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Definition ConstructiveDivisionRig := βˆ‘ X : rig, βˆ‘ R : tightap X, isapbinop (X := (pr1 (pr1 X)) ,, R) BinaryOperations.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
ConstructiveDivisionRig
700
Definition ConstructiveDivisionRig_rig : ConstructiveDivisionRig β†’ rig := pr1.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
ConstructiveDivisionRig_rig
701
Definition ConstructiveDivisionRig_apsetwith2binop : ConstructiveDivisionRig β†’ apsetwith2binop. Proof. intros X. exists (pr1 (pr1 (pr1 X)),,(pr1 (pr2 X))). split. exact (_,,(pr1 (pr2 (pr2 X)))). exact (_,,(pr1 (pr2 (pr2 (pr2 X))))). Defined.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
ConstructiveDivisionRig_apsetwith2binop
702
Definition CDRap {X : ConstructiveDivisionRig} : hrel X := Ξ» x y : X, (pr1 (pr2 X)) x y.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CDRap
703
Definition CDRzero {X : ConstructiveDivisionRig} : X := 0%rig.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CDRzero
704
Definition CDRone {X : ConstructiveDivisionRig} : X := 1%rig.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CDRone
705
Definition CDRplus {X : ConstructiveDivisionRig} : binop X := Ξ» x y : X, op1 (X := ConstructiveDivisionRig_apsetwith2binop X) x y.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CDRplus
706
Definition CDRmult {X : ConstructiveDivisionRig} : binop X := Ξ» x y : X, op2 (X := ConstructiveDivisionRig_apsetwith2binop X) x y.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CDRmult
707
Definition CDRinv {X : ConstructiveDivisionRig} (x : X) (Hx0 : x β‰  0) : X := (pr1 (pr2 (pr2 (pr2 (pr2 (pr2 X)))) x Hx0)).
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CDRinv
708
Definition CDRdiv {X : ConstructiveDivisionRig} (x y : X) (Hy0 : y β‰  0) : X := CDRmult x (CDRinv y Hy0).
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CDRdiv
709
Lemma isirrefl_CDRap : ∏ x : X, Β¬ (x β‰  x). Proof. exact (pr1 (pr1 (pr2 (pr1 (pr2 X))))). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isirrefl_CDRap
710
Lemma issymm_CDRap : ∏ x y : X, x β‰  y β†’ y β‰  x. Proof. exact (pr1 (pr2 (pr1 (pr2 (pr1 (pr2 X)))))). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
issymm_CDRap
711
Lemma iscotrans_CDRap : ∏ x y z : X, x β‰  z β†’ x β‰  y ∨ y β‰  z. Proof. exact (pr2 (pr2 (pr1 (pr2 (pr1 (pr2 X)))))). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
iscotrans_CDRap
712
Lemma istight_CDRap : ∏ x y : X, Β¬ (x β‰  y) β†’ x = y. Proof. exact (pr2 (pr2 (pr1 (pr2 X)))). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
istight_CDRap
713
Lemma isnonzeroCDR : (1 : X) β‰  (0 : X). Proof. exact (pr1 (pr2 (pr2 (pr2 (pr2 X))))). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isnonzeroCDR
714
Lemma islunit_CDRzero_CDRplus : ∏ x : X, 0 + x = x. Proof. now apply riglunax1. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
islunit_CDRzero_CDRplus
715
Lemma isrunit_CDRzero_CDRplus : ∏ x : X, x + 0 = x. Proof. now apply rigrunax1. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isrunit_CDRzero_CDRplus
716
Lemma isassoc_CDRplus : ∏ x y z : X, x + y + z = x + (y + z). Proof. now apply rigassoc1. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isassoc_CDRplus
717
Lemma iscomm_CDRplus : ∏ x y : X, x + y = y + x. Proof. now apply rigcomm1. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
iscomm_CDRplus
718
Lemma islunit_CDRone_CDRmult : ∏ x : X, 1 * x = x. Proof. now apply riglunax2. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
islunit_CDRone_CDRmult
719
Lemma isrunit_CDRone_CDRmult : ∏ x : X, x * 1 = x. Proof. now apply rigrunax2. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isrunit_CDRone_CDRmult
720
Lemma isassoc_CDRmult : ∏ x y z : X, x * y * z = x * (y * z). Proof. now apply rigassoc2. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isassoc_CDRmult
721
