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EQ <> GT. Proof. intros E. change ((fun r => match r with EQ => Unit | _ => Empty end) GT). rewrite <-E. split. Qed. | Lemma | Require Import HoTT.Classes.interfaces.canonical_names. | Classes\theory\additional_operations.v | EQ_GT | 3,200 |
GT <> EQ. Proof. apply symmetric_neq, EQ_GT. Qed. | Lemma | Require Import HoTT.Classes.interfaces.canonical_names. | Classes\theory\additional_operations.v | GT_EQ | 3,201 |
`{Compare A} : forall a b : A, a =? b = true -> a ?= b = EQ. Proof. unfold eqb,compare_eqb;simpl. intros a b. destruct (a ?= b);trivial;intros E;destruct (false_ne_true E). Qed. | Lemma | Require Import HoTT.Classes.interfaces.canonical_names. | Classes\theory\additional_operations.v | compare_eqb_eq | 3,202 |
`{Trichotomy A R} : forall a b : A, compare a b = EQ -> a = b. Proof. unfold compare,tricho_compare. intros a b;destruct (trichotomy R a b) as [E|[E|E]];auto. - intros E1;destruct (LT_EQ E1). - intros E1;destruct (GT_EQ E1). Qed. | Lemma | Require Import HoTT.Classes.interfaces.canonical_names. | Classes\theory\additional_operations.v | tricho_compare_eq | 3,203 |
`{Trichotomy A R} `{Irreflexive A R} : forall a b : A, compare a b = EQ <-> a = b. Proof. unfold compare,tricho_compare. intros a b;destruct (trichotomy R a b) as [E1|[E1|E1]];split;auto. - intros E2;destruct (LT_EQ E2). - intros E2;rewrite E2 in E1. destruct (irreflexivity R _ E1). - intros E2;destruct (GT_EQ E2). - intros E2;rewrite E2 in E1. destruct (irreflexivity R _ E1). Qed. | Lemma | Require Import HoTT.Classes.interfaces.canonical_names. | Classes\theory\additional_operations.v | tricho_compare_ok | 3,204 |
`{Abs A} `{!TotalRelation le} : forall x : A, (0 <= x /\ abs x = x) |_| (x <= 0 /\ abs x = - x). Proof. intros x. destruct (total le 0 x) as [E|E]. - left. split;trivial. apply ((abs_sig x).2);trivial. - right. split;trivial. apply ((abs_sig x).2);trivial. Qed. | Lemma | Require Import HoTT.Classes.interfaces.canonical_names. | Classes\theory\additional_operations.v | total_abs_either | 3,205 |
x y : PropHolds (x ≶ y) -> PropHolds (x <> y). Proof. unfold PropHolds. intros ap e;revert ap. apply tight_apart. assumption. Qed. | Lemma | Require Import HoTT.Classes.interfaces.abstract_algebra. | Classes\theory\apartness.v | apart_ne | 3,206 |
`{IsApart B} `{Apart A} `{IsHSet A} `{is_mere_relation A apart} (f: A -> B) (eq_correct : forall x y, x = y <-> f x = f y) (apart_correct : forall x y, x ≶ y <-> f x ≶ f y) : IsApart A. Proof. split. - apply _. - apply _. - intros x y ap. apply apart_correct, symmetry, apart_correct. assumption. - intros x y ap z. apply apart_correct in ap. apply (merely_destruct (cotransitive ap (f z))). intros [?|?];apply tr;[left|right];apply apart_correct;assumption. - intros x y;split. + intros nap. apply eq_correct. apply tight_apart. intros ap. apply nap. apply apart_correct;assumption. + intros e ap. apply apart_correct in ap;revert ap. apply tight_apart. apply eq_correct;assumption. Qed. | Lemma | Require Import HoTT.Classes.interfaces.abstract_algebra. | Classes\theory\apartness.v | projected_strong_setoid | 3,207 |
`{!IsApart A} `{!IsApart B} `{!IsApart C} {f : A -> B -> C} `{forall z, StrongExtensionality (f z)} `{forall z, StrongExtensionality (fun x => f x z)} : StrongBinaryExtensionality f. Proof. intros x₁ y₁ x₂ y₂ E. apply (merely_destruct (cotransitive E (f x₂ y₁))). intros [?|?];apply tr. - left. apply (strong_extensionality (fun x => f x y₁));trivial. - right. apply (strong_extensionality (f x₂));trivial. Qed. | Lemma | Require Import HoTT.Classes.interfaces.abstract_algebra. | Classes\theory\apartness.v | strong_binary_setoid_morphism_both_coordinates | 3,208 |
