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Coq-MathComp

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unit R Ux
:= (@Unit R%type _ Ux).
Notation
unit
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ComUnitRing_to_baseFinGroup (R : finComUnitRingType)
:= FinStarMonoid.clone R _.
Coercion
ComUnitRing_to_baseFinGroup
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ComUnitRing_to_finGroup (R : finComUnitRingType)
:= FinGroup.clone R _.
Coercion
ComUnitRing_to_finGroup
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
IntegralDomain_to_baseFinGroup (R : finIdomainType)
:= FinStarMonoid.clone R _.
Coercion
IntegralDomain_to_baseFinGroup
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
IntegralDomain_to_finGroup (R : finIdomainType)
:= FinGroup.clone R _.
Coercion
IntegralDomain_to_finGroup
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Field_to_baseFinGroup (R : finFieldType)
:= FinStarMonoid.clone R _.
Coercion
Field_to_baseFinGroup
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Field_to_finGroup (R : finFieldType)
:= FinGroup.clone R _.
Coercion
Field_to_finGroup
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sat e f
:= match f with | GRing.Bool b => b | t1 == t2 => (GRing.eval e t1 == GRing.eval e t2)%bool | GRing.Unit t => GRing.eval e t \is a GRing.unit | f1 /\ f2 => sat e f1 && sat e f2 | f1 \/ f2 => sat e f1 || sat e f2 | f1 ==> f2 => (sat e f1 ==> sat e f2)%bool | ~ f1 => ~~ sat e f1 | ('ex...
Fixpoint
sat
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "Bool", "eval", "f1", "f2", "set_nth", "unit" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
decidable : GRing.decidable_field_axiom sat.
Proof. move=> e f; elim: f e; try by move=> f1 IH1 f2 IH2 e /=; case IH1; case IH2; constructor; tauto. - by move=> b e; apply: idP. - by move=> t1 t2 e; apply: eqP. - by move=> t e; apply: idP. - by move=> f IH e /=; case: IH; constructor. - by move=> i f IH e; apply: (iffP existsP) => [] [x fx]; exists ...
Lemma
decidable
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "apply", "decidable_field_axiom", "existsP", "f1", "f2", "forallP", "sat" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Lmodule_to_baseFinGroup (R : nzRingType) (M : finLmodType R)
:= FinStarMonoid.clone M _.
Coercion
Lmodule_to_baseFinGroup
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Lmodule_to_finGroup (R : nzRingType) (M : finLmodType R) : finGroupType
:= FinGroup.clone (M : Type) _.
Coercion
Lmodule_to_finGroup
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Lalgebra_to_baseFinGroup (R : nzRingType) (M : finNzLalgType R)
:= FinStarMonoid.clone M _.
Coercion
Lalgebra_to_baseFinGroup
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Lalgebra_to_finGroup (R : nzRingType) (M : finNzLalgType R)
:= FinGroup.clone M _.
Coercion
Lalgebra_to_finGroup
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Algebra_to_baseFinGroup (R : nzRingType) (M : finNzAlgType R)
:= FinStarMonoid.clone M _.
Coercion
Algebra_to_baseFinGroup
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Algebra_to_finGroup (R : nzRingType) (M : finNzAlgType R)
:= FinGroup.clone M _.
Coercion
Algebra_to_finGroup
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
UnitAlgebra_to_baseFinGroup (R : unitRingType) (M : finUnitAlgType R)
:= FinStarMonoid.clone M _.
Coercion
UnitAlgebra_to_baseFinGroup
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
UnitAlgebra_to_finGroup (R : unitRingType) (M : finUnitAlgType R)
:= FinGroup.clone M _.
Coercion
UnitAlgebra_to_finGroup
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "clone" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
zmod1gE
:= zmod1gE.
Definition
zmod1gE
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
zmodVgE
:= zmodVgE.
Definition
zmodVgE
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
zmodMgE
:= zmodMgE.
Definition
zmodMgE
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
zmodXgE
:= zmodXgE.
Definition
zmodXgE
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
zmod_mulgC
:= zmod_mulgC.
Definition
zmod_mulgC
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
zmod_abelian
:= zmod_abelian.
Definition
zmod_abelian
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
val_unit1
:= val_unit1.
Definition
val_unit1
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
val_unitM
:= val_unitM.
Definition
val_unitM
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
val_unitX
:= val_unitX.
