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
"x *- n"	:= (opp (x *+ n)) : ring_scope.	Notation	x *- n	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "opp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"s `_ i"	:= (seq.nth 0%R s%R i) : ring_scope.	Notation	s `_ i	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "nth", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
support	:= 0.-support.	Notation	support	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"- 1"	:= (opp 1) : ring_scope.	Notation	- 1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "opp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"n %:R"	:= (natmul 1 n) : ring_scope.	Notation	n %:R	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "natmul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'pchar' R ]"	:= (GRing.pchar R) : ring_scope.	Notation	[ 'pchar' R ]	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "pchar" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
has_pchar0 R	:= (GRing.pchar R =i pred0).	Notation	has_pchar0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "pchar" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pFrobenius_aut chRp	:= (pFrobenius_aut chRp).	Notation	pFrobenius_aut	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"a *: m"	:= (scale a m) : ring_scope.	Notation	a *: m	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"k %:A"	:= (scale k 1) : ring_scope.	Notation	k %:A	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\0"	:= (null_fun _) : ring_scope.	Notation	\0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "null_fun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"f \+ g"	:= (add_fun f g) : ring_scope.	Notation	f \+ g	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "add_fun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"f \- g"	:= (sub_fun f g) : ring_scope.	Notation	f \- g	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "sub_fun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\- f"	:= (opp_fun f) : ring_scope.	Notation	\- f	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "opp_fun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"a \*: f"	:= (scale_fun a f) : ring_scope.	Notation	a \*: f	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "scale_fun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x \*o f"	:= (mull_fun x f) : ring_scope.	Notation	x \*o f	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mull_fun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x \o* f"	:= (mulr_fun x f) : ring_scope.	Notation	x \o* f	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulr_fun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"f \* g"	:= (mul_fun f g) : ring_scope.	Notation	f \* g	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mul_fun" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i <- r \| P ) F"	:= (\big[+%R/0%R]_(i <- r \| P%B) F%R) : ring_scope.	Notation	\sum_ ( i <- r \| P ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i <- r ) F"	:= (\big[+%R/0%R]_(i <- r) F%R) : ring_scope.	Notation	\sum_ ( i <- r ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( m <= i < n \| P ) F"	:= (\big[+%R/0%R]_(m <= i < n \| P%B) F%R) : ring_scope.	Notation	\sum_ ( m <= i < n \| P ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( m <= i < n ) F"	:= (\big[+%R/0%R]_(m <= i < n) F%R) : ring_scope.	Notation	\sum_ ( m <= i < n ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i \| P ) F"	:= (\big[+%R/0%R]_(i \| P%B) F%R) : ring_scope.	Notation	\sum_ ( i \| P ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ i F"	:= (\big[+%R/0%R]_i F%R) : ring_scope.	Notation	\sum_ i F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i : t \| P ) F"	:= (\big[+%R/0%R]_(i : t \| P%B) F%R) (only parsing) : ring_scope.	Notation	\sum_ ( i : t \| P ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i : t ) F"	:= (\big[+%R/0%R]_(i : t) F%R) (only parsing) : ring_scope.	Notation	\sum_ ( i : t ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i < n \| P ) F"	:= (\big[+%R/0%R]_(i < n \| P%B) F%R) : ring_scope.	Notation	\sum_ ( i < n \| P ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i < n ) F"	:= (\big[+%R/0%R]_(i < n) F%R) : ring_scope.	Notation	\sum_ ( i < n ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i 'in' A \| P ) F"	:= (\big[+%R/0%R]_(i in A \| P%B) F%R) : ring_scope.	Notation	\sum_ ( i 'in' A \| P ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\sum_ ( i 'in' A ) F"	:= (\big[+%R/0%R]_(i in A) F%R) : ring_scope.	Notation	\sum_ ( i 'in' A ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i <- r \| P ) F"	:= (\big[*%R/1%R]_(i <- r \| P%B) F%R) : ring_scope.	Notation	\prod_ ( i <- r \| P ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i <- r ) F"	:= (\big[*%R/1%R]_(i <- r) F%R) : ring_scope.	Notation	\prod_ ( i <- r ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( m <= i < n \| P ) F"	:= (\big[*%R/1%R]_(m <= i < n \| P%B) F%R) : ring_scope.	