{f : A -> A -> B} `{!Commutative f} `{forall z, StrongExtensionality (f z)} : StrongBinaryExtensionality f. Proof. apply @strong_binary_setoid_morphism_both_coordinates;try apply _. intros z x y. rewrite !(commutativity _ z). apply (strong_extensionality (f z)). Qed. | Lemma | Require Import HoTT.Classes.interfaces.abstract_algebra. | Classes\theory\apartness.v | strong_binary_setoid_morphism_commutative | 3,209 |
Apart A | 20 := fun x y => match dec (x = y) with | inl _ => false | inr _ => true end = true. | Instance | Require Import HoTT.Classes.interfaces.abstract_algebra. | Classes\theory\apartness.v | default_apart | 3,210 |
TrivialApart A (Aap:=default_apart). Proof. split. - unfold apart,default_apart. apply _. - intros x y;unfold apart,default_apart;split. + intros E. destruct (dec (x=y)). * destruct (false_ne_true E). * trivial. + intros E;destruct (dec (x=y)) as [e|_]. * destruct (E e). * split. Qed. | Instance | Require Import HoTT.Classes.interfaces.abstract_algebra. | Classes\theory\apartness.v | default_apart_trivial | 3,211 |
x y : PropHolds (x <> y) -> PropHolds (x ≶ y). Proof. intros ap. apply trivial_apart. assumption. Qed. | Instance | Require Import HoTT.Classes.interfaces.abstract_algebra. | Classes\theory\apartness.v | ne_apart | 3,212 |
(f : A -> B) : StrongExtensionality f. Proof. intros x y E. apply trivial_apart. intros e. apply tight_apart in E;auto. Qed. | Instance | Require Import HoTT.Classes.interfaces.abstract_algebra. | Classes\theory\apartness.v | dec_strong_morphism | 3,213 |
(f : A -> B) `{!IsInjective f} : IsStrongInjective f. Proof. split; try apply _. intros x y. intros ap. apply trivial_apart in ap. apply trivial_apart. intros e. apply ap. apply (injective f). assumption. Qed. | Instance | Require Import HoTT.Classes.interfaces.abstract_algebra. | Classes\theory\apartness.v | dec_strong_injective | 3,214 |
(f : A -> B -> C) : StrongBinaryExtensionality f. Proof. intros x1 y1 x2 y2 hap. apply (merely_destruct (cotransitive hap (f x2 y1)));intros [h|h];apply tr. - left. apply trivial_apart. intros e. apply tight_apart in h;auto. exact (ap (fun x => f x y1) e). - right. apply trivial_apart. intros e. apply tight_apart in h;auto. Qed. | Instance | Require Import HoTT.Classes.interfaces.abstract_algebra. | Classes\theory\apartness.v | dec_strong_binary_morphism | 3,215 |
/ 1 = 1. Proof. rewrite <-(rings.mult_1_l (/1)). apply dec_recip_inverse. solve_propholds. Qed. | Lemma | Require Import | Classes\theory\dec_fields.v | dec_recip_1 | 3,216 |
(x y: F): / (x * y) = / x * / y. Proof. destruct (dec (x = 0)) as [Ex|Ex]. - rewrite Ex, left_absorb, dec_recip_0. apply symmetry,mult_0_l. - destruct (dec (y = 0)) as [Ey|Ey]. + rewrite Ey, dec_recip_0, !mult_0_r. apply dec_recip_0. + assert (x * y <> 0) as Exy by (apply mult_ne_0;trivial). apply (left_cancellation_ne_0 (.*.) (x * y)); trivial. transitivity (x / x * (y / y)). * rewrite !dec_recip_inverse by assumption. rewrite mult_1_l;apply reflexivity. * rewrite !dec_recip_inverse by assumption. rewrite mult_assoc, (mult_comm x), <-(mult_assoc y). rewrite dec_recip_inverse by assumption. rewrite (mult_comm y), <-mult_assoc. rewrite dec_recip_inverse by assumption. reflexivity. Qed. | Lemma | Require Import | Classes\theory\dec_fields.v | dec_recip_distr | 3,217 |
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. apply mult_0_r. - rewrite E. apply dec_recip_0. Qed. | Lemma | Require Import | Classes\theory\dec_fields.v | dec_recip_zero | 3,218 |
x : / x <> 0 <-> x <> 0. Proof. split; intros E1 E2; destruct E1; apply dec_recip_zero;trivial. do 2 apply (snd (dec_recip_zero _)). trivial. Qed. | Lemma | Require Import | Classes\theory\dec_fields.v | dec_recip_ne_0_iff | 3,219 |
x : PropHolds (x <> 0) -> PropHolds (/x <> 0). Proof. intro. apply (snd (dec_recip_ne_0_iff _)). trivial. Qed. | Instance | Require Import | Classes\theory\dec_fields.v | dec_recip_ne_0 | 3,220 |
