phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
op_incr : forall x y, le x (op x y).		Hypothesis	op_incr	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
sub_le_big I [s] (P P' : {pred I}) (F : I -> R) : (forall i, P i -> P' i) -> le (\big[op/x]_(i <- s \| P i) F i) (\big[op/x]_(i <- s \| P' i) F i).	Proof. move=> PP'; rewrite [X in le _ X](big_AC_mk_monoid opA opC) (bigID P P') /=. under [in X in le _ X]eq_bigl do rewrite (andb_idl (PP' _)). rewrite [X in le X _](big_AC_mk_monoid opA opC). case: (bigop _ _ _) (bigop _ _ _) => [y\|] [z\|]//=. by rewrite -opA [_ y x]opC opA op_incr. by rewrite opC op_incr. Qed.	Lemma	sub_le_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" ]	[ "bigID", "big_AC_mk_monoid", "eq_bigl", "le", "opA", "opC", "op_incr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_le_big_seq (I : eqType) s s' P (F : I -> R) : (forall i, count_mem i s <= count_mem i s')%N -> le (\big[op/x]_(i <- s \| P i) F i) (\big[op/x]_(i <- s' \| P i) F i).	Proof. rewrite (big_AC_mk_monoid opA opC) => /count_subseqP[_ /subseqP[m sm ->]]. move/(perm_big _)->; rewrite big_mask [X in le _ X]big_tnth. by rewrite -!(big_AC_mk_monoid opA opC) sub_le_big // => j /andP[]. Qed.	Lemma	sub_le_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_AC_mk_monoid", "big_mask", "big_tnth", "count_mem", "count_subseqP", "le", "opA", "opC", "perm_big", "sub_le_big", "subseqP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_le_big_seq_cond (I : eqType) s s' P P' (F : I -> R) : (forall i, count_mem i (filter P s) <= count_mem i (filter P' s'))%N -> le (\big[op/x]_(i <- s \| P i) F i) (\big[op/x]_(i <- s' \| P' i) F i).	Proof. by move=> /(sub_le_big_seq xpredT F); rewrite !big_filter. Qed.	Lemma	sub_le_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" ]	[ "big_filter", "count_mem", "filter", "le", "sub_le_big_seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
uniq_sub_le_big (I : eqType) s s' P (F : I -> R) : uniq s -> uniq s' -> {subset s <= s'} -> le (\big[op/x]_(i <- s \| P i) F i) (\big[op/x]_(i <- s' \| P i) F i).	Proof. move=> us us' ss'; rewrite sub_le_big_seq => // i; rewrite !count_uniq_mem//. by have /implyP := ss' i; case: (_ \in s) (_ \in s') => [] []. Qed.	Lemma	uniq_sub_le_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" ]	[ "count_uniq_mem", "le", "sub_le_big_seq", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
uniq_sub_le_big_cond (I : eqType) s s' P P' (F : I -> R) : uniq (filter P s) -> uniq (filter P' s') -> {subset [seq i <- s \| P i] <= [seq i <- s' \| P' i]} -> le (\big[op/x]_(i <- s \| P i) F i) (\big[op/x]_(i <- s' \| P' i) F i).	Proof. by move=> u v /(uniq_sub_le_big xpredT F u v); rewrite !big_filter. Qed.	Lemma	uniq_sub_le_big_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_filter", "filter", "le", "seq", "uniq", "uniq_sub_le_big" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
opK : idempotent_op op.		Hypothesis	opK	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" ]	[ "idempotent_op" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
idem_sub_le_big (I : eqType) s s' P (F : I -> R) : {subset s <= s'} -> le (\big[op/x]_(i <- s \| P i) F i) (\big[op/x]_(i <- s' \| P i) F i).	Proof. move=> ss'; rewrite -big_undup// -[X in le _ X]big_undup//. by rewrite uniq_sub_le_big ?undup_uniq// => i; rewrite !mem_undup; apply: ss'. Qed.	Lemma	idem_sub_le_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", "big_undup", "le", "mem_undup", "undup_uniq", "uniq_sub_le_big" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
idem_sub_le_big_cond (I : eqType) s s' P P' (F : I -> R) : {subset [seq i <- s \| P i] <= [seq i <- s' \| P' i]} -> le (\big[op/x]_(i <- s \| P i) F i) (\big[op/x]_(i <- s' \| P' i) F i).	Proof. by move=> /(idem_sub_le_big xpredT F); rewrite !big_filter. Qed.	Lemma	idem_sub_le_big_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_filter", "idem_sub_le_big", "le", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_in_le_big [I : eqType] (s : seq I) (P P' : {pred I}) (F : I -> R) : {in s, forall i, P i -> P' i} -> le (\big[op/x]_(i <- s \| P i) F i) (\big[op/x]_(i <- s \| P' i) F i).	Proof. move=> PP'; apply: sub_le_big_seq_cond => i; rewrite leq_count_subseq//. rewrite subseq_filter filter_subseq andbT; apply/allP => j. by rewrite !mem_filter => /andP[/PP'/[apply]->]. Qed.	Lemma	sub_in_le_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" ]	[ "allP", "apply", "filter_subseq", "le", "leq_count_subseq", "mem_filter", "seq", "sub_le_big_seq_cond", "subseq_filter" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_big_ord n m [P : {pred nat}] [F : nat -> R] : (n <= m)%N -> le (\big[op/x]_(i < n \| P i) F i) (\big[op/x]_(i < m \| P i) F i).	Proof. by move=> nm; rewrite (big_ord_widen_cond m)// sub_le_big => //= ? /andP[]. Qed.	Lemma	le_big_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_ord_widen_cond", "le", "nat", "sub_le_big" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_le_big [I : finType] [A A' P : {pred I}] (F : I -> R) : A \subset A' -> le (\big[op/x]_(i in A \| P i) F i) (\big[op/x]_(i in A' \| P i) F i).	Proof. move=> AA'; apply: sub_le_big => y /andP[yA yP]; apply/andP; split => //. exact: subsetP yA. Qed.	Lemma	subset_le_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" ]	[ "A'", "apply", "le", "split", "sub_le_big", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_big_nat_cond n m n' m' (P P' : {pred nat}) (F : nat -> R) : (n' <= n)%N -> (m <= m')%N -> (forall i, (n <= i < m)%N -> P i -> P' i) -> le (\big[op/x]_(n <= i < m \| P i) F i) (\big[op/x]_(n' <= i < m' \| P' i) F i).	Proof. move=> len'n lemm' PP'i; rewrite uniq_sub_le_big_cond ?filter_uniq ?iota_uniq//. move=> i; rewrite !mem_filter !mem_index_iota => /and3P[Pi ni im]. by rewrite PP'i ?ni//= (leq_trans _ ni)// (leq_trans im). Qed.	Lemma	le_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" ]	[ "filter_uniq", "iota_uniq", "le", "leq_trans", "mem_filter", "mem_index_iota", "n'", "nat", "uniq_sub_le_big_cond" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_big_nat n m n' m' [P] [F : nat -> R] : (n' <= n)%N -> (m <= m')%N -> le (\big[op/x]_(n <= i < m \| P i) F i) (\big[op/x]_(n' <= i < m' \| P i) F i).	Proof. by move=> len'n lemm'; rewrite le_big_nat_cond. Qed.	Lemma	le_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" ]	[ "le", "le_big_nat_cond", "n'", "nat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_big_ord_cond n m (P P' : {pred nat}) (F : nat -> R) : (n <= m)%N -> (forall i : 'I_n, P i -> P' i) -> le (\big[op/x]_(i < n \| P i) F i) (\big[op/x]_(i < m \| P' i) F i).	Proof. move=> nm PP'; rewrite -!big_mkord le_big_nat_cond//= => i ni. by have := PP' (Ordinal ni). Qed.	Lemma	le_big_ord_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", "le", "le_big_nat_cond", "nat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_bigl_supp (r : seq I) (P1 : pred I) (P2 : pred I) (F : I -> R) : {in [pred x \| F x != idx], P1 =1 P2} -> \big[op/idx]_(i <- r \| P1 i) F i = \big[op/idx]_(i <- r \| P2 i) F i.	Proof. move=> P12; rewrite big_mkcond [RHS]big_mkcond; apply: eq_bigr => i _. by case: (eqVneq (F i) idx) => [->\|/P12->]; rewrite ?if_same. Qed.	Lemma	eq_bigl_supp	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_mkcond", "eqVneq", "eq_bigr", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_big_supp_cond [r s : seq I] [P : pred I] (F : I -> R) : perm_eq [seq i <- r \| P i && (F i != idx)] [seq i <- s \| P i && (F i != idx)] -> \big[op/idx]_(i <- r \| P i) F i = \big[op/idx]_(i <- s \| P i) F i.	Proof. move=> prs; rewrite !(bigID [pred i \| F i == idx] P F)/=. rewrite big1 ?Monoid.mul1m; first by move=> i /andP[_ /eqP->]. rewrite [in RHS]big1 ?Monoid.mul1m; first by move=> i /andP[_ /eqP->]. by rewrite -[in LHS]big_filter -[in RHS]big_filter; apply perm_big. Qed.	Lemma	perm_big_supp_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", "big1", "bigID", "big_filter", "mul1m", "perm_big", "perm_eq", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
perm_big_supp [r s : seq I] [P : pred I] (F : I -> R) : perm_eq [seq i <- r \| F i != idx] [seq i <- s \| F i != idx] -> \big[op/idx]_(i <- r \| P i) F i = \big[op/idx]_(i <- s \| P i) F i.	Proof. by move=> ?; apply: perm_big_supp_cond; rewrite !filter_predI perm_filter. Qed.	Lemma	perm_big_supp	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", "filter_predI", "perm_big_supp_cond", "perm_eq", "perm_filter", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"0"	:= zero.	Notation	0	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" ]	[ "zero" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"*%M"	:= times.	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"	:= (times 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
"+%M"	:= plus.	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"	:= (plus 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
big_distrl I r a (P : pred I) F : \big[+%M/0]_(i <- r \| P i) F i * a = \big[+%M/0]_(i <- r \| P i) (F i * a).	Proof. by rewrite (big_endo ( *%M^~ a)) ?mul0m // => x y; apply: mulmDl. Qed.	Lemma	big_distrl	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_endo", "mul0m", "mulmDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
