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 |
|---|---|---|---|---|---|---|---|---|---|---|
(f_op : {morph f : x y / op2 x y >-> op1 x y}) (f_id : f id2 = id1). | Hypotheses | f_op | 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"
] | [
"id1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
big_morph I r (P : pred I) F :
f (\big[op2/id2]_(i <- r | P i) F i) = \big[op1/id1]_(i <- r | P i) f (F i). | Proof. by rewrite unlock; elim: r => //= i r <-; rewrite -f_op -fun_if. Qed. | Lemma | big_morph | 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"
] | [
"f_op",
"id1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Kid : K idx. | Hypothesis | Kid | 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_rec I r (P : pred I) F
(Kop : forall i x, P i -> K x -> K (op (F i) x)) :
K (\big[op/idx]_(i <- r | P i) F i). | Proof. by rewrite unlock; elim: r => //= i r; case: ifP => //; apply: Kop. Qed. | Lemma | big_rec | 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"
] | [
"Kop",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Kop : forall x y, K x -> K y -> K (op x y). | Hypothesis | Kop | 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_ind I r (P : pred I) F (K_F : forall i, P i -> K (F i)) :
K (\big[op/idx]_(i <- r | P i) F i). | Proof. by apply: big_rec => // i x /K_F /Kop; apply. Qed. | Lemma | big_ind | 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"
] | [
"K_F",
"Kop",
"apply",
"big_rec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Kop' : forall x y, K x -> K y -> op x y = op' x y. | Hypothesis | Kop' | 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 | ||
eq_big_op I r (P : pred I) F (K_F : forall i, P i -> K (F i)) :
\big[op/idx]_(i <- r | P i) F i = \big[op'/idx]_(i <- r | P i) F i. | Proof.
by elim/(big_load K): _; elim/big_rec2: _ => // i _ y Pi [Ky <-]; auto.
Qed. | Lemma | eq_big_op | 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"
] | [
"K_F",
"big_load",
"big_rec2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(fM : {morph f : x y / op x y}) (f_id : f idx = idx). | Hypotheses | fM | 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_endo I r (P : pred I) F :
f (\big[op/idx]_(i <- r | P i) F i) = \big[op/idx]_(i <- r | P i) f (F i). | Proof. exact: big_morph. Qed. | Lemma | big_endo | 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_morph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_morph_in (R1 R2 : Type) (Q : {pred R2}) (f : R2 -> R1)
(id1 : R1) (op1 : R1 -> R1 -> R1)
(id2 : R2) (op2 : R2 -> R2 -> R2) :
{in Q &, forall x y, op2 x y \in Q} ->
id2 \in Q ->
{in Q &, {morph f : x y / op2 x y >-> op1 x y}} ->
f id2 = id1 ->
forall [I : Type] (r : seq I) (P : pred I) (F... | Proof.
move=> Qop Qid fop fid I r P F QF; elim/(big_load Q): _.
by elim/big_rec2: _ => // j x y Pj [Qx <-]; rewrite [Q _]Qop ?fop ?QF.
Qed. | Lemma | big_morph_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"
] | [
"R1",
"R2",
"big_load",
"big_rec2",
"id1",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oAC & associative op & commutative op | :=
fun x => oapp (fun y => Some (oapp (op^~ y) y x)) x. | Definition | oAC | 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 | |
(opA : associative op) (opC : commutative op). | Hypothesis | opA | 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"
] | [
"opC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
oop | := (oAC opA opC). | Notation | oop | 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"
] | [
"oAC",
"opA",
"opC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oACE x y : oop (Some x) (Some y) = some (op x y). | Proof. by []. Qed. | Lemma | oACE | 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"
] | [
"oop"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oopA_subdef : associative oop. | Proof. by move=> [x|] [y|] [z|]//; rewrite /oAC/= opA. Qed. | Lemma | oopA_subdef | 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"
] | [
"oAC",
"oop",
"opA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oopx1_subdef : left_id None oop. | Proof. by case. Qed. | Lemma | oopx1_subdef | 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"
] | [
"oop"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oop1x_subdef : right_id None oop. | Proof. by []. Qed. | Lemma | oop1x_subdef | 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"
] | [
"oop"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oopC_subdef : commutative oop. | Proof. by move=> [x|] [y|]//; rewrite /oAC/= opC. Qed. | Lemma | oopC_subdef | 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"
] | [
"oAC",
"oop",
"opC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
some_big_AC_mk_monoid [I : Type] r P (F : I -> T) :
Some (\big[op/x]_(i <- r | P i) F i) =
oop (\big[oop/None]_(i <- r | P i) Some (F i)) (Some x). | Proof. by elim/big_rec2 : _ => //= i [y|] _ Pi [] -> //=; rewrite opA. Qed. | Lemma | some_big_AC_mk_monoid | 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_rec2",
"oop",
"opA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_AC_mk_monoid [I : Type] r P (F : I -> T) :
\big[op/x]_(i <- r | P i) F i =
odflt x (oop (\big[oop/None]_(i <- r | P i) Some (F i)) (Some x)). | Proof. by apply: Some_inj; rewrite some_big_AC_mk_monoid. Qed. | Lemma | big_AC_mk_monoid | 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",
"oop",
"some_big_AC_mk_monoid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
foldrE r : foldr op idx r = \big[op/idx]_(x <- r) x. | Proof. by rewrite unlock. Qed. | Lemma | foldrE | 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"
] | [
"foldr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_filter r (P : pred I) F :
\big[op/idx]_(i <- filter P r) F i = \big[op/idx]_(i <- r | P i) F i. | Proof. by rewrite unlock; elim: r => //= i r <-; case (P i). Qed. | Lemma | big_filter | 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"
] | [
"filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_filter_cond r (P1 P2 : pred I) F :
\big[op/idx]_(i <- filter P1 r | P2 i) F i
= \big[op/idx]_(i <- r | P1 i && P2 i) F i. | Proof.
