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mul_fun f g x
:= f x * g x.
Definition
mul_fun
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
scale_fun a (f : U -> V) x
:= a *: f x.
Definition
scale_fun
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
in_alg k : A
:= k%:A.
Definition
in_alg
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 _) : function_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) : function_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) : function_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) : function_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) : function_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) : function_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) : function_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) : function_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
raddfMnat n x : f (n%:R * x) = n%:R * f x.
Proof. by rewrite !mulr_natl raddfMn. Qed.
Lemma
raddfMnat
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_natl", "raddfMn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
raddfZnat n u : h (n%:R *: u) = n%:R *: h u.
Proof. by rewrite !scaler_nat raddfMn. Qed.
Lemma
raddfZnat
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" ]
[ "raddfMn", "scaler_nat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mull_fun_is_nmod_morphism : nmod_morphism (a \*o f).
Proof. by split=> [|x y]; rewrite /= ?raddf0 ?mulr0// raddfD mulrDr. Qed.
Fact
mull_fun_is_nmod_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" ]
[ "mulr0", "mulrDr", "nmod_morphism", "raddf0", "raddfD", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulr_fun_is_nmod_morphism : nmod_morphism (a \o* f).
Proof. by split=> [|x y]; rewrite /= ?raddf0 ?mul0r// raddfD mulrDl. Qed.
Fact
mulr_fun_is_nmod_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" ]
[ "mul0r", "mulrDl", "nmod_morphism", "raddf0", "raddfD", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
raddfN : {morph f : x / - x}.
Proof. exact: raddfN. Qed.
Lemma
raddfN
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
raddfB : {morph f : x y / x - y}.
Proof. exact: raddfB. Qed.
Lemma
raddfB
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
raddf_inj : (forall x, f x = 0 -> x = 0) -> injective f.
Proof. exact: raddf_inj. Qed.
Lemma
raddf_inj
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
raddfMNn n : {morph f : x / x *- n}.
Proof. exact: raddfMNn. Qed.
Lemma
raddfMNn
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
raddfMsign n x : f ((-1) ^+ n * x) = (-1) ^+ n * f x.
Proof. by rewrite !(mulr_sign, =^~ signr_odd) (fun_if f) raddfN. Qed.
Lemma
raddfMsign
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_sign", "raddfN", "signr_odd" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
raddfZsign n u : h ((-1) ^+ n *: u) = (-1) ^+ n *: h u.
Proof. by rewrite !(scaler_sign, =^~ signr_odd) (fun_if h) raddfN. Qed.
Lemma
raddfZsign
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" ]
[ "raddfN", "scaler_sign", "signr_odd" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
monoid_morphism (R S : pzSemiRingType) (f : R -> S) : Prop
:= (f 1 = 1) * {morph f : x y / x * y}%R.
Definition
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" ]
[]
FIXME: remove once PzSemiRing extends Monoid.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'rmorphism' U -> V }"
:= (RMorphism.type U%type V%type) : type_scope.
Notation
{ 'rmorphism' U -> V }
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" ]
[ "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorph0 : f 0 = 0.
Proof. exact: raddf0. Qed.
Lemma
rmorph0
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" ]
[ "raddf0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorphD : {morph f : x y / x + y}.
Proof. exact: raddfD. Qed.
Lemma
rmorphD
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" ]
[ "raddfD" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorphMn n : {morph f : x / x *+ n}.
Proof. exact: raddfMn. Qed.
Lemma
rmorphMn
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" ]
[ "raddfMn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorph_sum I r (P : pred I) E : f (\sum_(i <- r | P i) E i) = \sum_(i <- r | P i) f (E i).
Proof. exact: raddf_sum. Qed.
Lemma
rmorph_sum
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" ]
[ "raddf_sum" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorphism_monoidP : monoid_morphism f.
Proof. exact: monoid_morphism_subproof. Qed.
Lemma
rmorphism_monoidP
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
rmorph1 : f 1 = 1.
Proof. by case: rmorphism_monoidP. Qed.
Lemma
rmorph1
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" ]
[ "rmorphism_monoidP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorphM : {morph f: x y / x * y}.
Proof. by case: rmorphism_monoidP. Qed.
Lemma
rmorphM
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" ]
[ "rmorphism_monoidP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorph_prod I r (P : pred I) E : f (\prod_(i <- r | P i) E i) = \prod_(i <- r | P i) f (E i).
Proof. exact: (big_morph f rmorphM rmorph1). Qed.
Lemma
rmorph_prod
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_morph", "rmorph1", "rmorphM" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorphXn n : {morph f : x / x ^+ n}.
Proof. by elim: n => [|n IHn] x; rewrite ?rmorph1 // !exprS rmorphM IHn. Qed.