Lemma islinv_CDRinv : ∏ (x : X) (Hx0 : x β‰  (0 : X)), (CDRinv x Hx0) * x = 1. Proof. intros x Hx0. apply (pr1 (pr2 (pr2 (pr2 (pr2 (pr2 (pr2 X)))) x Hx0))). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
islinv_CDRinv
722
Lemma isrinv_CDRinv : ∏ (x : X) (Hx0 : x β‰  (0 : X)), x * (CDRinv x Hx0) = 1. Proof. intros x Hx0. apply (pr2 (pr2 (pr2 (pr2 (pr2 (pr2 (pr2 X)))) x Hx0))). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isrinv_CDRinv
723
Lemma islabsorb_CDRzero_CDRmult : ∏ x : X, 0 * x = 0. Proof. now apply rigmult0x. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
islabsorb_CDRzero_CDRmult
724
Lemma israbsorb_CDRzero_CDRmult : ∏ x : X, x * 0 = 0. Proof. now apply rigmultx0. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
israbsorb_CDRzero_CDRmult
725
Lemma isldistr_CDRplus_CDRmult : ∏ x y z : X, z * (x + y) = z * x + z * y. Proof. now apply rigdistraxs. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isldistr_CDRplus_CDRmult
726
Lemma apCDRplus : ∏ x x' y y' : X, x + y β‰  x' + y' β†’ x β‰  x' ∨ y β‰  y'. Proof. exact (isapbinop_op1 (X := ConstructiveDivisionRig_apsetwith2binop X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
apCDRplus
727
Lemma CDRplus_apcompat_l : ∏ x y z : X, y + x β‰  z + x β†’ y β‰  z. Proof. intros x y z. exact (islapbinop_op1 (X := ConstructiveDivisionRig_apsetwith2binop X) _ _ _). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CDRplus_apcompat_l
728
Lemma CDRplus_apcompat_r : ∏ x y z : X, x + y β‰  x + z β†’ y β‰  z. Proof. exact (israpbinop_op1 (X := ConstructiveDivisionRig_apsetwith2binop X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CDRplus_apcompat_r
729
Lemma apCDRmult : ∏ x x' y y' : X, x * y β‰  x' * y' β†’ x β‰  x' ∨ y β‰  y'. Proof. exact (isapbinop_op2 (X := ConstructiveDivisionRig_apsetwith2binop X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
apCDRmult
730
Lemma CDRmult_apcompat_l : ∏ x y z : X, y * x β‰  z * x β†’ y β‰  z. Proof. intros x y z. exact (islapbinop_op2 (X := ConstructiveDivisionRig_apsetwith2binop X) _ _ _). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CDRmult_apcompat_l
731
Lemma CDRmult_apcompat_l' : ∏ x y z : X, x β‰  0 β†’ y β‰  z β†’ y * x β‰  z * x. Proof. intros x y z Hx Hap. refine (CDRmult_apcompat_l (CDRinv x Hx) _ _ _). rewrite !isassoc_CDRmult, isrinv_CDRinv, !isrunit_CDRone_CDRmult. exact Hap. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CDRmult_apcompat_l'
732
Lemma CDRmult_apcompat_r : ∏ x y z : X, x * y β‰  x * z β†’ y β‰  z. Proof. exact (israpbinop_op2 (X := ConstructiveDivisionRig_apsetwith2binop X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CDRmult_apcompat_r
733
Lemma CDRmult_apcompat_r' : ∏ x y z : X, x β‰  0 β†’ y β‰  z β†’ x * y β‰  x * z. Proof. intros x y z Hx Hap. refine (CDRmult_apcompat_r (CDRinv x Hx) _ _ _). rewrite <- !isassoc_CDRmult, islinv_CDRinv, !islunit_CDRone_CDRmult. exact Hap. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CDRmult_apcompat_r'
734
Lemma CDRmultapCDRzero : ∏ x y : X, x * y β‰  0 β†’ x β‰  0 ∧ y β‰  0. Proof. intros x y Hmult. split. - apply CDRmult_apcompat_l with y. rewrite islabsorb_CDRzero_CDRmult. exact Hmult. - apply CDRmult_apcompat_r with x. rewrite israbsorb_CDRzero_CDRmult. exact Hmult. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CDRmultapCDRzero
735
Definition ConstructiveCommutativeDivisionRig := βˆ‘ X : commrig, βˆ‘ R : tightap X, isapbinop (X := (pr1 (pr1 X)) ,, R) BinaryOperations.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
ConstructiveCommutativeDivisionRig
736
Definition ConstructiveCommutativeDivisionRig_commrig : ConstructiveCommutativeDivisionRig β†’ commrig := pr1.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
ConstructiveCommutativeDivisionRig_commrig
737
Definition ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig : ConstructiveCommutativeDivisionRig β†’ ConstructiveDivisionRig := Ξ» X, (pr1 (pr1 X),,pr1 (pr2 (pr1 X))) ,, (pr2 X).