Notation	\prod_ ( m <= i < n \| P ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( m <= i < n ) F"	:= (\big[*%R/1%R]_(m <= i < n) F%R) : ring_scope.	Notation	\prod_ ( m <= i < n ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i \| P ) F"	:= (\big[*%R/1%R]_(i \| P%B) F%R) : ring_scope.	Notation	\prod_ ( i \| P ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ i F"	:= (\big[*%R/1%R]_i F%R) : ring_scope.	Notation	\prod_ i F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i : t \| P ) F"	:= (\big[*%R/1%R]_(i : t \| P%B) F%R) (only parsing) : ring_scope.	Notation	\prod_ ( i : t \| P ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i : t ) F"	:= (\big[*%R/1%R]_(i : t) F%R) (only parsing) : ring_scope.	Notation	\prod_ ( i : t ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i < n \| P ) F"	:= (\big[*%R/1%R]_(i < n \| P%B) F%R) : ring_scope.	Notation	\prod_ ( i < n \| P ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i < n ) F"	:= (\big[*%R/1%R]_(i < n) F%R) : ring_scope.	Notation	\prod_ ( i < n ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i 'in' A \| P ) F"	:= (\big[*%R/1%R]_(i in A \| P%B) F%R) : ring_scope.	Notation	\prod_ ( i 'in' A \| P ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i 'in' A ) F"	:= (\big[*%R/1%R]_(i in A) F%R) : ring_scope.	Notation	\prod_ ( i 'in' A ) F	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum_ffunE x : (\sum_(i <- r \| P i) F i) x = \sum_(i <- r \| P i) F i x.	Proof. by elim/big_rec2: _ => // [\|i _ y _ <-]; rewrite !ffunE. Qed.	Lemma	sum_ffunE	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "big_rec2", "ffunE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum_ffun : \sum_(i <- r \| P i) F i = [ffun x => \sum_(i <- r \| P i) F i x].	Proof. by apply/ffunP=> i; rewrite sum_ffunE ffunE. Qed.	Lemma	sum_ffun	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "ffunE", "ffunP", "sum_ffunE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_one : {ffun aT -> R}	:= [ffun => 1].	Definition	ffun_one	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "aT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_mul (f g : {ffun aT -> R})	:= [ffun x => f x * g x].	Definition	ffun_mul	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "aT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_mulA : associative ffun_mul.	Proof. by move=> f1 f2 f3; apply/ffunP=> i; rewrite !ffunE mulrA. Qed.	Fact	ffun_mulA	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "f1", "f2", "f3", "ffunE", "ffunP", "ffun_mul", "mulrA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_mul_1l : left_id ffun_one ffun_mul.	Proof. by move=> f; apply/ffunP=> i; rewrite !ffunE mul1r. Qed.	Fact	ffun_mul_1l	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "ffunE", "ffunP", "ffun_mul", "ffun_one", "mul1r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_mul_1r : right_id ffun_one ffun_mul.	Proof. by move=> f; apply/ffunP=> i; rewrite !ffunE mulr1. Qed.	Fact	ffun_mul_1r	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "ffunE", "ffunP", "ffun_mul", "ffun_one", "mulr1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_mul_addl : left_distributive ffun_mul (@ffun_add _ _).	Proof. by move=> f1 f2 f3; apply/ffunP=> i; rewrite !ffunE mulrDl. Qed.	Fact	ffun_mul_addl	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "f1", "f2", "f3", "ffunE", "ffunP", "ffun_add", "ffun_mul", "mulrDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_mul_addr : right_distributive ffun_mul (@ffun_add _ _).	Proof. by move=> f1 f2 f3; apply/ffunP=> i; rewrite !ffunE mulrDr. Qed.	Fact	ffun_mul_addr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "f1", "f2", "f3", "ffunE", "ffunP", "ffun_add", "ffun_mul", "mulrDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_mul_0l : left_zero (@ffun_zero _ _) ffun_mul.	Proof. by move=> f; apply/ffunP=> i; rewrite !ffunE mul0r. Qed.	Fact	ffun_mul_0l	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "ffunE", "ffunP", "ffun_mul", "ffun_zero", "mul0r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_mul_0r : right_zero (@ffun_zero _ _) ffun_mul.	Proof. by move=> f; apply/ffunP=> i; rewrite !ffunE mulr0. Qed.	Fact	ffun_mul_0r	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "ffunE", "ffunP", "ffun_mul", "ffun_zero", "mulr0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_semiring : pzSemiRingType	:= {ffun aT -> R}.	Definition	ffun_semiring	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "aT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun1_nonzero : ffun_one aT R != 0.	Proof. by apply/eqP => /ffunP/(_ a)/eqP; rewrite !ffunE oner_eq0. Qed.	Fact	ffun1_nonzero	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "aT", "apply", "ffunE", "ffunP", "ffun_one", "oner_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_ring : nzRingType	:= {ffun aT -> R}.	