Datasets:

phanerozoic

/

Coq-HoTT

like 2

F_ind2@{i j} (P : F -> F -> Type@{i}) {sP : forall x y, IsHProp (P x y)} (dclass : forall x y : Frac R, P (' x) (' y)) : forall x y, P x y. Proof. apply (@F_ind (fun x => forall y, _)). - intros;apply istrunc_forall@{UR i j}. - intros x. apply (F_ind _);intros y. apply dclass. Qed.

Definition

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

F_ind2@

F_ind3@{i j} (P : F -> F -> F -> Type@{i}) {sP : forall x y z, IsHProp (P x y z)} (dclass : forall x y z : Frac R, P (' x) (' y) (' z)) : forall x y z, P x y z. Proof. apply (@F_ind (fun x => forall y z, _)). - intros;apply istrunc_forall@{UR j j}. - intros x. apply (F_ind2@{i j} _). auto. Qed.

Definition

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

F_ind3@

F_rec@{i} {T : Type@{i} } {sT : IsHSet T} : forall (dclass : Frac R -> T) (dequiv : forall x y, equiv x y -> dclass x = dclass y), F -> T := quotient_rec equiv.

Definition

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

F_rec@

T sT dclass dequiv x : @F_rec T sT dclass dequiv (' x) = dclass x := 1.

Definition

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

F_rec_compute

F_rec2@{i j} {T:Type@{i} } {sT : IsHSet T} : forall (dclass : Frac R -> Frac R -> T) (dequiv : forall x1 x2, equiv x1 x2 -> forall y1 y2, equiv y1 y2 -> dclass x1 y1 = dclass x2 y2), F -> F -> T := @quotient_rec2@{UR UR UR j i} _ _ _ _ _ (Build_HSet _).

Definition

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

F_rec2@

{T sT} dclass dequiv x y : @F_rec2 T sT dclass dequiv (' x) (' y) = dclass x y := 1. Global Instance F0@{} : Zero F := ('0 : F). Global Instance F1@{} : One F := ('1 : F). Global Instance Fplus@{} : Plus F. Proof. refine (F_rec2 (fun x y => ' (Frac.pl _ x y)) _). intros. apply path. apply Frac.pl_respect;trivial. Defined.

Definition

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

F_rec2_compute

Fplus_compute@{} q r : (' q) + (' r) = ' (Frac.pl _ q r) := 1. Global Instance Fneg@{} : Negate F. Proof. refine (F_rec (fun x => ' (Frac.neg _ x)) _). intros;apply path; eapply Frac.neg_respect;try apply _. trivial. Defined.

Definition

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

Fplus_compute@

Fneg_compute@{} q : - (' q) = ' (Frac.neg _ q) := 1. Global Instance Fmult@{} : Mult F. Proof. refine (F_rec2 (fun x y => ' (Frac.ml _ x y)) _). intros. apply path. apply Frac.ml_respect;trivial. Defined.

Definition

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

Fneg_compute@

Fmult_compute@{} q r : (' q) * (' r) = ' (Frac.

Definition

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

Fmult_compute@

Fmult_comm@{} : Commutative Fplus. Proof. hnf. apply (F_ind2 _). intros;apply path, Frac.pl_comm. Qed.

Instance

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

Fmult_comm@

F_ring@{} : IsCRing F. Proof. repeat split; first [change sg_op with mult; change mon_unit with 1| change sg_op with plus; change mon_unit with 0]. - apply _. - hnf. apply (F_ind3 _). intros;apply path. apply Frac.pl_assoc. - hnf. apply (F_ind _). intros;apply path, Frac.pl_0_l. - hnf. apply (F_ind _). intros;apply path, Frac.pl_0_r. - hnf. apply (F_ind _). intros;apply path, Frac.pl_neg_l. - hnf;intros. rewrite (commutativity (f:=plus)). revert x;apply (F_ind _). intros;apply path, Frac.pl_neg_l. - apply _. - apply _. - hnf; apply (F_ind3 _). intros;apply path, Frac.ml_assoc. - hnf. apply (F_ind _). intros;apply path. red;simpl. rewrite 2!mult_1_l. reflexivity. - hnf. apply (F_ind _). intros;apply path. red;simpl. rewrite 2!mult_1_r. reflexivity. - hnf; apply (F_ind2 _). intros;apply path. red;simpl. rewrite (mult_comm (num y)), (mult_comm (den y)). reflexivity. - hnf. apply (F_ind3 _). intros a b c;apply path. red;simpl. rewrite <-!(mult_assoc (num a)). rewrite <-plus_mult_distr_l. rewrite <-(mult_assoc (num a)). apply ap. rewrite (mult_comm (den a) (den c)), (mult_comm (den a) (den b)). rewrite (mult_assoc (num b)), (mult_assoc (num c)). rewrite <-plus_mult_distr_r. rewrite <-(mult_assoc _ (den a) (den a * _)). apply ap. rewrite (mult_comm (den b)), <-mult_assoc. apply ap. rewrite (mult_comm (den a)). apply associativity. Qed.

