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
sqrtr	:= numfield.Num.Def.sqrtr.	Notation	sqrtr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/ssrnum.v	[ "mathcomp", "orderedzmod", "numdomain", "numfield", "orderedzmod.Num", "numdomain.Num", "numfield.Num", "Num.Theory", "Num.Def" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
com_law	:= ComLaw.type.	Notation	com_law	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "type" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulmA : associative mul.	Proof. exact: opA. Qed.	Lemma	mulmA	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "mul", "opA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulmC : commutative mul.	Proof. exact: opC. Qed.	Lemma	mulmC	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "mul", "opC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulmCA : left_commutative mul.	Proof. by move=> x y z; rewrite !mulmA [_ x _]mulmC. Qed.	Lemma	mulmCA	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "mul", "mulmA", "mulmC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulmAC : right_commutative mul.	Proof. by move=> x y z; rewrite -!mulmA [_ y _]mulmC. Qed.	Lemma	mulmAC	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "mul", "mulmA", "mulmC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulmACA : interchange mul mul.	Proof. by move=> x y z t; rewrite -!mulmA [_ y _]mulmCA. Qed.	Lemma	mulmACA	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "mul", "mulmA", "mulmCA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
opm1 : right_id idm op.	Proof. by move=> x; rewrite opC op1m. Qed.	Lemma	opm1	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "idm", "opC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_law	:= MulLaw.type.	Notation	mul_law	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "type" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
add_law	:= AddLaw.type.	Notation	add_law	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "type" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulC : commutative mul.		Hypothesis	mulC	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "mul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulC_id : left_id one mul -> right_id one mul.	Proof. by move=> mul1x x; rewrite mulC. Qed.	Lemma	mulC_id	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "mul", "mulC", "one" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulC_zero : left_zero zero mul -> right_zero zero mul.	Proof. by move=> mul0x x; rewrite mulC. Qed.	Lemma	mulC_zero	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "mul", "mulC", "zero" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulC_dist : left_distributive mul add -> right_distributive mul add.	Proof. by move=> mul_addl x y z; rewrite !(mulC x). Qed.	Lemma	mulC_dist	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "add", "mul", "mulC", "mul_addl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul1m : left_id idm mul.	Proof. exact: op1m. Qed.	Lemma	mul1m	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "idm", "mul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulm1 : right_id idm mul.	Proof. exact: opm1. Qed.	Lemma	mulm1	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "idm", "mul", "opm1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
iteropE n x : iterop n mul x idm = iter n (mul x) idm.	Proof. by case: n => // n; rewrite iterSr mulm1 iteropS. Qed.	Lemma	iteropE	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "idm", "iter", "iterSr", "iterop", "iteropS", "mul", "mulm1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul0m : left_zero idm mul.	Proof. exact: mul_zerol. Qed.	Lemma	mul0m	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "idm", "mul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulm0 : right_zero idm mul.	Proof. exact: mul_zeror. Qed.	Lemma	mulm0	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "idm", "mul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addmA : associative add.	Proof. exact: mulmA. Qed.	Lemma	addmA	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "add", "mulmA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addmC : commutative add.	Proof. exact: mulmC. Qed.	Lemma	addmC	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "add", "mulmC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addmCA : left_commutative add.	Proof. exact: mulmCA. Qed.	Lemma	addmCA	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "add", "mulmCA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addmAC : right_commutative add.	Proof. exact: mulmAC. Qed.	Lemma	addmAC	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "add", "mulmAC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
add0m : left_id idm add.	Proof. exact: mul1m. Qed.	Lemma	add0m	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "add", "idm", "mul1m" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addm0 : right_id idm add.	Proof. exact: mulm1. Qed.	Lemma	addm0	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "add", "idm", "mulm1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulmDl : left_distributive mul add.	Proof. exact: mul_op_Dl. Qed.	Lemma	mulmDl	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "add", "mul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulmDr : right_distributive mul add.	Proof. exact: mul_op_Dr. Qed.	Lemma	mulmDr	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "add", "mul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