Definition	ffun_ring	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "aT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_mulC : commutative (@ffun_mul aT R).	Proof. by move=> f1 f2; apply/ffunP=> i; rewrite !ffunE mulrC. Qed.	Fact	ffun_mulC	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "aT", "apply", "f1", "f2", "ffunE", "ffunP", "ffun_mul", "mulrC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_scale k f	:= [ffun a => k *: f a].	Definition	ffun_scale	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_scaleA k1 k2 f : ffun_scale k1 (ffun_scale k2 f) = ffun_scale (k1 * k2) f.	Proof. by apply/ffunP=> a; rewrite !ffunE scalerA. Qed.	Fact	ffun_scaleA	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "ffunE", "ffunP", "ffun_scale", "scalerA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_scale0r f : ffun_scale 0 f = 0.	Proof. by apply/ffunP=> a; rewrite !ffunE scale0r. Qed.	Fact	ffun_scale0r	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "ffunE", "ffunP", "ffun_scale", "scale0r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_scale1 : left_id 1 ffun_scale.	Proof. by move=> f; apply/ffunP=> a; rewrite !ffunE scale1r. Qed.	Fact	ffun_scale1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "ffunE", "ffunP", "ffun_scale", "scale1r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_scale_addr k : {morph (ffun_scale k) : x y / x + y}.	Proof. by move=> f g; apply/ffunP=> a; rewrite !ffunE scalerDr. Qed.	Fact	ffun_scale_addr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "ffunE", "ffunP", "ffun_scale", "scalerDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_scale_addl u : {morph (ffun_scale)^~ u : k1 k2 / k1 + k2}.	Proof. by move=> k1 k2; apply/ffunP=> a; rewrite !ffunE scalerDl. Qed.	Fact	ffun_scale_addl	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "ffunE", "ffunP", "ffun_scale", "scalerDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_pair (x y : R1 * R2)	:= (x.1 * y.1, x.2 * y.2).	Definition	mul_pair	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "R1", "R2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_mulA : associative mul_pair.	Proof. by move=> x y z; congr (_, _); apply: mulrA. Qed.	Fact	pair_mulA	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "mul_pair", "mulrA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_mul1l : left_id (1, 1) mul_pair.	Proof. by case=> x1 x2; congr (_, _); apply: mul1r. Qed.	Fact	pair_mul1l	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "mul1r", "mul_pair" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_mul1r : right_id (1, 1) mul_pair.	Proof. by case=> x1 x2; congr (_, _); apply: mulr1. Qed.	Fact	pair_mul1r	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "mul_pair", "mulr1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_mulDl : left_distributive mul_pair +%R.	Proof. by move=> x y z; congr (_, _); apply: mulrDl. Qed.	Fact	pair_mulDl	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "mul_pair", "mulrDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_mulDr : right_distributive mul_pair +%R.	Proof. by move=> x y z; congr (_, _); apply: mulrDr. Qed.	Fact	pair_mulDr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "mul_pair", "mulrDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_mul0r : left_zero 0 mul_pair.	Proof. by move=> x; congr (_, _); apply: mul0r. Qed.	Fact	pair_mul0r	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "mul0r", "mul_pair" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_mulr0 : right_zero 0 mul_pair.	Proof. by move=> x; congr (_, _); apply: mulr0. Qed.	Fact	pair_mulr0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "mul_pair", "mulr0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fst_is_monoid_morphism : monoid_morphism fst.	Proof. by []. Qed.	Fact	fst_is_monoid_morphism	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "monoid_morphism" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
snd_is_monoid_morphism : monoid_morphism snd.	Proof. by []. Qed.	Fact	snd_is_monoid_morphism	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "monoid_morphism" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_one_neq0 : 1 != 0 :> R1 * R2.	Proof. by rewrite xpair_eqE oner_eq0. Qed.	Fact	pair_one_neq0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "R1", "R2", "oner_eq0", "xpair_eqE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_mulC : commutative (@mul_pair R1 R2).	Proof. by move=> x y; congr (_, _); apply: mulrC. Qed.	Fact	pair_mulC	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "R1", "R2", "apply", "mul_pair", "mulrC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scale_pair a (v : V1 * V2) : V1 * V2	:= (a : v.1, a : v.2).	Definition	scale_pair	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_scaleA a b u : scale_pair a (scale_pair b u) = scale_pair (a * b) u.	Proof. by congr (_, _); apply: scalerA. Qed.	Fact	pair_scaleA	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "scale_pair", "scalerA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_scale0 u : scale_pair 0 u = 0.	