Definition
val_unitX
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
val_unitV
:= val_unitV.
Definition
val_unitV
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
unit_actE
:= unit_actE.
Definition
unit_actE
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
finSemiRingType
:= (finNzSemiRingType) (only parsing).
Notation
finSemiRingType
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
finRingType
:= (finNzRingType) (only parsing).
Notation
finRingType
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
finComSemiRingType
:= (finComNzSemiRingType) (only parsing).
Notation
finComSemiRingType
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
finComRingType
:= (finComNzRingType) (only parsing).
Notation
finComRingType
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
finLalgType
:= (finNzLalgType) (only parsing).
Notation
finLalgType
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
finAlgType
:= (finNzAlgType) (only parsing).
Notation
finAlgType
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_finNzRing_gt1 (R : finNzRingType) : 1 < #|R|.
Proof. by rewrite (cardD1 0) (cardD1 1) !inE GRing.oner_neq0. Qed.
Lemma
card_finNzRing_gt1
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "cardD1", "inE", "oner_neq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_finRing_gt1
:= (card_finNzRing_gt1) (only parsing).
Notation
card_finRing_gt1
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "card_finNzRing_gt1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'unit' R }"
:= (unit_of R) (format "{ 'unit' R }") : type_scope.
Notation
{ 'unit' R }
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "unit_of" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''U'"
:= (unit_action _) : action_scope.
Notation
''U'
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "unit_action" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''U'"
:= (unit_groupAction _) : groupAction_scope.
Notation
''U'
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "unit_groupAction" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_finField_unit (F : finFieldType) : #|[set: {unit F}]| = #|F|.-1.
Proof. rewrite -(cardC1 0) cardsT card_sub; apply: eq_card => x. by rewrite GRing.unitfE. Qed.
Lemma
card_finField_unit
algebra
algebra/finalg.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "finset", "fingroup", "morphism", "perm", "action", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "countalg", "GRing.Theory", "FinRing" ]
[ "apply", "cardC1", "card_sub", "cardsT", "eq_card", "unit", "unitfE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ratio
:= mkRatio { frac :> R * R; _ : frac.2 != 0 }.
Inductive
ratio
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "frac" ]
ratios are pairs of R, such that the second member is nonzero
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
denom_ratioP : forall f : ratio, f.2 != 0.
Proof. by case. Qed.
Lemma
denom_ratioP
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "ratio" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ratio0
:= (@mkRatio (0, 1) (oner_neq0 _)).
Definition
ratio0
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "oner_neq0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ratio x y : ratio
:= insubd ratio0 (x, y).
Definition
Ratio
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "insubd", "ratio", "ratio0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
numer_Ratio x y : y != 0 -> (Ratio x y).1 = x.
Proof. by move=> ny0; rewrite /Ratio /insubd insubT. Qed.
Lemma
numer_Ratio
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "Ratio", "insubT", "insubd" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
denom_Ratio x y : y != 0 -> (Ratio x y).2 = y.
Proof. by move=> ny0; rewrite /Ratio /insubd insubT. Qed.
Lemma
denom_Ratio
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "Ratio", "insubT", "insubd" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
numden_Ratio
:= (numer_Ratio, denom_Ratio).
Definition
numden_Ratio
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "denom_Ratio", "numer_Ratio" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ratio_spec (n d : R) : ratio -> R -> R -> Type
:= | RatioNull of d = 0 : Ratio_spec n d ratio0 n 0 | RatioNonNull (d_neq0 : d != 0) : Ratio_spec n d (@mkRatio (n, d) d_neq0) n d.
Variant
Ratio_spec
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "ratio", "ratio0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
RatioP n d : Ratio_spec n d (Ratio n d) n d.
Proof. rewrite /Ratio /insubd; case: insubP=> /= [x /= d_neq0 hx|]. have ->: x = @mkRatio (n, d) d_neq0 by apply: val_inj. by constructor. by rewrite negbK=> /eqP hx; rewrite {2}hx; constructor. Qed.
Lemma
RatioP
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "Ratio", "Ratio_spec", "apply", "insubP", "insubd", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ratio0 x : Ratio x 0 = ratio0.
Proof. by rewrite /Ratio /insubd; case: insubP; rewrite //= eqxx. Qed.
Lemma
Ratio0
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "Ratio", "eqxx", "insubP", "insubd", "ratio0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'ratio' T }"
:= (ratio T) : type_scope.