(x y : F) : x / y = 1 -> x = y. Proof. intro Exy. destruct (dec (y = 0)) as [Ey|Ey]. - destruct (is_ne_0 1). rewrite <- Exy, Ey, dec_recip_0. apply mult_0_r. - apply (right_cancellation_ne_0 (.*.) (/y)). + apply dec_recip_ne_0. trivial. + rewrite dec_recip_inverse;trivial. Qed. | Lemma | Require Import | Classes\theory\dec_fields.v | equal_by_one_quotient | 3,221 |
(a b c d : F) : b <> 0 -> d <> 0 -> (a * d = c * b <-> a / b = c / d). Proof. split; intro E. - apply (right_cancellation_ne_0 (.*.) b);trivial. apply (right_cancellation_ne_0 (.*.) d);trivial. transitivity (a * d * (b * /b));[| transitivity (c * b * (d * /d))]. + rewrite <-!(mult_assoc a). apply ap. rewrite (mult_comm d), (mult_comm _ b). reflexivity. + rewrite E, dec_recip_inverse, dec_recip_inverse;trivial. + rewrite <-!(mult_assoc c). apply ap. rewrite (mult_comm d), mult_assoc, (mult_comm b). reflexivity. - transitivity (a * d * 1);[rewrite mult_1_r;reflexivity|]. rewrite <-(dec_recip_inverse b);trivial. transitivity (c * b * 1);[|rewrite mult_1_r;reflexivity]. rewrite <-(dec_recip_inverse d);trivial. rewrite mult_comm, <-mult_assoc, (mult_assoc _ a), (mult_comm _ a), E. rewrite <-mult_assoc. rewrite (mult_comm _ d). rewrite mult_assoc, (mult_comm c). reflexivity. Qed. | Lemma | Require Import | Classes\theory\dec_fields.v | equal_dec_quotients | 3,222 |
(a c b d : F) : b <> 0 -> d <> 0 -> a / b + c / d = (a * d + c * b) / (b * d). Proof. intros A B. assert (a / b = (a * d) / (b * d)) as E1. - apply equal_dec_quotients;auto. + solve_propholds. + rewrite (mult_comm b);apply associativity. - assert (c / d = (b * c) / (b * d)) as E2. + apply equal_dec_quotients;trivial. * solve_propholds. * rewrite mult_assoc, (mult_comm c). reflexivity. + rewrite E1, E2. rewrite (mult_comm c b). apply symmetry, simple_distribute_r. Qed. | Lemma | Require Import | Classes\theory\dec_fields.v | dec_quotients | 3,223 |
x y: x / y = / (/ x * y). Proof. rewrite dec_recip_distr, involutive. reflexivity. Qed. | Lemma | Require Import | Classes\theory\dec_fields.v | dec_recip_swap_l | 3,224 |
x y: / x * y = / (x / y). Proof. rewrite dec_recip_distr, involutive. reflexivity. Qed. | Lemma | Require Import | Classes\theory\dec_fields.v | dec_recip_swap_r | 3,225 |
x : -(/ x) = / (-x). Proof. destruct (dec (x = 0)) as [Ex|Ex]. - rewrite Ex, negate_0, dec_recip_0, negate_0. reflexivity. - apply (left_cancellation_ne_0 (.*.) (-x)). + apply (snd (flip_negate_ne_0 _)). trivial. + rewrite dec_recip_inverse. * rewrite negate_mult_negate. apply dec_recip_inverse. trivial. * apply (snd (flip_negate_ne_0 _)). trivial. Qed. | Lemma | Require Import | Classes\theory\dec_fields.v | dec_recip_negate | 3,226 |
(x : F) Px : / x = // (x;Px). Proof. apply (left_cancellation_ne_0 (.*.) x). - apply trivial_apart. trivial. - rewrite dec_recip_inverse, reciperse_alt by (apply trivial_apart;trivial). reflexivity. Qed. | Lemma | Require Import | Classes\theory\dec_fields.v | dec_recip_correct | 3,227 |
@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. | Definition | Require Import | Classes\theory\dec_fields.v | from_stdlib_field_theory | 3,228 |
`{IsDecField F2} `{forall x y: F2, Decidable (x = y)} `{!IsSemiRingPreserving (f : F -> F2)} x : f (/ x) = / f x. Proof. case (dec (x = 0)) as [E | E]. - rewrite E, dec_recip_0, preserves_0, dec_recip_0. reflexivity. - intros. apply (left_cancellation_ne_0 (.*.) (f x)). + apply isinjective_ne_0. trivial. + rewrite <-preserves_mult, 2!dec_recip_inverse. * apply preserves_1. * apply isinjective_ne_0. trivial. * trivial. Qed. | Lemma | Require Import | Classes\theory\dec_fields.v | preserves_dec_recip | 3,229 |