Lemma
rmorphXn
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" ]
[ "exprS", "rmorph1", "rmorphM" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorph_nat n : f n%:R = n%:R.
Proof. by rewrite rmorphMn rmorph1. Qed.
Lemma
rmorph_nat
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" ]
[ "rmorph1", "rmorphMn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorph_eq_nat x n : injective f -> (f x == n%:R) = (x == n%:R).
Proof. by move/inj_eq <-; rewrite rmorph_nat. Qed.
Lemma
rmorph_eq_nat
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" ]
[ "inj_eq", "rmorph_nat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorph_eq1 x : injective f -> (f x == 1) = (x == 1).
Proof. exact: rmorph_eq_nat 1%N. Qed.
Lemma
rmorph_eq1
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" ]
[ "rmorph_eq_nat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
can2_monoid_morphism f' : cancel f f' -> cancel f' f -> monoid_morphism f'.
Proof. move=> fK f'K. by split=> [|x y]; apply: (canLR fK); rewrite /= (rmorph1, rmorphM) ?f'K. Qed.
Lemma
can2_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" ]
[ "apply", "fK", "monoid_morphism", "rmorph1", "rmorphM", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
can2_rmorphism f' (cff' : cancel f f')
:= (fun p => (p.2, p.1)) \o (can2_monoid_morphism cff').
Definition
can2_rmorphism
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" ]
[ "can2_monoid_morphism" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorph_pchar (R S : nzSemiRingType) (f : {rmorphism R -> S}) p : p \in pchar R -> p \in pchar S.
Proof. by rewrite !inE -(rmorph_nat f) => /andP[-> /= /eqP->]; rewrite rmorph0. Qed.
Lemma
rmorph_pchar
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" ]
[ "inE", "pchar", "rmorph0", "rmorph_nat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
idfun_is_monoid_morphism : monoid_morphism (@idfun R).
Proof. by []. Qed.
Fact
idfun_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
comp_is_monoid_morphism : monoid_morphism (f \o g).
Proof. by split=> [|x y] /=; rewrite ?rmorph1 ?rmorphM. Qed.
Fact
comp_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", "rmorph1", "rmorphM", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorphN : {morph f : x / - x}.
Proof. exact: raddfN. Qed.
Lemma
rmorphN
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" ]
[ "raddfN" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorphB : {morph f: x y / x - y}.
Proof. exact: raddfB. Qed.
Lemma
rmorphB
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" ]
[ "raddfB" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorphMNn n : {morph f : x / x *- n}.
Proof. exact: raddfMNn. Qed.
Lemma
rmorphMNn
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" ]
[ "raddfMNn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorphMsign n : {morph f : x / (- 1) ^+ n * x}.
Proof. exact: raddfMsign. Qed.
Lemma
rmorphMsign
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" ]
[ "raddfMsign" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorphN1 : f (- 1) = (- 1).
Proof. by rewrite rmorphN rmorph1. Qed.
Lemma
rmorphN1
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" ]
[ "rmorph1", "rmorphN" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorph_sign n : f ((- 1) ^+ n) = (- 1) ^+ n.
Proof. by rewrite rmorphXn /= rmorphN1. Qed.
Lemma
rmorph_sign
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" ]
[ "rmorphN1", "rmorphXn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
in_alg_is_nmod_morphism : nmod_morphism (in_alg A).
Proof. by split; [exact: scale0r | exact: scalerDl]. Qed.
Fact
in_alg_is_nmod_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" ]
[ "in_alg", "nmod_morphism", "scale0r", "scalerDl", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
in_alg_is_monoid_morphism : monoid_morphism (in_alg A).
Proof. by split=> [|x y]; rewrite /= ?scale1r // mulr_algl scalerA. Qed.
Fact
in_alg_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" ]
[ "in_alg", "monoid_morphism", "mulr_algl", "scale1r", "scalerA", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
in_algE a : in_alg A a = a%:A.
Proof. by []. Qed.
Lemma
in_algE
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" ]
[ "in_alg" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
preLaw
:= PreLaw.type.
Definition
preLaw
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" ]
[ "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
semiLaw
:= SemiLaw.type.
Definition
semiLaw
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" ]
[ "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
law
:= Law.type.
Definition
law
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" ]
[ "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_op0v v : (nu \; s) 0 v = 0.
Proof. by rewrite /= rmorph0 op0v. Qed.
Fact
comp_op0v
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" ]
[ "rmorph0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_op1v : (nu \; s) 1 =1 id.
Proof. by move=> v; rewrite /= rmorph1 op1v. Qed.
Fact
comp_op1v
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" ]
[ "id", "rmorph1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_opA a b v : (nu \; s) a ((nu \; s) b v) = (nu \; s) (a * b) v.