Proof. by case: u => u1 u2; congr (_, _); apply: scale0r. Qed.	Fact	pair_scale0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "scale0r", "scale_pair" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_scale1 u : scale_pair 1 u = u.	Proof. by case: u => u1 u2; congr (_, _); apply: scale1r. Qed.	Fact	pair_scale1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "scale1r", "scale_pair" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_scaleDr : right_distributive scale_pair +%R.	Proof. by move=> a u v; congr (_, _); apply: scalerDr. Qed.	Fact	pair_scaleDr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "scale_pair", "scalerDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_scaleDl u : {morph scale_pair^~ u: a b / a + b}.	Proof. by move=> a b; congr (_, _); apply: scalerDl. Qed.	Fact	pair_scaleDl	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "scale_pair", "scalerDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fst_is_scalable : scalable fst.	Proof. by []. Qed.	Fact	fst_is_scalable	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "scalable" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
snd_is_scalable : scalable snd.	Proof. by []. Qed.	Fact	snd_is_scalable	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "scalable" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_scaleAl a (u v : A1 * A2) : a : (u v) = (a : u) v.	Proof. by congr (_, _); apply: scalerAl. Qed.	Fact	pair_scaleAl	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "scalerAl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_scaleAr a (u v : A1 * A2) : a : (u v) = u * (a *: v).	Proof. by congr (_, _); apply: scalerAr. Qed.	Fact	pair_scaleAr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "scalerAr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pairMnE (M1 M2 : zmodType) (x : M1 * M2) n : x + n = (x.1 + n, x.2 *+ n).	Proof. by case: x => x y; elim: n => //= n; rewrite !mulrS => ->. Qed.	Lemma	pairMnE	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulrS" ]	/TODO	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
natr0E : 0 = 0%N.	Proof. by []. Qed.	Lemma	natr0E	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
natr1E : 1 = 1%N.	Proof. by []. Qed.	Lemma	natr1E	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
natn n : n%:R = n.	Proof. by elim: n => [//\|n IHn]; rewrite -nat1r IHn. Qed.	Lemma	natn	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "nat1r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
natrDE n m : n + m = (n + m)%N.	Proof. by []. Qed.	Lemma	natrDE	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
natrME n m : n * m = (n * m)%N.	Proof. by []. Qed.	Lemma	natrME	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
natrXE n m : n ^+ m = (n ^ m)%N.	Proof. by []. Qed.	Lemma	natrXE	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
natrE	:= (natr0E, natr1E, natn, natrDE, natrME, natrXE).	Definition	natrE	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "natn", "natr0E", "natr1E", "natrDE", "natrME", "natrXE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
LSemiAlgebra R	:= (NzLSemiAlgebra R) (only parsing).	Notation	LSemiAlgebra	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/ssralg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing", "GRing.Theory", "AllExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sort	:= (NzLSemiAlgebra.sort) (only parsing).	Notation	sort	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/ssralg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing", "GRing.Theory", "AllExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
on R	:= (NzLSemiAlgebra.on R) (only parsing).	Notation	on	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/ssralg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing", "GRing.Theory", "AllExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
copy T U	:= (NzLSemiAlgebra.copy T U) (only parsing).	Notation	copy	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/ssralg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing", "GRing.Theory", "AllExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
SemiAlgebra R	:= (NzSemiAlgebra R) (only parsing).	Notation	SemiAlgebra	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/ssralg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing", "GRing.Theory", "AllExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sort	:= (NzSemiAlgebra.sort) (only parsing).	Notation	sort	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/ssralg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing", "GRing.Theory", "AllExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
on R	:= (NzSemiAlgebra.on R) (only parsing).	Notation	on	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/ssralg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing", "GRing.Theory", "AllExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x *- n"