Instance

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

F_ring@

classes_eq_related@{} : forall q r, ' q = ' r -> equiv q r. Proof. apply classes_eq_related@{UR UR Ularge UR Ularge};apply _. Qed.

Lemma

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

classes_eq_related@

class_neq@{} : forall q r, ~ (equiv q r) -> ' q <> ' r. Proof. intros q r E1 E2;apply E1;apply classes_eq_related, E2. Qed.

Lemma

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

class_neq@

classes_neq_related@{} : forall q r, ' q <> ' r -> ~ (equiv q r). Proof. intros q r E1 E2;apply E1,path,E2. Qed.

Lemma

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

classes_neq_related@

dec_recip_0@{} : / 0 = 0. Proof. unfold dec_recip. simpl. unfold Frac.dec_rec;simpl. destruct (decide_rel paths 0 0) as [_|E]. - reflexivity. - destruct E;reflexivity. Qed.

Lemma

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

dec_recip_0@

dec_recip_nonzero_aux@{} : forall q, ' q <> 0 -> num q <> 0. Proof. intros q E;apply classes_neq_related in E. apply Frac.nonzero_num in E. trivial. Qed.

Lemma

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

dec_recip_nonzero_aux@

dec_recip_nonzero@{} : forall q (E : ' q <> 0), / (' q) = ' (frac (den q) (num q) (dec_recip_nonzero_aux q E)). Proof. intros. apply path. red;simpl. unfold Frac.dec_rec. apply classes_neq_related, Frac.nonzero_num in E. destruct (decide_rel paths (num q) 0) as [E'|?];[destruct E;apply E'|]. simpl. reflexivity. Qed.

Lemma

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

dec_recip_nonzero@

dec_class@{} : forall q r, Decidable (class q = class r). Proof. intros q r. destruct (dec (equiv q r)) as [E|E]. - left. apply path,E. - right. intros E'. apply E. apply (classes_eq_related _ _ E'). Defined.

Lemma

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

dec_class@

mult_num_den@{} q : ' q = (' num q) / ' den q. Proof. apply path. red. simpl. rewrite mult_1_l. unfold Frac.dec_rec. simpl. destruct (decide_rel paths (den q) 0) as [E|E];simpl. - destruct (den_ne_0 q E). - rewrite mult_1_r. reflexivity. Qed.

Lemma

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

mult_num_den@

recip_den_num@{} q : / ' q = (' den q) / 'num q. Proof. apply path;red;simpl. unfold Frac.dec_rec;simpl. destruct (decide_rel paths (num q) 0) as [E|E];simpl. - rewrite (mult_0_r (Azero:=Azero)), 2!mult_0_l. reflexivity. - rewrite mult_1_l,mult_1_r. reflexivity. Qed.

Lemma

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

recip_den_num@

lift@{} : F R1 -> F R2. Proof. apply (F_rec (fun x => class (Frac.lift f x))). intros;apply path,Frac.lift_respects;trivial. Defined.

Definition

Require Import HoTT.HIT.quotient. Require Import Import Frac.

Classes\implementations\field_of_fractions.v

lift@

(X Y : HProp) : HProp := Build_HProp (X * Y).

Definition

Require Import

Classes\implementations\hprop_lattice.v

hand

Commutative lor. Proof. intros ??. apply path_iff_hprop; hor_intros; apply tr; auto. Defined.

Instance

Require Import

Classes\implementations\hprop_lattice.v

commutative_hor

Commutative hand. Proof. intros ??. apply path_hprop. apply equiv_prod_symm. Defined.

Instance

Require Import

Classes\implementations\hprop_lattice.v

commutative_hand

Associative lor. Proof. intros ???. apply path_iff_hprop; hor_intros; apply tr; ((by auto) || (left; apply tr) || (right; apply tr)); auto. Defined.

Instance

Require Import

Classes\implementations\hprop_lattice.v

associative_hor

Associative hand. Proof. intros ???. apply path_hprop. apply equiv_prod_assoc. Defined.

Instance

Require Import

Classes\implementations\hprop_lattice.v

associative_hand

BinaryIdempotent lor. Proof. intros ?. compute. apply path_iff_hprop; hor_intros; auto. by apply tr, inl. Defined.

Instance

Require Import

Classes\implementations\hprop_lattice.v

idempotent_hor

BinaryIdempotent hand. Proof. intros ?. apply path_iff_hprop. - intros [a _] ; apply a. - intros a; apply (pair a a). Defined.

