ge => /eqP <-.
Qed. | Lemma | big_ord1_eq | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"big_pred0",
"big_pred1_eq",
"contra_leqF",
"ltnP",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_ord1_cond_eq (F : nat -> R) (P : pred nat) i n :
\big[op/idx]_(j < n | P j && (j == i :> nat)) F j =
if (i < n) && P i then F i else idx. | Proof.
by rewrite big_mkcondl if_and (big_ord1_eq (fun j => if P j then F j else _)).
Qed. | Lemma | big_ord1_cond_eq | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_mkcondl",
"big_ord1_eq",
"if_and",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_cat_nat n m p (P : pred nat) F : m <= n -> n <= p ->
\big[*%M/1]_(m <= i < p | P i) F i =
(\big[*%M/1]_(m <= i < n | P i) F i) * (\big[*%M/1]_(n <= i < p | P i) F i). | Proof.
move=> le_mn le_np; rewrite -big_cat -{2}(subnKC le_mn) -iotaD subnDA.
by rewrite subnKC // leq_sub.
Qed. | Lemma | big_cat_nat | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_cat",
"iotaD",
"leq_sub",
"nat",
"subnDA",
"subnKC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_nat_widenl (m1 m2 n : nat) (P : pred nat) F :
m2 <= m1 ->
\big[op/idx]_(m1 <= i < n | P i) F i =
\big[op/idx]_(m2 <= i < n | P i && (m1 <= i)) F i. | Proof.
move=> le_m21; have [le_nm1|lt_m1n] := leqP n m1.
rewrite big_geq// big_nat_cond big1//.
by move=> i /and3P[/andP[_ /leq_trans/(_ le_nm1)/ltn_geF->]].
rewrite big_mkcond big_mkcondl (big_cat_nat _ _ le_m21) 1?ltnW//.
rewrite [X in op X]big_nat_cond [X in op X]big_pred0.
by move=> k; case: ltnP; rewrite and... | Lemma | big_nat_widenl | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"big1",
"big_cat_nat",
"big_geq",
"big_mkcond",
"big_mkcondl",
"big_nat_cond",
"big_pred0",
"congr_big_nat",
"leqP",
"leq_trans",
"ltnP",
"ltnW",
"ltn_geF",
"mul1m",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_geq_mkord (m n : nat) (P : pred nat) F :
\big[op/idx]_(m <= i < n | P i) F i =
\big[op/idx]_(i < n | P i && (m <= i)) F i. | Proof. by rewrite (@big_nat_widenl _ 0)// big_mkord. Qed. | Lemma | big_geq_mkord | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_mkord",
"big_nat_widenl",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_nat1_eq (F : nat -> R) i m n :
\big[op/idx]_(m <= j < n | j == i) F j = if m <= i < n then F i else idx. | Proof. by rewrite big_geq_mkord big_andbC big_ord1_cond_eq andbC. Qed. | Lemma | big_nat1_eq | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_andbC",
"big_geq_mkord",
"big_ord1_cond_eq",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_nat1_cond_eq (F : nat -> R) (P : pred nat) i m n :
\big[op/idx]_(m <= j < n | P j && (j == i)) F j =
if (m <= i < n) && P i then F i else idx. | Proof. by rewrite big_mkcondl big_nat1_eq -if_and. Qed. | Lemma | big_nat1_cond_eq | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_mkcondl",
"big_nat1_eq",
"if_and",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_nat1 n F : \big[*%M/1]_(n <= i < n.+1) F i = F n. | Proof. by rewrite big_ltn // big_geq // mulm1. Qed. | Lemma | big_nat1 | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_geq",
"big_ltn",
"mulm1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_nat_recr n m F : m <= n ->
\big[*%M/1]_(m <= i < n.+1) F i = (\big[*%M/1]_(m <= i < n) F i) * F n. | Proof. by move=> lemn; rewrite (@big_cat_nat n) ?leqnSn // big_nat1. Qed. | Lemma | big_nat_recr | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_cat_nat",
"big_nat1",
"leqnSn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_nat_mul n k F :
\big[*%M/1]_(0 <= i < n * k) F i =
\big[*%M/1]_(0 <= i < n) \big[*%M/1]_(i * k <= j < i.+1 * k) F j. | Proof.
elim: n => [|n ih]; first by rewrite mul0n 2!big_nil.
rewrite [in RHS]big_nat_recr//= -ih mulSn addnC [in LHS]/index_iota subn0 iotaD.
rewrite big_cat /= [in X in _ = X * _]/index_iota subn0; congr (_ * _).
by rewrite add0n /index_iota (addnC _ k) addnK.