:= (opp (x *+ n)) : ring_scope.

Notation

x *- n

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "opp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"s `_ i"

:= (seq.nth 0%R s%R i) : ring_scope.

Notation

s `_ i

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "nth", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

support

:= 0.-support.

Notation

support

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"- 1"

:= (opp 1) : ring_scope.

Notation

- 1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "opp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"n %:R"

:= (natmul 1 n) : ring_scope.

Notation

n %:R

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "natmul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'pchar' R ]"

:= (GRing.pchar R) : ring_scope.

Notation

[ 'pchar' R ]

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "pchar" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

has_pchar0 R

:= (GRing.pchar R =i pred0).

Notation

has_pchar0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "pchar" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pFrobenius_aut chRp

:= (pFrobenius_aut chRp).

Notation

pFrobenius_aut

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"a *: m"

:= (scale a m) : ring_scope.

Notation

a *: m

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"k %:A"

:= (scale k 1) : ring_scope.

Notation

k %:A

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\0"

:= (null_fun _) : ring_scope.

Notation

\0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "null_fun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"f \+ g"

:= (add_fun f g) : ring_scope.

Notation

f \+ g

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "add_fun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"f \- g"

:= (sub_fun f g) : ring_scope.

Notation

f \- g

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "sub_fun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\- f"

:= (opp_fun f) : ring_scope.

Notation

\- f

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "opp_fun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"a \*: f"

:= (scale_fun a f) : ring_scope.

Notation

a \*: f

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "scale_fun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x \*o f"

:= (mull_fun x f) : ring_scope.

Notation

x \*o f

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mull_fun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x \o* f"

:= (mulr_fun x f) : ring_scope.

Notation

x \o* f

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulr_fun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"f \* g"

:= (mul_fun f g) : ring_scope.

Notation

f \* g

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mul_fun" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Notation

\sum_ ( i <- r | P ) F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\sum_ ( i <- r ) F"

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

Notation

\sum_ ( i <- r ) F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Notation

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

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Notation

\sum_ ( m <= i < n ) F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\sum_ ( i | P ) F"

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

Notation

\sum_ ( i | P ) F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\sum_ i F"

:= (\big[+%R/0%R]_i F%R) : ring_scope.

Notation

\sum_ i F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Notation

\sum_ ( i : t | P ) F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\sum_ ( i : t ) F"

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

Notation

\sum_ ( i : t ) F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Notation

\sum_ ( i < n | P ) F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\sum_ ( i < n ) F"

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

Notation

\sum_ ( i < n ) F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Notation

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

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Notation

\sum_ ( i 'in' A ) F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Notation

\prod_ ( i <- r | P ) F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ ( i <- r ) F"

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

Notation

\prod_ ( i <- r ) F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Notation

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

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Notation

\prod_ ( m <= i < n ) F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ ( i | P ) F"

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

Notation

\prod_ ( i | P ) F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ i F"

:= (\big[*%R/1%R]_i F%R) : ring_scope.