simpm	:= (mulm1, mulm0, mul1m, mul0m, mulmA).	Definition	simpm	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "mul0m", "mul1m", "mulm0", "mulm1", "mulmA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigbody R I	:= BigBody of I & (R -> R -> R) & bool & R.	Variant	bigbody	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[]	the <general_term>, thus allowing for the insertion of coercions.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
applybig {R I} (body : bigbody R I) x	:= let: BigBody _ op b v := body in if b then op v x else x.	Definition	applybig	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "bigbody", "body" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
reducebig R I idx r (body : I -> bigbody R I)	:= foldr (applybig \o body) idx r.	Definition	reducebig	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "applybig", "bigbody", "body", "foldr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigop_unlock	:= Unlockable bigop.unlock.	Canonical	bigop_unlock	boot	boot/bigop.v	[ "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
index_iota m n	:= iota m (n - m).	Definition	index_iota	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "iota" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_index_iota m n i : (i \in index_iota m n) = (m <= i < n).	Proof. rewrite mem_iota; case le_m_i: (m <= i) => //=. by rewrite -leq_subLR subSn // -subn_gt0 -subnDA subnKC // subn_gt0. Qed.	Lemma	mem_index_iota	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "index_iota", "leq_subLR", "mem_iota", "subSn", "subnDA", "subnKC", "subn_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
index_enum_key : unit.	Proof. split. Defined.	Fact	index_enum_key	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "split", "unit" ]	definition below.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
index_enum (T : finType)	:= locked_with index_enum_key (Finite.enum T).	Definition	index_enum	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "enum", "index_enum_key" ]	Qed.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
deprecated_filter_index_enum T P : filter P (index_enum T) = enum P.	Proof. by rewrite [index_enum T]unlock. Qed.	Lemma	deprecated_filter_index_enum	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "enum", "filter", "index_enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_index_enum T i : i \in index_enum T.	Proof. by rewrite [index_enum T]unlock -enumT mem_enum. Qed.	Lemma	mem_index_enum	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "enumT", "index_enum", "mem_enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
index_enum_uniq T : uniq (index_enum T).	Proof. by rewrite [index_enum T]unlock -enumT enum_uniq. Qed.	Lemma	index_enum_uniq	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "enumT", "enum_uniq", "index_enum", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\big [ op / idx ]_ ( i <- r \| P ) F"	:= (bigop idx r (fun i => BigBody i op P%B F)) : big_scope.	Notation	\big [ op / idx ]_ ( i <- r \| P ) F	boot	boot/bigop.v	[ "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 [ op / idx ]_ ( i <- r ) F"	:= (bigop idx r (fun i => BigBody i op true F)) : big_scope.	Notation	\big [ op / idx ]_ ( i <- r ) F	boot	boot/bigop.v	[ "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 [ op / idx ]_ ( m <= i < n \| P ) F"	:= (bigop idx (index_iota m n) (fun i : nat => BigBody i op P%B F)) : big_scope.	Notation	\big [ op / idx ]_ ( m <= i < n \| P ) F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "index_iota", "nat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\big [ op / idx ]_ ( m <= i < n ) F"	:= (bigop idx (index_iota m n) (fun i : nat => BigBody i op true F)) : big_scope.	Notation	\big [ op / idx ]_ ( m <= i < n ) F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "index_iota", "nat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\big [ op / idx ]_ ( i \| P ) F"	:= (bigop idx (index_enum _) (fun i => BigBody i op P%B F)) : big_scope.	Notation	\big [ op / idx ]_ ( i \| P ) F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "index_enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\big [ op / idx ]_ i F"	:= (bigop idx (index_enum _) (fun i => BigBody i op true F)) : big_scope.	Notation	\big [ op / idx ]_ i F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "index_enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\big [ op / idx ]_ ( i : t \| P ) F"	:= (bigop idx (index_enum _) (fun i : t => BigBody i op P%B F)) (only parsing) : big_scope.	Notation	\big [ op / idx ]_ ( i : t \| P ) F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "index_enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\big [ op / idx ]_ ( i : t ) F"	:= (bigop idx (index_enum _) (fun i : t => BigBody i op true F)) (only parsing) : big_scope.	Notation	\big [ op / idx ]_ ( i : t ) F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "index_enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\big [ op / idx ]_ ( i < n \| P ) F"	:= (\big[op/idx]_(i : ordinal n \| P%B) F) : big_scope.	Notation	\big [ op / idx ]_ ( i < n \| P ) F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "ordinal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\big [ op / idx ]_ ( i < n ) F"	:= (\big[op/idx]_(i : ordinal n) F) : big_scope.	Notation	\big [ op / idx ]_ ( i < n ) F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "ordinal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\big [ op / idx ]_ ( i 'in' A \| P ) F"	:= (\big[op/idx]_(i \| (i \in A) && P) F) : big_scope.	Notation	\big [ op / idx ]_ ( i 'in' A \| P ) F	boot	boot/bigop.v	[ "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 [ op / idx ]_ ( i 'in' A ) F"	:= (\big[op/idx]_(i \| i \in A) F) : big_scope.	Notation	\big [ op / idx ]_ ( i 'in' A ) F	boot	boot/bigop.v	[ "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_F	:= (F in \big[_/_]_(i <- _ \| _) F i)%pattern.	Notation	BIG_F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "pattern" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