Qed. | Lemma | big_nat_mul | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"add0n",
"addnC",
"addnK",
"big_cat",
"big_nat_recr",
"big_nil",
"index_iota",
"iotaD",
"mul0n",
"mulSn",
"subn0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_ord_recr n F :
\big[*%M/1]_(i < n.+1) F i =
(\big[*%M/1]_(i < n) F (widen_ord (leqnSn n) i)) * F ord_max. | Proof.
transitivity (\big[*%M/1]_(0 <= i < n.+1) F (inord i)).
by rewrite big_mkord; apply: eq_bigr=> i _; rewrite inord_val.
rewrite big_nat_recr // big_mkord; congr (_ * F _); last first.
by apply: val_inj; rewrite /= inordK.
by apply: eq_bigr => [] i _; congr F; apply: ord_inj; rewrite inordK //= leqW.
Qed. | Lemma | big_ord_recr | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"big_mkord",
"big_nat_recr",
"eq_bigr",
"inord",
"inordK",
"inord_val",
"last",
"leqW",
"leqnSn",
"ord_inj",
"ord_max",
"val_inj",
"widen_ord"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_sumType (I1 I2 : finType) (P : pred (I1 + I2)) F :
\big[*%M/1]_(i | P i) F i =
(\big[*%M/1]_(i | P (inl _ i)) F (inl _ i))
* (\big[*%M/1]_(i | P (inr _ i)) F (inr _ i)). | Proof.
by rewrite ![index_enum _]unlock [@Finite.enum in LHS]unlock/= big_cat !big_map.
Qed. | Lemma | big_sumType | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_cat",
"big_map",
"enum",
"index_enum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_split_ord m n (P : pred 'I_(m + n)) F :
\big[*%M/1]_(i | P i) F i =
(\big[*%M/1]_(i | P (lshift n i)) F (lshift n i))
* (\big[*%M/1]_(i | P (rshift m i)) F (rshift m i)). | Proof.
rewrite -(big_map _ _ (lshift n) _ P F) -(big_map _ _ (@rshift m _) _ P F).
rewrite -big_cat; congr bigop; apply: (inj_map val_inj).
rewrite map_cat -!map_comp (map_comp (addn m)) /=.
by rewrite ![index_enum _]unlock unlock !val_ord_enum -iotaDl addn0 iotaD.
Qed. | Lemma | big_split_ord | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"addn",
"addn0",
"apply",
"big_cat",
"big_map",
"index_enum",
"inj_map",
"iotaD",
"iotaDl",
"lshift",
"map_cat",
"map_comp",
"rshift",
"val_inj",
"val_ord_enum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_fcat n m (F : R ^ n) (G : R ^ m) :
\big[*%M/1]_(i < n + m) (F +++ G) i =
(\big[*%M/1]_(i < n) F i) * (\big[*%M/1]_(i < m) G i). | Proof.
rewrite big_split_ord; congr (_ * _); apply: eq_bigr => i _.
by rewrite fcat_lshift.
by rewrite fcat_rshift.
Qed. | Lemma | big_fcat | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"big_split_ord",
"eq_bigr",
"fcat_lshift",
"fcat_rshift"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_flatten I rr (P : pred I) F :
\big[*%M/1]_(i <- flatten rr | P i) F i
= \big[*%M/1]_(r <- rr) \big[*%M/1]_(i <- r | P i) F i. | Proof.
by elim: rr => [|r rr IHrr]; rewrite ?big_nil //= big_cat big_cons -IHrr.