big_distrr I r a (P : pred I) F : a * \big[+%M/0]_(i <- r \| P i) F i = \big[+%M/0]_(i <- r \| P i) (a * F i).	Proof. by rewrite big_endo ?mulm0 // => x y; apply: mulmDr. Qed.	Lemma	big_distrr	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_endo", "mulm0", "mulmDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
big_distrlr I J rI rJ (pI : pred I) (pJ : pred J) F G : (\big[+%M/0]_(i <- rI \| pI i) F i) * (\big[+%M/0]_(j <- rJ \| pJ j) G j) = \big[+%M/0]_(i <- rI \| pI i) \big[+%M/0]_(j <- rJ \| pJ j) (F i * G j).	Proof. by rewrite big_distrl; under eq_bigr do rewrite big_distrr. Qed.	Lemma	big_distrlr	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_distrl", "big_distrr", "eq_bigr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
big_distr_big_dep (I J : finType) j0 (P : pred I) (Q : I -> pred J) F : \big[%M/1]_(i \| P i) \big[+%M/0]_(j \| Q i j) F i j = \big[+%M/0]_(f in pfamily j0 P Q) \big[%M/1]_(i \| P i) F i (f i).	Proof. pose fIJ := {ffun I -> J}; pose Pf := pfamily j0 (_ : seq I) Q. have [r big_r [Ur mem_r] _] := big_enumP P. symmetry; transitivity (\big[+%M/0]_(f in Pf r) \big[*%M/1]_(i <- r) F i (f i)). by apply: eq_big => // f; apply: eq_forallb => i; rewrite /= mem_r. rewrite -{P mem_r}big_r; elim: r Ur => /= [_ \| i r IHr...	Lemma	big_distr_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" ]	[ "apply", "big_cons", "big_distrl", "big_distrr", "big_enumP", "big_nil", "big_pred1", "big_seq", "eq_big", "eq_bigr", "eq_f", "eq_forallb", "eqxx", "familyP", "ffunE", "ffunP", "inE", "last", "partition_big", "pfamily", "reindex_onto", "seq", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
big_distr_big (I J : finType) j0 (P : pred I) (Q : pred J) F : \big[%M/1]_(i \| P i) \big[+%M/0]_(j \| Q j) F i j = \big[+%M/0]_(f in pffun_on j0 P Q) \big[%M/1]_(i \| P i) F i (f i).	Proof. rewrite (big_distr_big_dep j0); apply: eq_bigl => f. by apply/familyP/familyP=> Pf i; case: ifP (Pf i). Qed.	Lemma	big_distr_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", "big_distr_big_dep", "eq_bigl", "familyP", "pffun_on" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigA_distr_big_dep (I J : finType) (Q : I -> pred J) F : \big[%M/1]_i \big[+%M/0]_(j \| Q i j) F i j = \big[+%M/0]_(f in family Q) \big[%M/1]_i F i (f i).	Proof. have [j _ \| J0] := pickP J; first by rewrite (big_distr_big_dep j). have Q0 i: Q i =i pred0 by move=> /J0/esym/notF[]. transitivity (iter #\|I\| ( *%M 0) 1). by rewrite -big_const; apply/eq_bigr=> i; have /(big_pred0 _)-> := Q0 i. have [i _ \| I0] := pickP I. rewrite (cardD1 i) //= mul0m big_pred0 // => f. by...	Lemma	bigA_distr_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" ]	[ "apply", "big_const", "big_distr_big_dep", "big_pred0", "big_pred1", "cardD1", "eq_bigr", "eq_card0", "family", "familyP", "ffunP", "iter", "mul0m", "pickP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigA_distr_big (I J : finType) (Q : pred J) (F : I -> J -> R) : \big[%M/1]_i \big[+%M/0]_(j \| Q j) F i j = \big[+%M/0]_(f in ffun_on Q) \big[%M/1]_i F i (f i).	Proof. exact: bigA_distr_big_dep. Qed.	Lemma	bigA_distr_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" ]	[ "bigA_distr_big_dep", "ffun_on" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigA_distr_bigA (I J : finType) F : \big[%M/1]_(i : I) \big[+%M/0]_(j : J) F i j = \big[+%M/0]_(f : {ffun I -> J}) \big[%M/1]_i F i (f i).	Proof. by rewrite bigA_distr_big; apply: eq_bigl => ?; apply/familyP. Qed.	Lemma	bigA_distr_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" ]	[ "apply", "bigA_distr_big", "eq_bigl", "familyP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
big_has : \big[orb/false]_(i <- r) B i = has B r.	Proof. by rewrite unlock. Qed.	Lemma	big_has	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" ]	[ "has" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
big_all : \big[andb/true]_(i <- r) B i = all B r.	Proof. by rewrite unlock. Qed.	Lemma	big_all	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" ]	[ "all" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
big_has_cond : \big[orb/false]_(i <- r \| P i) B i = has (predI P B) r.	Proof. by rewrite big_mkcond unlock. Qed.	Lemma	big_has_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", "has" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
big_all_cond : \big[andb/true]_(i <- r \| P i) B i = all [pred i \| P i ==> B i] r.	Proof. by rewrite big_mkcond unlock. Qed.	Lemma	big_all_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" ]	[ "all", "big_mkcond" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
big_bool R (idx : R) (op : Monoid.com_law idx) (F : bool -> R): \big[op/idx]_(i : bool) F i = op (F true) (F false).	Proof. by rewrite /index_enum !unlock /= Monoid.mulm1. Qed.	Lemma	big_bool	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" ]	[ "com_law", "index_enum", "mulm1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
big_orE : \big[orb/false]_(i \| P i) B i = [exists (i \| P i), B i].	Proof. by rewrite big_has_cond; apply/hasP/existsP=> [] [i]; exists i. Qed.	Lemma	big_orE	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_has_cond", "existsP", "hasP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
big_andE : \big[andb/true]_(i \| P i) B i = [forall (i \| P i), B i].	Proof. rewrite big_all_cond; apply/allP/forallP=> /= allB i; rewrite allB //. exact: mem_index_enum. Qed.	Lemma	big_andE	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" ]	[ "allP", "apply", "big_all_cond", "forallP", "mem_index_enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum_nat_const n : \sum_(i in A) n = #\|A\| * n.	Proof. by rewrite big_const iter_addn_0 mulnC. Qed.	Lemma	sum_nat_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", "iter_addn_0", "mulnC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum1_card : \sum_(i in A) 1 = #\|A\|.	Proof. by rewrite sum_nat_const muln1. Qed.	Lemma	sum1_card	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" ]	[ "muln1", "sum_nat_const" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum1_count J (r : seq J) (a : pred J) : \sum_(j <- r \| a j) 1 = count a r.	Proof. by rewrite big_const_seq iter_addn_0 mul1n. Qed.	Lemma	sum1_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" ]	[ "big_const_seq", "count", "iter_addn_0", "mul1n", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum1_size J (r : seq J) : \sum_(j <- r) 1 = size r.	Proof. by rewrite sum1_count count_predT. Qed.	Lemma	sum1_size	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_predT", "seq", "size", "sum1_count" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prod_nat_const n : \prod_(i in A) n = n ^ #\|A\|.	Proof. by rewrite big_const -Monoid.iteropE. Qed.	Lemma	prod_nat_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", "iteropE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum_nat_const_nat n1 n2 n : \sum_(n1 <= i < n2) n = (n2 - n1) * n.	Proof. by rewrite big_const_nat iter_addn_0 mulnC. Qed.	Lemma	sum_nat_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_nat", "iter_addn_0", "mulnC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prod_nat_const_nat n1 n2 n : \prod_(n1 <= i < n2) n = n ^ (n2 - n1).	Proof. by rewrite big_const_nat -Monoid.iteropE. Qed.	Lemma	prod_nat_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_nat", "iteropE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
telescope_sumn_in n m f : n <= m -> (forall i, n <= i < m -> f i <= f i.+1) -> \sum_(n <= k < m) (f k.+1 - f k) = f m - f n.	Proof. move=> nm fle; rewrite (telescope_big (fun i j => f j - f i)); last first. by case: ltngtP nm => // ->; rewrite subnn. move=> k /andP[nk km]; rewrite /= addnBAC ?subnKC ?fle ?(ltnW nk)//. elim: k nk km => [//\| k IHk /[!ltnS]/[1!leq_eqVlt]+ km]. move=> /predU1P[/[dup]nk -> \| nk]; first by rewrite fle ?nk ?leq...	Lemma	telescope_sumn_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" ]	[ "addnBAC", "last", "leq_eqVlt", "leq_trans", "leqnn", "ltnS", "ltnW", "ltngtP", "predU1P", "subnKC", "subnn", "telescope_big" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