Proof. by rewrite /= opA rmorphM. Qed.
Fact
comp_opA
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" ]
[ "opA", "rmorphM" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
compN1op (R : pzRingType) (V : zmodType) (s : law R V) (aR : pzRingType) (nu : {rmorphism aR -> R}) : (nu \; s) (-1) =1 -%R.
Proof. by move=> v; rewrite /= rmorphN1 N1op. Qed.
Fact
compN1op
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" ]
[ "law", "rmorphN1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
scalable_for (R : pzSemiRingType) (U : lSemiModType R) (V : nmodType) (s : R -> V -> V) (f : U -> V)
:= forall a, {morph f : u / a *: u >-> s a u}.
Definition
scalable_for
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
semilinear_for (R : pzSemiRingType) (U : lSemiModType R) (V : nmodType) (s : R -> V -> V) (f : U -> V) : Type
:= scalable_for s f * {morph f : x y / x + y}.
Definition
semilinear_for
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_for" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nmod_morphism_semilinear (R : pzSemiRingType) (U : lSemiModType R) (V : nmodType) (s : Scale.semiLaw R V) (f : U -> V) : semilinear_for s f -> nmod_morphism f.
Proof. by case=> sf Df; split => //; rewrite -[0 in LHS](scale0r 0) sf Scale.op0v. Qed.
Lemma
nmod_morphism_semilinear
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" ]
[ "nmod_morphism", "scale0r", "semiLaw", "semilinear_for", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
additive_semilinear
:= nmod_morphism_semilinear.
Definition
additive_semilinear
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" ]
[ "nmod_morphism_semilinear" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
scalable_semilinear (R : pzSemiRingType) (U : lSemiModType R) (V : nmodType) (s : Scale.preLaw R V) (f : U -> V) : semilinear_for s f -> scalable_for s f.
Proof. by case. Qed.
Lemma
scalable_semilinear
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" ]
[ "preLaw", "scalable_for", "semilinear_for" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
linear_for (R : pzSemiRingType) (U : lSemiModType R) (V : nmodType) (s : R -> V -> V) (f : U -> V)
:= forall a, {morph f : u v / a *: u + v >-> s a u + v}.
Definition
linear_for
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
zmod_morphism_linear (R : pzRingType) (U : lmodType R) V (s : Scale.law R V) (f : U -> V) : linear_for s f -> zmod_morphism f.
Proof. by move=> Lsf x y; rewrite -scaleN1r addrC Lsf Scale.N1op addrC. Qed.
Lemma
zmod_morphism_linear
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" ]
[ "addrC", "law", "linear_for", "scaleN1r", "zmod_morphism" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
scalable_linear (R : pzRingType) (U : lmodType R) V (s : Scale.law R V) (f : U -> V) : linear_for s f -> scalable_for s f.
Proof. by move=> Lsf a v; rewrite -[a *:v](addrK v) (zmod_morphism_linear Lsf) Lsf addrK. Qed.
Lemma
scalable_linear
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" ]
[ "addrK", "law", "linear_for", "scalable_for", "zmod_morphism_linear" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
semilinear_linear (R : pzRingType) (U : lmodType R) V (s : Scale.law R V) (f : U -> V) : linear_for s f -> semilinear_for s f.
Proof. move=> Lsf; split=> [a x|x y]; first exact: (scalable_linear Lsf). have f0: f 0 = 0 by rewrite -[0 in LHS]subr0 (zmod_morphism_linear Lsf) subrr. by rewrite -[y in LHS]opprK -[- y]add0r !(zmod_morphism_linear Lsf) f0 sub0r opprK. Qed.
Lemma
semilinear_linear
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" ]
[ "add0r", "law", "linear_for", "opprK", "scalable_linear", "semilinear_for", "split", "sub0r", "subr0", "subrr", "zmod_morphism_linear" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
scalable f
:= (scalable_for *:%R f).
Notation
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_for" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
semilinear f
:= (semilinear_for *:%R f).
Notation
semilinear
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" ]
[ "semilinear_for" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
semiscalar f
:= (semilinear_for *%R f).
Notation
semiscalar
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" ]
[ "semilinear_for" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
linear f
:= (linear_for *:%R f).
Notation
linear
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" ]
[ "linear_for" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
scalar f
:= (linear_for *%R f).
Notation
scalar
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" ]
[ "linear_for" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mapUV
:= (@Linear.type R U V s).
Notation
mapUV
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" ]
[ "Linear", "type" ]
Support for right-to-left rewriting with the generic linearZ rule.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_class
:= mapUV.
Definition
map_class
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" ]
[ "mapUV" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_at (a : R)
:= mapUV.
Definition
map_at
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" ]
[ "mapUV" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
map_for a s_a
:= MapFor {map_for_map : mapUV; _ : s a = s_a}.