rewrite -big_filter -(big_filter r); congr bigop.
by rewrite -filter_predI; apply: eq_filter => i; apply: andbC.
Qed. | Lemma | big_filter_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"
] | [
"P1",
"apply",
"big_filter",
"eq_filter",
"filter",
"filter_predI"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_bigl r (P1 P2 : pred I) F :
P1 =1 P2 ->
\big[op/idx]_(i <- r | P1 i) F i = \big[op/idx]_(i <- r | P2 i) F i. | Proof. by move=> eqP12; rewrite -!(big_filter r) (eq_filter eqP12). Qed. | Lemma | eq_bigl | 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"
] | [
"P1",
"big_filter",
"eq_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_andbC r (P Q : pred I) F :
\big[op/idx]_(i <- r | P i && Q i) F i
= \big[op/idx]_(i <- r | Q i && P i) F i. | Proof. by apply: eq_bigl => i; apply: andbC. Qed. | Lemma | big_andbC | 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"
] | A lemma to permute aggregate conditions. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
eq_bigr r (P : pred I) F1 F2 : (forall i, P i -> F1 i = F2 i) ->
\big[op/idx]_(i <- r | P i) F1 i = \big[op/idx]_(i <- r | P i) F2 i. | Proof. by move=> eqF12; elim/big_rec2: _ => // i x _ /eqF12-> ->. Qed. | Lemma | eq_bigr | 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_rec2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_big r (P1 P2 : pred I) F1 F2 :
P1 =1 P2 -> (forall i, P1 i -> F1 i = F2 i) ->
\big[op/idx]_(i <- r | P1 i) F1 i = \big[op/idx]_(i <- r | P2 i) F2 i. | Proof. by move/eq_bigl <-; move/eq_bigr->. Qed. | Lemma | eq_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"
] | [
"F1",
"F2",
"P1",
"eq_bigl",
"eq_bigr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
congr_big r1 r2 (P1 P2 : pred I) F1 F2 :
r1 = r2 -> P1 =1 P2 -> (forall i, P1 i -> F1 i = F2 i) ->
\big[op/idx]_(i <- r1 | P1 i) F1 i = \big[op/idx]_(i <- r2 | P2 i) F2 i. | Proof. by move=> <-{r2}; apply: eq_big. Qed. | Lemma | congr_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"
] | [
"F1",
"F2",
"P1",
"apply",
"eq_big",
"r1",
"r2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_nil (P : pred I) F : \big[op/idx]_(i <- [::] | P i) F i = idx. | Proof. by rewrite unlock. Qed. | Lemma | big_nil | 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_cons i r (P : pred I) F :
let x := \big[op/idx]_(j <- r | P j) F j in
\big[op/idx]_(j <- i :: r | P j) F j = if P i then op (F i) x else x. | Proof. by rewrite unlock. Qed. | Lemma | big_cons | 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_rcons_op i r (P : pred I) F :
let idx' := if P i then op (F i) idx else idx in
\big[op/idx]_(j <- rcons r i | P j) F j = \big[op/idx']_(j <- r | P j) F j. | Proof.
by elim: r => /= [|j r]; rewrite !(big_nil, big_cons, unlock)// => ->.