Notation

\prod_ i F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Notation

\prod_ ( i : t | P ) F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ ( i : t ) F"

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

Notation

\prod_ ( i : t ) F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Notation

\prod_ ( i < n | P ) F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ ( i < n ) F"

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

Notation

\prod_ ( i < n ) F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Notation

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

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Notation

\prod_ ( i 'in' A ) F

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum_ffunE x : (\sum_(i <- r | P i) F i) x = \sum_(i <- r | P i) F i x.

Proof. by elim/big_rec2: _ => // [|i _ y _ <-]; rewrite !ffunE. Qed.

Lemma

sum_ffunE

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "big_rec2", "ffunE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum_ffun : \sum_(i <- r | P i) F i = [ffun x => \sum_(i <- r | P i) F i x].

Proof. by apply/ffunP=> i; rewrite sum_ffunE ffunE. Qed.

Lemma

sum_ffun

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "ffunE", "ffunP", "sum_ffunE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_one : {ffun aT -> R}

:= [ffun => 1].

Definition

ffun_one

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "aT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_mul (f g : {ffun aT -> R})

:= [ffun x => f x * g x].

Definition

ffun_mul

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "aT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_mulA : associative ffun_mul.

Proof. by move=> f1 f2 f3; apply/ffunP=> i; rewrite !ffunE mulrA. Qed.

Fact

ffun_mulA

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "f1", "f2", "f3", "ffunE", "ffunP", "ffun_mul", "mulrA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_mul_1l : left_id ffun_one ffun_mul.

Proof. by move=> f; apply/ffunP=> i; rewrite !ffunE mul1r. Qed.

Fact

ffun_mul_1l

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "ffunE", "ffunP", "ffun_mul", "ffun_one", "mul1r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_mul_1r : right_id ffun_one ffun_mul.

Proof. by move=> f; apply/ffunP=> i; rewrite !ffunE mulr1. Qed.

Fact

ffun_mul_1r

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "ffunE", "ffunP", "ffun_mul", "ffun_one", "mulr1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_mul_addl : left_distributive ffun_mul (@ffun_add _ _).

Proof. by move=> f1 f2 f3; apply/ffunP=> i; rewrite !ffunE mulrDl. Qed.

Fact

ffun_mul_addl

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "f1", "f2", "f3", "ffunE", "ffunP", "ffun_add", "ffun_mul", "mulrDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_mul_addr : right_distributive ffun_mul (@ffun_add _ _).

Proof. by move=> f1 f2 f3; apply/ffunP=> i; rewrite !ffunE mulrDr. Qed.

Fact

ffun_mul_addr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "f1", "f2", "f3", "ffunE", "ffunP", "ffun_add", "ffun_mul", "mulrDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_mul_0l : left_zero (@ffun_zero _ _) ffun_mul.

Proof. by move=> f; apply/ffunP=> i; rewrite !ffunE mul0r. Qed.

Fact

ffun_mul_0l

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "ffunE", "ffunP", "ffun_mul", "ffun_zero", "mul0r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_mul_0r : right_zero (@ffun_zero _ _) ffun_mul.

Proof. by move=> f; apply/ffunP=> i; rewrite !ffunE mulr0. Qed.

Fact

ffun_mul_0r

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "ffunE", "ffunP", "ffun_mul", "ffun_zero", "mulr0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_semiring : pzSemiRingType

:= {ffun aT -> R}.

Definition

ffun_semiring

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "aT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun1_nonzero : ffun_one aT R != 0.

Proof. by apply/eqP => /ffunP/(_ a)/eqP; rewrite !ffunE oner_eq0. Qed.

Fact

ffun1_nonzero

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "aT", "apply", "ffunE", "ffunP", "ffun_one", "oner_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_ring : nzRingType

:= {ffun aT -> R}.

Definition

ffun_ring

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "aT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_mulC : commutative (@ffun_mul aT R).

Proof. by move=> f1 f2; apply/ffunP=> i; rewrite !ffunE mulrC. Qed.