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig
738
Definition CCDRap {X : ConstructiveCommutativeDivisionRig} : hrel X := Ξ» x y : X, CDRap (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X) x y.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CCDRap
739
Definition CCDRzero {X : ConstructiveCommutativeDivisionRig} : X := 0%rig.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CCDRzero
740
Definition CCDRone {X : ConstructiveCommutativeDivisionRig} : X := 1%rig.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CCDRone
741
Definition CCDRplus {X : ConstructiveCommutativeDivisionRig} : binop X := Ξ» x y : X, CDRplus (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X) x y.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CCDRplus
742
Definition CCDRmult {X : ConstructiveCommutativeDivisionRig} : binop X := Ξ» x y : X, CDRmult (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X) x y.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CCDRmult
743
Definition CCDRinv {X : ConstructiveCommutativeDivisionRig} (x : X) (Hx0 : x β‰  CCDRzero) : X := CDRinv (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X) x Hx0.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CCDRinv
744
Definition CCDRdiv {X : ConstructiveCommutativeDivisionRig} (x y : X) (Hy0 : y β‰  CCDRzero) : X := CCDRmult x (CCDRinv y Hy0).
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CCDRdiv
745
Lemma isirrefl_CCDRap : ∏ x : X, Β¬ (x β‰  x). Proof. exact (isirrefl_CDRap (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isirrefl_CCDRap
746
Lemma issymm_CCDRap : ∏ x y : X, x β‰  y β†’ y β‰  x. Proof. exact (issymm_CDRap (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
issymm_CCDRap
747
Lemma iscotrans_CCDRap : ∏ x y z : X, x β‰  z β†’ x β‰  y ∨ y β‰  z. Proof. exact (iscotrans_CDRap (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
iscotrans_CCDRap
748
Lemma istight_CCDRap : ∏ x y : X, Β¬ (x β‰  y) β†’ x = y. Proof. exact (istight_CDRap (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
istight_CCDRap
749
Lemma isnonzeroCCDR : (1 : X) β‰  (0 : X). Proof. exact isnonzeroCDR. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isnonzeroCCDR
750
Lemma islunit_CCDRzero_CCDRplus : ∏ x : X, 0 + x = x. Proof. now apply riglunax1. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
islunit_CCDRzero_CCDRplus
751
Lemma isrunit_CCDRzero_CCDRplus : ∏ x : X, x + 0 = x. Proof. now apply rigrunax1. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isrunit_CCDRzero_CCDRplus
752
Lemma isassoc_CCDRplus : ∏ x y z : X, x + y + z = x + (y + z). Proof. now apply rigassoc1. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isassoc_CCDRplus
753
Lemma iscomm_CCDRplus : ∏ x y : X, x + y = y + x. Proof. now apply rigcomm1. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
iscomm_CCDRplus
754
Lemma islunit_CCDRone_CCDRmult : ∏ x : X, 1 * x = x. Proof. now apply riglunax2. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
islunit_CCDRone_CCDRmult
755
Lemma isrunit_CCDRone_CCDRmult : ∏ x : X, x * 1 = x. Proof. now apply rigrunax2. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isrunit_CCDRone_CCDRmult
756
Lemma isassoc_CCDRmult : ∏ x y z : X, x * y * z = x * (y * z). Proof. now apply rigassoc2. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isassoc_CCDRmult
757
Lemma iscomm_CCDRmult : ∏ x y : X, x * y = y * x. Proof. now apply rigcomm2. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