telescope_sumn n m f : {homo f : x y / x <= y} -> \sum_(n <= k < m) (f k.+1 - f k) = f m - f n.	Proof. move=> fle; case: (ltnP n m) => nm. by apply: (telescope_sumn_in (ltnW nm)) => ? ?; apply: fle. by apply/esym/eqP; rewrite big_geq// subn_eq0 fle. Qed.	Lemma	telescope_sumn	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_geq", "ltnP", "ltnW", "subn_eq0", "telescope_sumn_in" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sumnE r : sumn r = \sum_(i <- r) i.	Proof. exact: foldrE. Qed.	Lemma	sumnE	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" ]	[ "foldrE", "sumn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_bseq n (T : finType) : #\|{bseq n of T}\| = \sum_(i < n.+1) #\|T\| ^ i.	Proof. rewrite (bij_eq_card bseq_tagged_tuple_bij) card_tagged sumnE big_map big_enum. by under eq_bigr do rewrite card_tuple. Qed.	Lemma	card_bseq	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", "bij_eq_card", "bseq", "bseq_tagged_tuple_bij", "card_tagged", "card_tuple", "eq_bigr", "sumnE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leqif_sum (I : finType) (P C : pred I) (E1 E2 : I -> nat) : (forall i, P i -> E1 i <= E2 i ?= iff C i) -> \sum_(i \| P i) E1 i <= \sum_(i \| P i) E2 i ?= iff [forall (i \| P i), C i].	Proof. move=> leE12; rewrite -big_andE. by elim/big_rec3: _ => // i Ci m1 m2 /leE12; apply: leqif_add. Qed.	Lemma	leqif_sum	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_andE", "big_rec3", "leqif_add", "nat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leq_sum I r (P : pred I) (E1 E2 : I -> nat) : (forall i, P i -> E1 i <= E2 i) -> \sum_(i <- r \| P i) E1 i <= \sum_(i <- r \| P i) E2 i.	Proof. by move=> leE12; elim/big_ind2: _ => // m1 m2 n1 n2; apply: leq_add. Qed.	Lemma	leq_sum	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_ind2", "leq_add", "nat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sumnB I r (P : pred I) (E1 E2 : I -> nat) : (forall i, P i -> E1 i <= E2 i) -> \sum_(i <- r \| P i) (E2 i - E1 i) = \sum_(i <- r \| P i) E2 i - \sum_(i <- r \| P i) E1 i.	Proof. by move=> /(_ _ _)/subnK-/(eq_bigr _)<-; rewrite big_split addnK. Qed.	Lemma	sumnB	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" ]	[ "addnK", "big_split", "eq_bigr", "nat", "subnK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum_nat_eq0 (I : finType) (P : pred I) (E : I -> nat) : (\sum_(i \| P i) E i == 0)%N = [forall (i \| P i), E i == 0%N].	Proof. by rewrite eq_sym -(@leqif_sum I P _ (fun _ => 0%N) E) ?big1_eq. Qed.	Lemma	sum_nat_eq0	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_eq", "eq_sym", "leqif_sum", "nat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum_nat_seq_eq0 I r (P : pred I) F : (\sum_(i <- r \| P i) F i == 0)%N = all (fun i => P i ==> (F i == 0%N)) r.	Proof. by rewrite (big_morph _ (id1:=true) addn_eq0)// big_all_cond. Qed.	Lemma	sum_nat_seq_eq0	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_eq0", "all", "big_all_cond", "big_morph", "id1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum_nat_seq_neq0 I r (P : pred I) F : (\sum_(i <- r \| P i) F i != 0)%N = has (fun i => P i && (F i != 0)%N) r.	Proof. by rewrite sum_nat_seq_eq0// -has_predC; apply: eq_has => x /=; case Px: (P x). Qed.	Lemma	sum_nat_seq_neq0	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" ]	[ "Px", "apply", "eq_has", "has", "has_predC", "sum_nat_seq_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum_nat_eq1 (I : finType) (P : pred I) (F : I -> nat) : reflect (exists i : I, [/\ P i, F i = 1 & forall j, j != i -> P j -> F j = 0]%N) (\sum_(i \| P i) F i == 1)%N.	Proof. apply/(iffP idP) => [sumF_eq1 \| [i [Pi Fi1 zFj]]]; last first. rewrite (bigD1 i)//= Fi1 addn_eq1//= orbF sum_nat_eq0. by apply/forall_inP => j /andP[Pj ji]; apply/eqP/zFj. have /forall_inPn [i Pi FiN0]: ~~ [forall i in P, F i == 0]. by apply: contraTN sumF_eq1 => /'forall_in_eqP F0; rewrite big1. move: sum...	Lemma	sum_nat_eq1	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" ]	[ "F0", "addn_eq1", "apply", "big1", "bigD1", "conj", "forall_inP", "forall_inPn", "last", "nat", "split", "sum_nat_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum_nat_seq_eq1 (I : eqType) r (P : pred I) (F : I -> nat) : (\sum_(i <- r \| P i) F i = 1)%N -> exists i, [/\ i \in r, P i, F i = 1 & forall j, j != i -> j \in r -> P j -> F j = 0]%N.	Proof. rewrite big_tnth/= => /eqP/sum_nat_eq1[/= i [Pi Fi FNi]]. exists (tnth (in_tuple r) i); split; rewrite //= ?mem_tnth// => j. move=> /[swap] /(tnthP (in_tuple r))[{} j -> Nij /FNi->//]. by apply: contra_neq Nij => ->. Qed.	Lemma	sum_nat_seq_eq1	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", "contra_neq", "in_tuple", "mem_tnth", "nat", "split", "sum_nat_eq1", "tnth", "tnthP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prod_nat_seq_eq0 I r (P : pred I) F : (\prod_(i <- r \| P i) F i == 0)%N = has (fun i => P i && (F i == 0%N)) r.	Proof. by rewrite (big_morph _ (id1 := false) muln_eq0)// big_has_cond. Qed.	Lemma	prod_nat_seq_eq0	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_has_cond", "big_morph", "has", "id1", "muln_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prod_nat_seq_neq0 I r (P : pred I) F : (\prod_(i <- r \| P i) F i != 0)%N = all (fun i => P i ==> (F i != 0%N)) r.	Proof. by rewrite prod_nat_seq_eq0 -all_predC; apply: eq_all => i /=; case: (P i). Qed.	Lemma	prod_nat_seq_neq0	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" ]	[ "all", "all_predC", "apply", "eq_all", "prod_nat_seq_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prod_nat_seq_eq1 I r (P : pred I) F : (\prod_(i <- r \| P i) F i == 1)%N = all (fun i => P i ==> (F i == 1%N)) r.	Proof. by rewrite (big_morph _ (id1:=true) muln_eq1)// big_all_cond. Qed.	Lemma	prod_nat_seq_eq1	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" ]	[ "all", "big_all_cond", "big_morph", "id1", "muln_eq1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prod_nat_seq_neq1 I r (P : pred I) F : (\prod_(i <- r \| P i) F i != 1)%N = has (fun i => P i && (F i != 1%N)) r.	Proof. by rewrite prod_nat_seq_eq1 -has_predC; apply: eq_has => i /=; case: (P i). Qed.	Lemma	prod_nat_seq_neq1	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_has", "has", "has_predC", "prod_nat_seq_eq1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leq_prod I r (P : pred I) (E1 E2 : I -> nat) : (forall i, P i -> E1 i <= E2 i) -> \prod_(i <- r \| P i) E1 i <= \prod_(i <- r \| P i) E2 i.	Proof. by move=> leE12; elim/big_ind2: _ => // m1 m2 n1 n2; apply: leq_mul. Qed.	Lemma	leq_prod	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_ind2", "leq_mul", "nat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodn_cond_gt0 I r (P : pred I) F : (forall i, P i -> 0 < F i) -> 0 < \prod_(i <- r \| P i) F i.	Proof. by move=> Fpos; elim/big_ind: _ => // n1 n2; rewrite muln_gt0 => ->. Qed.	Lemma	prodn_cond_gt0	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", "muln_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodn_gt0 I r (P : pred I) F : (forall i, 0 < F i) -> 0 < \prod_(i <- r \| P i) F i.	Proof. by move=> Fpos; apply: prodn_cond_gt0. Qed.	Lemma	prodn_gt0	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", "prodn_cond_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gt0_prodn_seq (I : eqType) r (P : pred I) F : 0 < \prod_(i <- r \| P i) F i -> forall i, i \in r -> P i -> 0 < F i.	Proof. move=> + i ri Pi; rewrite !lt0n; apply: contra_neq => Fi_eq0. by case: (path.splitP ri) => *; rewrite big_cat big_rcons Pi Fi_eq0/= muln0. Qed.	Lemma	gt0_prodn_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" ]	[ "apply", "big_cat", "big_rcons", "contra_neq", "lt0n", "muln0", "path", "splitP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gt0_prodn (I : finType) (P : pred I) F : 0 < \prod_(i \| P i) F i -> forall i, P i -> 0 < F i.	Proof. by move=> /gt0_prodn_seq + i => /[apply]; apply. Qed.	Lemma	gt0_prodn	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", "gt0_prodn_seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leq_bigmax_seq (I : eqType) r (P : pred I) F i0 : i0 \in r -> P i0 -> F i0 <= \max_(i <- r \| P i) F i.	Proof. move=> + Pi0; elim: r => // h t ih; rewrite inE big_cons. move=> /predU1P[<-\|i0t]; first by rewrite Pi0 leq_maxl. by case: ifPn => Ph; [rewrite leq_max ih// orbT\|rewrite ih]. Qed.	Lemma	leq_bigmax_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" ]	[ "Pi0", "big_cons", "i0", "inE", "leq_max", "leq_maxl", "predU1P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leq_bigmax_cond (I : finType) (P : pred I) F i0 : P i0 -> F i0 <= \max_(i \| P i) F i.	Proof. exact: leq_bigmax_seq. Qed.	Lemma	leq_bigmax_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" ]	[ "i0", "leq_bigmax_seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leq_bigmax (I : finType) F (i0 : I) : F i0 <= \max_i F i.	Proof. exact: leq_bigmax_cond. Qed.	Lemma	leq_bigmax	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" ]	[ "i0", "leq_bigmax_cond" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_leqP (I : finType) (P : pred I) m F : reflect (forall i, P i -> F i <= m) (\max_(i \| P i) F i <= m).	Proof. apply: (iffP idP) => leFm => [i Pi\|]. by apply: leq_trans leFm; apply: leq_bigmax_cond. by elim/big_ind: _ => // m1 m2; rewrite geq_max => ->. Qed.	Lemma	bigmax_leqP	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_ind", "geq_max", "leq_bigmax_cond", "leq_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_leqP_seq (I : eqType) r (P : pred I) m F : reflect (forall i, i \in r -> P i -> F i <= m) (\max_(i <- r \| P i) F i <= m).	Proof. apply: (iffP idP) => leFm => [i ri Pi\|]. exact/(leq_trans _ leFm)/leq_bigmax_seq. rewrite big_seq_cond; elim/big_ind: _ => // [m1 m2\|i /andP[ri]]. by rewrite geq_max => ->. exact: leFm. Qed.	Lemma	bigmax_leqP_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" ]	[ "apply", "big_ind", "big_seq_cond", "geq_max", "leq_bigmax_seq", "leq_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_sup (I : finType) i0 (P : pred I) m F : P i0 -> m <= F i0 -> m <= \max_(i \| P i) F i.	Proof. by move=> Pi0 le_m_Fi0; apply: leq_trans (leq_bigmax_cond i0 Pi0). Qed.	Lemma	bigmax_sup	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", "i0", "leq_bigmax_cond", "leq_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmaxn_sup_seq (I : eqType) r i0 (P : pred I) m F : i0 \in r -> P i0 -> m <= F i0 -> m <= \max_(i <- r \| P i) F i.	