BIG_P	:= (P in \big[_/_]_(i <- _ \| P i) _)%pattern.	Notation	BIG_P	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "pattern" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"+%N"	:= addn (only parsing).	Notation	+%N	boot	boot/bigop.v	[ "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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i <- r \| P ) F"	:= (\big[+%N/0%N]_(i <- r \| P%B) F%N) : nat_scope.	Notation	\sum_ ( i <- r \| P ) F	boot	boot/bigop.v	[ "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
"\sum_ ( i <- r ) F"	:= (\big[+%N/0%N]_(i <- r) F%N) : nat_scope.	Notation	\sum_ ( i <- r ) F	boot	boot/bigop.v	[ "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
"\sum_ ( m <= i < n \| P ) F"	:= (\big[+%N/0%N]_(m <= i < n \| P%B) F%N) : nat_scope.	Notation	\sum_ ( m <= i < n \| P ) F	boot	boot/bigop.v	[ "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
"\sum_ ( m <= i < n ) F"	:= (\big[+%N/0%N]_(m <= i < n) F%N) : nat_scope.	Notation	\sum_ ( m <= i < n ) F	boot	boot/bigop.v	[ "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
"\sum_ ( i \| P ) F"	:= (\big[+%N/0%N]_(i \| P%B) F%N) : nat_scope.	Notation	\sum_ ( i \| P ) F	boot	boot/bigop.v	[ "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
"\sum_ i F"	:= (\big[+%N/0%N]_i F%N) : nat_scope.	Notation	\sum_ i F	boot	boot/bigop.v	[ "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
"\sum_ ( i : t \| P ) F"	:= (\big[+%N/0%N]_(i : t \| P%B) F%N) (only parsing) : nat_scope.	Notation	\sum_ ( i : t \| P ) F	boot	boot/bigop.v	[ "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
"\sum_ ( i : t ) F"	:= (\big[+%N/0%N]_(i : t) F%N) (only parsing) : nat_scope.	Notation	\sum_ ( i : t ) F	boot	boot/bigop.v	[ "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
"\sum_ ( i < n \| P ) F"	:= (\big[+%N/0%N]_(i < n \| P%B) F%N) : nat_scope.	Notation	\sum_ ( i < n \| P ) F	boot	boot/bigop.v	[ "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
"\sum_ ( i < n ) F"	:= (\big[+%N/0%N]_(i < n) F%N) : nat_scope.	Notation	\sum_ ( i < n ) F	boot	boot/bigop.v	[ "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
"\sum_ ( i 'in' A \| P ) F"	:= (\big[+%N/0%N]_(i in A \| P%B) F%N) : nat_scope.	Notation	\sum_ ( i 'in' A \| P ) F	boot	boot/bigop.v	[ "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
"\sum_ ( i 'in' A ) F"	:= (\big[+%N/0%N]_(i in A) F%N) : nat_scope.	Notation	\sum_ ( i 'in' A ) F	boot	boot/bigop.v	[ "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
"*%N"	:= muln (only parsing).	Notation	*%N	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "muln" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i <- r \| P ) F"	:= (\big[*%N/1%N]_(i <- r \| P%B) F%N) : nat_scope.	Notation	\prod_ ( i <- r \| P ) F	boot	boot/bigop.v	[ "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
"\prod_ ( i <- r ) F"	:= (\big[*%N/1%N]_(i <- r) F%N) : nat_scope.	Notation	\prod_ ( i <- r ) F	boot	boot/bigop.v	[ "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
"\prod_ ( m <= i < n \| P ) F"	:= (\big[*%N/1%N]_(m <= i < n \| P%B) F%N) : nat_scope.	Notation	\prod_ ( m <= i < n \| P ) F	boot	boot/bigop.v	[ "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
"\prod_ ( m <= i < n ) F"	:= (\big[*%N/1%N]_(m <= i < n) F%N) : nat_scope.	Notation	\prod_ ( m <= i < n ) F	boot	boot/bigop.v	[ "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
"\prod_ ( i \| P ) F"	:= (\big[*%N/1%N]_(i \| P%B) F%N) : nat_scope.	Notation	\prod_ ( i \| P ) F	boot	boot/bigop.v	[ "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
"\prod_ i F"	:= (\big[*%N/1%N]_i F%N) : nat_scope.	Notation	\prod_ i F	boot	boot/bigop.v	[ "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
"\prod_ ( i : t \| P ) F"	:= (\big[*%N/1%N]_(i : t \| P%B) F%N) (only parsing) : nat_scope.	Notation	\prod_ ( i : t \| P ) F	boot	boot/bigop.v	[ "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
"\prod_ ( i : t ) F"	:= (\big[*%N/1%N]_(i : t) F%N) (only parsing) : nat_scope.	Notation	\prod_ ( i : t ) F	boot	boot/bigop.v	[ "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
"\prod_ ( i < n \| P ) F"	:= (\big[*%N/1%N]_(i < n \| P%B) F%N) : nat_scope.	Notation	\prod_ ( i < n \| P ) F	boot	boot/bigop.v	[ "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
"\prod_ ( i < n ) F"	:= (\big[*%N/1%N]_(i < n) F%N) : nat_scope.	Notation	\prod_ ( i < n ) F	boot	boot/bigop.v	[ "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
"\prod_ ( i 'in' A \| P ) F"	:= (\big[*%N/1%N]_(i in A \| P%B) F%N) : nat_scope.	Notation	\prod_ ( i 'in' A \| P ) F	boot	boot/bigop.v	[ "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