`{IsField F2} `{!IsSemiRingStrongPreserving (f : F -> F2)} x Pfx : f (/ x) = // (f x;Pfx). Proof. assert (x <> 0). - intros Ex. destruct (apart_ne (f x) 0 Pfx). rewrite Ex, (preserves_0 (f:=f)). reflexivity. - apply (left_cancellation_ne_0 (.*.) (f x)). + apply isinjective_ne_0. trivial. + rewrite <-preserves_mult, dec_recip_inverse, reciperse_alt by assumption. apply preserves_1. Qed. | Lemma | Require Import | Classes\theory\dec_fields.v | dec_recip_to_recip | 3,230 |
recip' (x : F) (apx : x ≶ 0) : F := //(x;apx). | Definition | Require Import | Classes\theory\fields.v | recip' | 3,231 |
recip_inverse' (x : F) (Px : x ≶ 0) : x // (x; Px) = 1. Proof. apply (recip_inverse (x;Px)). Qed. | Lemma | Require Import | Classes\theory\fields.v | recip_inverse' | 3,232 |
(x : F) Px : x // (x;Px) = 1. Proof. rewrite <-(recip_inverse (x;Px)). trivial. Qed. | Lemma | Require Import | Classes\theory\fields.v | reciperse_alt | 3,233 |
x y Px Py : x = y -> // (x;Px) = // (y;Py). Proof. intro E. apply ap. apply Sigma.path_sigma with E. apply path_ishprop. Qed. | Lemma | Require Import | Classes\theory\fields.v | recip_proper_alt | 3,234 |
x y Py : x // (y;Py) = 1 -> x = y. Proof. intros eqxy. rewrite <- (mult_1_r y). rewrite <- eqxy. rewrite (mult_assoc y x (//(y;Py))). rewrite (mult_comm y x). rewrite <- (mult_assoc x y (//(y;Py))). rewrite (recip_inverse (y;Py)). rewrite (mult_1_r x). reflexivity. Qed. | Lemma | Require Import | Classes\theory\fields.v | recip_proper | 3,235 |
x Px1 Px2 : // (x;Px1) = // (x;Px2). Proof. apply recip_proper_alt. reflexivity. Qed. | Lemma | Require Import | Classes\theory\fields.v | recip_irrelevant | 3,236 |
{x y} : x ≶ 0 -> x = y -> y ≶ 0. Proof. intros ? E. rewrite <-E. trivial. Qed. | Lemma | Require Import | Classes\theory\fields.v | apart_0_proper | 3,237 |
x y : x * y ≶ 0 -> x ≶ 0. Proof. intros. apply (strong_extensionality (.* y)). rewrite mult_0_l. trivial. Qed. | Lemma | Require Import | Classes\theory\fields.v | mult_apart_zero_l | 3,238 |
x y : x * y ≶ 0 -> y ≶ 0. Proof. intros. apply (strong_extensionality (x *.)). rewrite mult_0_r. trivial. Qed. | Lemma | Require Import | Classes\theory\fields.v | mult_apart_zero_r | 3,239 |
x y : PropHolds (x ≶ 0) -> PropHolds (y ≶ 0) -> PropHolds (x * y ≶ 0). Proof. intros Ex Ey. apply (strong_extensionality (.* // (y;(Ey : (≶0) y)))). rewrite <-simple_associativity, reciperse_alt, mult_1_r, mult_0_l. trivial. Qed. | Instance | Require Import | Classes\theory\fields.v | mult_apart_zero | 3,240 |
x : PropHolds (// x ≶ 0). Proof. red. apply mult_apart_zero_r with (x.1). rewrite recip_inverse. solve_propholds. Qed. | Instance | Require Import | Classes\theory\fields.v | recip_apart_zero | 3,241 |
x y : x = 0 -> x // y = 0. Proof. intros E. rewrite E. apply left_absorb. Qed. | Lemma | Require Import | Classes\theory\fields.v | field_div_0_l | 3,242 |
x y : x = y.1 -> x // y = 1. Proof. intros E. rewrite E. apply recip_inverse. Qed. | Lemma | Require Import | Classes\theory\fields.v | field_div_diag | 3,243 |
(a c: F) b d : a * d.1 = c * b.1 <-> a // b = c // d. Proof. split; intro E. - rewrite <-(mult_1_l (a // b)), <- (recip_inverse d), (commutativity (f:=mult) d.1 (// d)), <-simple_associativity, (simple_associativity d.1), (commutativity (f:=mult) d.1 a), E, <-simple_associativity, simple_associativity, recip_inverse, mult_1_r. apply commutativity. - rewrite <-(mult_1_r (a * d.1)), <- (recip_inverse b), <-simple_associativity, (commutativity (f:=mult) b.1 (// b)), (simple_associativity d.1), (commutativity (f:=mult) d.1), !simple_associativity, E, <-(simple_associativity c), (commutativity (f:=mult) (// d)), recip_inverse, mult_1_r. reflexivity. Qed. | Lemma | Require Import | Classes\theory\fields.v | equal_quotients | 3,244 |
