Library mathcomp.algebra.fraction
+ +
+(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.
+ Distributed under the terms of CeCILL-B. *)
+Require Import mathcomp.ssreflect.ssreflect.
+ +
+
+
++ Distributed under the terms of CeCILL-B. *)
+Require Import mathcomp.ssreflect.ssreflect.
+ +
+
+ This file builds the field of fraction of any integral domain.
+ The main result of this file is the existence of the field
+ and of the tofrac function which is a injective ring morphism from R
+ to its fraction field {fraction R}
+
+
+
+
+Set Implicit Arguments.
+ +
+Import GRing.Theory.
+Local Open Scope ring_scope.
+Local Open Scope quotient_scope.
+ +
+Reserved Notation "{ 'ratio' T }" (at level 0, format "{ 'ratio' T }").
+Reserved Notation "{ 'fraction' T }" (at level 0, format "{ 'fraction' T }").
+Reserved Notation "x %:F" (at level 2, format "x %:F").
+ +
+Section FracDomain.
+Variable R : ringType.
+ +
+
+
++Set Implicit Arguments.
+ +
+Import GRing.Theory.
+Local Open Scope ring_scope.
+Local Open Scope quotient_scope.
+ +
+Reserved Notation "{ 'ratio' T }" (at level 0, format "{ 'ratio' T }").
+Reserved Notation "{ 'fraction' T }" (at level 0, format "{ 'fraction' T }").
+Reserved Notation "x %:F" (at level 2, format "x %:F").
+ +
+Section FracDomain.
+Variable R : ringType.
+ +
+
+ ratios are pairs of R, such that the second member is nonzero
+
+
+Inductive ratio := mkRatio { frac :> R × R; _ : frac.2 != 0 }.
+Definition ratio_of of phant R := ratio.
+ +
+Canonical ratio_subType := Eval hnf in [subType for frac].
+Canonical ratio_of_subType := Eval hnf in [subType of {ratio R}].
+Definition ratio_EqMixin := [eqMixin of ratio by <:].
+Canonical ratio_eqType := EqType ratio ratio_EqMixin.
+Canonical ratio_of_eqType := Eval hnf in [eqType of {ratio R}].
+Definition ratio_ChoiceMixin := [choiceMixin of ratio by <:].
+Canonical ratio_choiceType := ChoiceType ratio ratio_ChoiceMixin.
+Canonical ratio_of_choiceType := Eval hnf in [choiceType of {ratio R}].
+ +
+Lemma denom_ratioP : ∀ f : ratio, f.2 != 0.
+ +
+Definition ratio0 := (@mkRatio (0, 1) (oner_neq0 _)).
+Definition Ratio x y : {ratio R} := insubd ratio0 (x, y).
+ +
+Lemma numer_Ratio x y : y != 0 → (Ratio x y).1 = x.
+ +
+Lemma denom_Ratio x y : y != 0 → (Ratio x y).2 = y.
+ +
+Definition numden_Ratio := (numer_Ratio, denom_Ratio).
+ +
+CoInductive Ratio_spec (n d : R) : {ratio R} → 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.
+ +
+Lemma RatioP n d : Ratio_spec n d (Ratio n d) n d.
+ +
+Lemma Ratio0 x : Ratio x 0 = ratio0.
+ +
+End FracDomain.
+ +
+Notation "{ 'ratio' T }" := (ratio_of (Phant T)).
+Identity Coercion type_fracdomain_of : ratio_of >-> ratio.
+ +
+Notation "'\n_' x" := (frac x).1
+ (at level 8, x at level 2, format "'\n_' x").
+Notation "'\d_' x" := (frac x).2
+ (at level 8, x at level 2, format "'\d_' x").
+ +
+Module FracField.
+Section FracField.
+ +
+Variable R : idomainType.
+ +
+ +
+Implicit Types x y z : dom.
+ +
+
+
++Definition ratio_of of phant R := ratio.
+ +
+Canonical ratio_subType := Eval hnf in [subType for frac].
+Canonical ratio_of_subType := Eval hnf in [subType of {ratio R}].
+Definition ratio_EqMixin := [eqMixin of ratio by <:].
+Canonical ratio_eqType := EqType ratio ratio_EqMixin.
+Canonical ratio_of_eqType := Eval hnf in [eqType of {ratio R}].
+Definition ratio_ChoiceMixin := [choiceMixin of ratio by <:].
+Canonical ratio_choiceType := ChoiceType ratio ratio_ChoiceMixin.
+Canonical ratio_of_choiceType := Eval hnf in [choiceType of {ratio R}].