Instance

Require Import

Classes\implementations\hprop_lattice.v

idempotent_hand

LeftIdentity lor False_hp. Proof. intros ?. apply path_iff_hprop; hor_intros; try contradiction || assumption. by apply tr, inr. Defined.

Instance

Require Import

Classes\implementations\hprop_lattice.v

leftidentity_hor

RightIdentity lor False_hp. Proof. intros ?. apply path_iff_hprop; hor_intros; try contradiction || assumption. by apply tr, inl. Defined.

Instance

Require Import

Classes\implementations\hprop_lattice.v

rightidentity_hor

LeftIdentity hand Unit_hp. Proof. intros ?. apply path_trunctype, prod_unit_l. Defined.

Instance

Require Import

Classes\implementations\hprop_lattice.v

leftidentity_hand

RightIdentity hand Unit_hp. Proof. intros ?. apply path_trunctype, prod_unit_r. Defined.

Instance

Require Import

Classes\implementations\hprop_lattice.v

rightidentity_hand

Absorption lor hand. Proof. intros ??. apply path_iff_hprop. - intros X; strip_truncations. destruct X as [? | [? _]]; assumption. - intros ?. by apply tr, inl. Defined.

Instance

Require Import

Classes\implementations\hprop_lattice.v

absorption_hor_hand

Absorption hand lor. Proof. intros ??. apply path_iff_hprop. - intros [? _]; assumption. - intros ?. split. * assumption. * by apply tr, inl. Defined.

Instance

Require Import

Classes\implementations\hprop_lattice.v

absorption_hand_hor

(N : Type) := C { pos : N ; neg : N }.

Record

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

T

fun x y => pos x + neg y = pos y + neg x. Global Instance equiv_is_equiv_rel@{} : EquivRel . Proof. split. - hnf. reflexivity. - hnf. unfold . intros ??;apply symmetry. - hnf. unfold . intros a b c E1 E2. apply (left_cancellation (+) (neg b)). rewrite (plus_assoc (neg b) (pos a)). rewrite (plus_comm (neg b) (pos a)), E1. rewrite (plus_comm (pos b)). rewrite <-plus_assoc. rewrite E2. rewrite (plus_comm (pos c) (neg b)). rewrite plus_assoc. rewrite (plus_comm (neg a)). rewrite <-plus_assoc. rewrite (plus_comm (neg a)). reflexivity. Qed.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

equiv

Plus (T N) := fun x y => C (pos x + pos y) (neg x + neg y).

Instance

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

pl

Mult (T N) := fun x y => C (pos x * pos y + neg x * neg y) (pos x * neg y + neg x * pos y).

Instance

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

ml

Negate (T N) := fun x => C (neg x) (pos x).

Instance

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

opp

Zero (T N) := C 0 0.

Instance

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

SR0

One (T N) := C 1 0.

Instance

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

SR1

forall q1 q2, equiv q1 q2 -> forall r1 r2, equiv r1 r2 -> equiv (q1 + r1) (q2 + r2). Proof. unfold equiv;simpl. intros q1 q2 Eq r1 r2 Er. rewrite (plus_assoc _ (neg q2)). rewrite <-(plus_assoc (pos q1)). rewrite (plus_comm (pos r1)). rewrite (plus_assoc (pos q1)). rewrite Eq. rewrite <-(plus_assoc _ (pos r1)). rewrite Er. rewrite plus_assoc. rewrite <-(plus_assoc (pos q2)). rewrite (plus_comm (neg q1)). rewrite !plus_assoc. reflexivity. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

pl_respects

forall q1 q2, equiv q1 q2 -> forall r1 r2, equiv r1 r2 -> equiv (q1 * r1) (q2 * r2). Proof. intros q1 q2 Eq r1 r2 Er. transitivity (q1 * r2);unfold equiv in *;simpl. - transitivity (pos q1 * (pos r1 + neg r2) + neg q1 * (neg r1 + pos r2)). + rewrite 2!plus_mult_distr_l. rewrite <-2!plus_assoc. apply ap. rewrite 2!plus_assoc. rewrite (plus_comm (neg q1 * neg r1)). reflexivity. + rewrite Er. rewrite plus_mult_distr_l. rewrite (plus_comm (neg r1)). rewrite <-Er. rewrite plus_mult_distr_l. rewrite <-2!plus_assoc. apply ap. rewrite (plus_comm (neg q1 * pos r1)). rewrite 2!plus_assoc. rewrite (plus_comm (pos q1 * neg r1)). reflexivity. - transitivity ((pos q1 + neg q2) * pos r2 + (neg q1 + pos q2) * neg r2). + rewrite 2!plus_mult_distr_r. rewrite <-2!plus_assoc;apply ap. rewrite (plus_comm (pos q2 * neg r2)). rewrite 2!plus_assoc. rewrite (plus_comm (neg q1 * neg r2)). reflexivity. + rewrite Eq,plus_mult_distr_r. rewrite (plus_comm (neg q1)),<-Eq,plus_mult_distr_r. rewrite <-2!plus_assoc. apply ap. rewrite plus_assoc,(plus_comm (neg q1 * pos r2)). apply plus_comm. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