Notation
{ 'ratio' T }
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "ratio" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'\n_' x"
:= (frac x).1 (at level 8, x at level 2, format "'\n_' x").
Notation
'\n_' x
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "frac" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'\d_' x"
:= (frac x).2 (at level 8, x at level 2, format "'\d_' x").
Notation
'\d_' x
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "frac" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
frac
:= (R * R).
Notation
frac
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
dom
:= (ratio R).
Notation
dom
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "ratio" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
domP
:= denom_ratioP.
Notation
domP
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "denom_ratioP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
equivf_notation x y
:= (\n_x * \d_y == \d_x * \n_y).
Notation
equivf_notation
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[]
We define a relation in ratios
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
equivf x y
:= equivf_notation x y.
Definition
equivf
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "equivf_notation" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
equivfE x y : equivf x y = equivf_notation x y.
Proof. by []. Qed.
Lemma
equivfE
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "equivf", "equivf_notation" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
equivf_refl : reflexive equivf.
Proof. by move=> x; rewrite /equivf mulrC. Qed.
Lemma
equivf_refl
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "equivf", "mulrC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
equivf_sym : symmetric equivf.
Proof. by move=> x y; rewrite /equivf eq_sym; congr (_==_); rewrite mulrC. Qed.
Lemma
equivf_sym
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "eq_sym", "equivf", "mulrC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
equivf_trans : transitive equivf.
Proof. move=> [x Px] [y Py] [z Pz]; rewrite /equivf /= mulrC => /eqP xy /eqP yz. by rewrite -(inj_eq (mulfI Px)) mulrA xy -mulrA yz mulrCA. Qed.
Lemma
equivf_trans
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "Px", "equivf", "inj_eq", "mulfI", "mulrA", "mulrC", "mulrCA" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
equivf_equiv
:= EquivRel equivf equivf_refl equivf_sym equivf_trans.
Canonical
equivf_equiv
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "EquivRel", "equivf", "equivf_refl", "equivf_sym", "equivf_trans" ]
we show that equivf is an equivalence
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
type
:= {eq_quot equivf}.
Definition
type
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "equivf" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
equivf_def (x y : ratio R) : (x == y %[mod type]) = (\n_x * \d_y == \d_x * \n_y).
Proof. by rewrite eqmodE. Qed.
Lemma
equivf_def
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "eqmodE", "ratio", "type" ]
we explain what was the equivalence on the quotient
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
equivf_r x : \n_x * \d_(repr (\pi_type x)) = \d_x * \n_(repr (\pi_type x)).
Proof. by apply/eqP; rewrite -equivf_def reprK. Qed.
Lemma
equivf_r
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "apply", "equivf_def", "n_", "repr", "reprK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
equivf_l x : \n_(repr (\pi_type x)) * \d_x = \d_(repr (\pi_type x)) * \n_x.
Proof. by apply/eqP; rewrite -equivf_def reprK. Qed.
Lemma
equivf_l
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "apply", "equivf_def", "n_", "repr", "reprK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
numer0 x : (\n_x == 0) = (x == (ratio0 R) %[mod_eq equivf]).
Proof. by rewrite eqmodE /= !equivfE // mulr1 mulr0. Qed.
Lemma
numer0
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "eqmodE", "equivf", "equivfE", "mulr0", "mulr1", "ratio0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ratio_numden : forall x, Ratio \n_x \d_x = x.
Proof. case=> [[n d] /= nd]; rewrite /Ratio /insubd; apply: val_inj=> /=. by case: insubP=> //=; rewrite nd. Qed.
Lemma
Ratio_numden
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "Ratio", "apply", "insubP", "insubd", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
tofrac
:= lift_embed type (fun x : R => Ratio x 1).
Definition
tofrac
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "Ratio", "lift_embed", "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
tofrac_pi_morph
:= PiEmbed tofrac.
Canonical
tofrac_pi_morph
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "PiEmbed", "tofrac" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x %:F"
:= (@tofrac x).
Notation
x %:F
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "tofrac" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
addf x y : dom
:= Ratio (\n_x * \d_y + \n_y * \d_x) (\d_x * \d_y).
Definition
addf
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "Ratio", "dom" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
add
:= lift_op2 type addf.
Definition
add
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "addf", "lift_op2", "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_add : {morph \pi : x y / addf x y >-> add x y}.