+ +
+Lemma denom_ratioP : ∀ f : ratio, f.2 != 0.
+ +
+Definition ratio0 := (@mkRatio (0, 1) (oner_neq0 _)).
+Definition Ratio x y : {ratio R} := insubd ratio0 (x, y).
+ +
+Lemma numer_Ratio x y : y != 0 → (Ratio x y).1 = x.
+ +
+Lemma denom_Ratio x y : y != 0 → (Ratio x y).2 = y.
+ +
+Definition numden_Ratio := (numer_Ratio, denom_Ratio).
+ +
+CoInductive Ratio_spec (n d : R) : {ratio R} → 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.
+ +
+Lemma RatioP n d : Ratio_spec n d (Ratio n d) n d.
+ +
+Lemma Ratio0 x : Ratio x 0 = ratio0.
+ +
+End FracDomain.
+ +
+Notation "{ 'ratio' T }" := (ratio_of (Phant T)).
+Identity Coercion type_fracdomain_of : ratio_of >-> ratio.
+ +
+Notation "'\n_' x" := (frac x).1
+ (at level 8, x at level 2, format "'\n_' x").
+Notation "'\d_' x" := (frac x).2
+ (at level 8, x at level 2, format "'\d_' x").
+ +
+Module FracField.
+Section FracField.
+ +
+Variable R : idomainType.
+ +
+ +
+Implicit Types x y z : dom.
+ +
+
+ We define a relation in ratios
+
+
+Definition equivf x y := equivf_notation x y.
+ +
+Lemma equivfE x y : equivf x y = equivf_notation x y.
+ +
+Lemma equivf_refl : reflexive equivf.
+ +
+Lemma equivf_sym : symmetric equivf.
+ +
+Lemma equivf_trans : transitive equivf.
+ +
+
+
++ +
+Lemma equivfE x y : equivf x y = equivf_notation x y.
+ +
+Lemma equivf_refl : reflexive equivf.
+ +
+Lemma equivf_sym : symmetric equivf.
+ +
+Lemma equivf_trans : transitive equivf.
+ +
+
+ we show that equivf is an equivalence
+
+
+Canonical equivf_equiv := EquivRel equivf equivf_refl equivf_sym equivf_trans.
+ +
+Definition type := {eq_quot equivf}.
+Definition type_of of phant R := type.
+Notation "{ 'fraction' T }" := (type_of (Phant T)).
+ +
+
+
++ +
+Definition type := {eq_quot equivf}.
+Definition type_of of phant R := type.
+Notation "{ 'fraction' T }" := (type_of (Phant T)).
+ +
+
+ we recover some structure for the quotient
+
+
+Canonical frac_quotType := [quotType of type].
+Canonical frac_eqType := [eqType of type].
+Canonical frac_choiceType := [choiceType of type].
+Canonical frac_eqQuotType := [eqQuotType equivf of type].
+ +
+Canonical frac_of_quotType := [quotType of {fraction R}].
+Canonical frac_of_eqType := [eqType of {fraction R}].
+Canonical frac_of_choiceType := [choiceType of {fraction R}].
+Canonical frac_of_eqQuotType := [eqQuotType equivf of {fraction R}].
+ +
+
+
++Canonical frac_eqType := [eqType of type].
+Canonical frac_choiceType := [choiceType of type].
+Canonical frac_eqQuotType := [eqQuotType equivf of type].
+ +
+Canonical frac_of_quotType := [quotType of {fraction R}].
+Canonical frac_of_eqType := [eqType of {fraction R}].
+Canonical frac_of_choiceType := [choiceType of {fraction R}].
+Canonical frac_of_eqQuotType := [eqQuotType equivf of {fraction R}].
+ +
+
+ we explain what was the equivalence on the quotient
+
+
+Lemma equivf_def (x y : ratio R) : x == y %[mod type]
+ = (\n_x × \d_y == \d_x × \n_y).
+ +
+Lemma equivf_r x : \n_x × \d_(repr (\pi_type x)) = \d_x × \n_(repr (\pi_type x)).
+ +
+Lemma equivf_l x : \n_(repr (\pi_type x)) × \d_x = \d_(repr (\pi_type x)) × \n_x.
+ +
+Lemma numer0 x : (\n_x == 0) = (x == (ratio0 R) %[mod_eq equivf]).
+ +
+Lemma Ratio_numden : ∀ x, Ratio \n_x \d_x = x.
+ +
+Definition tofrac := lift_embed {fraction R} (fun x : R ⇒ Ratio x 1).