Qed. | Lemma | big_rcons_op | 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",
"rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_map J (h : J -> I) r (P : pred I) F :
\big[op/idx]_(i <- map h r | P i) F i
= \big[op/idx]_(j <- r | P (h j)) F (h j). | Proof. by rewrite unlock; elim: r => //= j r ->. Qed. | Lemma | big_map | 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"
] | [
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_nth x0 r (P : pred I) F :
\big[op/idx]_(i <- r | P i) F i
= \big[op/idx]_(0 <= i < size r | P (nth x0 r i)) (F (nth x0 r i)). | Proof. by rewrite -[r in LHS](mkseq_nth x0) big_map /index_iota subn0. Qed. | Lemma | big_nth | 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_map",
"index_iota",
"mkseq_nth",
"nth",
"size",
"subn0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_hasC r (P : pred I) F :
~~ has P r -> \big[op/idx]_(i <- r | P i) F i = idx. | Proof.
by rewrite -big_filter has_count -size_filter -eqn0Ngt unlock => /nilP->.
Qed. | Lemma | big_hasC | 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",
"eqn0Ngt",
"has",
"has_count",
"nilP",
"size_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_pred0_eq (r : seq I) F : \big[op/idx]_(i <- r | false) F i = idx. | Proof. by rewrite big_hasC // has_pred0. Qed. | Lemma | big_pred0_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_hasC",
"has_pred0",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_pred0 r (P : pred I) F :
P =1 xpred0 -> \big[op/idx]_(i <- r | P i) F i = idx. | Proof. by move/eq_bigl->; apply: big_pred0_eq. Qed. | Lemma | big_pred0 | 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_eq",
"eq_bigl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_cat_nested r1 r2 (P : pred I) F :
let x := \big[op/idx]_(i <- r2 | P i) F i in
\big[op/idx]_(i <- r1 ++ r2 | P i) F i = \big[op/x]_(i <- r1 | P i) F i. | Proof. by rewrite unlock /reducebig foldr_cat. Qed. | Lemma | big_cat_nested | 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"
] | [
"foldr_cat",
"r1",
"r2",
"reducebig"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_catl r1 r2 (P : pred I) F :
~~ has P r2 ->
\big[op/idx]_(i <- r1 ++ r2 | P i) F i = \big[op/idx]_(i <- r1 | P i) F i. | Proof. by rewrite big_cat_nested => /big_hasC->. Qed. | Lemma | big_catl | 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_hasC",
"has",
"r1",
"r2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_catr r1 r2 (P : pred I) F :
~~ has P r1 ->
\big[op/idx]_(i <- r1 ++ r2 | P i) F i = \big[op/idx]_(i <- r2 | P i) F i. | Proof.
rewrite -big_filter -(big_filter r2) filter_cat.
by rewrite has_count -size_filter; case: filter.
Qed. | Lemma | big_catr | 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",
"filter",
"filter_cat",
"has",
"has_count",
"r1",
"r2",
"size_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_map_id J (h : J -> R) r (P : pred R) :
\big[op/idx]_(i <- map h r | P i) i
= \big[op/idx]_(j <- r | P (h j)) h j. | Proof. exact: big_map. Qed. | Lemma | big_map_id | 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_map",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_condT (J : finType) (A : {pred J}) F :
\big[op/idx]_(i in A | true) F i = \big[op/idx]_(i in A) F i. | Proof. by apply: eq_bigl => i; exact: andbT. Qed. | Lemma | big_condT | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_seq_cond (I : eqType) r (P : pred I) F :
\big[op/idx]_(i <- r | P i) F i
= \big[op/idx]_(i <- r | (i \in r) && P i) F i. | Proof.
by rewrite -!(big_filter r); congr bigop; apply: eq_in_filter => i ->.
Qed. | Lemma | big_seq_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_filter",
"eq_in_filter"
] | congruence or induction lemmas. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
big_seq (I : eqType) (r : seq I) F :
\big[op/idx]_(i <- r) F i = \big[op/idx]_(i <- r | i \in r) F i. | Proof. by rewrite big_seq_cond big_andbC. Qed. | Lemma | big_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_andbC",
"big_seq_cond",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_big_seq (I : eqType) (r : seq I) F1 F2 :
{in r, F1 =1 F2} -> \big[op/idx]_(i <- r) F1 i = \big[op/idx]_(i <- r) F2 i. | Proof. by move=> eqF; rewrite !big_seq (eq_bigr _ eqF). Qed. | Lemma | eq_big_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"
] | [
"F1",
"F2",
"big_seq",
"eq_bigr",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_nat_cond m n (P : pred nat) F :
\big[op/idx]_(m <= i < n | P i) F i
= \big[op/idx]_(m <= i < n | (m <= i < n) && P i) F i. | Proof.
by rewrite big_seq_cond; apply: eq_bigl => i; rewrite mem_index_iota.