(x y : F) Px Py Pxy : // (x * y ; Pxy) = // (x;Px) * // (y;Py). Proof. apply (left_cancellation_ne_0 (.*.) (x * y)). - apply apart_ne;trivial. - transitivity ((x // (x;Px)) * (y // (y;Py))). + rewrite 3!reciperse_alt,mult_1_r. reflexivity. + rewrite <-simple_associativity,<-simple_associativity. apply ap. rewrite simple_associativity. rewrite (commutativity (f:=mult) _ y). rewrite <-simple_associativity. reflexivity. Qed. | Lemma | Require Import | Classes\theory\fields.v | recip_distr_alt | 3,245 |
(x : F) (Px : x ≶ 0) : (-x) ≶ 0. Proof. assert (ap : x + 0 ≶ x - x). { rewrite (plus_0_r x). rewrite (plus_negate_r x). assumption. } refine (Trunc_rec _ (field_plus_ext F x 0 x (-x) ap)). intros [apxx|ap0x]. - destruct (apart_ne x x apxx); reflexivity. - symmetry; assumption. Qed. | Lemma | Require Import | Classes\theory\fields.v | apart_negate | 3,246 |
ApartZero F -> ApartZero F. Proof. intros [x Px]. exists (-x). exact ((apart_negate x Px)). Defined. | Definition | Require Import | Classes\theory\fields.v | negate_apart | 3,247 |
(x : F) (Px : x ≶ 0) : (-//(x;Px))=//(negate_apart(x;Px)). Proof. apply (left_cancellation (.*.) x). rewrite <- negate_mult_distr_r. rewrite reciperse_alt. apply flip_negate. rewrite negate_mult_distr_l. refine (_^). apply reciperse_alt. Qed. | Lemma | Require Import | Classes\theory\fields.v | recip_negate | 3,248 |
(x : F) (Px : x ≶ 0) : // (x;Px) ≶ 0. Proof. apply (strong_extensionality (x*.) (// (x; Px)) 0). rewrite (recip_inverse (x;Px)). rewrite mult_0_r. solve_propholds. Qed. | Lemma | Require Import | Classes\theory\fields.v | recip_apart | 3,249 |
(x : ApartZero F) : ApartZero F. Proof. exists (//x). apply recip_apart. Defined. | Definition | Require Import | Classes\theory\fields.v | recip_on_apart | 3,250 |
(forall x, x ≶ 0 -> f x ≶ 0) -> IsStrongInjective f. Proof. intros E1. split; try apply _. intros x y E2. apply (strong_extensionality (+ -f y)). rewrite plus_negate_r, <-preserves_minus. apply E1. apply (strong_extensionality (+ y)). rewrite <-simple_associativity,left_inverse,plus_0_l,plus_0_r. trivial. Qed. | Lemma | Require Import | Classes\theory\fields.v | strong_injective_preserves_0 | 3,251 |
x Px Pfx : f (// (x;Px)) = // (f x;Pfx). Proof. apply (left_cancellation_ne_0 (.*.) (f x)). - apply apart_ne;trivial. - rewrite <-rings.preserves_mult. rewrite !reciperse_alt. apply preserves_1. Qed. | Lemma | Require Import | Classes\theory\fields.v | preserves_recip | 3,252 |
- mon_unit = mon_unit. Proof. change ((fun x => - mon_unit = x) mon_unit). apply (transport _ (left_inverse mon_unit)). apply symmetry, right_identity. Qed. | Lemma | Require Import | Classes\theory\groups.v | negate_mon_unit | 3,253 |
x y : - (x * y) = -y * -x. Proof. rewrite <- (left_identity (-y * -x)). rewrite <- (left_inverse (unit:=mon_unit) (x * y)). rewrite <- simple_associativity. rewrite <- simple_associativity. rewrite (associativity y). rewrite right_inverse. rewrite (left_identity (-x)). rewrite right_inverse. apply symmetry, right_identity. Qed. | Lemma | Require Import | Classes\theory\groups.v | negate_sg_op | 3,254 |
`{IsAbGroup G} x y : -(x * y) = -x * -y. Proof. path_via (-y * -x). - apply negate_sg_op. - apply commutativity. Qed. | Lemma | Require Import | Classes\theory\groups.v | negate_sg_op_distr | 3,255 |
x : f (-x) = -f x. Proof. apply (left_cancellation (.*.) (f x)). rewrite <-preserves_sg_op. rewrite 2!right_inverse. apply preserves_mon_unit. Qed. | Lemma | Require Import | Classes\theory\groups.v | preserves_negate | 3,256 |
IsSemiGroup B. Proof. split. - apply _. - repeat intro; apply (injective f). rewrite !op_correct. apply associativity. Qed. | Lemma | Require Import | Classes\theory\groups.v | projected_sg | 3,257 |
Commutative (A:=B) sg_op. Proof. intros x y. apply (injective f). rewrite 2!op_correct. apply commutativity. Qed. | Lemma | Require Import | Classes\theory\groups.v | projected_comm | 3,258 |
IsCommutativeSemiGroup B. Proof. split. - apply (projected_sg f);assumption. - apply (projected_comm f);assumption. Qed. | Lemma | Require Import | Classes\theory\groups.v | projected_com_sg | 3,259 |