Proof. move=> x y; unlock add; apply/eqmodP; rewrite /= equivfE /addf /=. rewrite !numden_Ratio ?mulf_neq0 ?domP // mulrDr mulrDl; apply/eqP. symmetry; rewrite (AC (2*2) (3*1*2*4)) (AC (2*2) (3*2*1*4))/=. by rewrite !equivf_l (ACl ((2*3)*(1*4))) (ACl ((2*3)*(4*1)))/=. Qed.
Lemma
pi_add
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "AC", "ACl", "add", "addf", "apply", "domP", "eqmodP", "equivfE", "equivf_l", "mulf_neq0", "mulrDl", "mulrDr", "numden_Ratio", "pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_add_morph
:= PiMorph2 pi_add.
Canonical
pi_add_morph
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "PiMorph2", "pi_add" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
oppf x : dom
:= Ratio (- \n_x) \d_x.
Definition
oppf
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "Ratio", "dom" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
opp
:= lift_op1 type oppf.
Definition
opp
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "lift_op1", "oppf", "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_opp : {morph \pi : x / oppf x >-> opp x}.
Proof. move=> x; unlock opp; apply/eqmodP; rewrite /= /equivf /oppf /=. by rewrite !numden_Ratio ?(domP,mulf_neq0) // mulNr mulrN -equivf_r. Qed.
Lemma
pi_opp
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "apply", "domP", "eqmodP", "equivf", "equivf_r", "mulNr", "mulf_neq0", "mulrN", "numden_Ratio", "opp", "oppf", "pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_opp_morph
:= PiMorph1 pi_opp.
Canonical
pi_opp_morph
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "PiMorph1", "pi_opp" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulf x y : dom
:= Ratio (\n_x * \n_y) (\d_x * \d_y).
Definition
mulf
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "Ratio", "dom" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul
:= lift_op2 type mulf.
Definition
mul
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "lift_op2", "mulf", "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_mul : {morph \pi : x y / mulf x y >-> mul x y}.
Proof. move=> x y; unlock mul; apply/eqmodP=> /=. rewrite equivfE /= /addf /= !numden_Ratio ?mulf_neq0 ?domP //. by rewrite mulrACA !equivf_r mulrACA. Qed.
Lemma
pi_mul
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "addf", "apply", "domP", "eqmodP", "equivfE", "equivf_r", "mul", "mulf", "mulf_neq0", "mulrACA", "numden_Ratio", "pi" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_mul_morph
:= PiMorph2 pi_mul.
Canonical
pi_mul_morph
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "PiMorph2", "pi_mul" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
invf x : dom
:= Ratio \d_x \n_x.
Definition
invf
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "Ratio", "dom" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
inv
:= lift_op1 type invf.
Definition
inv
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "invf", "lift_op1", "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_inv : {morph \pi : x / invf x >-> inv x}.
Proof. move=> x; unlock inv; apply/eqmodP=> /=; rewrite equivfE /invf eq_sym. do 2?case: RatioP=> /= [/eqP|]; rewrite ?mul0r ?mul1r -?equivf_def ?numer0 ?reprK //. by move=> hx /eqP hx'; rewrite hx' eqxx in hx. by move=> /eqP ->; rewrite eqxx. Qed.
Lemma
pi_inv
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "RatioP", "apply", "eq_sym", "eqmodP", "equivfE", "equivf_def", "eqxx", "inv", "invf", "mul0r", "mul1r", "numer0", "pi", "reprK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_inv_morph
:= PiMorph1 pi_inv.
Canonical
pi_inv_morph
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "PiMorph1", "pi_inv" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
addA : associative add.
Proof. elim/quotW=> x; elim/quotW=> y; elim/quotW=> z; rewrite !piE. rewrite /addf /= !numden_Ratio ?mulf_neq0 ?domP // !mulrDl. by rewrite !mulrA !addrA ![_ * _ * \d_x]mulrAC. Qed.
Lemma
addA
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "add", "addf", "addrA", "domP", "mulf_neq0", "mulrA", "mulrAC", "mulrDl", "numden_Ratio", "piE", "quotW" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
addC : commutative add.
Proof. by elim/quotW=> x; elim/quotW=> y; rewrite !piE /addf addrC [\d__ * _]mulrC. Qed.
Lemma
addC
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "add", "addf", "addrC", "mulrC", "piE", "quotW" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
add0_l : left_id 0%:F add.