Qed. | Lemma | big_nat_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_seq_cond",
"eq_bigl",
"mem_index_iota",
"nat"
] | Similar lemmas for exposing integer indexing in the predicate. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
big_nat m n F :
\big[op/idx]_(m <= i < n) F i = \big[op/idx]_(m <= i < n | m <= i < n) F i. | Proof. by rewrite big_nat_cond big_andbC. Qed. | Lemma | 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"
] | [
"big_andbC",
"big_nat_cond"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
congr_big_nat m1 n1 m2 n2 P1 P2 F1 F2 :
m1 = m2 -> n1 = n2 ->
(forall i, m1 <= i < n2 -> P1 i = P2 i) ->
(forall i, P1 i && (m1 <= i < n2) -> F1 i = F2 i) ->
\big[op/idx]_(m1 <= i < n1 | P1 i) F1 i
= \big[op/idx]_(m2 <= i < n2 | P2 i) F2 i. | Proof.
move=> <- <- eqP12 eqF12; rewrite big_seq_cond (big_seq_cond _ P2).
apply: eq_big => i; rewrite ?inE /= !mem_index_iota.
by apply: andb_id2l; apply: eqP12.
by rewrite andbC; apply: eqF12.
Qed. | Lemma | congr_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"
] | [
"F1",
"F2",
"P1",
"apply",
"big_seq_cond",
"eq_big",
"inE",
"mem_index_iota"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_big_nat m n F1 F2 :
(forall i, m <= i < n -> F1 i = F2 i) ->
\big[op/idx]_(m <= i < n) F1 i = \big[op/idx]_(m <= i < n) F2 i. | Proof. by move=> eqF; apply: congr_big_nat. Qed. | Lemma | eq_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"
] | [
"F1",
"F2",
"apply",
"congr_big_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_geq m n (P : pred nat) F :
m >= n -> \big[op/idx]_(m <= i < n | P i) F i = idx. | Proof. by move=> ge_m_n; rewrite /index_iota (eqnP ge_m_n) big_nil. Qed. | Lemma | big_geq | 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_nil",
"eqnP",
"index_iota",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_ltn_cond m n (P : pred nat) F :
m < n -> let x := \big[op/idx]_(m.+1 <= i < n | P i) F i in
\big[op/idx]_(m <= i < n | P i) F i = if P m then op (F m) x else x. | Proof. by case: n => [//|n] le_m_n; rewrite /index_iota subSn // big_cons. Qed. | Lemma | big_ltn_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_cons",
"index_iota",
"nat",
"subSn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_ltn m n F :
m < n ->
\big[op/idx]_(m <= i < n) F i = op (F m) (\big[op/idx]_(m.+1 <= i < n) F i). | Proof. by move=> lt_mn; apply: big_ltn_cond. Qed. | Lemma | big_ltn | 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_ltn_cond"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_addn m n a (P : pred nat) F :
\big[op/idx]_(m + a <= i < n | P i) F i =
\big[op/idx]_(m <= i < n - a | P (i + a)) F (i + a). | Proof.
rewrite /index_iota -subnDA addnC iotaDl big_map.
by apply: eq_big => ? *; rewrite addnC.
Qed. | Lemma | big_addn | 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"
] | [
"addnC",
"apply",
"big_map",
"eq_big",
"index_iota",
"iotaDl",
"nat",
"subnDA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_add1 m n (P : pred nat) F :
\big[op/idx]_(m.+1 <= i < n | P i) F i =
\big[op/idx]_(m <= i < n.-1 | P (i.+1)) F (i.+1). | Proof.
by rewrite -addn1 big_addn subn1; apply: eq_big => ? *; rewrite addn1.
Qed. | Lemma | big_add1 | 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"
] | [
"addn1",
"apply",
"big_addn",
"eq_big",
"nat",
"subn1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_nat_recl n m F : m <= n ->
\big[op/idx]_(m <= i < n.+1) F i =
op (F m) (\big[op/idx]_(m <= i < n) F i.+1). | Proof. by move=> lemn; rewrite big_ltn // big_add1. Qed. | Lemma | big_nat_recl | 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_add1",
"big_ltn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_mkord n (P : pred nat) F :
\big[op/idx]_(0 <= i < n | P i) F i = \big[op/idx]_(i < n | P i) F i. | Proof.
rewrite /index_iota subn0 -(big_map (@nat_of_ord n)).
by congr bigop; rewrite /index_enum 2!unlock val_ord_enum.