IsMonoid B. Proof. split. - apply (projected_sg f). assumption. - repeat intro; apply (injective f). rewrite op_correct, unit_correct, left_identity. reflexivity. - repeat intro; apply (injective f). rewrite op_correct, unit_correct, right_identity. reflexivity. Qed. | Lemma | Require Import | Classes\theory\groups.v | projected_monoid | 3,260 |
IsCommutativeMonoid B. Proof. split. - apply (projected_monoid f);assumption. - apply (projected_comm f);assumption. Qed. | Lemma | Require Import | Classes\theory\groups.v | projected_com_monoid | 3,261 |
IsGroup B. Proof. split. - apply (projected_monoid f);assumption. - repeat intro; apply (injective f). rewrite op_correct, negate_correct, unit_correct, left_inverse. apply reflexivity. - repeat intro; apply (injective f). rewrite op_correct, negate_correct, unit_correct, right_inverse. reflexivity. Qed. | Lemma | Require Import | Classes\theory\groups.v | projected_group | 3,262 |
IsAbGroup B. Proof. split. - apply (projected_group f);assumption. - apply (projected_comm f);assumption. Qed. | Lemma | Require Import | Classes\theory\groups.v | projected_ab_group | 3,263 |
`{Integers Z} `{IsCRing R} (f: Z -> R) {h: IsSemiRingPreserving f} x : f x = integers_to_ring Z R x. Proof. symmetry. apply integers_initial. Qed. | Lemma | Require Import Require Import Import NatPair.Instances. | Classes\theory\integers.v | to_ring_unique | 3,264 |
`{Integers Z} `{IsCRing R} (f g: Z -> R) `{!IsSemiRingPreserving f} `{!IsSemiRingPreserving g} x : f x = g x. Proof. rewrite (to_ring_unique f), (to_ring_unique g);reflexivity. Qed. | Lemma | Require Import Require Import Import NatPair.Instances. | Classes\theory\integers.v | to_ring_unique_alt | 3,265 |
Z `{Integers Z} Z2 `{Integers Z2} x : integers_to_ring Z2 Z (integers_to_ring Z Z2 x) = x. Proof. change (Compose (integers_to_ring Z2 Z) (integers_to_ring Z Z2) x = id x). apply to_ring_unique_alt;apply _. Qed. | Lemma | Require Import Require Import Import NatPair.Instances. | Classes\theory\integers.v | to_ring_involutive | 3,266 |
`{Integers Z} `{IsCRing R} (f: R -> Z) (g: Z -> R) `{!IsSemiRingPreserving f} `{!IsSemiRingPreserving g} x : f (g x) = x. Proof. exact (to_ring_unique_alt (f ∘ g) id _). Qed. | Lemma | Require Import Require Import Import NatPair.Instances. | Classes\theory\integers.v | morphisms_involutive | 3,267 |
`{Integers Z} `{IsCRing R1} `{IsCRing R2} (f : R1 -> R2) (g : Z -> R1) (h : Z -> R2) `{!IsSemiRingPreserving f} `{!IsSemiRingPreserving g} `{!IsSemiRingPreserving h} x : f (g x) = h x. Proof. exact (to_ring_unique_alt (f ∘ g) h _). Qed. | Lemma | Require Import Require Import Import NatPair.Instances. | Classes\theory\integers.v | to_ring_twice | 3,268 |
`{Integers Z} (f : Z -> Z) `{!IsSemiRingPreserving f} x : f x = x. Proof. exact (to_ring_unique_alt f id _). Qed. | Lemma | Require Import Require Import Import NatPair.Instances. | Classes\theory\integers.v | to_ring_self | 3,269 |
`{Integers Z} `{IsCRing R} (f: R -> Z) (g: Z -> R) `{!IsSemiRingPreserving f} `{!IsSemiRingPreserving g} : IsInjective g. Proof. intros x y E. change (id x = id y). rewrite <-(to_ring_twice f g id x), <-(to_ring_twice f g id y). apply ap,E. Qed. | Lemma | Require Import Require Import Import NatPair.Instances. | Classes\theory\integers.v | to_ring_injective | 3,270 |
IntegersToRing Z2 := fun Z2 _ _ _ _ _ _ => integers_to_ring Z Z2 ∘ f^-1. | Definition | Require Import Require Import Import NatPair.Instances. | Classes\theory\integers.v | retract_is_int_to_ring | 3,271 |
x : (integers_to_ring Z R ∘ f^-1) x = h x. Proof. transitivity ((h ∘ (f ∘ f^-1)) x). - symmetry. apply (to_ring_unique (h ∘ f)). - unfold Compose. apply ap. apply eisretr. Qed. | Lemma | Require Import Require Import Import NatPair.Instances. | Classes\theory\integers.v | same_morphism | 3,272 |
