fact
stringlengths 13
15.5k
| type
stringclasses 9
values | library
stringclasses 15
values | imports
stringlengths 14
7.64k
β | filename
stringlengths 12
97
| symbolic_name
stringlengths 1
78
| index_level
int64 0
38.7k
|
|---|---|---|---|---|---|---|
Definition ConstructiveDivisionRig := β X : rig, β 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 : β 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 : β 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 : β x y z : X, x β z β x β y β¨ 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 : β 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 : β 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 : β 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 : β 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 : β 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 : β 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 : β 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 : β 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 : β (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 : β (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 : β 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 : β 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 : β 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 : β x x' y y' : X, x + y β x' + y' β x β x' β¨ 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 : β 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 : β 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 : β x x' y y' : X, x * y β x' * y' β x β x' β¨ 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 : β 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' : β 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 : β 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' : β 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 : β x y : X, x * y β 0 β x β 0 β§ 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 := β X : commrig, β 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 : β 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 : β 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 : β x y z : X, x β z β x β y β¨ 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 : β 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 : β 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 : β 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 : β 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 : β 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 : β 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 : β 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 : β 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 : β 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 : β (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 : β (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 : β 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 : β 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 : β 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 : β 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 : β x x' y y' : X, x + y β x' + y' β x β x' β¨ 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 : β 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 : β 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 : β x x' y y' : X, x * y β x' * y' β x β x' β¨ 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 : β 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' : β 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 : β 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' : β 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 : β x y : X, x * y β 0 β x β 0 β§ 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 := β X : commring, β 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 : β 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 : β 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 : β x y z : X, x β z β x β y β¨ 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 : β 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 : β 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 : β 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 : β 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 : β 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 : β 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 : β 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 : β 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 : β 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 : β 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 |
Subsets and Splits