Qed. | Lemma | big_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_map",
"index_enum",
"index_iota",
"nat",
"nat_of_ord",
"subn0",
"val_ord_enum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_mknat n (P : pred 'I_n.+1) F :
\big[op/idx]_(i < n.+1 | P i) F i
= \big[op/idx]_(0 <= i < n.+1 | P (inord i)) F (inord i). | Proof. by rewrite big_mkord; apply: eq_big => ?; rewrite inord_val. Qed. | Lemma | big_mknat | 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",
"eq_big",
"inord",
"inord_val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_nat_widen m n1 n2 (P : pred nat) F :
n1 <= n2 ->
\big[op/idx]_(m <= i < n1 | P i) F i
= \big[op/idx]_(m <= i < n2 | P i && (i < n1)) F i. | Proof.
move=> len12; symmetry; rewrite -big_filter filter_predI big_filter.
have [ltn_trans eq_by_mem] := (ltn_trans, irr_sorted_eq ltn_trans ltnn).
congr bigop; apply: eq_by_mem; rewrite ?sorted_filter ?iota_ltn_sorted // => i.
rewrite mem_filter !mem_index_iota andbCA andbA andb_idr => // /andP[_].
by move/leq_trans-... | Lemma | big_nat_widen | 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_filter",
"filter_predI",
"iota_ltn_sorted",
"irr_sorted_eq",
"leq_trans",
"ltn_trans",
"ltnn",
"mem_filter",
"mem_index_iota",
"nat",
"sorted_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_ord_widen_cond n1 n2 (P : pred nat) (F : nat -> R) :
n1 <= n2 ->
\big[op/idx]_(i < n1 | P i) F i
= \big[op/idx]_(i < n2 | P i && (i < n1)) F i. | Proof. by move/big_nat_widen=> len12; rewrite -big_mkord len12 big_mkord. Qed. | Lemma | big_ord_widen_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_mkord",
"big_nat_widen",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_ord_widen n1 n2 (F : nat -> R) :
n1 <= n2 ->
\big[op/idx]_(i < n1) F i = \big[op/idx]_(i < n2 | i < n1) F i. | Proof. by move=> le_n12; apply: (big_ord_widen_cond (predT)). Qed. | Lemma | big_ord_widen | 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_ord_widen_cond",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_ord_widen_leq n1 n2 (P : pred 'I_(n1.+1)) F :
n1 < n2 ->
\big[op/idx]_(i < n1.+1 | P i) F i
= \big[op/idx]_(i < n2 | P (inord i) && (i <= n1)) F (inord i). | Proof.
move=> len12; pose g G i := G (inord i : 'I_(n1.+1)).
rewrite -(big_ord_widen_cond (g _ P) (g _ F) len12) {}/g.
by apply: eq_big => i *; rewrite inord_val.
Qed. | Lemma | big_ord_widen_leq | 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_ord_widen_cond",
"eq_big",
"inord",
"inord_val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_ord0 P F : \big[op/idx]_(i < 0 | P i) F i = idx. | Proof. by rewrite big_pred0 => [[]|]. Qed. | Lemma | big_ord0 | 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_pred0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_mask_tuple I n m (t : n.-tuple I) (P : pred I) F :
\big[op/idx]_(i <- mask m t | P i) F i
= \big[op/idx]_(i < n | nth false m i && P (tnth t i)) F (tnth t i). | Proof.
rewrite [t in LHS]tuple_map_ord/= -map_mask big_map.
by rewrite mask_enum_ord big_filter_cond/= enumT.
Qed. | Lemma | big_mask_tuple | 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_cond",
"big_map",
"enumT",
"map_mask",
"mask",
"mask_enum_ord",
"nth",
"tnth",
"tuple",
"tuple_map_ord"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_mask I r m (P : pred I) (F : I -> R) (r_ := tnth (in_tuple r)) :
\big[op/idx]_(i <- mask m r | P i) F i
= \big[op/idx]_(i < size r | nth false m i && P (r_ i)) F (r_ i). | Proof. exact: (big_mask_tuple _ (in_tuple r)). Qed. | Lemma | big_mask | 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_mask_tuple",
"in_tuple",
"mask",
"nth",
"size",
"tnth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_tnth I r (P : pred I) F (r_ := tnth (in_tuple r)) :
\big[op/idx]_(i <- r | P i) F i
= \big[op/idx]_(i < size r | P (r_ i)) (F (r_ i)). | Proof.
rewrite /= -[r in LHS](mask_true (leqnn (size r))) big_mask//.
by apply: eq_bigl => i /=; rewrite nth_nseq ltn_ord.
Qed. | Lemma | big_tnth | 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_mask",
"eq_bigl",
"in_tuple",
"leqnn",
"ltn_ord",
"mask_true",
"nth_nseq",
"size",
"tnth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_index_uniq (I : eqType) (r : seq I) (E : 'I_(size r) -> R) :
uniq r ->
\big[op/idx]_i E i = \big[op/idx]_(x <- r) oapp E idx (insub (index x r)). | Proof.
move=> Ur; apply/esym; rewrite big_tnth.
by under [LHS]eq_bigr do rewrite index_uniq// valK.