Proof. by move=> i0r Pi0 ?; apply: leq_trans (leq_bigmax_seq i0 _ _). Qed.	Lemma	bigmaxn_sup_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" ]	[ "Pi0", "apply", "i0", "leq_bigmax_seq", "leq_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_sup_seq	:= bigmaxn_sup_seq.	Notation	bigmax_sup_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" ]	[ "bigmaxn_sup_seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigmax_eq_arg (I : finType) i0 (P : pred I) F : P i0 -> \max_(i \| P i) F i = F [arg max_(i > i0 \| P i) F i].	Proof. move=> Pi0; case: arg_maxnP => //= i Pi maxFi. by apply/eqP; rewrite eqn_leq leq_bigmax_cond // andbT; apply/bigmax_leqP. Qed.	Lemma	bigmax_eq_arg	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", "arg_maxnP", "bigmax_leqP", "eqn_leq", "i0", "leq_bigmax_cond" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_bigmax_cond (I : finType) (A : pred I) F : #\|A\| > 0 -> {i0 \| i0 \in A & \max_(i in A) F i = F i0}.	Proof. case: (pickP A) => [i0 Ai0 _ \| ]; last by move/eq_card0->. by exists [arg max_(i > i0 in A) F i]; [case: arg_maxnP \| apply: bigmax_eq_arg]. Qed.	Lemma	eq_bigmax_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", "arg_maxnP", "bigmax_eq_arg", "eq_card0", "i0", "last", "pickP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_bigmax (I : finType) F : #\|I\| > 0 -> {i0 : I \| \max_i F i = F i0}.	Proof. by case/(eq_bigmax_cond F) => x _ ->; exists x. Qed.	Lemma	eq_bigmax	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_bigmax_cond", "i0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expn_sum m I r (P : pred I) F : (m ^ (\sum_(i <- r \| P i) F i) = \prod_(i <- r \| P i) m ^ F i)%N.	Proof. exact: (big_morph _ (expnD m)). Qed.	Lemma	expn_sum	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", "expnD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dvdn_biglcmP (I : finType) (P : pred I) F m : reflect (forall i, P i -> F i %\| m) (\big[lcmn/1%N]_(i \| P i) F i %\| m).	Proof. apply: (iffP idP) => [dvFm i Pi \| dvFm]. by rewrite (bigD1 i) // dvdn_lcm in dvFm; case/andP: dvFm. by elim/big_ind: _ => // p q p_m; rewrite dvdn_lcm p_m. Qed.	Lemma	dvdn_biglcmP	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", "big_ind", "dvdn_lcm", "lcmn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
biglcmn_sup (I : finType) i0 (P : pred I) F m : P i0 -> m %\| F i0 -> m %\| \big[lcmn/1%N]_(i \| P i) F i.	Proof. by move=> Pi0 m_Fi0; rewrite (dvdn_trans m_Fi0) // (bigD1 i0) ?dvdn_lcml. Qed.	Lemma	biglcmn_sup	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", "bigD1", "dvdn_lcml", "dvdn_trans", "i0", "lcmn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dvdn_biggcdP (I : finType) (P : pred I) F m : reflect (forall i, P i -> m %\| F i) (m %\| \big[gcdn/0]_(i \| P i) F i).	Proof. apply: (iffP idP) => [dvmF i Pi \| dvmF]. by rewrite (bigD1 i) // dvdn_gcd in dvmF; case/andP: dvmF. by elim/big_ind: _ => // p q m_p; rewrite dvdn_gcd m_p. Qed.	Lemma	dvdn_biggcdP	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", "big_ind", "dvdn_gcd", "gcdn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
biggcdn_inf (I : finType) i0 (P : pred I) F m : P i0 -> F i0 %\| m -> \big[gcdn/0]_(i \| P i) F i %\| m.	Proof. by move=> Pi0; apply: dvdn_trans; rewrite (bigD1 i0) ?dvdn_gcdl. Qed.	Lemma	biggcdn_inf	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", "bigD1", "dvdn_gcdl", "dvdn_trans", "gcdn", "i0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fact_prod n : n`! = \prod_(1 <= i < n.+1) i.	Proof. elim: n => [\|n IHn] //; first by rewrite big_nil. by apply/esym; rewrite factS IHn // !big_add1 big_nat_recr //= mulnC. Qed.	Lemma	fact_prod	boot	boot/binomial.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]	[ "apply", "big_add1", "big_nat_recr", "big_nil", "factS", "mulnC" ]	More properties of the factorial *	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fact_split n m : m <= n -> n`! = m`! * \prod_(m.+1 <= k < n.+1) k.	Proof. by move=> leq_mn; rewrite !fact_prod -big_cat_nat. Qed.	Lemma	fact_split	boot	boot/binomial.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]	[ "big_cat_nat", "fact_prod" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
logn_fact p n : prime p -> logn p n`! = \sum_(1 <= k < n.+1) n %/ p ^ k.	Proof. move=> p_prime; transitivity (\sum_(1 <= i < n.+1) logn p i). rewrite big_add1; elim: n => /= [\|n IHn]; first by rewrite logn1 big_geq. by rewrite big_nat_recr // -IHn /= factS mulnC lognM ?fact_gt0. transitivity (\sum_(1 <= i < n.+1) \sum_(1 <= k < n.+1) (p ^ k %\| i)). apply: eq_big_nat => i /andP[i_gt0 l...	Lemma	logn_fact	boot	boot/binomial.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]	[ "apply", "big_add1", "big_geq", "big_mkcond", "big_nat_recr", "big_nat_widen", "divn_count_dvd", "dvdn_leq", "eq_big_nat", "eq_bigl", "eq_bigr", "exchange_big_nat", "factS", "fact_gt0", "leq_trans", "logn", "logn1", "lognM", "logn_count_dvd", "ltnW", "ltn_expl", "mulnC", ...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Wilson p : p > 1 -> prime p = (p %\| ((p.-1)`!).+1).	Proof. have dFact n: 0 < n -> (n.-1)`! = \prod_(0 <= i < n \| i != 0) i. move=> n_gt0; rewrite -big_filter fact_prod; symmetry; apply: congr_big => //. rewrite /index_iota subn1 -[n]prednK //=; apply/all_filterP. by rewrite all_predC has_pred1 mem_iota. move=> lt1p; have p_gt0 := ltnW lt1p. apply/idP/idP=> [pr_p \|...	Theorem	Wilson	boot	boot/binomial.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]	[ "Build", "Euclid_dvdM", "addKn", "addn1", "addnA", "addnBA", "addnCA", "addnS", "all_filterP", "all_predC", "apply", "big1", "bigD1", "bigID", "big_filter", "big_mkord", "big_morph", "big_split", "com_law", "congr_big", "contra_eq_neq", "coprime", "coprime_sym", "dvdn",...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffact_rec n m	:= if m is m'.+1 then n * ffact_rec n.-1 m' else 1.	Fixpoint	ffact_rec	boot	boot/binomial.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]	[]	The falling factorial	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
falling_factorial	:= ffact_rec.	Definition	falling_factorial	boot	boot/binomial.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]	[ "ffact_rec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"n ^_ m"	:= (falling_factorial n m) (at level 30, right associativity) : nat_scope.	Notation	n ^_ m	boot	boot/binomial.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]	[ "falling_factorial" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffactE : falling_factorial = ffact_rec.	Proof. by []. Qed.	Lemma	ffactE	boot	boot/binomial.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]	[ "falling_factorial", "ffact_rec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffactn0 n : n ^_ 0 = 1.	Proof. by []. Qed.	Lemma	ffactn0	boot	boot/binomial.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffact0n m : 0 ^_ m = (m == 0).	Proof. by case: m. Qed.	Lemma	ffact0n	boot	boot/binomial.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffactnS n m : n ^_ m.+1 = n * n.-1 ^_ m.	Proof. by []. Qed.	Lemma	ffactnS	boot	boot/binomial.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffactSS n m : n.+1 ^_ m.+1 = n.+1 * n ^_ m.	Proof. by []. Qed.	Lemma	ffactSS	boot	boot/binomial.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffactn1 n : n ^_ 1 = n.	Proof. exact: muln1. Qed.	Lemma	ffactn1	boot	boot/binomial.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]	[ "muln1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffactnSr n m : n ^_ m.+1 = n ^_ m * (n - m).	Proof. elim: n m => [\|n IHn] [\|m] //=; first by rewrite ffactn1 mul1n. by rewrite !ffactSS IHn mulnA. Qed.	Lemma	ffactnSr	boot	boot/binomial.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]	[ "ffactSS", "ffactn1", "mul1n", "mulnA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffact_prod n m : n ^_ m = \prod_(i < m) (n - i).	Proof. elim: m n => [n \| m IH [\|n] //]; first by rewrite ffactn0 big_ord0. by rewrite big_ord_recr /= sub0n muln0. by rewrite ffactSS IH big_ord_recl subn0. Qed.	Lemma	ffact_prod	boot	boot/binomial.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]	[ "big_ord0", "big_ord_recl", "big_ord_recr", "ffactSS", "ffactn0", "muln0", "sub0n", "subn0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffact_gt0 n m : (0 < n ^_ m) = (m <= n).	Proof. by elim: n m => [\|n IHn] [\|m] //=; rewrite ffactSS muln_gt0 IHn. Qed.	Lemma	ffact_gt0	boot	boot/binomial.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]	[ "ffactSS", "muln_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffact_small n m : n < m -> n ^_ m = 0.	Proof. by rewrite ltnNge -ffact_gt0; case: posnP. Qed.	Lemma	ffact_small	boot	boot/binomial.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]	[ "ffact_gt0", "ltnNge", "posnP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffactnn n : n ^_ n = n`!.	Proof. by elim: n => [\|n IHn] //; rewrite ffactnS IHn. Qed.	Lemma	ffactnn	boot	boot/binomial.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]	[ "ffactnS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