+Canonical tofrac_pi_morph := PiEmbed tofrac.
+ +
+Notation "x %:F" := (@tofrac x).
+ +
+Implicit Types a b c : type.
+ +
+Definition addf x y : dom := Ratio (\n_x × \d_y + \n_y × \d_x) (\d_x × \d_y).
+Definition add := lift_op2 {fraction R} addf.
+ +
+Lemma pi_add : {morph \pi : x y / addf x y >-> add x y}.
+Canonical pi_add_morph := PiMorph2 pi_add.
+ +
+Definition oppf x : dom := Ratio (- \n_x) \d_x.
+Definition opp := lift_op1 {fraction R} oppf.
+Lemma pi_opp : {morph \pi : x / oppf x >-> opp x}.
+Canonical pi_opp_morph := PiMorph1 pi_opp.
+ +
+Definition mulf x y : dom := Ratio (\n_x × \n_y) (\d_x × \d_y).
+Definition mul := lift_op2 {fraction R} mulf.
+ +
+Lemma pi_mul : {morph \pi : x y / mulf x y >-> mul x y}.
+Canonical pi_mul_morph := PiMorph2 pi_mul.
+ +
+Definition invf x : dom := Ratio \d_x \n_x.
+Definition inv := lift_op1 {fraction R} invf.
+ +
+Lemma pi_inv : {morph \pi : x / invf x >-> inv x}.
+Canonical pi_inv_morph := PiMorph1 pi_inv.
+ +
+Lemma addA : associative add.
+ +
+Lemma addC : commutative add.
+ +
+Lemma add0_l : left_id 0%:F add.
+ +
+Lemma addN_l : left_inverse 0%:F opp add.
+ +
+
+
++ = (\n_x × \d_y == \d_x × \n_y).
+ +
+Lemma equivf_r x : \n_x × \d_(repr (\pi_type x)) = \d_x × \n_(repr (\pi_type x)).
+ +
+Lemma equivf_l x : \n_(repr (\pi_type x)) × \d_x = \d_(repr (\pi_type x)) × \n_x.
+ +
+Lemma numer0 x : (\n_x == 0) = (x == (ratio0 R) %[mod_eq equivf]).
+ +
+Lemma Ratio_numden : ∀ x, Ratio \n_x \d_x = x.
+ +
+Definition tofrac := lift_embed {fraction R} (fun x : R ⇒ Ratio x 1).
+Canonical tofrac_pi_morph := PiEmbed tofrac.
+ +
+Notation "x %:F" := (@tofrac x).
+ +
+Implicit Types a b c : type.
+ +
+Definition addf x y : dom := Ratio (\n_x × \d_y + \n_y × \d_x) (\d_x × \d_y).
+Definition add := lift_op2 {fraction R} addf.
+ +
+Lemma pi_add : {morph \pi : x y / addf x y >-> add x y}.
+Canonical pi_add_morph := PiMorph2 pi_add.
+ +
+Definition oppf x : dom := Ratio (- \n_x) \d_x.
+Definition opp := lift_op1 {fraction R} oppf.
+Lemma pi_opp : {morph \pi : x / oppf x >-> opp x}.
+Canonical pi_opp_morph := PiMorph1 pi_opp.
+ +
+Definition mulf x y : dom := Ratio (\n_x × \n_y) (\d_x × \d_y).
+Definition mul := lift_op2 {fraction R} mulf.
+ +
+Lemma pi_mul : {morph \pi : x y / mulf x y >-> mul x y}.
+Canonical pi_mul_morph := PiMorph2 pi_mul.
+ +
+Definition invf x : dom := Ratio \d_x \n_x.
+Definition inv := lift_op1 {fraction R} invf.
+ +
+Lemma pi_inv : {morph \pi : x / invf x >-> inv x}.
+Canonical pi_inv_morph := PiMorph1 pi_inv.
+ +
+Lemma addA : associative add.
+ +
+Lemma addC : commutative add.
+ +
+Lemma add0_l : left_id 0%:F add.
+ +
+Lemma addN_l : left_inverse 0%:F opp add.
+ +
+
+ fracions form an abelian group
+
+
+Definition frac_zmodMixin := ZmodMixin addA addC add0_l addN_l.
+Canonical frac_zmodType := Eval hnf in ZmodType type frac_zmodMixin.
+ +
+Lemma mulA : associative mul.
+ +
+Lemma mulC : commutative mul.
+ +
+Lemma mul1_l : left_id 1%:F mul.
+ +
+Lemma mul_addl : left_distributive mul add.