ml_respects

forall q1 q2, equiv q1 q2 -> equiv (opp q1) (opp q2). Proof. unfold equiv;simpl. intros q1 q2 E. rewrite !(plus_comm (neg _)). symmetry. apply E. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

opp_respects

Le (T N) := fun a b => pos a + neg b <= pos b + neg a.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Tle

Lt (T N) := fun a b => pos a + neg b < pos b + neg a.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Tlt

Apart (T N) := fun a b => apart (pos a + neg b) (pos b + neg a). Global Instance Tle_hprop@{} : is_mere_relation (T N) Tle. Proof. intros;unfold Tle. apply full_pseudo_srorder_le_hprop. Qed.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Tapart

le_respects_aux@{} : forall q1 q2, equiv q1 q2 -> forall r1 r2, equiv r1 r2 -> Tle q1 r1 -> Tle q2 r2. Proof. unfold equiv,Tle;intros [pa na] [pb nb] Eq [pc nc] [pd nd] Er E;simpl in *. apply (order_reflecting (+ (pc + na))). assert (Erw : pb + nd + (pc + na) = (pb + na) + (pc + nd)) by ring_with_nat. rewrite Erw,<-Eq,Er;clear Erw. assert (Erw : pa + nb + (pd + nc) = pd + nb + (pa + nc)) by ring_with_nat. rewrite Erw. apply (order_preserving _), E. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

le_respects_aux@

le_respects@{} : forall q1 q2, equiv q1 q2 -> forall r1 r2, equiv r1 r2 -> Tle q1 r1 <~> Tle q2 r2. Proof. intros. apply equiv_iff_hprop_uncurried. split;apply le_respects_aux; trivial;apply symmetry;trivial. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

le_respects@

lt_respects_aux@{} : forall q1 q2, equiv q1 q2 -> forall r1 r2, equiv r1 r2 -> Tlt q1 r1 -> Tlt q2 r2. Proof. unfold equiv,Tlt;intros [pa na] [pb nb] Eq [pc nc] [pd nd] Er E;simpl in *. apply (strictly_order_reflecting (+ (pc + na))). assert (Erw : pb + nd + (pc + na) = (pb + na) + (pc + nd)) by ring_with_nat. rewrite Erw,<-Eq,Er;clear Erw. assert (Erw : pa + nb + (pd + nc) = pd + nb + (pa + nc)) by ring_with_nat. rewrite Erw. apply (strictly_order_preserving _), E. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

lt_respects_aux@

lt_respects@{} : forall q1 q2, equiv q1 q2 -> forall r1 r2, equiv r1 r2 -> Tlt q1 r1 <~> Tlt q2 r2. Proof. intros. apply equiv_iff_hprop_uncurried. split;apply lt_respects_aux; trivial;apply symmetry;trivial. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

lt_respects@

apart_cotrans@{} : CoTransitive Tapart. Proof. hnf. unfold Tapart. intros q1 q2 Eq r. apply (strong_left_cancellation (+) (neg r)) in Eq. apply (merely_destruct (cotransitive Eq (pos r + neg q1 + neg q2))); intros [E|E];apply tr. - left. apply (strong_extensionality (+ (neg q2))). assert (Hrw : pos q1 + neg r + neg q2 = neg r + (pos q1 + neg q2)) by ring_with_nat. rewrite Hrw;clear Hrw. trivial. - right. apply (strong_extensionality (+ (neg q1))). assert (Hrw : pos r + neg q2 + neg q1 = pos r + neg q1 + neg q2) by ring_with_nat. rewrite Hrw;clear Hrw. assert (Hrw : pos q2 + neg r + neg q1 = neg r + (pos q2 + neg q1)) by ring_with_nat. rewrite Hrw;clear Hrw. trivial. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