op_incr : forall x y, le x (op x y).

Hypothesis

op_incr

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

sub_le_big I [s] (P P' : {pred I}) (F : I -> R) : (forall i, P i -> P' i) -> le (\big[op/x]_(i <- s | P i) F i) (\big[op/x]_(i <- s | P' i) F i).

Proof. move=> PP'; rewrite [X in le _ X](big_AC_mk_monoid opA opC) (bigID P P') /=. under [in X in le _ X]eq_bigl do rewrite (andb_idl (PP' _)). rewrite [X in le X _](big_AC_mk_monoid opA opC). case: (bigop _ _ _) (bigop _ _ _) => [y|] [z|]//=. by rewrite -opA [_ y x]opC opA op_incr. by rewrite opC op_incr. Qed.

Lemma

sub_le_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" ]

[ "bigID", "big_AC_mk_monoid", "eq_bigl", "le", "opA", "opC", "op_incr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_le_big_seq (I : eqType) s s' P (F : I -> R) : (forall i, count_mem i s <= count_mem i s')%N -> le (\big[op/x]_(i <- s | P i) F i) (\big[op/x]_(i <- s' | P i) F i).

Proof. rewrite (big_AC_mk_monoid opA opC) => /count_subseqP[_ /subseqP[m sm ->]]. move/(perm_big _)->; rewrite big_mask [X in le _ X]big_tnth. by rewrite -!(big_AC_mk_monoid opA opC) sub_le_big // => j /andP[]. Qed.

Lemma

sub_le_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_AC_mk_monoid", "big_mask", "big_tnth", "count_mem", "count_subseqP", "le", "opA", "opC", "perm_big", "sub_le_big", "subseqP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_le_big_seq_cond (I : eqType) s s' P P' (F : I -> R) : (forall i, count_mem i (filter P s) <= count_mem i (filter P' s'))%N -> le (\big[op/x]_(i <- s | P i) F i) (\big[op/x]_(i <- s' | P' i) F i).

Proof. by move=> /(sub_le_big_seq xpredT F); rewrite !big_filter. Qed.

Lemma

sub_le_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" ]

[ "big_filter", "count_mem", "filter", "le", "sub_le_big_seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

uniq_sub_le_big (I : eqType) s s' P (F : I -> R) : uniq s -> uniq s' -> {subset s <= s'} -> le (\big[op/x]_(i <- s | P i) F i) (\big[op/x]_(i <- s' | P i) F i).

Proof. move=> us us' ss'; rewrite sub_le_big_seq => // i; rewrite !count_uniq_mem//. by have /implyP := ss' i; case: (_ \in s) (_ \in s') => [] []. Qed.

Lemma

uniq_sub_le_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" ]

[ "count_uniq_mem", "le", "sub_le_big_seq", "uniq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

uniq_sub_le_big_cond (I : eqType) s s' P P' (F : I -> R) : uniq (filter P s) -> uniq (filter P' s') -> {subset [seq i <- s | P i] <= [seq i <- s' | P' i]} -> le (\big[op/x]_(i <- s | P i) F i) (\big[op/x]_(i <- s' | P' i) F i).

Proof. by move=> u v /(uniq_sub_le_big xpredT F u v); rewrite !big_filter. Qed.

Lemma

uniq_sub_le_big_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_filter", "filter", "le", "seq", "uniq", "uniq_sub_le_big" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

opK : idempotent_op op.

Hypothesis

opK

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" ]

[ "idempotent_op" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

idem_sub_le_big (I : eqType) s s' P (F : I -> R) : {subset s <= s'} -> le (\big[op/x]_(i <- s | P i) F i) (\big[op/x]_(i <- s' | P i) F i).

Proof. move=> ss'; rewrite -big_undup// -[X in le _ X]big_undup//. by rewrite uniq_sub_le_big ?undup_uniq// => i; rewrite !mem_undup; apply: ss'. Qed.

Lemma

idem_sub_le_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", "big_undup", "le", "mem_undup", "undup_uniq", "uniq_sub_le_big" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

idem_sub_le_big_cond (I : eqType) s s' P P' (F : I -> R) : {subset [seq i <- s | P i] <= [seq i <- s' | P' i]} -> le (\big[op/x]_(i <- s | P i) F i) (\big[op/x]_(i <- s' | P' i) F i).

Proof. by move=> /(idem_sub_le_big xpredT F); rewrite !big_filter. Qed.

Lemma

idem_sub_le_big_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_filter", "idem_sub_le_big", "le", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_in_le_big [I : eqType] (s : seq I) (P P' : {pred I}) (F : I -> R) : {in s, forall i, P i -> P' i} -> le (\big[op/x]_(i <- s | P i) F i) (\big[op/x]_(i <- s | P' i) F i).

Proof. move=> PP'; apply: sub_le_big_seq_cond => i; rewrite leq_count_subseq//. rewrite subseq_filter filter_subseq andbT; apply/allP => j. by rewrite !mem_filter => /andP[/PP'/[apply]->]. Qed.

Lemma

sub_in_le_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" ]

[ "allP", "apply", "filter_subseq", "le", "leq_count_subseq", "mem_filter", "seq", "sub_le_big_seq_cond", "subseq_filter" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_big_ord n m [P : {pred nat}] [F : nat -> R] : (n <= m)%N -> le (\big[op/x]_(i < n | P i) F i) (\big[op/x]_(i < m | P i) F i).

Proof. by move=> nm; rewrite (big_ord_widen_cond m)// sub_le_big => //= ? /andP[]. Qed.

Lemma

le_big_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_ord_widen_cond", "le", "nat", "sub_le_big" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_le_big [I : finType] [A A' P : {pred I}] (F : I -> R) : A \subset A' -> le (\big[op/x]_(i in A | P i) F i) (\big[op/x]_(i in A' | P i) F i).

Proof. move=> AA'; apply: sub_le_big => y /andP[yA yP]; apply/andP; split => //. exact: subsetP yA. Qed.

Lemma

subset_le_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" ]

[ "A'", "apply", "le", "split", "sub_le_big", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_big_nat_cond n m n' m' (P P' : {pred nat}) (F : nat -> R) : (n' <= n)%N -> (m <= m')%N -> (forall i, (n <= i < m)%N -> P i -> P' i) -> le (\big[op/x]_(n <= i < m | P i) F i) (\big[op/x]_(n' <= i < m' | P' i) F i).

Proof. move=> len'n lemm' PP'i; rewrite uniq_sub_le_big_cond ?filter_uniq ?iota_uniq//. move=> i; rewrite !mem_filter !mem_index_iota => /and3P[Pi ni im]. by rewrite PP'i ?ni//= (leq_trans _ ni)// (leq_trans im). Qed.

Lemma

le_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" ]

[ "filter_uniq", "iota_uniq", "le", "leq_trans", "mem_filter", "mem_index_iota", "n'", "nat", "uniq_sub_le_big_cond" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_big_nat n m n' m' [P] [F : nat -> R] : (n' <= n)%N -> (m <= m')%N -> le (\big[op/x]_(n <= i < m | P i) F i) (\big[op/x]_(n' <= i < m' | P i) F i).

Proof. by move=> len'n lemm'; rewrite le_big_nat_cond. Qed.

Lemma

le_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" ]

[ "le", "le_big_nat_cond", "n'", "nat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_big_ord_cond n m (P P' : {pred nat}) (F : nat -> R) : (n <= m)%N -> (forall i : 'I_n, P i -> P' i) -> le (\big[op/x]_(i < n | P i) F i) (\big[op/x]_(i < m | P' i) F i).

Proof. move=> nm PP'; rewrite -!big_mkord le_big_nat_cond//= => i ni. by have := PP' (Ordinal ni). Qed.

Lemma

le_big_ord_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", "le", "le_big_nat_cond", "nat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_bigl_supp (r : seq I) (P1 : pred I) (P2 : pred I) (F : I -> R) : {in [pred x | F x != idx], P1 =1 P2} -> \big[op/idx]_(i <- r | P1 i) F i = \big[op/idx]_(i <- r | P2 i) F i.

Proof. move=> P12; rewrite big_mkcond [RHS]big_mkcond; apply: eq_bigr => i _. by case: (eqVneq (F i) idx) => [->|/P12->]; rewrite ?if_same. Qed.

Lemma

eq_bigl_supp

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_mkcond", "eqVneq", "eq_bigr", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_big_supp_cond [r s : seq I] [P : pred I] (F : I -> R) : perm_eq [seq i <- r | P i && (F i != idx)] [seq i <- s | P i && (F i != idx)] -> \big[op/idx]_(i <- r | P i) F i = \big[op/idx]_(i <- s | P i) F i.

Proof. move=> prs; rewrite !(bigID [pred i | F i == idx] P F)/=. rewrite big1 ?Monoid.mul1m; first by move=> i /andP[_ /eqP->]. rewrite [in RHS]big1 ?Monoid.mul1m; first by move=> i /andP[_ /eqP->]. by rewrite -[in LHS]big_filter -[in RHS]big_filter; apply perm_big. Qed.

Lemma

perm_big_supp_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", "big1", "bigID", "big_filter", "mul1m", "perm_big", "perm_eq", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

perm_big_supp [r s : seq I] [P : pred I] (F : I -> R) : perm_eq [seq i <- r | F i != idx] [seq i <- s | F i != idx] -> \big[op/idx]_(i <- r | P i) F i = \big[op/idx]_(i <- s | P i) F i.

Proof. by move=> ?; apply: perm_big_supp_cond; rewrite !filter_predI perm_filter. Qed.

Lemma

perm_big_supp

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", "filter_predI", "perm_big_supp_cond", "perm_eq", "perm_filter", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"0"

:= zero.

Notation

0

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" ]

[ "zero" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"*%M"

:= times.

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"

:= (times 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

"+%M"

:= plus.

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"

:= (plus 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

big_distrl I r a (P : pred I) F : \big[+%M/0]_(i <- r | P i) F i * a = \big[+%M/0]_(i <- r | P i) (F i * a).