+ +
+Lemma nonzero1 : 1%:F != 0%:F :> type.
+ +
+
+
++Canonical frac_zmodType := Eval hnf in ZmodType type frac_zmodMixin.
+ +
+Lemma mulA : associative mul.
+ +
+Lemma mulC : commutative mul.
+ +
+Lemma mul1_l : left_id 1%:F mul.
+ +
+Lemma mul_addl : left_distributive mul add.
+ +
+Lemma nonzero1 : 1%:F != 0%:F :> type.
+ +
+
+ fracions form a commutative ring
+
+
+Definition frac_comRingMixin := ComRingMixin mulA mulC mul1_l mul_addl nonzero1.
+Canonical frac_ringType := Eval hnf in RingType type frac_comRingMixin.
+Canonical frac_comRingType := Eval hnf in ComRingType type mulC.
+ +
+Lemma mulV_l : ∀ a, a != 0%:F → mul (inv a) a = 1%:F.
+ +
+Lemma inv0 : inv 0%:F = 0%:F.
+ +
+
+
++Canonical frac_ringType := Eval hnf in RingType type frac_comRingMixin.
+Canonical frac_comRingType := Eval hnf in ComRingType type mulC.
+ +
+Lemma mulV_l : ∀ a, a != 0%:F → mul (inv a) a = 1%:F.
+ +
+Lemma inv0 : inv 0%:F = 0%:F.
+ +
+
+ fractions form a ring with explicit unit
+
+
+Definition RatFieldUnitMixin := FieldUnitMixin mulV_l inv0.
+Canonical frac_unitRingType := Eval hnf in UnitRingType type RatFieldUnitMixin.
+Canonical frac_comUnitRingType := [comUnitRingType of type].
+ +
+Lemma field_axiom : GRing.Field.mixin_of frac_unitRingType.
+ +
+
+
++Canonical frac_unitRingType := Eval hnf in UnitRingType type RatFieldUnitMixin.
+Canonical frac_comUnitRingType := [comUnitRingType of type].
+ +
+Lemma field_axiom : GRing.Field.mixin_of frac_unitRingType.
+ +
+
+ fractions form a field
+
+
+Definition RatFieldIdomainMixin := (FieldIdomainMixin field_axiom).
+Canonical frac_idomainType :=
+ Eval hnf in IdomainType type (FieldIdomainMixin field_axiom).
+Canonical frac_fieldType := FieldType type field_axiom.
+ +
+End FracField.
+End FracField.
+ +
+Notation "{ 'fraction' T }" := (FracField.type_of (Phant T)).
+Notation equivf := (@FracField.equivf _).
+Hint Resolve denom_ratioP.
+ +
+Section Canonicals.
+ +
+Variable R : idomainType.
+ +
+
+
++Canonical frac_idomainType :=
+ Eval hnf in IdomainType type (FieldIdomainMixin field_axiom).
+Canonical frac_fieldType := FieldType type field_axiom.
+ +
+End FracField.
+End FracField.
+ +
+Notation "{ 'fraction' T }" := (FracField.type_of (Phant T)).
+Notation equivf := (@FracField.equivf _).
+Hint Resolve denom_ratioP.
+ +
+Section Canonicals.
+ +
+Variable R : idomainType.
+ +
+
+ reexporting the structures
+
+
+Canonical FracField.frac_quotType.
+Canonical FracField.frac_eqType.
+Canonical FracField.frac_choiceType.
+Canonical FracField.frac_zmodType.
+Canonical FracField.frac_ringType.
+Canonical FracField.frac_comRingType.
+Canonical FracField.frac_unitRingType.
+Canonical FracField.frac_comUnitRingType.
+Canonical FracField.frac_idomainType.
+Canonical FracField.frac_fieldType.
+Canonical FracField.tofrac_pi_morph.
+Canonical frac_of_quotType := Eval hnf in [quotType of {fraction R}].
+Canonical frac_of_eqType := Eval hnf in [eqType of {fraction R}].
+Canonical frac_of_choiceType := Eval hnf in [choiceType of {fraction R}].
+Canonical frac_of_zmodType := Eval hnf in [zmodType of {fraction R}].
+Canonical frac_of_ringType := Eval hnf in [ringType of {fraction R}].
+Canonical frac_of_comRingType := Eval hnf in [comRingType of {fraction R}].
+Canonical frac_of_unitRingType := Eval hnf in [unitRingType of {fraction R}].
+Canonical frac_of_comUnitRingType := Eval hnf in [comUnitRingType of {fraction R}].