apart_cotrans@

apart_respects_aux@{} : forall q1 q2, equiv q1 q2 -> forall r1 r2, equiv r1 r2 -> Tapart q1 r1 -> Tapart q2 r2. Proof. assert (forall q1 q2, equiv q1 q2 -> forall r, Tapart q1 r -> Tapart q2 r) as E. - intros q1 q2 Eq r Er. unfold Tapart,equiv in *. apply (strong_extensionality (+ (neg q1))). assert (Hrw : pos q2 + neg r + neg q1 = (pos q2 + neg q1) + neg r) by ring_with_nat. rewrite Hrw;clear Hrw. rewrite <-Eq. assert (Hrw : pos q1 + neg q2 + neg r = neg q2 + (pos q1 + neg r)) by ring_with_nat. rewrite Hrw;clear Hrw. assert (Hrw : pos r + neg q2 + neg q1 = neg q2 + (pos r + neg q1)) by ring_with_nat;rewrite Hrw;clear Hrw. apply (strong_left_cancellation _ _),Er. - intros ?? Eq ?? Er E'. apply E with q1;trivial. apply symmetry;apply E with r1;apply symmetry;trivial. apply symmetry;trivial. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

apart_respects_aux@

forall q1 q2, equiv q1 q2 -> forall r1 r2, equiv r1 r2 -> Tapart q1 r1 <~> Tapart q2 r2. Proof. intros ?? Eq ?? Er. apply equiv_iff_hprop_uncurried. split;apply apart_respects_aux; trivial;apply symmetry;trivial. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

apart_respects

to_ring@{} : T N -> B. Proof. intros p. exact (naturals_to_semiring N B (pos p) - naturals_to_semiring N B (neg p)). Defined.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

to_ring@

to_ring_respects@{} : forall a b, equiv a b -> to_ring a = to_ring b. Proof. unfold equiv;intros [pa na] [pb nb] E. unfold to_ring;simpl in *. apply (left_cancellation (+) (naturals_to_semiring N B na + naturals_to_semiring N B nb)). path_via (naturals_to_semiring N B pa + naturals_to_semiring N B nb + 0); [rewrite <-(plus_negate_r (naturals_to_semiring N B na));ring_with_nat |rewrite plus_0_r]. path_via (naturals_to_semiring N B pb + naturals_to_semiring N B na + 0); [rewrite plus_0_r| rewrite <-(plus_negate_r (naturals_to_semiring N B nb));ring_with_nat]. rewrite <-2!preserves_plus. apply ap,E. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

to_ring_respects@

FullPartialOrder Ale Alt := fullpseudo_fullpartial@{UN UN UN UN UN UN UN Ularge}.

Instance

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

N_fullpartial

Z@{} : Type@{UN} := @quotient _ PairT.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z@

{x y} : PairT.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_path

{x y} : Z_of_pair x = Z_of_pair y -> PairT.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

related_path

Z_rect@{i} (P : Z -> Type@{i}) {sP : forall x, IsHSet (P x)} (dclass : forall x : PairT.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_rect@

P {sP} dclass dequiv x : @Z_rect P sP dclass dequiv (Z_of_pair x) = dclass x := 1.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_compute

P {sP} dclass dequiv q r (E : PairT.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_compute_path

Z_ind@{i} (P : Z -> Type@{i}) {sP : forall x : Z, IsHProp (P x)} (dclass : forall x : PairT.T N, P (cast (PairT.T N) Z x)) : forall x : Z, P x. Proof. apply (Z_rect@{i} P dclass). intros;apply path_ishprop@{i}. Defined.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_ind@

(P : Z -> Z -> Type) {sP : forall x y, IsHProp (P x y)} (dclass : forall x y : PairT.T N, P (' x) (' y)) : forall x y, P x y. Proof. apply (Z_ind (fun x => forall y, _));intros x. apply (Z_ind _);intros y. apply dclass. Defined.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_ind2

Z_ind3@{i j} (P : Z -> Z -> Z -> Type@{i}) {sP : forall x y z : Z, IsHProp (P x y z)} (dclass : forall x y z : PairT.T N, P (' x) (' y) (' z)) : forall x y z : Z, P x y z. Proof. apply (@Z_ind (fun x => forall y z, _));intros x. 2:apply (Z_ind2@{i j} _);auto. apply (@istrunc_forall@{UN j j} _). intros. apply istrunc_forall@{UN i j}. Defined.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_ind3@

Z_rec@{i} {T : Type@{i} } {sT : IsHSet T} : forall (dclass : PairT.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_rec@

T sT dclass dequiv x : @Z_rec T sT dclass dequiv (' x) = dclass x := 1.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_rec_compute

Z_rec2@{i j} {T:Type@{i} } {sT : IsHSet T} : forall (dclass : PairT.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_rec2@

{T sT} dclass dequiv x y : @Z_rec2 T sT dclass dequiv (' x) (' y) = dclass x y := 1.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_rec2_compute