Proof. by rewrite (big_endo ( *%M^~ a)) ?mul0m // => x y; apply: mulmDl. Qed.

Lemma

big_distrl

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_endo", "mul0m", "mulmDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

big_distrr I r a (P : pred I) F : a * \big[+%M/0]_(i <- r | P i) F i = \big[+%M/0]_(i <- r | P i) (a * F i).

Proof. by rewrite big_endo ?mulm0 // => x y; apply: mulmDr. Qed.

Lemma

big_distrr

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_endo", "mulm0", "mulmDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

big_distrlr I J rI rJ (pI : pred I) (pJ : pred J) F G : (\big[+%M/0]_(i <- rI | pI i) F i) * (\big[+%M/0]_(j <- rJ | pJ j) G j) = \big[+%M/0]_(i <- rI | pI i) \big[+%M/0]_(j <- rJ | pJ j) (F i * G j).

Proof. by rewrite big_distrl; under eq_bigr do rewrite big_distrr. Qed.

Lemma

big_distrlr

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_distrl", "big_distrr", "eq_bigr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

big_distr_big_dep (I J : finType) j0 (P : pred I) (Q : I -> pred J) F : \big[*%M/1]_(i | P i) \big[+%M/0]_(j | Q i j) F i j = \big[+%M/0]_(f in pfamily j0 P Q) \big[*%M/1]_(i | P i) F i (f i).

Proof. pose fIJ := {ffun I -> J}; pose Pf := pfamily j0 (_ : seq I) Q. have [r big_r [Ur mem_r] _] := big_enumP P. symmetry; transitivity (\big[+%M/0]_(f in Pf r) \big[*%M/1]_(i <- r) F i (f i)). by apply: eq_big => // f; apply: eq_forallb => i; rewrite /= mem_r. rewrite -{P mem_r}big_r; elim: r Ur => /= [_ | i r IHr...

Lemma

big_distr_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" ]

[ "apply", "big_cons", "big_distrl", "big_distrr", "big_enumP", "big_nil", "big_pred1", "big_seq", "eq_big", "eq_bigr", "eq_f", "eq_forallb", "eqxx", "familyP", "ffunE", "ffunP", "inE", "last", "partition_big", "pfamily", "reindex_onto", "seq", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

big_distr_big (I J : finType) j0 (P : pred I) (Q : pred J) F : \big[*%M/1]_(i | P i) \big[+%M/0]_(j | Q j) F i j = \big[+%M/0]_(f in pffun_on j0 P Q) \big[*%M/1]_(i | P i) F i (f i).

Proof. rewrite (big_distr_big_dep j0); apply: eq_bigl => f. by apply/familyP/familyP=> Pf i; case: ifP (Pf i). Qed.

Lemma

big_distr_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", "big_distr_big_dep", "eq_bigl", "familyP", "pffun_on" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigA_distr_big_dep (I J : finType) (Q : I -> pred J) F : \big[*%M/1]_i \big[+%M/0]_(j | Q i j) F i j = \big[+%M/0]_(f in family Q) \big[*%M/1]_i F i (f i).

Proof. have [j _ | J0] := pickP J; first by rewrite (big_distr_big_dep j). have Q0 i: Q i =i pred0 by move=> /J0/esym/notF[]. transitivity (iter #|I| ( *%M 0) 1). by rewrite -big_const; apply/eq_bigr=> i; have /(big_pred0 _)-> := Q0 i. have [i _ | I0] := pickP I. rewrite (cardD1 i) //= mul0m big_pred0 // => f. by...

Lemma

bigA_distr_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" ]

[ "apply", "big_const", "big_distr_big_dep", "big_pred0", "big_pred1", "cardD1", "eq_bigr", "eq_card0", "family", "familyP", "ffunP", "iter", "mul0m", "pickP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigA_distr_big (I J : finType) (Q : pred J) (F : I -> J -> R) : \big[*%M/1]_i \big[+%M/0]_(j | Q j) F i j = \big[+%M/0]_(f in ffun_on Q) \big[*%M/1]_i F i (f i).

Proof. exact: bigA_distr_big_dep. Qed.

Lemma

bigA_distr_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" ]

[ "bigA_distr_big_dep", "ffun_on" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigA_distr_bigA (I J : finType) F : \big[*%M/1]_(i : I) \big[+%M/0]_(j : J) F i j = \big[+%M/0]_(f : {ffun I -> J}) \big[*%M/1]_i F i (f i).

Proof. by rewrite bigA_distr_big; apply: eq_bigl => ?; apply/familyP. Qed.

Lemma

bigA_distr_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" ]

[ "apply", "bigA_distr_big", "eq_bigl", "familyP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

big_has : \big[orb/false]_(i <- r) B i = has B r.

Proof. by rewrite unlock. Qed.

Lemma

big_has

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" ]

[ "has" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

big_all : \big[andb/true]_(i <- r) B i = all B r.

Proof. by rewrite unlock. Qed.

Lemma

big_all

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" ]

[ "all" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

big_has_cond : \big[orb/false]_(i <- r | P i) B i = has (predI P B) r.

Proof. by rewrite big_mkcond unlock. Qed.

Lemma

big_has_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", "has" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

big_all_cond : \big[andb/true]_(i <- r | P i) B i = all [pred i | P i ==> B i] r.

Proof. by rewrite big_mkcond unlock. Qed.

Lemma

big_all_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" ]

[ "all", "big_mkcond" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

big_bool R (idx : R) (op : Monoid.com_law idx) (F : bool -> R): \big[op/idx]_(i : bool) F i = op (F true) (F false).

Proof. by rewrite /index_enum !unlock /= Monoid.mulm1. Qed.

Lemma

big_bool

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" ]

[ "com_law", "index_enum", "mulm1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

big_orE : \big[orb/false]_(i | P i) B i = [exists (i | P i), B i].

Proof. by rewrite big_has_cond; apply/hasP/existsP=> [] [i]; exists i. Qed.

Lemma

big_orE

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_has_cond", "existsP", "hasP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

big_andE : \big[andb/true]_(i | P i) B i = [forall (i | P i), B i].

Proof. rewrite big_all_cond; apply/allP/forallP=> /= allB i; rewrite allB //. exact: mem_index_enum. Qed.

Lemma

big_andE

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" ]

[ "allP", "apply", "big_all_cond", "forallP", "mem_index_enum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum_nat_const n : \sum_(i in A) n = #|A| * n.

Proof. by rewrite big_const iter_addn_0 mulnC. Qed.

Lemma

sum_nat_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", "iter_addn_0", "mulnC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum1_card : \sum_(i in A) 1 = #|A|.

Proof. by rewrite sum_nat_const muln1. Qed.

Lemma

sum1_card

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" ]

[ "muln1", "sum_nat_const" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum1_count J (r : seq J) (a : pred J) : \sum_(j <- r | a j) 1 = count a r.

Proof. by rewrite big_const_seq iter_addn_0 mul1n. Qed.

Lemma

sum1_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" ]

[ "big_const_seq", "count", "iter_addn_0", "mul1n", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum1_size J (r : seq J) : \sum_(j <- r) 1 = size r.

Proof. by rewrite sum1_count count_predT. Qed.

Lemma

sum1_size

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_predT", "seq", "size", "sum1_count" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prod_nat_const n : \prod_(i in A) n = n ^ #|A|.

Proof. by rewrite big_const -Monoid.iteropE. Qed.

Lemma

prod_nat_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", "iteropE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum_nat_const_nat n1 n2 n : \sum_(n1 <= i < n2) n = (n2 - n1) * n.

Proof. by rewrite big_const_nat iter_addn_0 mulnC. Qed.

Lemma

sum_nat_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_nat", "iter_addn_0", "mulnC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prod_nat_const_nat n1 n2 n : \prod_(n1 <= i < n2) n = n ^ (n2 - n1).

Proof. by rewrite big_const_nat -Monoid.iteropE. Qed.

Lemma

prod_nat_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_nat", "iteropE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

telescope_sumn_in n m f : n <= m -> (forall i, n <= i < m -> f i <= f i.+1) -> \sum_(n <= k < m) (f k.+1 - f k) = f m - f n.

Proof. move=> nm fle; rewrite (telescope_big (fun i j => f j - f i)); last first. by case: ltngtP nm => // ->; rewrite subnn. move=> k /andP[nk km]; rewrite /= addnBAC ?subnKC ?fle ?(ltnW nk)//. elim: k nk km => [//| k IHk /[!ltnS]/[1!leq_eqVlt]+ km]. move=> /predU1P[/[dup]nk -> | nk]; first by rewrite fle ?nk ?leq...

Lemma

telescope_sumn_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" ]

[ "addnBAC", "last", "leq_eqVlt", "leq_trans", "leqnn", "ltnS", "ltnW", "ltngtP", "predU1P", "subnKC", "subnn", "telescope_big" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

telescope_sumn n m f : {homo f : x y / x <= y} -> \sum_(n <= k < m) (f k.+1 - f k) = f m - f n.

Proof. move=> fle; case: (ltnP n m) => nm. by apply: (telescope_sumn_in (ltnW nm)) => ? ?; apply: fle. by apply/esym/eqP; rewrite big_geq// subn_eq0 fle. Qed.

Lemma

telescope_sumn

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_geq", "ltnP", "ltnW", "subn_eq0", "telescope_sumn_in" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sumnE r : sumn r = \sum_(i <- r) i.

Proof. exact: foldrE. Qed.

Lemma

sumnE

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" ]

[ "foldrE", "sumn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_bseq n (T : finType) : #|{bseq n of T}| = \sum_(i < n.+1) #|T| ^ i.

Proof. rewrite (bij_eq_card bseq_tagged_tuple_bij) card_tagged sumnE big_map big_enum. by under eq_bigr do rewrite card_tuple. Qed.

Lemma

card_bseq

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", "bij_eq_card", "bseq", "bseq_tagged_tuple_bij", "card_tagged", "card_tuple", "eq_bigr", "sumnE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leqif_sum (I : finType) (P C : pred I) (E1 E2 : I -> nat) : (forall i, P i -> E1 i <= E2 i ?= iff C i) -> \sum_(i | P i) E1 i <= \sum_(i | P i) E2 i ?= iff [forall (i | P i), C i].