"\prod_ ( i 'in' A ) F"	:= (\big[*%N/1%N]_(i in A) F%N) : nat_scope.	Notation	\prod_ ( i 'in' A ) F	boot	boot/bigop.v	[ "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
"\max_ ( i <- r \| P ) F"	:= (\big[maxn/0%N]_(i <- r \| P%B) F%N) : nat_scope.	Notation	\max_ ( i <- r \| P ) F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "maxn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\max_ ( i <- r ) F"	:= (\big[maxn/0%N]_(i <- r) F%N) : nat_scope.	Notation	\max_ ( i <- r ) F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "maxn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\max_ ( i \| P ) F"	:= (\big[maxn/0%N]_(i \| P%B) F%N) : nat_scope.	Notation	\max_ ( i \| P ) F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "maxn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\max_ i F"	:= (\big[maxn/0%N]_i F%N) : nat_scope.	Notation	\max_ i F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "maxn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\max_ ( i : I \| P ) F"	:= (\big[maxn/0%N]_(i : I \| P%B) F%N) (only parsing) : nat_scope.	Notation	\max_ ( i : I \| P ) F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "maxn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\max_ ( i : I ) F"	:= (\big[maxn/0%N]_(i : I) F%N) (only parsing) : nat_scope.	Notation	\max_ ( i : I ) F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "maxn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\max_ ( m <= i < n \| P ) F"	:= (\big[maxn/0%N]_(m <= i < n \| P%B) F%N) : nat_scope.	Notation	\max_ ( m <= i < n \| P ) F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "maxn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\max_ ( m <= i < n ) F"	:= (\big[maxn/0%N]_(m <= i < n) F%N) : nat_scope.	Notation	\max_ ( m <= i < n ) F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "maxn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\max_ ( i < n \| P ) F"	:= (\big[maxn/0%N]_(i < n \| P%B) F%N) : nat_scope.	Notation	\max_ ( i < n \| P ) F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "maxn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\max_ ( i < n ) F"	:= (\big[maxn/0%N]_(i < n) F%N) : nat_scope.	Notation	\max_ ( i < n ) F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "maxn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\max_ ( i 'in' A \| P ) F"	:= (\big[maxn/0%N]_(i in A \| P%B) F%N) : nat_scope.	Notation	\max_ ( i 'in' A \| P ) F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "maxn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\max_ ( i 'in' A ) F"	:= (\big[maxn/0%N]_(i in A) F%N) : nat_scope.	Notation	\max_ ( i 'in' A ) F	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "maxn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
big_load R (K K' : R -> Type) idx op I r (P : pred I) F : K (\big[op/idx]_(i <- r \| P i) F i) * K' (\big[op/idx]_(i <- r \| P i) F i) -> K' (\big[op/idx]_(i <- r \| P i) F i).	Proof. by case. Qed.	Lemma	big_load	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[]	Induction loading	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Kid : K id1 id2 id3.		Hypothesis	Kid	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "id1", "id3" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
big_rec3 I r (P : pred I) F1 F2 F3 (K_F : forall i y1 y2 y3, P i -> K y1 y2 y3 -> K (op1 (F1 i) y1) (op2 (F2 i) y2) (op3 (F3 i) y3)) : K (\big[op1/id1]_(i <- r \| P i) F1 i) (\big[op2/id2]_(i <- r \| P i) F2 i) (\big[op3/id3]_(i <- r \| P i) F3 i).	Proof. by rewrite unlock; elim: r => //= i r; case: ifP => //; apply: K_F. Qed.	Lemma	big_rec3	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "F1", "F2", "F3", "K_F", "apply", "id1", "id3" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Kop : forall x1 x2 x3 y1 y2 y3, K x1 x2 x3 -> K y1 y2 y3-> K (op1 x1 y1) (op2 x2 y2) (op3 x3 y3).		Hypothesis	Kop	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
big_ind3 I r (P : pred I) F1 F2 F3 (K_F : forall i, P i -> K (F1 i) (F2 i) (F3 i)) : K (\big[op1/id1]_(i <- r \| P i) F1 i) (\big[op2/id2]_(i <- r \| P i) F2 i) (\big[op3/id3]_(i <- r \| P i) F3 i).	Proof. by apply: big_rec3 => i x1 x2 x3 /K_F; apply: Kop. Qed.	Lemma	big_ind3	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "F1", "F2", "F3", "K_F", "Kop", "apply", "big_rec3", "id1", "id3" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Kid : K id1 id2.		Hypothesis	Kid	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "id1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
big_rec2 I r (P : pred I) F1 F2 (K_F : forall i y1 y2, P i -> K y1 y2 -> K (op1 (F1 i) y1) (op2 (F2 i) y2)) : K (\big[op1/id1]_(i <- r \| P i) F1 i) (\big[op2/id2]_(i <- r \| P i) F2 i).	Proof. by rewrite unlock; elim: r => //= i r; case: ifP => //; apply: K_F. Qed.	Lemma	big_rec2	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "F1", "F2", "K_F", "apply", "id1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Kop : forall x1 x2 y1 y2, K x1 x2 -> K y1 y2 -> K (op1 x1 y1) (op2 x2 y2).		Hypothesis	Kop	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
big_ind2 I r (P : pred I) F1 F2 (K_F : forall i, P i -> K (F1 i) (F2 i)) : K (\big[op1/id1]_(i <- r \| P i) F1 i) (\big[op2/id2]_(i <- r \| P i) F2 i).	Proof. by apply: big_rec2 => i x1 x2 /K_F; apply: Kop. Qed.	Lemma	big_ind2	boot	boot/bigop.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]	[ "F1", "F2", "K_F", "Kop", "apply", "big_rec2", "id1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