`{DecidablePaths N} : forall q r : PairT.T N, Decidable (' q = ' r). Proof. intros q r. destruct (dec (PairT.equiv q r)) as [E|E]. - left. apply Z_path,E. - right. intros E'. apply E. apply (related_path E'). Defined.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

dec_Z_of_pair

q r : (' q) + (' r) = ' (PairT.pl _ q r) := 1. Global Instance Z_mult@{} : Mult Z. Proof. refine (Z_rec2 (fun x y => ' (PairT.ml@{UN UNalt} _ x y)) _). intros;apply Z_path;eapply PairT.ml_respects;trivial. Defined.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_plus_compute

q r : (' q) * (' r) = ' (PairT.ml _ q r) := 1. Global Instance Z_negate@{} : Negate Z. Proof. red. apply (Z_rec (fun x => ' (PairT.opp@{UN UNalt} _ x))). intros;apply Z_path;eapply PairT.opp_respects;trivial. Defined.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_mult_compute

q : - (' q) = ' (PairT.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_negate_compute

Z_ring@{} : IsCRing Z. Proof. repeat split. 1,8: exact _. all: first [change sg_op with mult; change mon_unit with 1 | change sg_op with plus; change mon_unit with 0]; hnf. - apply (Z_ind3 _). intros a b c;apply Z_path;red;simpl. rewrite !plus_assoc. reflexivity. - apply (Z_ind _). intros a;apply Z_path;red;simpl. rewrite !plus_0_l. reflexivity. - apply (Z_ind _). intros a;apply Z_path;red;simpl. rewrite !plus_0_r. reflexivity. - apply (Z_ind _). intros a;apply Z_path;red;simpl. rewrite plus_0_l,plus_0_r. apply plus_comm. - apply (Z_ind _). intros a;apply Z_path;red;simpl. rewrite plus_0_l,plus_0_r. apply plus_comm. - apply (Z_ind2 _). intros a b;apply Z_path;red;simpl. apply ap011;apply plus_comm. - apply (Z_ind3 _). intros [pa na] [pb nb] [pc nc];apply Z_path;red;simpl. ring_with_nat. - apply (Z_ind _). intros;apply Z_path;red;simpl. ring_with_nat. - apply (Z_ind _). intros;apply Z_path;red;simpl. ring_with_nat. - apply (Z_ind2 _). intros;apply Z_path;red;simpl. ring_with_nat. - apply (Z_ind3 _). intros [pa na] [pb nb] [pc nc];apply Z_path;red;simpl. ring_with_nat. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_ring@

Z_of_N_morphism@{} : IsSemiRingPreserving (cast N Z). Proof. repeat (constructor; try apply _). - intros x y. apply Z_path. red. simpl. ring_with_nat. - intros x y. apply Z_path. red;simpl. ring_with_nat. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_of_N_morphism@

Npair_splits@{} : forall n m : N, ' (PairT.C n m) = ' n + - ' m. Proof. intros. apply Z_path;red;simpl. ring_with_nat. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Npair_splits@

Zle_HProp@{} : Z -> Z -> HProp@{UN}. Proof. apply (@Z_rec2@{Ularge Ularge} _ (@trunctype_istrunc@{Ularge} _ _) (fun q r => Build_HProp (PairT.Tle q r))). intros. apply path_hprop. simpl. apply (PairT.le_respects _);trivial. Defined.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Zle_HProp@

Zle_def@{} : forall a b : PairT.T N, @paths@{Uhuge} Type@{UN} (' a <= ' b) (PairT.Tle@{UN UNalt} a b). Proof. intros; exact idpath. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Zle_def@

Z_partial_order' : PartialOrder Zle. Proof. split;[apply _|apply _|split|]. - hnf. apply (Z_ind _). intros. change (PairT.Tle x x). red. reflexivity. - hnf. apply (Z_ind3 (fun _ _ _ => _ -> _ -> _)). intros [pa na] [pb nb] [pc nc]. rewrite !Zle_def;unfold PairT.Tle;simpl. intros E1 E2. apply (order_reflecting (+ (nb + pb))). assert (Hrw : pa + nc + (nb + pb) = (pa + nb) + (pb + nc)) by abstract ring_with_nat. rewrite Hrw;clear Hrw. assert (Hrw : pc + na + (nb + pb) = (pb + na) + (pc + nb)) by abstract ring_with_nat. rewrite Hrw;clear Hrw. apply plus_le_compat;trivial. - hnf. apply (Z_ind2 (fun _ _ => _ -> _ -> _)). intros [pa na] [pb nb];rewrite !Zle_def;unfold PairT.Tle;simpl. intros E1 E2;apply Z_path;red;simpl. apply (antisymmetry le);trivial. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_partial_order'

Z_partial_order@{} : PartialOrder Zle := ltac:(first [exact Z_partial_order'@{Ularge Ularge Ularge Ularge Ularge}| exact Z_partial_order']).