Integers Z2 (U:=retract_is_int_to_ring). Proof. split;try apply _. - unfold integers_to_ring, retract_is_int_to_ring. apply _. - intros;apply same_morphism;apply _. Qed. | Lemma | Require Import Require Import Import NatPair.Instances. | Classes\theory\integers.v | retract_is_int | 3,273 |
(Z':Type@{U}) `{Integers@{U U U U U U U U} Z'} : forall (P : Rings.Operations -> Type), P (Rings.BuildOperations Z') -> P (Rings.BuildOperations Z). Proof. apply Rings.iso_leibnitz with (integers_to_ring Z' Z);apply _. Qed. | Lemma | Require Import Require Import Import NatPair.Instances. | Classes\theory\integers.v | from_int_stmt | 3,274 |
PropHolds ((1:Z) <>0). Proof. intros E. apply (rings.is_ne_0 (1:nat)). apply (injective (naturals_to_semiring nat Z)). exact E. Qed. | Instance | Require Import Require Import Import NatPair.Instances. | Classes\theory\integers.v | int_nontrivial | 3,275 |
(a b : IntAbs Z N) (z : Z) : int_abs Z N (ia:=a) z = int_abs Z N (ia:=b) z. Proof. unfold int_abs. destruct (int_abs_sig Z N (IntAbs:=a) z) as [[n1 E1]|[n1 E1]]; destruct (int_abs_sig Z N (IntAbs:=b) z) as [[n2 E2]|[n2 E2]]. - apply (injective (naturals_to_semiring N Z)). path_via z. - assert (E : n1 + n2 = 0);[|path_via 0;[|symmetry]; apply (naturals.zero_sum _ _ E)]. apply (injective (naturals_to_semiring N Z)). rewrite preserves_0,preserves_plus. rewrite E1,E2. apply plus_negate_r. - assert (E : n1 + n2 = 0);[|path_via 0;[|symmetry]; apply (naturals.zero_sum _ _ E)]. apply (injective (naturals_to_semiring N Z)). rewrite preserves_0,preserves_plus. rewrite E1,E2. apply plus_negate_l. - apply (injective (naturals_to_semiring N Z)). path_via (- z). Qed. | Lemma | Require Import | Classes\theory\int_abs.v | int_abs_unique | 3,276 |
x : (0 ≤ x /\ f (int_abs Z N x) = x) |_| (x ≤ 0 /\ f (int_abs Z N x) = -x). Proof. unfold int_abs. destruct (int_abs_sig Z N x) as [[n E]|[n E]]. - left. rewrite <-E. split. + eapply @to_semiring_nonneg;apply _. + apply (naturals.to_semiring_unique_alt _ _). - right. split. + apply flip_nonpos_negate. rewrite <-E. eapply @to_semiring_nonneg;apply _. + rewrite <-E. apply (naturals.to_semiring_unique_alt _ _). Qed. | Lemma | Require Import | Classes\theory\int_abs.v | int_abs_spec | 3,277 |
x : (sig (fun n : N => f n = x)) |_| (sig (fun n : N => f n = - x)). Proof. destruct (int_abs_spec x) as [[??]|[??]]; eauto. Qed. | Lemma | Require Import | Classes\theory\int_abs.v | int_abs_sig_alt | 3,278 |
n : int_abs Z N (f n) = n. Proof. apply (injective f). destruct (int_abs_spec (f n)) as [[? E]|[? E]];trivial. apply naturals.negate_to_ring. rewrite E, involutive. trivial. Qed. | Lemma | Require Import | Classes\theory\int_abs.v | int_abs_nat | 3,279 |
n : int_abs Z N (-f n) = n. Proof. apply (injective f). destruct (int_abs_spec (-f n)) as [[? E]|[? E]]. - symmetry. apply naturals.negate_to_ring. apply symmetry; trivial. - rewrite involutive in E. trivial. Qed. | Lemma | Require Import | Classes\theory\int_abs.v | int_abs_negate_nat | 3,280 |
x : int_abs Z N (-x) = int_abs Z N x. Proof. destruct (int_abs_spec x) as [[_ E]|[_ E]]. - path_via (int_abs Z N (- f (int_abs Z N x))). apply int_abs_negate_nat. - rewrite <-E. apply int_abs_nat. Qed. | Lemma | Require Import | Classes\theory\int_abs.v | int_abs_negate | 3,281 |
x : int_abs Z N x = 0 <-> x = 0. Proof. split; intros E1. - destruct (int_abs_spec x) as [[_ E2]|[_ E2]];[|apply flip_negate_0]; rewrite <-E2, E1, (preserves_0 (f:=f)); trivial. - rewrite E1, <-(preserves_0 (f:=f)). apply int_abs_nat. Qed. | Lemma | Require Import | Classes\theory\int_abs.v | int_abs_0_alt | 3,282 |
x : int_abs Z N x <> 0 <-> x <> 0. Proof. destruct (int_abs_0_alt x). split;intros E1 E2;auto. Qed. | Lemma | Require Import | Classes\theory\int_abs.v | int_abs_ne_0 | 3,283 |