sqrtr

:= numfield.Num.Def.sqrtr.

Notation

sqrtr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/ssrnum.v

[ "mathcomp", "orderedzmod", "numdomain", "numfield", "orderedzmod.Num", "numdomain.Num", "numfield.Num", "Num.Theory", "Num.Def" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

com_law

:= ComLaw.type.

Notation

com_law

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulmA : associative mul.

Proof. exact: opA. Qed.

Lemma

mulmA

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "mul", "opA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulmC : commutative mul.

Proof. exact: opC. Qed.

Lemma

mulmC

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "mul", "opC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulmCA : left_commutative mul.

Proof. by move=> x y z; rewrite !mulmA [_ x _]mulmC. Qed.

Lemma

mulmCA

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "mul", "mulmA", "mulmC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulmAC : right_commutative mul.

Proof. by move=> x y z; rewrite -!mulmA [_ y _]mulmC. Qed.

Lemma

mulmAC

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "mul", "mulmA", "mulmC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulmACA : interchange mul mul.

Proof. by move=> x y z t; rewrite -!mulmA [_ y _]mulmCA. Qed.

Lemma

mulmACA

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "mul", "mulmA", "mulmCA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

opm1 : right_id idm op.

Proof. by move=> x; rewrite opC op1m. Qed.

Lemma

opm1

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "idm", "opC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mul_law

:= MulLaw.type.

Notation

mul_law

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

add_law

:= AddLaw.type.

Notation

add_law

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulC : commutative mul.

Hypothesis

mulC

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "mul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulC_id : left_id one mul -> right_id one mul.

Proof. by move=> mul1x x; rewrite mulC. Qed.

Lemma

mulC_id

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "mul", "mulC", "one" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulC_zero : left_zero zero mul -> right_zero zero mul.

Proof. by move=> mul0x x; rewrite mulC. Qed.

Lemma

mulC_zero

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "mul", "mulC", "zero" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulC_dist : left_distributive mul add -> right_distributive mul add.

Proof. by move=> mul_addl x y z; rewrite !(mulC x). Qed.

Lemma

mulC_dist

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "add", "mul", "mulC", "mul_addl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mul1m : left_id idm mul.

Proof. exact: op1m. Qed.

Lemma

mul1m

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "idm", "mul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulm1 : right_id idm mul.

Proof. exact: opm1. Qed.

Lemma

mulm1

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "idm", "mul", "opm1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

iteropE n x : iterop n mul x idm = iter n (mul x) idm.

Proof. by case: n => // n; rewrite iterSr mulm1 iteropS. Qed.

Lemma

iteropE

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "idm", "iter", "iterSr", "iterop", "iteropS", "mul", "mulm1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mul0m : left_zero idm mul.

Proof. exact: mul_zerol. Qed.

Lemma

mul0m

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "idm", "mul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulm0 : right_zero idm mul.

Proof. exact: mul_zeror. Qed.

Lemma

mulm0

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "idm", "mul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addmA : associative add.

Proof. exact: mulmA. Qed.

Lemma

addmA

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "add", "mulmA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addmC : commutative add.

Proof. exact: mulmC. Qed.

Lemma

addmC

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "add", "mulmC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addmCA : left_commutative add.

Proof. exact: mulmCA. Qed.

Lemma

addmCA

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "add", "mulmCA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addmAC : right_commutative add.

Proof. exact: mulmAC. Qed.

Lemma

addmAC

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "add", "mulmAC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

add0m : left_id idm add.

Proof. exact: mul1m. Qed.

Lemma

add0m

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "add", "idm", "mul1m" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addm0 : right_id idm add.

Proof. exact: mulm1. Qed.

Lemma

addm0

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "add", "idm", "mulm1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulmDl : left_distributive mul add.

Proof. exact: mul_op_Dl. Qed.

Lemma

mulmDl

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "add", "mul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulmDr : right_distributive mul add.

Proof. exact: mul_op_Dr. Qed.

Lemma

mulmDr

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "add", "mul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

simpm

:= (mulm1, mulm0, mul1m, mul0m, mulmA).

Definition

simpm

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "mul0m", "mul1m", "mulm0", "mulm1", "mulmA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigbody R I

:= BigBody of I & (R -> R -> R) & bool & R.

Variant

bigbody

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[]

the <general_term>, thus allowing for the insertion of coercions.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

applybig {R I} (body : bigbody R I) x

:= let: BigBody _ op b v := body in if b then op v x else x.

Definition

applybig

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "bigbody", "body" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

reducebig R I idx r (body : I -> bigbody R I)

:= foldr (applybig \o body) idx r.

Definition

reducebig

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "applybig", "bigbody", "body", "foldr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigop_unlock

:= Unlockable bigop.unlock.

Canonical

bigop_unlock

boot

boot/bigop.v

[ "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

index_iota m n

:= iota m (n - m).

Definition

index_iota

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "iota" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_index_iota m n i : (i \in index_iota m n) = (m <= i < n).

Proof. rewrite mem_iota; case le_m_i: (m <= i) => //=. by rewrite -leq_subLR subSn // -subn_gt0 -subnDA subnKC // subn_gt0. Qed.