Qed. | Lemma | big_index_uniq | 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_bigr",
"index",
"index_uniq",
"insub",
"seq",
"size",
"uniq",
"valK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_tuple I n (t : n.-tuple I) (P : pred I) F :
\big[op/idx]_(i <- t | P i) F i
= \big[op/idx]_(i < n | P (tnth t i)) F (tnth t i). | Proof. by rewrite big_tnth tvalK; case: _ / (esym _). Qed. | Lemma | big_tuple | 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_tnth",
"tnth",
"tuple",
"tvalK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_ord_narrow_cond n1 n2 (P : pred 'I_n2) F (le_n12 : n1 <= n2) :
let w := widen_ord le_n12 in
\big[op/idx]_(i < n2 | P i && (i < n1)) F i
= \big[op/idx]_(i < n1 | P (w i)) F (w i). | Proof.
case: n1 => [|n1] /= in le_n12 *.
by rewrite big_ord0 big_pred0 // => i; rewrite andbF.
rewrite (big_ord_widen_leq _ _ le_n12); apply: eq_big => i.
by apply: andb_id2r => le_i_n1; congr P; apply: val_inj; rewrite /= inordK.
by case/andP=> _ le_i_n1; congr F; apply: val_inj; rewrite /= inordK.
Qed. | Lemma | big_ord_narrow_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_ord0",
"big_ord_widen_leq",
"big_pred0",
"eq_big",
"inordK",
"val_inj",
"widen_ord"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_ord_narrow_cond_leq n1 n2 (P : pred _) F (le_n12 : n1 <= n2) :
let w := @widen_ord n1.+1 n2.+1 le_n12 in
\big[op/idx]_(i < n2.+1 | P i && (i <= n1)) F i
= \big[op/idx]_(i < n1.+1 | P (w i)) F (w i). | Proof. exact: (@big_ord_narrow_cond n1.+1 n2.+1). Qed. | Lemma | big_ord_narrow_cond_leq | 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_ord_narrow_cond",
"widen_ord"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_ord_narrow n1 n2 F (le_n12 : n1 <= n2) :
let w := widen_ord le_n12 in
\big[op/idx]_(i < n2 | i < n1) F i = \big[op/idx]_(i < n1) F (w i). | Proof. exact: (big_ord_narrow_cond (predT)). Qed. | Lemma | big_ord_narrow | 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_ord_narrow_cond",
"widen_ord"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_ord_narrow_leq n1 n2 F (le_n12 : n1 <= n2) :
let w := @widen_ord n1.+1 n2.+1 le_n12 in
\big[op/idx]_(i < n2.+1 | i <= n1) F i = \big[op/idx]_(i < n1.+1) F (w i). | Proof. exact: (big_ord_narrow_cond_leq (predT)). Qed. | Lemma | big_ord_narrow_leq | 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_ord_narrow_cond_leq",
"widen_ord"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_ord_recl n F :
\big[op/idx]_(i < n.+1) F i =
op (F ord0) (\big[op/idx]_(i < n) F (@lift n.+1 ord0 i)). | Proof.
pose G i := F (inord i); have eqFG i: F i = G i by rewrite /G inord_val.
under eq_bigr do rewrite eqFG; under [in RHS]eq_bigr do rewrite eqFG.
by rewrite -(big_mkord _ (fun _ => _) G) eqFG big_ltn // big_add1 /= big_mkord.
Qed. | Lemma | big_ord_recl | 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_add1",
"big_ltn",
"big_mkord",
"eq_bigr",
"inord",
"inord_val",
"lift",
"ord0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_nseq_cond I n a (P : pred I) F :
\big[op/idx]_(i <- nseq n a | P i) F i
= if P a then iter n (op (F a)) idx else idx. | Proof. by rewrite unlock; elim: n => /= [|n ->]; case: (P a). Qed. | Lemma | big_nseq_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"
] | [
"iter",
"nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_nseq I n a (F : I -> R):
\big[op/idx]_(i <- nseq n a) F i = iter n (op (F a)) idx. | Proof. exact: big_nseq_cond. Qed. | Lemma | big_nseq | 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_nseq_cond",
"iter",
"nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_enum_spec (I : finType) (P : pred I) : seq I -> Type | :=
BigEnumSpec e of
forall R idx op (F : I -> R),
\big[op/idx]_(i <- e) F i = \big[op/idx]_(i | P i) F i
& uniq e /\ (forall i, (i \in e) = P i)
& (let cP := [pred i | P i] in perm_eq e (enum cP) /\ size e = #|cP|)
: big_enum_spec P e. | Variant | big_enum_spec | 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"
] | [
"enum",
"perm_eq",
"seq",
"size",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_enumP I P : big_enum_spec P (filter P (index_enum I)). | Proof.
