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From Coq Require Import Field Ring.
Require Import
MathClasses.interfaces.abstract_algebra MathClasses.interfaces.vectorspace
MathClasses.theory.fields MathClasses.theory.strong_setoids.
Require Export
MathClasses.theory.rings.
Section contents.
Context `{DecField F} `{∀ x y: F, Decision (x = y)}.
Add Ring F : (stdlib_ring_theory F).
Global Instance: ZeroProduct F.
Proof with auto.
intros x y E.
destruct (decide (x = 0)) as [? | Ex]...
right.
rewrite <-(mult_1_r y), <-(dec_recip_inverse x) by assumption.
rewrite associativity, (commutativity y), E. ring.
Qed.
Global Instance: IntegralDomain F.
Proof. split; try apply _. Qed.
Lemma dec_recip_1: / 1 = 1.
Proof.
rewrite <-(rings.mult_1_l (/1)).
apply dec_recip_inverse.
solve_propholds.
Qed.
Lemma dec_recip_distr (x y: F): / (x * y) = / x * / y.
Proof.
destruct (decide (x = 0)) as [Ex|Ex].
rewrite Ex, left_absorb, dec_recip_0. ring.
destruct (decide (y = 0)) as [Ey|Ey].
rewrite Ey, right_absorb, dec_recip_0. ring.
assert (x * y ≠ 0) as Exy by now apply mult_ne_0.
apply (left_cancellation_ne_0 (.*.) (x * y)); trivial.
transitivity (x / x * (y / y)).
rewrite !dec_recip_inverse by assumption. ring.
ring.
Qed.
Lemma dec_recip_zero x : / x = 0 ↔ x = 0.
Proof.
split; intros E.
apply stable. intros Ex.
destruct (is_ne_0 1).
rewrite <-(dec_recip_inverse x), E by assumption. ring.
rewrite E. now apply dec_recip_0.
Qed.
Lemma dec_recip_ne_0_iff x : / x ≠ 0 ↔ x ≠ 0.
Proof. now split; intros E1 E2; destruct E1; apply dec_recip_zero. Qed.
Instance dec_recip_ne_0 x : PropHolds (x ≠ 0) → PropHolds (/x ≠ 0).
Proof. intro. now apply dec_recip_ne_0_iff. Qed.
Lemma equal_by_one_quotient (x y : F) : x / y = 1 → x = y.
Proof.
intro Exy.
destruct (decide (y = 0)) as [Ey|Ey].
exfalso. apply (is_ne_0 1).
rewrite <- Exy, Ey, dec_recip_0. ring.
apply (right_cancellation_ne_0 (.*.) (/y)).
now apply dec_recip_ne_0.
now rewrite dec_recip_inverse.
Qed.
Global Instance dec_recip_inj: Injective (/).
Proof.
repeat (split; try apply _).
intros x y E.
destruct (decide (y = 0)) as [Ey|Ey].
rewrite Ey in *. rewrite dec_recip_0 in E.
now apply dec_recip_zero.
apply (right_cancellation_ne_0 (.*.) (/y)).
now apply dec_recip_ne_0.
rewrite dec_recip_inverse by assumption.
rewrite <-E, dec_recip_inverse.
easy.
apply dec_recip_ne_0_iff. rewrite E.
now apply dec_recip_ne_0.
Qed.
Global Instance dec_recip_involutive: Involutive (/).
Proof.
intros x. destruct (decide (x = 0)) as [Ex|Ex].
now rewrite Ex, !dec_recip_0.
apply (right_cancellation_ne_0 (.*.) (/x)).
now apply dec_recip_ne_0.
rewrite dec_recip_inverse by assumption.
rewrite commutativity, dec_recip_inverse.
reflexivity.
now apply dec_recip_ne_0.
Qed.
Lemma equal_dec_quotients (a b c d : F) : b ≠ 0 → d ≠ 0 → (a * d = c * b ↔ a / b = c / d).
Proof with trivial; try ring.
split; intro E.
apply (right_cancellation_ne_0 (.*.) b)...
apply (right_cancellation_ne_0 (.*.) d)...
transitivity (a * d * (b * /b))...
transitivity (c * b * (d * /d))...
rewrite E, dec_recip_inverse, dec_recip_inverse...
transitivity (a * d * 1)...
rewrite <-(dec_recip_inverse b)...
transitivity (a * / b * d * b)...
rewrite E.
transitivity (c * (d * / d) * b)...
rewrite dec_recip_inverse...
Qed.
Lemma dec_quotients (a c b d : F) : b ≠ 0 → d ≠ 0 → a / b + c / d = (a * d + c * b) / (b * d).
Proof with auto.
intros A B.
assert (a / b = (a * d) / (b * d)) as E1.
apply ->equal_dec_quotients...
ring.
intros G. destruct (zero_product b d)...
assert (c / d = (b * c) / (b * d)) as E2.
apply ->equal_dec_quotients...
ring.
intros G. destruct (zero_product b d)...
rewrite E1, E2. ring.
Qed.
Lemma dec_recip_swap_l x y: x / y = / (/ x * y).
Proof. rewrite dec_recip_distr, involutive. ring. Qed.
Lemma dec_recip_swap_r x y: / x * y = / (x / y).
Proof. rewrite dec_recip_distr, involutive. ring. Qed.