Fact

ffun_mulC

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "aT", "apply", "f1", "f2", "ffunE", "ffunP", "ffun_mul", "mulrC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_scale k f

:= [ffun a => k *: f a].

Definition

ffun_scale

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_scaleA k1 k2 f : ffun_scale k1 (ffun_scale k2 f) = ffun_scale (k1 * k2) f.

Proof. by apply/ffunP=> a; rewrite !ffunE scalerA. Qed.

Fact

ffun_scaleA

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "ffunE", "ffunP", "ffun_scale", "scalerA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_scale0r f : ffun_scale 0 f = 0.

Proof. by apply/ffunP=> a; rewrite !ffunE scale0r. Qed.

Fact

ffun_scale0r

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "ffunE", "ffunP", "ffun_scale", "scale0r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_scale1 : left_id 1 ffun_scale.

Proof. by move=> f; apply/ffunP=> a; rewrite !ffunE scale1r. Qed.

Fact

ffun_scale1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "ffunE", "ffunP", "ffun_scale", "scale1r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_scale_addr k : {morph (ffun_scale k) : x y / x + y}.

Proof. by move=> f g; apply/ffunP=> a; rewrite !ffunE scalerDr. Qed.

Fact

ffun_scale_addr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "ffunE", "ffunP", "ffun_scale", "scalerDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_scale_addl u : {morph (ffun_scale)^~ u : k1 k2 / k1 + k2}.

Proof. by move=> k1 k2; apply/ffunP=> a; rewrite !ffunE scalerDl. Qed.

Fact

ffun_scale_addl

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "ffunE", "ffunP", "ffun_scale", "scalerDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mul_pair (x y : R1 * R2)

:= (x.1 * y.1, x.2 * y.2).

Definition

mul_pair

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "R1", "R2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pair_mulA : associative mul_pair.

Proof. by move=> x y z; congr (_, _); apply: mulrA. Qed.

Fact

pair_mulA

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "mul_pair", "mulrA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pair_mul1l : left_id (1, 1) mul_pair.

Proof. by case=> x1 x2; congr (_, _); apply: mul1r. Qed.

Fact

pair_mul1l

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "mul1r", "mul_pair" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pair_mul1r : right_id (1, 1) mul_pair.

Proof. by case=> x1 x2; congr (_, _); apply: mulr1. Qed.

Fact

pair_mul1r

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "mul_pair", "mulr1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pair_mulDl : left_distributive mul_pair +%R.

Proof. by move=> x y z; congr (_, _); apply: mulrDl. Qed.

Fact

pair_mulDl

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "mul_pair", "mulrDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pair_mulDr : right_distributive mul_pair +%R.

Proof. by move=> x y z; congr (_, _); apply: mulrDr. Qed.

Fact

pair_mulDr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "mul_pair", "mulrDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pair_mul0r : left_zero 0 mul_pair.

Proof. by move=> x; congr (_, _); apply: mul0r. Qed.

Fact

pair_mul0r

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "mul0r", "mul_pair" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pair_mulr0 : right_zero 0 mul_pair.

Proof. by move=> x; congr (_, _); apply: mulr0. Qed.

Fact

pair_mulr0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "mul_pair", "mulr0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fst_is_monoid_morphism : monoid_morphism fst.

Proof. by []. Qed.

Fact

fst_is_monoid_morphism

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "monoid_morphism" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

snd_is_monoid_morphism : monoid_morphism snd.

Proof. by []. Qed.

Fact

snd_is_monoid_morphism

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "monoid_morphism" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pair_one_neq0 : 1 != 0 :> R1 * R2.

Proof. by rewrite xpair_eqE oner_eq0. Qed.

Fact

pair_one_neq0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "R1", "R2", "oner_eq0", "xpair_eqE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pair_mulC : commutative (@mul_pair R1 R2).

Proof. by move=> x y; congr (_, _); apply: mulrC. Qed.

Fact

pair_mulC

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "R1", "R2", "apply", "mul_pair", "mulrC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scale_pair a (v : V1 * V2) : V1 * V2

:= (a *: v.1, a *: v.2).