set e := filter P _; have Ue: uniq e by apply/filter_uniq/index_enum_uniq.
have mem_e i: (i \in e) = P i by rewrite mem_filter mem_index_enum andbT.
split=> // [R idx op F | cP]; first by rewrite big_filter.
suffices De: perm_eq e (enum cP) by rewrite (perm_size De) cardE.
by apply/uniq_perm=> // [|i]; rewrite ?... | Lemma | big_enumP | 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_spec",
"big_filter",
"cardE",
"enum",
"enum_uniq",
"filter",
"filter_uniq",
"index_enum",
"index_enum_uniq",
"mem_enum",
"mem_filter",
"mem_index_enum",
"perm_eq",
"perm_size",
"split",
"uniq",
"uniq_perm"
] | does the same while remembering the definition of e. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
big_const_seq I r (P : pred I) x :
\big[op/idx]_(i <- r | P i) x = iter (count P r) (op x) idx. | Proof. by rewrite unlock; elim: r => //= i r ->; case: (P i). Qed. | Lemma | big_const_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"
] | [
"count",
"iter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_const (I : finType) (A : {pred I}) x :
\big[op/idx]_(i in A) x = iter #|A| (op x) idx. | Proof.
by have [e <- _ [_ <-]] := big_enumP A; rewrite big_const_seq count_predT.
Qed. | Lemma | big_const | 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_seq",
"big_enumP",
"count_predT",
"iter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_const_nat m n x :
\big[op/idx]_(m <= i < n) x = iter (n - m) (op x) idx. | Proof. by rewrite big_const_seq count_predT size_iota. Qed. | Lemma | big_const_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_const_seq",
"count_predT",
"iter",
"size_iota"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_const_ord n x :
\big[op/idx]_(i < n) x = iter n (op x) idx. | Proof. by rewrite big_const card_ord. Qed. | Lemma | big_const_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"
] | [
"big_const",
"card_ord",
"iter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_seq1_id I (i : I) (F : I -> R) :
\big[op/x]_(j <- [:: i]) F j = op (F i) x. | Proof. by rewrite big_cons big_nil. Qed. | Lemma | big_seq1_id | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_nat1_id n F : \big[op/x]_(n <= i < n.+1) F i = op (F n) x. | Proof. by rewrite big_ltn // big_geq // mulm1. Qed. | Lemma | big_nat1_id | 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_pred1_eq_id (I : finType) (i : I) F :
\big[op/x]_(j | j == i) F j = op (F i) x. | Proof.
have [e1 <- _ [e_enum _]] := big_enumP (pred1 i).
by rewrite (perm_small_eq _ e_enum) enum1 ?big_seq1_id.
Qed. | Lemma | big_pred1_eq_id | 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_enumP",
"big_seq1_id",
"enum1",
"perm_small_eq",
"pred1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_pred1_id (I : finType) i (P : pred I) F :
P =1 pred1 i -> \big[op/x]_(j | P j) F j = op (F i) x. | Proof. by move/(eq_bigl _ _)->; apply: big_pred1_eq_id. Qed. | Lemma | big_pred1_id | 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_id",
"eq_bigl",
"pred1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
opA | := SemiGroup.opA. | Notation | opA | 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 | |
opC | := SemiGroup.opC. | Notation | opC | 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 | |
opxx : op x x = x. | Hypothesis | opxx | 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_const_idem I (r : seq I) P : \big[op/x]_(i <- r | P i) x = x. | Proof. by elim/big_ind : _ => // _ _ -> ->. Qed. | Lemma | big_const_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_ind",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big1_idem I r (P : pred I) F :
(forall i, P i -> F i = x) -> \big[op/x]_(i <- r | P i) F i = x. | Proof.
move=> Fix; under eq_bigr => ? ? do rewrite Fix//; exact: big_const_idem.
Qed. | Lemma | big1_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_const_idem",
"eq_bigr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_id_idem I (r : seq I) P F :
op (\big[op/x]_(i <- r | P i) F i) x = \big[op/x]_(i <- r | P i) F i. | Proof. by elim/big_rec : _ => // ? ? ?; rewrite -opA => ->. Qed. | Lemma | big_id_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_rec",
"opA",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
opCA : left_commutative op. | Proof. by move=> x *; rewrite !opA /= (opC x). Qed. | Let | opCA | 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"
] | [
"opA",
"opC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_rem_AC (I : eqType) (r : seq I) z (P : pred I) F : z \in r ->
\big[op/x]_(y <- r | P y) F y
= if P z then op (F z) (\big[op/x]_(y <- rem z r | P y) F y)
else \big[op/x]_(y <- rem z r | P y) F y. | Proof.
elim: r =>// i r ih; rewrite big_cons rem_cons inE =>/predU1P[-> /[!eqxx]//|zr].
by case: eqP => [-> //|]; rewrite ih// big_cons; case: ifPn; case: ifPn.