Lemma

mem_index_iota

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "index_iota", "leq_subLR", "mem_iota", "subSn", "subnDA", "subnKC", "subn_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

index_enum_key : unit.

Proof. split. Defined.

Fact

index_enum_key

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "split", "unit" ]

definition below.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

index_enum (T : finType)

:= locked_with index_enum_key (Finite.enum T).

Definition

index_enum

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "enum", "index_enum_key" ]

Qed.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

deprecated_filter_index_enum T P : filter P (index_enum T) = enum P.

Proof. by rewrite [index_enum T]unlock. Qed.

Lemma

deprecated_filter_index_enum

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "enum", "filter", "index_enum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_index_enum T i : i \in index_enum T.

Proof. by rewrite [index_enum T]unlock -enumT mem_enum. Qed.

Lemma

mem_index_enum

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "enumT", "index_enum", "mem_enum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

index_enum_uniq T : uniq (index_enum T).

Proof. by rewrite [index_enum T]unlock -enumT enum_uniq. Qed.

Lemma

index_enum_uniq

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "enumT", "enum_uniq", "index_enum", "uniq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\big [ op / idx ]_ ( i <- r | P ) F"

:= (bigop idx r (fun i => BigBody i op P%B F)) : big_scope.

Notation

\big [ op / idx ]_ ( i <- r | P ) F

boot

boot/bigop.v

[ "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 [ op / idx ]_ ( i <- r ) F"

:= (bigop idx r (fun i => BigBody i op true F)) : big_scope.

Notation

\big [ op / idx ]_ ( i <- r ) F

boot

boot/bigop.v

[ "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 [ op / idx ]_ ( m <= i < n | P ) F"

:= (bigop idx (index_iota m n) (fun i : nat => BigBody i op P%B F)) : big_scope.

Notation

\big [ op / idx ]_ ( m <= i < n | P ) F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "index_iota", "nat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\big [ op / idx ]_ ( m <= i < n ) F"

:= (bigop idx (index_iota m n) (fun i : nat => BigBody i op true F)) : big_scope.

Notation

\big [ op / idx ]_ ( m <= i < n ) F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "index_iota", "nat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\big [ op / idx ]_ ( i | P ) F"

:= (bigop idx (index_enum _) (fun i => BigBody i op P%B F)) : big_scope.

Notation

\big [ op / idx ]_ ( i | P ) F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "index_enum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\big [ op / idx ]_ i F"

:= (bigop idx (index_enum _) (fun i => BigBody i op true F)) : big_scope.

Notation

\big [ op / idx ]_ i F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "index_enum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\big [ op / idx ]_ ( i : t | P ) F"

:= (bigop idx (index_enum _) (fun i : t => BigBody i op P%B F)) (only parsing) : big_scope.

Notation

\big [ op / idx ]_ ( i : t | P ) F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "index_enum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\big [ op / idx ]_ ( i : t ) F"

:= (bigop idx (index_enum _) (fun i : t => BigBody i op true F)) (only parsing) : big_scope.

Notation

\big [ op / idx ]_ ( i : t ) F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "index_enum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\big [ op / idx ]_ ( i < n | P ) F"

:= (\big[op/idx]_(i : ordinal n | P%B) F) : big_scope.

Notation

\big [ op / idx ]_ ( i < n | P ) F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "ordinal" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\big [ op / idx ]_ ( i < n ) F"

:= (\big[op/idx]_(i : ordinal n) F) : big_scope.

Notation

\big [ op / idx ]_ ( i < n ) F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "ordinal" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\big [ op / idx ]_ ( i 'in' A | P ) F"

:= (\big[op/idx]_(i | (i \in A) && P) F) : big_scope.

Notation

\big [ op / idx ]_ ( i 'in' A | P ) F

boot

boot/bigop.v

[ "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 [ op / idx ]_ ( i 'in' A ) F"

:= (\big[op/idx]_(i | i \in A) F) : big_scope.

Notation

\big [ op / idx ]_ ( i 'in' A ) F

boot

boot/bigop.v

[ "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_F

:= (F in \big[_/_]_(i <- _ | _) F i)%pattern.

Notation

BIG_F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "pattern" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

BIG_P

:= (P in \big[_/_]_(i <- _ | P i) _)%pattern.

Notation

BIG_P

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "pattern" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"+%N"

:= addn (only parsing).

Notation

+%N

boot

boot/bigop.v

[ "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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\sum_ ( i <- r | P ) F"

:= (\big[+%N/0%N]_(i <- r | P%B) F%N) : nat_scope.

Notation

\sum_ ( i <- r | P ) F

boot

boot/bigop.v

[ "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

"\sum_ ( i <- r ) F"

:= (\big[+%N/0%N]_(i <- r) F%N) : nat_scope.

Notation

\sum_ ( i <- r ) F

boot

boot/bigop.v

[ "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

"\sum_ ( m <= i < n | P ) F"

:= (\big[+%N/0%N]_(m <= i < n | P%B) F%N) : nat_scope.

Notation

\sum_ ( m <= i < n | P ) F

boot

boot/bigop.v

[ "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

"\sum_ ( m <= i < n ) F"

:= (\big[+%N/0%N]_(m <= i < n) F%N) : nat_scope.