Instance

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_partial_order@

Zle_cast_embedding' : OrderEmbedding (cast N Z). Proof. split;red. - intros. rewrite Zle_def. unfold PairT.Tle. simpl. rewrite 2!plus_0_r;trivial. - intros ??. rewrite Zle_def. unfold PairT.Tle. simpl. rewrite 2!plus_0_r;trivial. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Zle_cast_embedding'

Zle_plus_preserving_l' : forall z : Z, OrderPreserving ((+) z). Proof. red. apply (Z_ind3 (fun _ _ _ => _ -> _)). intros [pc nc] [pa na] [pb nb]. rewrite !Zle_def;unfold PairT.Tle;simpl. intros E. assert (Hrw : pc + pa + (nc + nb) = (pc + nc) + (pa + nb)) by ring_with_nat. rewrite Hrw;clear Hrw. assert (Hrw : pc + pb + (nc + na) = (pc + nc) + (pb + na)) by ring_with_nat. rewrite Hrw;clear Hrw. apply (order_preserving _),E. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Zle_plus_preserving_l'

Zle_plus_preserving_l@{} : forall z : Z, OrderPreserving ((+) z) := ltac:(first [exact Zle_plus_preserving_l'@{Ularge Ularge}| exact Zle_plus_preserving_l']).

Instance

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Zle_plus_preserving_l@

Zmult_nonneg' : forall x y : Z, PropHolds (0 ≤ x) -> PropHolds (0 ≤ y) -> PropHolds (0 ≤ x * y). Proof. unfold PropHolds. apply (Z_ind2 (fun _ _ => _ -> _ -> _)). intros [pa na] [pb nb]. rewrite !Zle_def;unfold PairT.Tle;simpl. rewrite !plus_0_l,!plus_0_r. intros E1 E2. destruct (decompose_le E1) as [a [Ea1 Ea2]], (decompose_le E2) as [b [Eb1 Eb2]]. rewrite Ea2, Eb2. apply compose_le with (a * b). - apply nonneg_mult_compat;trivial. - ring_with_nat. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Zmult_nonneg'

Zmult_nonneg@{} : forall x y : Z, PropHolds (0 ≤ x) -> PropHolds (0 ≤ y) -> PropHolds (0 ≤ x * y) := ltac:(first [exact Zmult_nonneg'@{Ularge Ularge Ularge}| exact Zmult_nonneg']). Global Z_order@{} : SemiRingOrder Zle. Proof. pose proof Z_ring; apply rings.from_ring_order; apply _. Qed.

Instance

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Zmult_nonneg@

Zlt_HProp@{} : Z -> Z -> HProp@{UN}. Proof. apply (@Z_rec2@{Ularge Ularge} _ (@trunctype_istrunc@{Ularge} _ _) (fun q r => Build_HProp (PairT.Tlt q r))). intros. apply path_hprop. simpl. apply (PairT.lt_respects _);trivial. Defined.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Zlt_HProp@

Zlt_def' : forall a b, ' a < ' b = PairT.Tlt a b. Proof. reflexivity. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Zlt_def'

Zlt_def@{i} := ltac:(first [exact Zlt_def'@{Uhuge i}|exact Zlt_def'@{i}]).

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Zlt_def@

Zlt_strict' : StrictOrder Zlt. Proof. split. - apply _. - change (forall x, x < x -> Empty). apply (Z_ind (fun _ => _ -> _)). intros [pa na];rewrite Zlt_def;unfold PairT.Tlt;simpl. apply irreflexivity,_. - hnf. apply (Z_ind3 (fun _ _ _ => _ -> _ -> _)). intros [pa na] [pb nb] [pc nc];rewrite !Zlt_def;unfold PairT.Tlt;simpl. intros E1 E2. apply (strictly_order_reflecting (+ (nb + pb))). assert (Hrw : pa + nc + (nb + pb) = (pa + nb) + (pb + nc)) by ring_with_nat. rewrite Hrw;clear Hrw. assert (Hrw : pc + na + (nb + pb) = (pb + na) + (pc + nb)) by ring_with_nat. rewrite Hrw;clear Hrw. apply plus_lt_compat;trivial. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Zlt_strict'

Zlt_strict@{} : StrictOrder Zlt := ltac:(first [exact Zlt_strict'@{Ularge Ularge Ularge Ularge Ularge}| exact Zlt_strict'@{}]).

Instance

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Zlt_strict@

plus_strict_order_preserving_l' : forall z : Z, StrictlyOrderPreserving ((+) z). Proof. red; apply (Z_ind3 (fun _ _ _ => _ -> _)). intros [pa na] [pb nb] [pc nc]. rewrite !Zlt_def;unfold PairT.Tlt;simpl. intros E. assert (Hrw : pa + pb + (na + nc) = (pa + na) + (pb + nc)) by ring_with_nat. rewrite Hrw;clear Hrw. assert (Hrw : pa + pc + (na + nb) = (pa + na) + (pc + nb)) by ring_with_nat. rewrite Hrw;clear Hrw. apply (strictly_order_preserving _),E. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

plus_strict_order_preserving_l'

Zplus_strict_order_preserving_l@{} : forall z : Z, StrictlyOrderPreserving ((+) z) := ltac:(first [exact plus_strict_order_preserving_l'@{Ularge Ularge}| exact plus_strict_order_preserving_l'@{}]).