Qed. | Lemma | big_rem_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_cons",
"eqxx",
"inE",
"predU1P",
"rem",
"rem_cons",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_undup (I : eqType) (r : seq I) (P : pred I) F :
idempotent_op op ->
\big[op/x]_(i <- undup r | P i) F i = \big[op/x]_(i <- r | P i) F i. | Proof.
move=> opxx; rewrite -!(big_filter _ _ _ P) filter_undup.
elim: {P r}(filter P r) => //= i r IHr.
case: ifP => [r_i | _]; rewrite !big_cons {}IHr //.
by rewrite (big_rem_AC _ _ r_i) opA /= opxx.
Qed. | Lemma | big_undup | 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_filter",
"big_rem_AC",
"filter",
"filter_undup",
"idempotent_op",
"opA",
"opxx",
"seq",
"undup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_big (I : eqType) r1 r2 (P : pred I) F :
perm_eq r1 r2 ->
\big[op/x]_(i <- r1 | P i) F i = \big[op/x]_(i <- r2 | P i) F i. | Proof.
elim: r1 r2 => [|i r1 IHr1] r2 eq_r12.
by case: r2 eq_r12 => [//|i r2] /[1!perm_sym] /perm_nilP.
have r2i: i \in r2 by rewrite -has_pred1 has_count -(permP eq_r12) /= eqxx.
rewrite big_cons (IHr1 (rem i r2)) -?big_rem_AC// -(perm_cons i).
exact: perm_trans (perm_to_rem _).
Qed. | Lemma | perm_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_cons",
"big_rem_AC",
"eqxx",
"has_count",
"has_pred1",
"permP",
"perm_cons",
"perm_eq",
"perm_nilP",
"perm_sym",
"perm_to_rem",
"perm_trans",
"r1",
"r2",
"rem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_enum_cond (I : finType) (A : {pred I}) (P : pred I) F :
\big[op/x]_(i <- enum A | P i) F i = \big[op/x]_(i in A | P i) F i. | Proof.
by rewrite -big_filter_cond; have [e _ _ [/perm_big->]] := big_enumP.
Qed. | Lemma | big_enum_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_enumP",
"big_filter_cond",
"enum",
"perm_big"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_enum (I : finType) (A : {pred I}) F :
\big[op/x]_(i <- enum A) F i = \big[op/x]_(i in A) F i. | Proof. by rewrite big_enum_cond big_andbC. Qed. | Lemma | big_enum | 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_enum_cond",
"enum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_uniq (I : finType) (r : seq I) F :
uniq r -> \big[op/x]_(i <- r) F i = \big[op/x]_(i in r) F i. | Proof.
move=> uniq_r; rewrite -big_enum; apply: perm_big.
by rewrite uniq_perm ?enum_uniq // => i; rewrite mem_enum.
Qed. | Lemma | big_uniq | 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",
"enum_uniq",
"mem_enum",
"perm_big",
"seq",
"uniq",
"uniq_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigD1 (I : finType) j (P : pred I) F :
P j -> \big[op/x]_(i | P i) F i
= op (F j) (\big[op/x]_(i | P i && (i != j)) F i). | Proof.
rewrite (big_rem_AC _ _ (mem_index_enum j)) => ->.
by rewrite rem_filter ?index_enum_uniq// big_filter_cond big_andbC.
Qed. | Lemma | bigD1 | 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_filter_cond",
"big_rem_AC",
"index_enum_uniq",
"mem_index_enum",
"rem_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigD1_seq (I : eqType) (r : seq I) j F :
j \in r -> uniq r ->
\big[op/x]_(i <- r) F i = op (F j) (\big[op/x]_(i <- r | i != j) F i). | Proof. by move=> /big_rem_AC-> /rem_filter->; rewrite big_filter. Qed. | Lemma | bigD1_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_filter",
"big_rem_AC",
"rem_filter",
"seq",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
big_image_cond I (J : finType) (h : J -> I) (A : pred J) (P : pred I) F :
\big[op/x]_(i <- [seq h j | j in A] | P i) F i
= \big[op/x]_(j in A | P (h j)) F (h j). | Proof. by rewrite big_map big_enum_cond. Qed. | Lemma | big_image_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_enum_cond",
"big_map",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.