int_abs Z N 0 = 0. Proof. apply int_abs_0_alt;trivial. Qed. | Lemma | Require Import | Classes\theory\int_abs.v | int_abs_0 | 3,284 |
x : 0 ≤ x -> f (int_abs Z N x) = x. Proof. intros E1. destruct (int_abs_spec x) as [[n E2]|[n E2]];trivial. assert (Hrw : x = 0). - apply (antisymmetry (<=));trivial. - rewrite Hrw,int_abs_0, (preserves_0 (f:=f)). trivial. Qed. | Lemma | Require Import | Classes\theory\int_abs.v | int_abs_nonneg | 3,285 |
x : x ≤ 0 -> f (int_abs Z N x) = -x. Proof. intros E. rewrite <-int_abs_negate, int_abs_nonneg; auto. apply flip_nonpos_negate. trivial. Qed. | Lemma | Require Import | Classes\theory\int_abs.v | int_abs_nonpos | 3,286 |
int_abs Z N 1 = 1. Proof. apply (injective f). rewrite (preserves_1 (f:=f)). apply int_abs_nonneg; solve_propholds. Qed. | Lemma | Require Import | Classes\theory\int_abs.v | int_abs_1 | 3,287 |
x y : 0 ≤ x -> 0 ≤ y -> int_abs Z N (x + y) = int_abs Z N x + int_abs Z N y. Proof. intros. apply (injective f). rewrite (preserves_plus (f:=f)), !int_abs_nonneg;auto. apply nonneg_plus_compat;trivial. Qed. | Lemma | Require Import | Classes\theory\int_abs.v | int_abs_nonneg_plus | 3,288 |
x y : int_abs Z N (x * y) = int_abs Z N x * int_abs Z N y. Proof. apply (injective f). rewrite (preserves_mult (f:=f)). destruct (int_abs_spec x) as [[? Ex]|[? Ex]], (int_abs_spec y) as [[? Ey]|[? Ey]]; rewrite Ex, Ey. - rewrite int_abs_nonneg;trivial. apply nonneg_mult_compat;trivial. - rewrite int_abs_nonpos. + apply negate_mult_distr_r. + apply nonneg_nonpos_mult;trivial. - rewrite int_abs_nonpos. + apply negate_mult_distr_l. + apply nonpos_nonneg_mult;trivial. - rewrite int_abs_nonneg. + symmetry;apply negate_mult_negate. + apply nonpos_mult;trivial. Qed. | Lemma | Require Import | Classes\theory\int_abs.v | int_abs_mult | 3,289 |
`{IsJoinPreserving L K f} x y : f (x ⊔ y) = f x ⊔ f y. Proof. apply preserves_sg_op. Qed. | Lemma | Require Import | Classes\theory\lattices.v | preserves_join | 3,290 |
`{IsBoundedJoinPreserving L K f} : f ⊥ = ⊥. Proof. apply preserves_mon_unit. Qed. | Lemma | Require Import | Classes\theory\lattices.v | preserves_bottom | 3,291 |
`{IsMeetPreserving L K f} x y : f (x ⊓ y) = f x ⊓ f y. Proof. apply preserves_sg_op. Qed. | Lemma | Require Import | Classes\theory\lattices.v | preserves_meet | 3,292 |
LeftIdentity (⊔) ⊥ := _. | Instance | Require Import | Classes\theory\lattices.v | join_bottom_l | 3,293 |
RightIdentity (⊔) ⊥ := _. | Instance | Require Import | Classes\theory\lattices.v | join_bottom_r | 3,294 |
x y : x ⊓ (x ⊔ y) = x := absorption x y. | Definition | Require Import | Classes\theory\lattices.v | meet_join_absorption | 3,295 |
x y : x ⊔ (x ⊓ y) = x := absorption x y. | Definition | Require Import | Classes\theory\lattices.v | join_meet_absorption | 3,296 |
LeftDistribute (⊔) (⊓). Proof. exact ( _). Qed. | Instance | Require Import | Classes\theory\lattices.v | join_meet_distr_l | 3,297 |
x y z : (x ⊓ y) ⊔ (x ⊓ z) ⊔ (y ⊓ z) = (x ⊔ y) ⊓ (x ⊔ z) ⊓ (y ⊔ z). Proof. rewrite (distribute_r x y (x ⊓ z)), join_meet_absorption. rewrite (distribute_r _ _ (y ⊓ z)). rewrite (distribute_l x y z). rewrite (commutativity y (x ⊓ z)), <-(simple_associativity _ y). rewrite join_meet_absorption. rewrite (distribute_r x z y). rewrite (commutativity (f:=join) z y). rewrite (commutativity (x ⊔ y) (x ⊔ z)). rewrite simple_associativity, <-(simple_associativity (x ⊔ z)). rewrite (idempotency _ _). rewrite (commutativity (x ⊔ z) (x ⊔ y)). reflexivity. Qed. | Lemma | Require Import | Classes\theory\lattices.v | distribute_alt | 3,298 |
IsSemiLattice B. Proof. split. - apply (projected_com_sg f). assumption. - repeat intro; apply (injective f). rewrite !op_correct, (idempotency (+) _). reflexivity. Qed. | Lemma | Require Import | Classes\theory\lattices.v | projected_sl | 3,299 |