iscomm_CCDRmult
758
Lemma islinv_CCDRinv : ∏ (x : X) (Hx0 : x β‰  (0 : X)), (CDRinv (X := X) x Hx0) * x = 1. Proof. exact (islinv_CDRinv (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
islinv_CCDRinv
759
Lemma isrinv_CCDRinv : ∏ (x : X) (Hx0 : x β‰  (0 : X)), x * (CDRinv (X := X) x Hx0) = 1. Proof. exact (isrinv_CDRinv (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isrinv_CCDRinv
760
Lemma islabsorb_CCDRzero_CCDRmult : ∏ x : X, 0 * x = 0. Proof. now apply rigmult0x. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
islabsorb_CCDRzero_CCDRmult
761
Lemma israbsorb_CCDRzero_CCDRmult : ∏ x : X, x * 0 = 0. Proof. now apply rigmultx0. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
israbsorb_CCDRzero_CCDRmult
762
Lemma isldistr_CCDRplus_CCDRmult : ∏ x y z : X, z * (x + y) = z * x + z * y. Proof. now apply rigdistraxs. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isldistr_CCDRplus_CCDRmult
763
Lemma isrdistr_CCDRplus_CCDRmult : ∏ x y z : X, (x + y) * z = x * z + y * z. Proof. intros x y z. rewrite !(iscomm_CCDRmult _ z). now apply rigdistraxs. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isrdistr_CCDRplus_CCDRmult
764
Lemma apCCDRplus : ∏ x x' y y' : X, x + y β‰  x' + y' β†’ x β‰  x' ∨ y β‰  y'. Proof. exact (apCDRplus (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
apCCDRplus
765
Lemma CCDRplus_apcompat_l : ∏ x y z : X, y + x β‰  z + x β†’ y β‰  z. Proof. exact (CDRplus_apcompat_l (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CCDRplus_apcompat_l
766
Lemma CCDRplus_apcompat_r : ∏ x y z : X, x + y β‰  x + z β†’ y β‰  z. Proof. exact (CDRplus_apcompat_r (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CCDRplus_apcompat_r
767
Lemma apCCDRmult : ∏ x x' y y' : X, x * y β‰  x' * y' β†’ x β‰  x' ∨ y β‰  y'. Proof. exact (apCDRmult (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
apCCDRmult
768
Lemma CCDRmult_apcompat_l : ∏ x y z : X, y * x β‰  z * x β†’ y β‰  z. Proof. exact (CDRmult_apcompat_l (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CCDRmult_apcompat_l
769
Lemma CCDRmult_apcompat_l' : ∏ x y z : X, x β‰  0 β†’ y β‰  z β†’ y * x β‰  z * x. Proof. exact (CDRmult_apcompat_l' (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CCDRmult_apcompat_l'
770
Lemma CCDRmult_apcompat_r : ∏ x y z : X, x * y β‰  x * z β†’ y β‰  z. Proof. exact (CDRmult_apcompat_r (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CCDRmult_apcompat_r
771
Lemma CCDRmult_apcompat_r' : ∏ x y z : X, x β‰  0 β†’ y β‰  z β†’ x * y β‰  x * z. Proof. exact (CDRmult_apcompat_r' (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CCDRmult_apcompat_r'
772
Lemma CCDRmultapCCDRzero : ∏ x y : X, x * y β‰  0 β†’ x β‰  0 ∧ y β‰  0. Proof. exact (CDRmultapCDRzero (X := ConstructiveCommutativeDivisionRig_ConstructiveDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CCDRmultapCCDRzero
773
Definition ConstructiveField := βˆ‘ X : commring, βˆ‘ R : tightap X, isapbinop (X := (pr1 (pr1 X)) ,, R) BinaryOperations.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
ConstructiveField
774
Definition ConstructiveField_commring : ConstructiveField β†’ commring := pr1.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
ConstructiveField_commring
775
Definition ConstructiveField_ConstructiveCommutativeDivisionRig : ConstructiveField β†’ ConstructiveCommutativeDivisionRig := Ξ» X, (commringtocommrig (pr1 X)) ,, (pr2 X).