Definition

scale_pair

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pair_scaleA a b u : scale_pair a (scale_pair b u) = scale_pair (a * b) u.

Proof. by congr (_, _); apply: scalerA. Qed.

Fact

pair_scaleA

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "scale_pair", "scalerA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pair_scale0 u : scale_pair 0 u = 0.

Proof. by case: u => u1 u2; congr (_, _); apply: scale0r. Qed.

Fact

pair_scale0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "scale0r", "scale_pair" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pair_scale1 u : scale_pair 1 u = u.

Proof. by case: u => u1 u2; congr (_, _); apply: scale1r. Qed.

Fact

pair_scale1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "scale1r", "scale_pair" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pair_scaleDr : right_distributive scale_pair +%R.

Proof. by move=> a u v; congr (_, _); apply: scalerDr. Qed.

Fact

pair_scaleDr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "scale_pair", "scalerDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pair_scaleDl u : {morph scale_pair^~ u: a b / a + b}.

Proof. by move=> a b; congr (_, _); apply: scalerDl. Qed.

Fact

pair_scaleDl

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "scale_pair", "scalerDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fst_is_scalable : scalable fst.

Proof. by []. Qed.

Fact

fst_is_scalable

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "scalable" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

snd_is_scalable : scalable snd.

Proof. by []. Qed.

Fact

snd_is_scalable

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "scalable" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pair_scaleAl a (u v : A1 * A2) : a *: (u * v) = (a *: u) * v.

Proof. by congr (_, _); apply: scalerAl. Qed.

Fact

pair_scaleAl

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "scalerAl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pair_scaleAr a (u v : A1 * A2) : a *: (u * v) = u * (a *: v).

Proof. by congr (_, _); apply: scalerAr. Qed.

Fact

pair_scaleAr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "scalerAr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pairMnE (M1 M2 : zmodType) (x : M1 * M2) n : x *+ n = (x.1 *+ n, x.2 *+ n).

Proof. by case: x => x y; elim: n => //= n; rewrite !mulrS => ->. Qed.

Lemma

pairMnE

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulrS" ]

/TODO

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

natr0E : 0 = 0%N.

Proof. by []. Qed.

Lemma

natr0E

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

natr1E : 1 = 1%N.

Proof. by []. Qed.

Lemma

natr1E

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

natn n : n%:R = n.

Proof. by elim: n => [//|n IHn]; rewrite -nat1r IHn. Qed.

Lemma

natn

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "nat1r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

natrDE n m : n + m = (n + m)%N.

Proof. by []. Qed.

Lemma

natrDE

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

natrME n m : n * m = (n * m)%N.

Proof. by []. Qed.

Lemma

natrME

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

natrXE n m : n ^+ m = (n ^ m)%N.

Proof. by []. Qed.

Lemma

natrXE

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

natrE

:= (natr0E, natr1E, natn, natrDE, natrME, natrXE).

Definition

natrE

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "natn", "natr0E", "natr1E", "natrDE", "natrME", "natrXE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

LSemiAlgebra R

:= (NzLSemiAlgebra R) (only parsing).

Notation

LSemiAlgebra

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/ssralg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing", "GRing.Theory", "AllExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sort

:= (NzLSemiAlgebra.sort) (only parsing).

Notation

sort

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/ssralg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing", "GRing.Theory", "AllExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

on R

:= (NzLSemiAlgebra.on R) (only parsing).

Notation

on

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/ssralg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing", "GRing.Theory", "AllExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

copy T U

:= (NzLSemiAlgebra.copy T U) (only parsing).

Notation

copy

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/ssralg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing", "GRing.Theory", "AllExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

SemiAlgebra R

:= (NzSemiAlgebra R) (only parsing).

Notation

SemiAlgebra

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/ssralg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing", "GRing.Theory", "AllExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sort

:= (NzSemiAlgebra.sort) (only parsing).

Notation

sort

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/ssralg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing", "GRing.Theory", "AllExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

on R

:= (NzSemiAlgebra.on R) (only parsing).

Notation

on

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/ssralg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing", "GRing.Theory", "AllExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d