Qed. | Lemma | big_flatten | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_cat",
"big_cons",
"big_nil",
"flatten"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_pmap J I (h : J -> option I) (r : seq J) F :
\big[op/idx]_(i <- pmap h r) F i = \big[op/idx]_(j <- r) oapp F idx (h j). | Proof.
elim: r => [| r0 r IHr]/=; first by rewrite !big_nil.
rewrite /= big_cons; case: (h r0) => [i|] /=; last by rewrite mul1m.
by rewrite big_cons IHr.
Qed. | Lemma | big_pmap | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_cons",
"big_nil",
"last",
"mul1m",
"pmap",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
telescope_big (f : nat -> nat -> R) (n m : nat) :
(forall k, n < k < m -> op (f n k) (f k k.+1) = f n k.+1) ->
\big[op/idx]_(n <= i < m) f i i.+1 = if n < m then f n m else idx. | Proof.
elim: m => [//| m IHm]; first by rewrite ltn0 big_geq.
move=> tm; rewrite ltnS; case: ltnP=> // mn; first by rewrite big_geq.
rewrite big_nat_recr// IHm//.
by move=> k /andP[nk /ltnW nm]; rewrite tm// nk.
by case: ltngtP mn=> //= [nm|<-]; rewrite ?mul1m// tm// nm leqnn.
Qed. | Lemma | telescope_big | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_geq",
"big_nat_recr",
"leqnn",
"ltn0",
"ltnP",
"ltnS",
"ltnW",
"ltngtP",
"mul1m",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"'*%M'" | := op. | Notation | '*%M' | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_rem (I : eqType) r x (P : pred I) F :
x \in r ->
\big[*%M/1]_(y <- r | P y) F y
= (if P x then F x else 1) * \big[*%M/1]_(y <- rem x r | P y) F y. | Proof.
by move/perm_to_rem/(perm_big _)->; rewrite !(big_mkcond _ _ P) big_cons.
Qed. | Lemma | big_rem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_cons",
"big_mkcond",
"perm_big",
"perm_to_rem",
"rem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_rev I (r : seq I) P F :
\big[*%M/1]_(i <- rev r | P i) F i = \big[*%M/1]_(i <- r | P i) F i. | Proof.
by rewrite rev_big_rev; apply: (eq_big_op (fun=> True)) => // *; apply: mulmC.
Qed. | Lemma | big_rev | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"True",
"apply",
"eq_big_op",
"mulmC",
"rev",
"rev_big_rev",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_big_idem (I : eqType) (r1 r2 : seq I) (P : pred I) F :
idempotent_op *%M -> r1 =i r2 ->
\big[*%M/1]_(i <- r1 | P i) F i = \big[*%M/1]_(i <- r2 | P i) F i. | Proof.
move=> idM eq_r; rewrite -big_undup // -(big_undup r2) //; apply/perm_big.
by rewrite uniq_perm ?undup_uniq // => i; rewrite !mem_undup eq_r.
Qed. | Lemma | eq_big_idem | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"big_undup",
"idempotent_op",
"mem_undup",
"perm_big",
"r1",
"r2",
"seq",
"undup_uniq",
"uniq_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_undup_iterop_count (I : eqType) (r : seq I) (P : pred I) F :
\big[*%M/1]_(i <- undup r | P i) iterop (count_mem i r) *%M (F i) 1
= \big[*%M/1]_(i <- r | P i) F i. | Proof.
rewrite -[RHS](perm_big _ F (perm_count_undup _)) big_flatten big_map.
by rewrite [LHS]big_mkcond; apply: eq_bigr=> i _; rewrite big_nseq_cond iteropE.