Notation

\sum_ ( m <= i < n ) F

boot

boot/bigop.v

[ "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

"\sum_ ( i | P ) F"

:= (\big[+%N/0%N]_(i | P%B) F%N) : nat_scope.

Notation

\sum_ ( i | P ) F

boot

boot/bigop.v

[ "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

"\sum_ i F"

:= (\big[+%N/0%N]_i F%N) : nat_scope.

Notation

\sum_ i F

boot

boot/bigop.v

[ "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

"\sum_ ( i : t | P ) F"

:= (\big[+%N/0%N]_(i : t | P%B) F%N) (only parsing) : nat_scope.

Notation

\sum_ ( i : t | P ) F

boot

boot/bigop.v

[ "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

"\sum_ ( i : t ) F"

:= (\big[+%N/0%N]_(i : t) F%N) (only parsing) : nat_scope.

Notation

\sum_ ( i : t ) F

boot

boot/bigop.v

[ "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

"\sum_ ( i < n | P ) F"

:= (\big[+%N/0%N]_(i < n | P%B) F%N) : nat_scope.

Notation

\sum_ ( i < n | P ) F

boot

boot/bigop.v

[ "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

"\sum_ ( i < n ) F"

:= (\big[+%N/0%N]_(i < n) F%N) : nat_scope.

Notation

\sum_ ( i < n ) F

boot

boot/bigop.v

[ "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

"\sum_ ( i 'in' A | P ) F"

:= (\big[+%N/0%N]_(i in A | P%B) F%N) : nat_scope.

Notation

\sum_ ( i 'in' A | P ) F

boot

boot/bigop.v

[ "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

"\sum_ ( i 'in' A ) F"

:= (\big[+%N/0%N]_(i in A) F%N) : nat_scope.

Notation

\sum_ ( i 'in' A ) F

boot

boot/bigop.v

[ "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

"*%N"

:= muln (only parsing).

Notation

*%N

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "muln" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ ( i <- r | P ) F"

:= (\big[*%N/1%N]_(i <- r | P%B) F%N) : nat_scope.

Notation

\prod_ ( i <- r | P ) F

boot

boot/bigop.v

[ "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

"\prod_ ( i <- r ) F"

:= (\big[*%N/1%N]_(i <- r) F%N) : nat_scope.

Notation

\prod_ ( i <- r ) F

boot

boot/bigop.v

[ "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

"\prod_ ( m <= i < n | P ) F"

:= (\big[*%N/1%N]_(m <= i < n | P%B) F%N) : nat_scope.

Notation

\prod_ ( m <= i < n | P ) F

boot

boot/bigop.v

[ "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

"\prod_ ( m <= i < n ) F"

:= (\big[*%N/1%N]_(m <= i < n) F%N) : nat_scope.

Notation

\prod_ ( m <= i < n ) F

boot

boot/bigop.v

[ "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

"\prod_ ( i | P ) F"

:= (\big[*%N/1%N]_(i | P%B) F%N) : nat_scope.

Notation

\prod_ ( i | P ) F

boot

boot/bigop.v

[ "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

"\prod_ i F"

:= (\big[*%N/1%N]_i F%N) : nat_scope.

Notation

\prod_ i F

boot

boot/bigop.v

[ "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

"\prod_ ( i : t | P ) F"

:= (\big[*%N/1%N]_(i : t | P%B) F%N) (only parsing) : nat_scope.

Notation

\prod_ ( i : t | P ) F

boot

boot/bigop.v

[ "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

"\prod_ ( i : t ) F"

:= (\big[*%N/1%N]_(i : t) F%N) (only parsing) : nat_scope.

Notation

\prod_ ( i : t ) F

boot

boot/bigop.v

[ "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

"\prod_ ( i < n | P ) F"

:= (\big[*%N/1%N]_(i < n | P%B) F%N) : nat_scope.

Notation

\prod_ ( i < n | P ) F

boot

boot/bigop.v

[ "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

"\prod_ ( i < n ) F"

:= (\big[*%N/1%N]_(i < n) F%N) : nat_scope.

Notation

\prod_ ( i < n ) F

boot

boot/bigop.v

[ "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

"\prod_ ( i 'in' A | P ) F"

:= (\big[*%N/1%N]_(i in A | P%B) F%N) : nat_scope.

Notation

\prod_ ( i 'in' A | P ) F

boot

boot/bigop.v

[ "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

"\prod_ ( i 'in' A ) F"

:= (\big[*%N/1%N]_(i in A) F%N) : nat_scope.

Notation

\prod_ ( i 'in' A ) F

boot

boot/bigop.v

[ "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

"\max_ ( i <- r | P ) F"

:= (\big[maxn/0%N]_(i <- r | P%B) F%N) : nat_scope.

Notation

\max_ ( i <- r | P ) F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "maxn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\max_ ( i <- r ) F"

:= (\big[maxn/0%N]_(i <- r) F%N) : nat_scope.

Notation

\max_ ( i <- r ) F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "maxn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\max_ ( i | P ) F"

:= (\big[maxn/0%N]_(i | P%B) F%N) : nat_scope.