Proof. elim/quotW=> x; rewrite !piE /addf !numden_Ratio ?oner_eq0 //. by rewrite mul0r mul1r mulr1 add0r Ratio_numden. Qed.
Lemma
add0_l
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "Ratio_numden", "add", "add0r", "addf", "mul0r", "mul1r", "mulr1", "numden_Ratio", "oner_eq0", "piE", "quotW" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
addN_l : left_inverse 0%:F opp add.
Proof. elim/quotW=> x; apply/eqP; rewrite piE /equivf. rewrite /addf /oppf !numden_Ratio ?(oner_eq0, mulf_neq0, domP) //. by rewrite mulr1 mulr0 mulNr addNr. Qed.
Lemma
addN_l
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "add", "addNr", "addf", "apply", "domP", "equivf", "mulNr", "mulf_neq0", "mulr0", "mulr1", "numden_Ratio", "oner_eq0", "opp", "oppf", "piE", "quotW" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulA : associative mul.
Proof. elim/quotW=> x; elim/quotW=> y; elim/quotW=> z; rewrite !piE. by rewrite /mulf !numden_Ratio ?mulf_neq0 ?domP // !mulrA. Qed.
Lemma
mulA
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "domP", "mul", "mulf", "mulf_neq0", "mulrA", "numden_Ratio", "piE", "quotW" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulC : commutative mul.
Proof. elim/quotW=> x; elim/quotW=> y; rewrite !piE /mulf. by rewrite [_ * (\d_x)]mulrC [_ * (\n_x)]mulrC. Qed.
Lemma
mulC
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "mul", "mulf", "mulrC", "piE", "quotW" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul1_l : left_id 1%:F mul.
Proof. elim/quotW=> x; rewrite !piE /mulf. by rewrite !numden_Ratio ?oner_eq0 // !mul1r Ratio_numden. Qed.
Lemma
mul1_l
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "Ratio_numden", "mul", "mul1r", "mulf", "numden_Ratio", "oner_eq0", "piE", "quotW" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_addl : left_distributive mul add.
Proof. elim/quotW=> x; elim/quotW=> y; elim/quotW=> z; apply/eqP. rewrite !piE /equivf /mulf /addf !numden_Ratio ?mulf_neq0 ?domP //; apply/eqP. rewrite !(mulrDr, mulrDl) (AC (3*(2*2)) (4*2*7*((1*3)*(6*5))))/=. by rewrite [X in _ + X](AC (3*(2*2)) (4*6*7*((1*3)*(2*5))))/=. Qed.
Lemma
mul_addl
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "AC", "add", "addf", "apply", "domP", "equivf", "mul", "mulf", "mulf_neq0", "mulrDl", "mulrDr", "numden_Ratio", "piE", "quotW" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nonzero1 : 1%:F != 0%:F :> type.
Proof. by rewrite piE equivfE !numden_Ratio ?mul1r ?oner_eq0. Qed.
Lemma
nonzero1
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "equivfE", "mul1r", "numden_Ratio", "oner_eq0", "piE", "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulV_l : forall a, a != 0%:F -> mul (inv a) a = 1%:F.
Proof. elim/quotW=> x /=; rewrite !piE. rewrite /equivf !numden_Ratio ?oner_eq0 // mulr1 mulr0=> nx0. apply/eqmodP; rewrite /= equivfE. by rewrite !numden_Ratio ?(oner_eq0, mulf_neq0, domP) // !mulr1 mulrC. Qed.
Lemma
mulV_l
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "apply", "domP", "eqmodP", "equivf", "equivfE", "inv", "mul", "mulf_neq0", "mulr0", "mulr1", "mulrC", "numden_Ratio", "oner_eq0", "piE", "quotW" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
inv0 : inv 0%:F = 0%:F.
Proof. rewrite !piE /invf !numden_Ratio ?oner_eq0 // /Ratio /insubd. do 2?case: insubP; rewrite //= ?eqxx ?oner_eq0 // => u _ hu _. by congr \pi; apply: val_inj; rewrite /= hu. Qed.
Lemma
inv0
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "Ratio", "apply", "eqxx", "insubP", "insubd", "inv", "invf", "numden_Ratio", "oner_eq0", "pi", "piE", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'fraction' T }"
:= (FracField.type T).
Notation
{ 'fraction' T }
algebra
algebra/fraction.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "ssrAC", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "FracField" ]
[ "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d