Instance

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Zplus_strict_order_preserving_l@

Zmult_pos' : forall x y : Z, PropHolds (0 < x) -> PropHolds (0 < y) -> PropHolds (0 < x * y). Proof. unfold PropHolds. apply (Z_ind2 (fun _ _ => _ -> _ -> _)). intros [pa na] [pb nb]. rewrite !Zlt_def;unfold PairT.Tlt;simpl. rewrite !plus_0_l,!plus_0_r. intros E1 E2. destruct (decompose_lt E1) as [a [Ea1 Ea2]], (decompose_lt E2) as [b [Eb1 Eb2]]. rewrite Ea2, Eb2. apply compose_lt with (a * b). - apply pos_mult_compat;trivial. - ring_with_nat. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Zmult_pos'

Zmult_pos@{} : forall x y : Z, PropHolds (0 < x) -> PropHolds (0 < y) -> PropHolds (0 < x * y) := ltac:(first [exact Zmult_pos'@{Ularge Ularge Ularge}| exact Zmult_pos'@{}]). Global Z_strict_srorder : StrictSemiRingOrder Zlt. Proof. pose proof Z_ring; apply from_strict_ring_order; apply _. Qed.

Instance

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Zmult_pos@

Zapart_HProp@{} : Z -> Z -> HProp@{UN}. Proof. apply (@Z_rec2@{Ularge Ularge} _ _ (fun q r => Build_HProp (PairT.Tapart q r))). intros. apply path_hprop. simpl. apply (PairT.apart_respects _);trivial. Defined.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Zapart_HProp@

Zapart_def' : forall a b, apart (' a) (' b) = PairT.Tapart a b. Proof. reflexivity. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Zapart_def'

Zapart_def@{i} := ltac:(first [exact Zapart_def'@{Uhuge i}| exact Zapart_def'@{i}]). Global Instance ishprop_Zapart : is_mere_relation _ Zapart. Proof. unfold Zapart;exact _. Qed.

Definition

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Zapart_def@

Z_trivial_apart' `{!TrivialApart N} : TrivialApart Z. Proof. split;[exact _|idtac]. apply (Z_ind2 _). intros [pa na] [pb nb];rewrite Zapart_def;unfold PairT.Tapart;simpl. split;intros E1. - intros E2. apply related_path in E2. red in E2;simpl in E2. apply trivial_apart in E1. auto. - apply trivial_apart. intros E2. apply E1,Z_path. red;simpl. trivial. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_trivial_apart'

Z_is_apart' : IsApart Z. Proof. split. - apply _. - apply _. - hnf. apply (Z_ind2 (fun _ _ => _ -> _)). intros [pa na] [pb nb];rewrite !Zapart_def;unfold PairT.Tapart;simpl. apply symmetry. - hnf. intros x y E z;revert x y z E. apply (Z_ind3 (fun _ _ _ => _ -> _)). intros a b c;rewrite !Zapart_def;unfold PairT.Tapart;simpl. intros E1. apply (strong_left_cancellation (+) (PairT.neg c)) in E1. eapply (merely_destruct (cotransitive E1 _));intros [E2|E2];apply tr. + left. apply (strong_extensionality (+ (PairT.neg b))). assert (Hrw : PairT.pos a + PairT.neg c + PairT.neg b = PairT.neg c + (PairT.pos a + PairT.neg b)) by ring_with_nat;rewrite Hrw;exact E2. + right. apply (strong_extensionality (+ (PairT.neg a))). assert (Hrw : PairT.pos c + PairT.neg b + PairT.neg a = PairT.pos c + PairT.neg a + PairT.neg b) by ring_with_nat;rewrite Hrw;clear Hrw. assert (Hrw : PairT.pos b + PairT.neg c + PairT.neg a = PairT.neg c + (PairT.pos b + PairT.neg a)) by ring_with_nat;rewrite Hrw;clear Hrw. trivial. - apply (Z_ind2 _). intros a b;rewrite Zapart_def;unfold PairT.Tapart. split;intros E. + apply Z_path;red. apply tight_apart,E. + apply related_path in E. apply tight_apart,E. Qed.

Lemma

Require Import HoTT.HIT.quotient Require Import Import ring_quote.Quoting.Instances. Module Import PairT.

Classes\implementations\natpair_integers.v

Z_is_apart'