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
ConstructiveField_ConstructiveCommutativeDivisionRig
776
Definition CFap {X : ConstructiveField} : hrel X := Ξ» x y : X, CCDRap (X := ConstructiveField_ConstructiveCommutativeDivisionRig X) x y.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CFap
777
Definition CFzero {X : ConstructiveField} : X := 0%ring.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CFzero
778
Definition CFone {X : ConstructiveField} : X := 1%ring.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CFone
779
Definition CFplus {X : ConstructiveField} : binop X := Ξ» x y : X, CCDRplus (X := ConstructiveField_ConstructiveCommutativeDivisionRig X) x y.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CFplus
780
Definition CFopp {X : ConstructiveField} : unop X := Ξ» x : X, (- x)%ring.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CFopp
781
Definition CFminus {X : ConstructiveField} : binop X := Ξ» x y : X, CFplus x (CFopp y).
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CFminus
782
Definition CFmult {X : ConstructiveField} : binop X := Ξ» x y : X, CCDRmult (X := ConstructiveField_ConstructiveCommutativeDivisionRig X) x y.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CFmult
783
Definition CFinv {X : ConstructiveField} (x : X) (Hx0 : x β‰  CFzero) : X := CCDRinv (X := ConstructiveField_ConstructiveCommutativeDivisionRig X) x Hx0.
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CFinv
784
Definition CFdiv {X : ConstructiveField} (x y : X) (Hy0 : y β‰  CFzero) : X := CFmult x (CFinv y Hy0).
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
CFdiv
785
Lemma isirrefl_CFap : ∏ x : X, Β¬ (x β‰  x). Proof. exact (isirrefl_CCDRap (X := ConstructiveField_ConstructiveCommutativeDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isirrefl_CFap
786
Lemma issymm_CFap : ∏ x y : X, x β‰  y β†’ y β‰  x. Proof. exact (issymm_CCDRap (X := ConstructiveField_ConstructiveCommutativeDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
issymm_CFap
787
Lemma iscotrans_CFap : ∏ x y z : X, x β‰  z β†’ x β‰  y ∨ y β‰  z. Proof. exact (iscotrans_CCDRap (X := ConstructiveField_ConstructiveCommutativeDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
iscotrans_CFap
788
Lemma istight_CFap : ∏ x y : X, Β¬ (x β‰  y) β†’ x = y. Proof. exact (istight_CCDRap (X := ConstructiveField_ConstructiveCommutativeDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
istight_CFap
789
Lemma isnonzeroCF : (1 : X) β‰  (0 : X). Proof. exact (isnonzeroCCDR (X := ConstructiveField_ConstructiveCommutativeDivisionRig X)). Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isnonzeroCF
790
Lemma islunit_CFzero_CFplus : ∏ x : X, 0 + x = x. Proof. now apply ringlunax1. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
islunit_CFzero_CFplus
791
Lemma isrunit_CFzero_CFplus : ∏ x : X, x + 0 = x. Proof. now apply ringrunax1. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isrunit_CFzero_CFplus
792
Lemma isassoc_CFplus : ∏ x y z : X, x + y + z = x + (y + z). Proof. now apply ringassoc1. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isassoc_CFplus
793
Lemma islinv_CFopp : ∏ x : X, - x + x = 0. Proof. now apply ringlinvax1. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
islinv_CFopp
794
Lemma isrinv_CFopp : ∏ x : X, x + - x = 0. Proof. now apply ringrinvax1. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isrinv_CFopp
795
Lemma iscomm_CFplus : ∏ x y : X, x + y = y + x. Proof. now apply ringcomm1. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
iscomm_CFplus
796
Lemma islunit_CFone_CFmult : ∏ x : X, 1 * x = x. Proof. now apply ringlunax2. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
islunit_CFone_CFmult
797
Lemma isrunit_CFone_CFmult : ∏ x : X, x * 1 = x. Proof. now apply ringrunax2. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isrunit_CFone_CFmult
798
Lemma isassoc_CFmult : ∏ x y z : X, x * y * z = x * (y * z). Proof. now apply ringassoc2. Qed.
Lemma
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isassoc_CFmult
799