Lemma dec_recip_negate x : -(/ x) = / (-x).
Proof.
destruct (decide (x = 0)) as [Ex|Ex].
now rewrite Ex, negate_0, dec_recip_0, negate_0.
apply (left_cancellation_ne_0 (.*.) (-x)).
now apply flip_negate_ne_0.
rewrite dec_recip_inverse.
ring_simplify.
now apply dec_recip_inverse.
now apply flip_negate_ne_0.
Qed.
Instance: @VectorSpace F F Ae Aplus Amult Azero Aone Anegate Adec_recip
Ae Aplus Azero Anegate Amult.
Proof.
repeat split; repeat (try apply _).
- apply dec_recip_0.
- now apply dec_recip_inverse.
Qed.
End contents.
(* Due to bug #2528 *)
#[global]
Hint Extern 7 (PropHolds (/ _ ≠ 0)) => eapply @dec_recip_ne_0 : typeclass_instances.
(* Given a decidable field we can easily construct a constructive field. *)
Section is_field.
Context `{DecField F} `{Apart F} `{!TrivialApart F} `{∀ x y: F, Decision (x = y)}.
Global Program Instance recip_dec_field: Recip F := λ x, /`x.
Instance: StrongSetoid F := dec_strong_setoid.
Global Instance: Field F.
Proof.
split; try apply _.
now apply (dec_strong_binary_morphism (+)).
now apply (dec_strong_binary_morphism (.*.)).
split; try apply _.
intros ? ? E. unfold recip, recip_dec_field.
now apply sm_proper.
intros [x Px]. rapply (dec_recip_inverse x).
now apply trivial_apart.
Qed.
Lemma dec_recip_correct (x : F) Px : / x = // x↾Px.
Proof with auto.
apply (left_cancellation_ne_0 (.*.) x).
now apply trivial_apart.
now rewrite dec_recip_inverse, reciperse_alt by now apply trivial_apart.
Qed.
End is_field.
Definition stdlib_field_theory F `{DecField F} :
Field_theory.field_theory 0 1 (+) (.*.) (λ x y, x - y) (-) (λ x y, x / y) (/) (=).
Proof with auto.
intros.
constructor.
apply (theory.rings.stdlib_ring_theory _).
apply (is_ne_0 1).
reflexivity.
intros.
rewrite commutativity. now apply dec_recip_inverse.
Qed.
Section from_stdlib_field_theory.
Context `(ftheory : @field_theory F Fzero Fone Fplus Fmult Fminus Fnegate Fdiv Frecip Fe)
(rinv_0 : Fe (Frecip Fzero) Fzero)
`{!@Setoid F Fe}
`{!Proper (Fe ==> Fe ==> Fe) Fplus}
`{!Proper (Fe ==> Fe ==> Fe) Fmult}
`{!Proper (Fe ==> Fe) Fnegate}
`{!Proper (Fe ==> Fe) Frecip}.
Add Field F2 : ftheory.
Definition from_stdlib_field_theory: @DecField F Fe Fplus Fmult Fzero Fone Fnegate Frecip.
Proof with auto.
destruct ftheory.
repeat (constructor; try assumption); repeat intro
; unfold equiv, mon_unit, sg_op, zero_is_mon_unit, plus_is_sg_op,
one_is_mon_unit, mult_is_sg_op, plus, mult, recip, negate; try field...
unfold recip, mult.
simpl.
assert (Fe (Fmult x (Frecip x)) (Fmult (Frecip x) x)) as E by ring.
rewrite E...
Qed.
End from_stdlib_field_theory.
Section morphisms.
Context `{DecField F} `{∀ x y: F, Decision (x = y)}.
Global Instance dec_field_to_domain_inj `{IntegralDomain R}
`{!SemiRing_Morphism (f : F → R)} : Injective f.
Proof.
apply injective_preserves_0.
intros x Efx.
apply stable. intros Ex.
destruct (is_ne_0 (1:R)).
rewrite <-(rings.preserves_1 (f:=f)).
rewrite <-(dec_recip_inverse x) by assumption.
rewrite rings.preserves_mult, Efx.
now apply left_absorb.
Qed.
Lemma preserves_dec_recip `{DecField F2} `{∀ x y: F2, Decision (x = y)}
`{!SemiRing_Morphism (f : F → F2)} x : f (/ x) = / f x.
Proof.
case (decide (x = 0)) as [E | E].
now rewrite E, dec_recip_0, preserves_0, dec_recip_0.
intros.
apply (left_cancellation_ne_0 (.*.) (f x)).
now apply injective_ne_0.
rewrite <-preserves_mult, 2!dec_recip_inverse.
now apply preserves_1.
now apply injective_ne_0.
easy.
Qed.
Lemma dec_recip_to_recip `{Field F2} `{!StrongSemiRing_Morphism (f : F → F2)} x Pfx :
f (/ x) = // (f x)↾Pfx.
Proof.
assert (x ≠ 0).
intros Ex.
destruct (apart_ne (f x) 0 Pfx).
now rewrite Ex, preserves_0.
apply (left_cancellation_ne_0 (.*.) (f x)).
now apply injective_ne_0.
rewrite <-preserves_mult, dec_recip_inverse, reciperse_alt by assumption.
now apply preserves_1.
Qed.
End morphisms.
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