Qed. | Lemma | big_undup_iterop_count | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"big_flatten",
"big_map",
"big_mkcond",
"big_nseq_cond",
"count_mem",
"eq_bigr",
"iterop",
"iteropE",
"perm_big",
"perm_count_undup",
"seq",
"undup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_split I r (P : pred I) F1 F2 :
\big[*%M/1]_(i <- r | P i) (F1 i * F2 i) =
\big[*%M/1]_(i <- r | P i) F1 i * \big[*%M/1]_(i <- r | P i) F2 i. | Proof. exact/big_split_idem/mul1m. Qed. | Lemma | big_split | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"F1",
"F2",
"big_split_idem",
"mul1m"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigID I r (a P : pred I) F :
\big[*%M/1]_(i <- r | P i) F i =
\big[*%M/1]_(i <- r | P i && a i) F i *
\big[*%M/1]_(i <- r | P i && ~~ a i) F i. | Proof. exact/bigID_idem/mul1m. Qed. | Lemma | bigID | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"bigID_idem",
"mul1m"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_if I r (P Q : pred I) F G :
\big[*%M/1]_(i <- r | P i) (if Q i then F i else G i) =
\big[*%M/1]_(i <- r | P i && Q i) F i *
\big[*%M/1]_(i <- r | P i && ~~ Q i) G i. | Proof.
rewrite (bigID Q); congr (_ * _); apply: eq_bigr => i /andP[_].
by move=> ->.
by move=> /negPf ->.
Qed. | Lemma | big_if | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"bigID",
"eq_bigr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigU (I : finType) (A B : pred I) F :
[disjoint A & B] ->
\big[*%M/1]_(i in [predU A & B]) F i =
(\big[*%M/1]_(i in A) F i) * (\big[*%M/1]_(i in B) F i). | Proof. exact/bigU_idem/mul1m. Qed. | Lemma | bigU | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"bigU_idem",
"disjoint",
"mul1m"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
partition_big I (s : seq I)
(J : finType) (P : pred I) (p : I -> J) (Q : pred J) F :
(forall i, P i -> Q (p i)) ->
\big[*%M/1]_(i <- s | P i) F i =
\big[*%M/1]_(j : J | Q j) \big[*%M/1]_(i <- s | (P i) && (p i == j)) F i. | Proof.
move=> Qp; transitivity (\big[*%M/1]_(i <- s | P i && Q (p i)) F i).
by apply: eq_bigl => i; case Pi: (P i); rewrite // Qp.
have [n leQn] := ubnP #|Q|; elim: n => // n IHn in Q {Qp} leQn *.
case: (pickP Q) => [j Qj | Q0]; last first.
by rewrite !big_pred0 // => i; rewrite Q0 andbF.
rewrite (bigD1 j) // -IHn;... | Lemma | partition_big | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"bigD1",
"bigID",
"big_pred0",
"cardD1x",
"eq_bigl",
"last",
"ltnS",
"pickP",
"seq",
"simpm",
"ubnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_enum_val (I : finType) (A : pred I) F :
\big[op/idx]_(x in A) F x = \big[op/idx]_(i < #|A|) F (enum_val i). | Proof. by rewrite -(big_enum_val_cond predT) big_mkcondr. Qed. | Lemma | big_enum_val | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_enum_val_cond",
"big_mkcondr",
"enum_val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_enum_rank (I : finType) (A : pred I) x (xA : x \in A) F
(h := enum_rank_in xA) :
\big[op/idx]_(i < #|A|) F i = \big[op/idx]_(s in A) F (h s). | Proof. by rewrite (big_enum_rank_cond xA) big_mkcondr. Qed. | Lemma | big_enum_rank | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_enum_rank_cond",
"big_mkcondr",
"enum_rank_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_sub_cond (I : finType) (A P : {pred I}) (F : I -> R) :
\big[*%M/1]_(i in A | P i) F i =
\big[*%M/1]_(x : {x in A} | P (val x)) F (val x). | Proof.
rewrite (reindex_omap (val : {x in A} -> I) insub).
by move=> i /andP[iA Pi]; rewrite insubT.
by apply: eq_bigl=> -[i iA]/=; rewrite insubT ?iA /= eqxx andbT.