Proof. move=> leE12; rewrite -big_andE. by elim/big_rec3: _ => // i Ci m1 m2 /leE12; apply: leqif_add. Qed.

Lemma

leqif_sum

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_andE", "big_rec3", "leqif_add", "nat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leq_sum I r (P : pred I) (E1 E2 : I -> nat) : (forall i, P i -> E1 i <= E2 i) -> \sum_(i <- r | P i) E1 i <= \sum_(i <- r | P i) E2 i.

Proof. by move=> leE12; elim/big_ind2: _ => // m1 m2 n1 n2; apply: leq_add. Qed.

Lemma

leq_sum

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_ind2", "leq_add", "nat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sumnB I r (P : pred I) (E1 E2 : I -> nat) : (forall i, P i -> E1 i <= E2 i) -> \sum_(i <- r | P i) (E2 i - E1 i) = \sum_(i <- r | P i) E2 i - \sum_(i <- r | P i) E1 i.

Proof. by move=> /(_ _ _)/subnK-/(eq_bigr _)<-; rewrite big_split addnK. Qed.

Lemma

sumnB

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" ]

[ "addnK", "big_split", "eq_bigr", "nat", "subnK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum_nat_eq0 (I : finType) (P : pred I) (E : I -> nat) : (\sum_(i | P i) E i == 0)%N = [forall (i | P i), E i == 0%N].

Proof. by rewrite eq_sym -(@leqif_sum I P _ (fun _ => 0%N) E) ?big1_eq. Qed.

Lemma

sum_nat_eq0

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_eq", "eq_sym", "leqif_sum", "nat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum_nat_seq_eq0 I r (P : pred I) F : (\sum_(i <- r | P i) F i == 0)%N = all (fun i => P i ==> (F i == 0%N)) r.

Proof. by rewrite (big_morph _ (id1:=true) addn_eq0)// big_all_cond. Qed.

Lemma

sum_nat_seq_eq0

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_eq0", "all", "big_all_cond", "big_morph", "id1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum_nat_seq_neq0 I r (P : pred I) F : (\sum_(i <- r | P i) F i != 0)%N = has (fun i => P i && (F i != 0)%N) r.

Proof. by rewrite sum_nat_seq_eq0// -has_predC; apply: eq_has => x /=; case Px: (P x). Qed.

Lemma

sum_nat_seq_neq0

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" ]

[ "Px", "apply", "eq_has", "has", "has_predC", "sum_nat_seq_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum_nat_eq1 (I : finType) (P : pred I) (F : I -> nat) : reflect (exists i : I, [/\ P i, F i = 1 & forall j, j != i -> P j -> F j = 0]%N) (\sum_(i | P i) F i == 1)%N.

Proof. apply/(iffP idP) => [sumF_eq1 | [i [Pi Fi1 zFj]]]; last first. rewrite (bigD1 i)//= Fi1 addn_eq1//= orbF sum_nat_eq0. by apply/forall_inP => j /andP[Pj ji]; apply/eqP/zFj. have /forall_inPn [i Pi FiN0]: ~~ [forall i in P, F i == 0]. by apply: contraTN sumF_eq1 => /'forall_in_eqP F0; rewrite big1. move: sum...

Lemma

sum_nat_eq1

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" ]

[ "F0", "addn_eq1", "apply", "big1", "bigD1", "conj", "forall_inP", "forall_inPn", "last", "nat", "split", "sum_nat_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum_nat_seq_eq1 (I : eqType) r (P : pred I) (F : I -> nat) : (\sum_(i <- r | P i) F i = 1)%N -> exists i, [/\ i \in r, P i, F i = 1 & forall j, j != i -> j \in r -> P j -> F j = 0]%N.

Proof. rewrite big_tnth/= => /eqP/sum_nat_eq1[/= i [Pi Fi FNi]]. exists (tnth (in_tuple r) i); split; rewrite //= ?mem_tnth// => j. move=> /[swap] /(tnthP (in_tuple r))[{} j -> Nij /FNi->//]. by apply: contra_neq Nij => ->. Qed.

Lemma

sum_nat_seq_eq1

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", "contra_neq", "in_tuple", "mem_tnth", "nat", "split", "sum_nat_eq1", "tnth", "tnthP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prod_nat_seq_eq0 I r (P : pred I) F : (\prod_(i <- r | P i) F i == 0)%N = has (fun i => P i && (F i == 0%N)) r.

Proof. by rewrite (big_morph _ (id1 := false) muln_eq0)// big_has_cond. Qed.

Lemma

prod_nat_seq_eq0

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_has_cond", "big_morph", "has", "id1", "muln_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prod_nat_seq_neq0 I r (P : pred I) F : (\prod_(i <- r | P i) F i != 0)%N = all (fun i => P i ==> (F i != 0%N)) r.

Proof. by rewrite prod_nat_seq_eq0 -all_predC; apply: eq_all => i /=; case: (P i). Qed.

Lemma

prod_nat_seq_neq0

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" ]

[ "all", "all_predC", "apply", "eq_all", "prod_nat_seq_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prod_nat_seq_eq1 I r (P : pred I) F : (\prod_(i <- r | P i) F i == 1)%N = all (fun i => P i ==> (F i == 1%N)) r.

Proof. by rewrite (big_morph _ (id1:=true) muln_eq1)// big_all_cond. Qed.

Lemma

prod_nat_seq_eq1

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" ]

[ "all", "big_all_cond", "big_morph", "id1", "muln_eq1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prod_nat_seq_neq1 I r (P : pred I) F : (\prod_(i <- r | P i) F i != 1)%N = has (fun i => P i && (F i != 1%N)) r.

Proof. by rewrite prod_nat_seq_eq1 -has_predC; apply: eq_has => i /=; case: (P i). Qed.

Lemma

prod_nat_seq_neq1

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_has", "has", "has_predC", "prod_nat_seq_eq1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leq_prod I r (P : pred I) (E1 E2 : I -> nat) : (forall i, P i -> E1 i <= E2 i) -> \prod_(i <- r | P i) E1 i <= \prod_(i <- r | P i) E2 i.

Proof. by move=> leE12; elim/big_ind2: _ => // m1 m2 n1 n2; apply: leq_mul. Qed.

Lemma

leq_prod

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_ind2", "leq_mul", "nat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodn_cond_gt0 I r (P : pred I) F : (forall i, P i -> 0 < F i) -> 0 < \prod_(i <- r | P i) F i.

Proof. by move=> Fpos; elim/big_ind: _ => // n1 n2; rewrite muln_gt0 => ->. Qed.

Lemma

prodn_cond_gt0

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", "muln_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodn_gt0 I r (P : pred I) F : (forall i, 0 < F i) -> 0 < \prod_(i <- r | P i) F i.

Proof. by move=> Fpos; apply: prodn_cond_gt0. Qed.

Lemma

prodn_gt0

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", "prodn_cond_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gt0_prodn_seq (I : eqType) r (P : pred I) F : 0 < \prod_(i <- r | P i) F i -> forall i, i \in r -> P i -> 0 < F i.

Proof. move=> + i ri Pi; rewrite !lt0n; apply: contra_neq => Fi_eq0. by case: (path.splitP ri) => *; rewrite big_cat big_rcons Pi Fi_eq0/= muln0. Qed.

Lemma

gt0_prodn_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" ]

[ "apply", "big_cat", "big_rcons", "contra_neq", "lt0n", "muln0", "path", "splitP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gt0_prodn (I : finType) (P : pred I) F : 0 < \prod_(i | P i) F i -> forall i, P i -> 0 < F i.

Proof. by move=> /gt0_prodn_seq + i => /[apply]; apply. Qed.

Lemma

gt0_prodn

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", "gt0_prodn_seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leq_bigmax_seq (I : eqType) r (P : pred I) F i0 : i0 \in r -> P i0 -> F i0 <= \max_(i <- r | P i) F i.

Proof. move=> + Pi0; elim: r => // h t ih; rewrite inE big_cons. move=> /predU1P[<-|i0t]; first by rewrite Pi0 leq_maxl. by case: ifPn => Ph; [rewrite leq_max ih// orbT|rewrite ih]. Qed.

Lemma

leq_bigmax_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" ]

[ "Pi0", "big_cons", "i0", "inE", "leq_max", "leq_maxl", "predU1P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leq_bigmax_cond (I : finType) (P : pred I) F i0 : P i0 -> F i0 <= \max_(i | P i) F i.

Proof. exact: leq_bigmax_seq. Qed.

Lemma

leq_bigmax_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" ]

[ "i0", "leq_bigmax_seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leq_bigmax (I : finType) F (i0 : I) : F i0 <= \max_i F i.

Proof. exact: leq_bigmax_cond. Qed.

Lemma

leq_bigmax

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" ]

[ "i0", "leq_bigmax_cond" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmax_leqP (I : finType) (P : pred I) m F : reflect (forall i, P i -> F i <= m) (\max_(i | P i) F i <= m).

Proof. apply: (iffP idP) => leFm => [i Pi|]. by apply: leq_trans leFm; apply: leq_bigmax_cond. by elim/big_ind: _ => // m1 m2; rewrite geq_max => ->. Qed.

Lemma

bigmax_leqP

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_ind", "geq_max", "leq_bigmax_cond", "leq_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmax_leqP_seq (I : eqType) r (P : pred I) m F : reflect (forall i, i \in r -> P i -> F i <= m) (\max_(i <- r | P i) F i <= m).

Proof. apply: (iffP idP) => leFm => [i ri Pi|]. exact/(leq_trans _ leFm)/leq_bigmax_seq. rewrite big_seq_cond; elim/big_ind: _ => // [m1 m2|i /andP[ri]]. by rewrite geq_max => ->. exact: leFm. Qed.

Lemma

bigmax_leqP_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" ]

[ "apply", "big_ind", "big_seq_cond", "geq_max", "leq_bigmax_seq", "leq_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmax_sup (I : finType) i0 (P : pred I) m F : P i0 -> m <= F i0 -> m <= \max_(i | P i) F i.

Proof. by move=> Pi0 le_m_Fi0; apply: leq_trans (leq_bigmax_cond i0 Pi0). Qed.