+Canonical frac_of_idomainType := Eval hnf in [idomainType of {fraction R}].
+Canonical frac_of_fieldType := Eval hnf in [fieldType of {fraction R}].
+ +
+End Canonicals.
+ +
+Section FracFieldTheory.
+ +
+Import FracField.
+ +
+Variable R : idomainType.
+ +
+Lemma Ratio_numden (x : {ratio R}) : Ratio \n_x \d_x = x.
+ +
+
+
++Canonical FracField.frac_eqType.
+Canonical FracField.frac_choiceType.
+Canonical FracField.frac_zmodType.
+Canonical FracField.frac_ringType.
+Canonical FracField.frac_comRingType.
+Canonical FracField.frac_unitRingType.
+Canonical FracField.frac_comUnitRingType.
+Canonical FracField.frac_idomainType.
+Canonical FracField.frac_fieldType.
+Canonical FracField.tofrac_pi_morph.
+Canonical frac_of_quotType := Eval hnf in [quotType of {fraction R}].
+Canonical frac_of_eqType := Eval hnf in [eqType of {fraction R}].
+Canonical frac_of_choiceType := Eval hnf in [choiceType of {fraction R}].
+Canonical frac_of_zmodType := Eval hnf in [zmodType of {fraction R}].
+Canonical frac_of_ringType := Eval hnf in [ringType of {fraction R}].
+Canonical frac_of_comRingType := Eval hnf in [comRingType of {fraction R}].
+Canonical frac_of_unitRingType := Eval hnf in [unitRingType of {fraction R}].
+Canonical frac_of_comUnitRingType := Eval hnf in [comUnitRingType of {fraction R}].
+Canonical frac_of_idomainType := Eval hnf in [idomainType of {fraction R}].
+Canonical frac_of_fieldType := Eval hnf in [fieldType of {fraction R}].
+ +
+End Canonicals.
+ +
+Section FracFieldTheory.
+ +
+Import FracField.
+ +
+Variable R : idomainType.
+ +
+Lemma Ratio_numden (x : {ratio R}) : Ratio \n_x \d_x = x.
+ +
+
+ exporting the embeding from R to {fraction R}
+
+
+
+
+Lemma tofrac_is_additive: additive tofrac.
+ +
+Canonical tofrac_additive := Additive tofrac_is_additive.
+ +
+Lemma tofrac_is_multiplicative: multiplicative tofrac.
+ +
+Canonical tofrac_rmorphism := AddRMorphism tofrac_is_multiplicative.
+ +
+
+
++Lemma tofrac_is_additive: additive tofrac.
+ +
+Canonical tofrac_additive := Additive tofrac_is_additive.
+ +
+Lemma tofrac_is_multiplicative: multiplicative tofrac.
+ +
+Canonical tofrac_rmorphism := AddRMorphism tofrac_is_multiplicative.
+ +
+
+ tests
+
+
+Lemma tofrac0 : 0%:F = 0.
+Lemma tofracN : {morph tofrac: x / - x}.
+Lemma tofracD : {morph tofrac: x y / x + y}.
+Lemma tofracB : {morph tofrac: x y / x - y}.
+Lemma tofracMn n : {morph tofrac: x / x *+ n}.
+Lemma tofracMNn n : {morph tofrac: x / x *- n}.
+Lemma tofrac1 : 1%:F = 1.
+Lemma tofracM : {morph tofrac: x y / x × y}.
+Lemma tofracX n : {morph tofrac: x / x ^+ n}.
+ +
+Lemma tofrac_eq (p q : R): (p%:F == q%:F) = (p == q).
+ +
+Lemma tofrac_eq0 (p : R): (p%:F == 0) = (p == 0).
+ End FracFieldTheory.
+
++Lemma tofracN : {morph tofrac: x / - x}.
+Lemma tofracD : {morph tofrac: x y / x + y}.
+Lemma tofracB : {morph tofrac: x y / x - y}.
+Lemma tofracMn n : {morph tofrac: x / x *+ n}.
+Lemma tofracMNn n : {morph tofrac: x / x *- n}.
+Lemma tofrac1 : 1%:F = 1.
+Lemma tofracM : {morph tofrac: x y / x × y}.
+Lemma tofracX n : {morph tofrac: x / x ^+ n}.
+ +
+Lemma tofrac_eq (p q : R): (p%:F == q%:F) = (p == q).
+ +
+Lemma tofrac_eq0 (p : R): (p%:F == 0) = (p == 0).
+ End FracFieldTheory.
+