Notation

\max_ ( i | P ) F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "maxn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\max_ i F"

:= (\big[maxn/0%N]_i F%N) : nat_scope.

Notation

\max_ i F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "maxn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\max_ ( i : I | P ) F"

:= (\big[maxn/0%N]_(i : I | P%B) F%N) (only parsing) : nat_scope.

Notation

\max_ ( i : I | P ) F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "maxn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\max_ ( i : I ) F"

:= (\big[maxn/0%N]_(i : I) F%N) (only parsing) : nat_scope.

Notation

\max_ ( i : I ) F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "maxn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\max_ ( m <= i < n | P ) F"

:= (\big[maxn/0%N]_(m <= i < n | P%B) F%N) : nat_scope.

Notation

\max_ ( m <= i < n | P ) F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "maxn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\max_ ( m <= i < n ) F"

:= (\big[maxn/0%N]_(m <= i < n) F%N) : nat_scope.

Notation

\max_ ( m <= i < n ) F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "maxn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\max_ ( i < n | P ) F"

:= (\big[maxn/0%N]_(i < n | P%B) F%N) : nat_scope.

Notation

\max_ ( i < n | P ) F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "maxn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\max_ ( i < n ) F"

:= (\big[maxn/0%N]_(i < n) F%N) : nat_scope.

Notation

\max_ ( i < n ) F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "maxn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\max_ ( i 'in' A | P ) F"

:= (\big[maxn/0%N]_(i in A | P%B) F%N) : nat_scope.

Notation

\max_ ( i 'in' A | P ) F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "maxn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\max_ ( i 'in' A ) F"

:= (\big[maxn/0%N]_(i in A) F%N) : nat_scope.

Notation

\max_ ( i 'in' A ) F

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "maxn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

big_load R (K K' : R -> Type) idx op I r (P : pred I) F : K (\big[op/idx]_(i <- r | P i) F i) * K' (\big[op/idx]_(i <- r | P i) F i) -> K' (\big[op/idx]_(i <- r | P i) F i).

Proof. by case. Qed.

Lemma

big_load

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[]

Induction loading

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Kid : K id1 id2 id3.

Hypothesis

Kid

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "id1", "id3" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

big_rec3 I r (P : pred I) F1 F2 F3 (K_F : forall i y1 y2 y3, P i -> K y1 y2 y3 -> K (op1 (F1 i) y1) (op2 (F2 i) y2) (op3 (F3 i) y3)) : K (\big[op1/id1]_(i <- r | P i) F1 i) (\big[op2/id2]_(i <- r | P i) F2 i) (\big[op3/id3]_(i <- r | P i) F3 i).

Proof. by rewrite unlock; elim: r => //= i r; case: ifP => //; apply: K_F. Qed.

Lemma

big_rec3

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "F1", "F2", "F3", "K_F", "apply", "id1", "id3" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Kop : forall x1 x2 x3 y1 y2 y3, K x1 x2 x3 -> K y1 y2 y3-> K (op1 x1 y1) (op2 x2 y2) (op3 x3 y3).

Hypothesis

Kop

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

big_ind3 I r (P : pred I) F1 F2 F3 (K_F : forall i, P i -> K (F1 i) (F2 i) (F3 i)) : K (\big[op1/id1]_(i <- r | P i) F1 i) (\big[op2/id2]_(i <- r | P i) F2 i) (\big[op3/id3]_(i <- r | P i) F3 i).

Proof. by apply: big_rec3 => i x1 x2 x3 /K_F; apply: Kop. Qed.

Lemma

big_ind3

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "F1", "F2", "F3", "K_F", "Kop", "apply", "big_rec3", "id1", "id3" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Kid : K id1 id2.

Hypothesis

Kid

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "id1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

big_rec2 I r (P : pred I) F1 F2 (K_F : forall i y1 y2, P i -> K y1 y2 -> K (op1 (F1 i) y1) (op2 (F2 i) y2)) : K (\big[op1/id1]_(i <- r | P i) F1 i) (\big[op2/id2]_(i <- r | P i) F2 i).

Proof. by rewrite unlock; elim: r => //= i r; case: ifP => //; apply: K_F. Qed.

Lemma

big_rec2

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "F1", "F2", "K_F", "apply", "id1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Kop : forall x1 x2 y1 y2, K x1 x2 -> K y1 y2 -> K (op1 x1 y1) (op2 x2 y2).

Hypothesis

Kop

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

big_ind2 I r (P : pred I) F1 F2 (K_F : forall i, P i -> K (F1 i) (F2 i)) : K (\big[op1/id1]_(i <- r | P i) F1 i) (\big[op2/id2]_(i <- r | P i) F2 i).

Proof. by apply: big_rec2 => i x1 x2 /K_F; apply: Kop. Qed.

Lemma

big_ind2

boot

boot/bigop.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "div", "fintype", "tuple", "finfun", "SemiGroup.Exports", "SemiGroup", "SemiGroup.Theory", "Monoid.Exports", "Monoid", "Monoid.Theory" ]

[ "F1", "F2", "K_F", "Kop", "apply", "big_rec2", "id1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d