Lemma

bigmax_sup

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", "i0", "leq_bigmax_cond", "leq_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmaxn_sup_seq (I : eqType) r i0 (P : pred I) m F : i0 \in r -> P i0 -> m <= F i0 -> m <= \max_(i <- r | P i) F i.

Proof. by move=> i0r Pi0 ?; apply: leq_trans (leq_bigmax_seq i0 _ _). Qed.

Lemma

bigmaxn_sup_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" ]

[ "Pi0", "apply", "i0", "leq_bigmax_seq", "leq_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmax_sup_seq

:= bigmaxn_sup_seq.

Notation

bigmax_sup_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" ]

[ "bigmaxn_sup_seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigmax_eq_arg (I : finType) i0 (P : pred I) F : P i0 -> \max_(i | P i) F i = F [arg max_(i > i0 | P i) F i].

Proof. move=> Pi0; case: arg_maxnP => //= i Pi maxFi. by apply/eqP; rewrite eqn_leq leq_bigmax_cond // andbT; apply/bigmax_leqP. Qed.

Lemma

bigmax_eq_arg

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", "arg_maxnP", "bigmax_leqP", "eqn_leq", "i0", "leq_bigmax_cond" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_bigmax_cond (I : finType) (A : pred I) F : #|A| > 0 -> {i0 | i0 \in A & \max_(i in A) F i = F i0}.

Proof. case: (pickP A) => [i0 Ai0 _ | ]; last by move/eq_card0->. by exists [arg max_(i > i0 in A) F i]; [case: arg_maxnP | apply: bigmax_eq_arg]. Qed.

Lemma

eq_bigmax_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", "arg_maxnP", "bigmax_eq_arg", "eq_card0", "i0", "last", "pickP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_bigmax (I : finType) F : #|I| > 0 -> {i0 : I | \max_i F i = F i0}.

Proof. by case/(eq_bigmax_cond F) => x _ ->; exists x. Qed.

Lemma

eq_bigmax

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_bigmax_cond", "i0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expn_sum m I r (P : pred I) F : (m ^ (\sum_(i <- r | P i) F i) = \prod_(i <- r | P i) m ^ F i)%N.

Proof. exact: (big_morph _ (expnD m)). Qed.

Lemma

expn_sum

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", "expnD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dvdn_biglcmP (I : finType) (P : pred I) F m : reflect (forall i, P i -> F i %| m) (\big[lcmn/1%N]_(i | P i) F i %| m).

Proof. apply: (iffP idP) => [dvFm i Pi | dvFm]. by rewrite (bigD1 i) // dvdn_lcm in dvFm; case/andP: dvFm. by elim/big_ind: _ => // p q p_m; rewrite dvdn_lcm p_m. Qed.

Lemma

dvdn_biglcmP

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", "big_ind", "dvdn_lcm", "lcmn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

biglcmn_sup (I : finType) i0 (P : pred I) F m : P i0 -> m %| F i0 -> m %| \big[lcmn/1%N]_(i | P i) F i.

Proof. by move=> Pi0 m_Fi0; rewrite (dvdn_trans m_Fi0) // (bigD1 i0) ?dvdn_lcml. Qed.

Lemma

biglcmn_sup

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", "bigD1", "dvdn_lcml", "dvdn_trans", "i0", "lcmn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dvdn_biggcdP (I : finType) (P : pred I) F m : reflect (forall i, P i -> m %| F i) (m %| \big[gcdn/0]_(i | P i) F i).

Proof. apply: (iffP idP) => [dvmF i Pi | dvmF]. by rewrite (bigD1 i) // dvdn_gcd in dvmF; case/andP: dvmF. by elim/big_ind: _ => // p q m_p; rewrite dvdn_gcd m_p. Qed.

Lemma

dvdn_biggcdP

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", "big_ind", "dvdn_gcd", "gcdn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

biggcdn_inf (I : finType) i0 (P : pred I) F m : P i0 -> F i0 %| m -> \big[gcdn/0]_(i | P i) F i %| m.

Proof. by move=> Pi0; apply: dvdn_trans; rewrite (bigD1 i0) ?dvdn_gcdl. Qed.

Lemma

biggcdn_inf

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", "bigD1", "dvdn_gcdl", "dvdn_trans", "gcdn", "i0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fact_prod n : n`! = \prod_(1 <= i < n.+1) i.

Proof. elim: n => [|n IHn] //; first by rewrite big_nil. by apply/esym; rewrite factS IHn // !big_add1 big_nat_recr //= mulnC. Qed.

Lemma

fact_prod

boot

boot/binomial.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]

[ "apply", "big_add1", "big_nat_recr", "big_nil", "factS", "mulnC" ]

More properties of the factorial *

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fact_split n m : m <= n -> n`! = m`! * \prod_(m.+1 <= k < n.+1) k.

Proof. by move=> leq_mn; rewrite !fact_prod -big_cat_nat. Qed.

Lemma

fact_split

boot

boot/binomial.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]

[ "big_cat_nat", "fact_prod" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

logn_fact p n : prime p -> logn p n`! = \sum_(1 <= k < n.+1) n %/ p ^ k.

Proof. move=> p_prime; transitivity (\sum_(1 <= i < n.+1) logn p i). rewrite big_add1; elim: n => /= [|n IHn]; first by rewrite logn1 big_geq. by rewrite big_nat_recr // -IHn /= factS mulnC lognM ?fact_gt0. transitivity (\sum_(1 <= i < n.+1) \sum_(1 <= k < n.+1) (p ^ k %| i)). apply: eq_big_nat => i /andP[i_gt0 l...

Lemma

logn_fact

boot

boot/binomial.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]

[ "apply", "big_add1", "big_geq", "big_mkcond", "big_nat_recr", "big_nat_widen", "divn_count_dvd", "dvdn_leq", "eq_big_nat", "eq_bigl", "eq_bigr", "exchange_big_nat", "factS", "fact_gt0", "leq_trans", "logn", "logn1", "lognM", "logn_count_dvd", "ltnW", "ltn_expl", "mulnC", ...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Wilson p : p > 1 -> prime p = (p %| ((p.-1)`!).+1).

Proof. have dFact n: 0 < n -> (n.-1)`! = \prod_(0 <= i < n | i != 0) i. move=> n_gt0; rewrite -big_filter fact_prod; symmetry; apply: congr_big => //. rewrite /index_iota subn1 -[n]prednK //=; apply/all_filterP. by rewrite all_predC has_pred1 mem_iota. move=> lt1p; have p_gt0 := ltnW lt1p. apply/idP/idP=> [pr_p |...

Theorem

Wilson

boot

boot/binomial.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]

[ "Build", "Euclid_dvdM", "addKn", "addn1", "addnA", "addnBA", "addnCA", "addnS", "all_filterP", "all_predC", "apply", "big1", "bigD1", "bigID", "big_filter", "big_mkord", "big_morph", "big_split", "com_law", "congr_big", "contra_eq_neq", "coprime", "coprime_sym", "dvdn",...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffact_rec n m

:= if m is m'.+1 then n * ffact_rec n.-1 m' else 1.

Fixpoint

ffact_rec

boot

boot/binomial.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]

[]

The falling factorial

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

falling_factorial

:= ffact_rec.

Definition

falling_factorial

boot

boot/binomial.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]

[ "ffact_rec" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"n ^_ m"

:= (falling_factorial n m) (at level 30, right associativity) : nat_scope.

Notation

n ^_ m

boot

boot/binomial.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]

[ "falling_factorial" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffactE : falling_factorial = ffact_rec.

Proof. by []. Qed.

Lemma

ffactE

boot

boot/binomial.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]

[ "falling_factorial", "ffact_rec" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffactn0 n : n ^_ 0 = 1.

Proof. by []. Qed.

Lemma

ffactn0

boot

boot/binomial.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffact0n m : 0 ^_ m = (m == 0).

Proof. by case: m. Qed.

Lemma

ffact0n

boot

boot/binomial.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffactnS n m : n ^_ m.+1 = n * n.-1 ^_ m.

Proof. by []. Qed.

Lemma

ffactnS

boot

boot/binomial.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffactSS n m : n.+1 ^_ m.+1 = n.+1 * n ^_ m.

Proof. by []. Qed.

Lemma

ffactSS

boot

boot/binomial.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffactn1 n : n ^_ 1 = n.

Proof. exact: muln1. Qed.

Lemma

ffactn1

boot

boot/binomial.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]

[ "muln1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffactnSr n m : n ^_ m.+1 = n ^_ m * (n - m).

Proof. elim: n m => [|n IHn] [|m] //=; first by rewrite ffactn1 mul1n. by rewrite !ffactSS IHn mulnA. Qed.

Lemma

ffactnSr

boot

boot/binomial.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]

[ "ffactSS", "ffactn1", "mul1n", "mulnA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffact_prod n m : n ^_ m = \prod_(i < m) (n - i).

Proof. elim: m n => [n | m IH [|n] //]; first by rewrite ffactn0 big_ord0. by rewrite big_ord_recr /= sub0n muln0. by rewrite ffactSS IH big_ord_recl subn0. Qed.

Lemma

ffact_prod

boot

boot/binomial.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]

[ "big_ord0", "big_ord_recl", "big_ord_recr", "ffactSS", "ffactn0", "muln0", "sub0n", "subn0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffact_gt0 n m : (0 < n ^_ m) = (m <= n).

Proof. by elim: n m => [|n IHn] [|m] //=; rewrite ffactSS muln_gt0 IHn. Qed.

Lemma

ffact_gt0

boot

boot/binomial.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]

[ "ffactSS", "muln_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffact_small n m : n < m -> n ^_ m = 0.

Proof. by rewrite ltnNge -ffact_gt0; case: posnP. Qed.

Lemma

ffact_small

boot

boot/binomial.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]

[ "ffact_gt0", "ltnNge", "posnP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffactnn n : n ^_ n = n`!.

Proof. by elim: n => [|n IHn] //; rewrite ffactnS IHn. Qed.

Lemma

ffactnn

boot

boot/binomial.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "bigop", "prime", "finset" ]

[ "ffactnS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d