Structure
map_for
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" ]
[ "mapUV" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
unify_map_at a (g : map_at a)
:= MapFor g (erefl (s a)).
Definition
unify_map_at
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" ]
[ "map_at" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
wrapped
:= Wrap {unwrap : mapUV}.
Structure
wrapped
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" ]
[ "mapUV" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'linear' U -> V | s }"
:= (@Linear.type _ U V s) : type_scope.
Notation
{ 'linear' U -> V | s }
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" ]
[ "Linear", "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'linear' U -> V }"
:= {linear U -> V | *:%R} : type_scope.
Notation
{ 'linear' U -> V }
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" ]
[ "linear" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'scalar' U }"
:= {linear U -> _ | *%R} (format "{ 'scalar' U }") : type_scope.
Notation
{ 'scalar' U }
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" ]
[ "linear" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Linear.map_for_map : Linear.map_for >-> Linear.type.
Coercion
Linear
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" ]
[ "map_for", "type" ]
Support for right-to-left rewriting with the generic linearZ rule.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Linear.unify_map_at : Linear.map_at >-> Linear.map_for.
Coercion
Linear
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" ]
[ "map_at", "map_for", "unify_map_at" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Linear.unify_map_at.
Canonical
Linear
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" ]
[ "unify_map_at" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Linear.unwrap : Linear.wrapped >-> Linear.type.
Coercion
Linear
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" ]
[ "type", "wrapped" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Linear.wrap : Linear.map_class >-> Linear.wrapped.
Coercion
Linear
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" ]
[ "map_class", "wrap", "wrapped" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Linear.wrap.
Canonical
Linear
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" ]
[ "wrap" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
linear0 : f 0 = 0.
Proof. exact: raddf0. Qed.
Lemma
linear0
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" ]
[ "raddf0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
linearD : {morph f : x y / x + y}.
Proof. exact: raddfD. Qed.
Lemma
linearD
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" ]
[ "raddfD" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
linearMn n : {morph f : x / x *+ n}.
Proof. exact: raddfMn. Qed.
Lemma
linearMn
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" ]
[ "raddfMn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
linear_sum I r (P : pred I) E : f (\sum_(i <- r | P i) E i) = \sum_(i <- r | P i) f (E i).
Proof. exact: raddf_sum. Qed.
Lemma
linear_sum
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" ]
[ "raddf_sum" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
linearZ_LR : scalable_for s f.
Proof. exact: semi_linear_subproof. Qed.
Lemma
linearZ_LR
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_for" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
semilinearP : semilinear_for s f.
Proof. split; [exact: linearZ_LR | exact: linearD]. Qed.
Lemma
semilinearP
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" ]
[ "linearD", "linearZ_LR", "semilinear_for", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
linearP : linear_for s f.
Proof. by move=> a u v /=; rewrite !semilinearP. Qed.
Lemma
linearP
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" ]
[ "linear_for", "semilinearP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
linearN : {morph f : x / - x}.
Proof. exact: raddfN. Qed.
Lemma
linearN
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" ]
[ "raddfN" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
linearB : {morph f : x y / x - y}.
Proof. exact: raddfB. Qed.
Lemma
linearB
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" ]
[ "raddfB" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
linearMNn n : {morph f : x / x *- n}.
Proof. exact: raddfMNn. Qed.
Lemma
linearMNn
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" ]
[ "raddfMNn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
linearZ (R : pzSemiRingType) (U : lSemiModType R) (V : nmodType) (s : R -> V -> V) (S : pzSemiRingType) (h : Scale.preLaw S V) (c : S) (a : R) (h_c := h c) (f : Linear.map_for U s a h_c) (u : U) : f (a *: u) = h_c (Linear.wrap f u).
Proof. by rewrite linearZ_LR; case: f => f /= ->. Qed.
Lemma
linearZ
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" ]
[ "Linear", "linearZ_LR", "map_for", "preLaw", "wrap" ]
the projections and default instances involved are declared as coercions.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
linearZZ : scalable f.
Proof. exact: linearZ_LR. Qed.
Lemma
linearZZ
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" ]
[ "linearZ_LR", "scalable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
semilinearPZ : semilinear f.
Proof. exact: semilinearP. Qed.
Lemma
semilinearPZ
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" ]
[ "semilinear", "semilinearP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
linearPZ : linear f.
Proof. exact: linearP. Qed.
Lemma
linearPZ
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" ]
[ "linear", "linearP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
can2_scalable f' : cancel f f' -> cancel f' f -> scalable f'.
Proof. by move=> fK f'K a x; apply: (canLR fK); rewrite linearZZ f'K. Qed.
Lemma
can2_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" ]
[ "apply", "fK", "linearZZ", "scalable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d