Qed. | Lemma | big_sub_cond | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"apply",
"eq_bigl",
"eqxx",
"insub",
"insubT",
"reindex_omap",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_sub (I : finType) (A : {pred I}) (F : I -> R) :
\big[*%M/1]_(i in A) F i = \big[*%M/1]_(x : {x in A}) F (val x). | Proof. by rewrite -(big_sub_cond A xpredT) big_mkcondr. Qed. | Lemma | big_sub | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"big_mkcondr",
"big_sub_cond",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sig_big_dep (I : finType) (J : I -> finType)
(P : pred I) (Q : forall {i}, pred (J i)) (F : forall {i}, J i -> R) :
\big[op/idx]_(i | P i) \big[op/idx]_(j : J i | Q j) F j =
\big[op/idx]_(p : {i : I & J i} | P (tag p) && Q (tagged p)) F (tagged p). | Proof. exact/sig_big_dep_idem/mul1m. Qed. | Lemma | sig_big_dep | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"mul1m",
"sig_big_dep_idem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_big_dep (I J : finType) (P : pred I) (Q : I -> pred J) F :
\big[*%M/1]_(i | P i) \big[*%M/1]_(j | Q i j) F i j =
\big[*%M/1]_(p | P p.1 && Q p.1 p.2) F p.1 p.2. | Proof. exact/pair_big_dep_idem/mul1m. Qed. | Lemma | pair_big_dep | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"mul1m",
"pair_big_dep_idem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_big (I J : finType) (P : pred I) (Q : pred J) F :
\big[*%M/1]_(i | P i) \big[*%M/1]_(j | Q j) F i j =
\big[*%M/1]_(p | P p.1 && Q p.2) F p.1 p.2. | Proof. exact/pair_big_idem/mul1m. Qed. | Lemma | pair_big | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"mul1m",
"pair_big_idem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_bigA (I J : finType) (F : I -> J -> R) :
\big[*%M/1]_i \big[*%M/1]_j F i j = \big[*%M/1]_p F p.1 p.2. | Proof. exact/pair_bigA_idem/mul1m. Qed. | Lemma | pair_bigA | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"mul1m",
"pair_bigA_idem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exchange_big_dep I J rI rJ (P : pred I) (Q : I -> pred J)
(xQ : pred J) F :
(forall i j, P i -> Q i j -> xQ j) ->
\big[*%M/1]_(i <- rI | P i) \big[*%M/1]_(j <- rJ | Q i j) F i j =
\big[*%M/1]_(j <- rJ | xQ j) \big[*%M/1]_(i <- rI | P i && Q i j) F i j. | Proof. exact/exchange_big_dep_idem/mul1m. Qed. | Lemma | exchange_big_dep | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"exchange_big_dep_idem",
"mul1m"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exchange_big I J rI rJ (P : pred I) (Q : pred J) F :
\big[*%M/1]_(i <- rI | P i) \big[*%M/1]_(j <- rJ | Q j) F i j =
\big[*%M/1]_(j <- rJ | Q j) \big[*%M/1]_(i <- rI | P i) F i j. | Proof. exact/exchange_big_idem/mul1m. Qed. | Lemma | exchange_big | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"exchange_big_idem",
"mul1m"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exchange_big_dep_nat m1 n1 m2 n2 (P : pred nat) (Q : rel nat)
(xQ : pred nat) F :
(forall i j, m1 <= i < n1 -> m2 <= j < n2 -> P i -> Q i j -> xQ j) ->
\big[*%M/1]_(m1 <= i < n1 | P i) \big[*%M/1]_(m2 <= j < n2 | Q i j) F i j =
\big[*%M/1]_(m2 <= j < n2 | xQ j)
\big[*%M/1]_(m... | Proof. exact/exchange_big_dep_nat_idem/mul1m. Qed. | Lemma | exchange_big_dep_nat | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"exchange_big_dep_nat_idem",
"mul1m",
"nat",
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exchange_big_nat m1 n1 m2 n2 (P Q : pred nat) F :
\big[*%M/1]_(m1 <= i < n1 | P i) \big[*%M/1]_(m2 <= j < n2 | Q j) F i j =
\big[*%M/1]_(m2 <= j < n2 | Q j) \big[*%M/1]_(m1 <= i < n1 | P i) F i j. | Proof. exact/exchange_big_nat_idem/mul1m. Qed. | Lemma | exchange_big_nat | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"exchange_big_nat_idem",
"mul1m",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_refl : reflexive le. | Hypothesis | le_refl | boot | boot/bigop.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"SemiGroup.Exports",
"SemiGroup",
"SemiGroup.Theory",
"Monoid.Exports",
"Monoid",
"Monoid.Theory"
] | [
"le"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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