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Lemma iscomprelfun_generated_binopeqrel {X Y : setwithbinop} {R : hrel X} (f : binopfun X Y) (H : iscomprelfun R f) : iscomprelfun (generated_binopeqrel R) f. Proof. intros x x' r. exact (r (binopeqrel_of_binopfun f) H). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | iscomprelfun_generated_binopeqrel | 600 |
Lemma iscomprelrelfun_generated_binopeqrel {X Y : setwithbinop} {R : hrel X} {S : hrel Y} (f : binopfun X Y) (H : iscomprelrelfun R S f) : iscomprelrelfun (generated_binopeqrel R) (generated_binopeqrel S) f. Proof. intros x x' r. apply (r (pullback_binopeqrel f (generated_binopeqrel S))). intros x1 x2 r' S' s. use s. apply H. exact r'. Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | iscomprelrelfun_generated_binopeqrel | 601 |
Lemma iscomprelrelfun_generated_binopeqrel_rev {X Y : setwithbinop} {R : hrel X} {S : hrel Y} (f : binopfun X (setwithbinop_rev Y)) (H : iscomprelrelfun R S f) : iscomprelrelfun (generated_binopeqrel R) (generated_binopeqrel S) f. Proof. intros x x' r. apply (r (pullback_binopeqrel_rev f (generated_binopeqrel S))). intros x1 x2 r' S' s. use s. apply H. exact r'. Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | iscomprelrelfun_generated_binopeqrel_rev | 602 |
Definition isinvbinophrel {X : setwithbinop} (R : hrel X) : UU := (β a b c : X, R (op c a) (op c b) β R a b) Γ (β a b c : X, R (op a c) (op b c) β R a b). | Definition | Algebra | null | Algebra\BinaryOperations.v | isinvbinophrel | 603 |
Definition isinvbinophrellogeqf {X : setwithbinop} {L R : hrel X} (lg : hrellogeq L R) (isl : isinvbinophrel L) : isinvbinophrel R. Proof. split. - intros a b c rab. apply ((pr1 (lg _ _) ((pr1 isl) _ _ _ (pr2 (lg _ _) rab)))). - intros a b c rab. apply ((pr1 (lg _ _) ((pr2 isl) _ _ _ (pr2 (lg _ _) rab)))). Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | isinvbinophrellogeqf | 604 |
Lemma isapropisinvbinophrel {X : setwithbinop} (R : hrel X) : isaprop (isinvbinophrel R). Proof. apply isapropdirprod. - apply impred. intro a. apply impred. intro b. apply impred. intro c. apply impred. intro r. apply (pr2 (R _ _)). - apply impred. intro a. apply impred. intro b. apply impred. intro c. apply impred. intro r. apply (pr2 (R _ _)). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isapropisinvbinophrel | 605 |
Lemma isinvbinophrelif {X : setwithbinop} (R : hrel X) (is : iscomm (@op X)) (isl : β a b c : X, R (op c a) (op c b) β R a b) : isinvbinophrel R. Proof. exists isl. intros a b c rab. induction (is c a). induction (is c b). apply (isl _ _ _ rab). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isinvbinophrelif | 606 |
Definition ispartinvbinophrel {X : setwithbinop} (S : hsubtype X) (R : hrel X) : UU := (β a b c : X, S c β R (op c a) (op c b) β R a b) Γ (β a b c : X, S c β R (op a c) (op b c) β R a b). | Definition | Algebra | null | Algebra\BinaryOperations.v | ispartinvbinophrel | 607 |
Definition isinvbinoptoispartinvbinop {X : setwithbinop} (S : hsubtype X) (L : hrel X) (d2 : isinvbinophrel L) : ispartinvbinophrel S L. Proof. unfold isinvbinophrel in d2. unfold ispartinvbinophrel. split. - intros a b c s. apply (pr1 d2 a b c). - intros a b c s. apply (pr2 d2 a b c). Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | isinvbinoptoispartinvbinop | 608 |
Definition ispartinvbinophrellogeqf {X : setwithbinop} (S : hsubtype X) {L R : hrel X} (lg : hrellogeq L R) (isl : ispartinvbinophrel S L) : ispartinvbinophrel S R. Proof. split. - intros a b c s rab. apply ((pr1 (lg _ _) ((pr1 isl) _ _ _ s (pr2 (lg _ _) rab)))). - intros a b c s rab. apply ((pr1 (lg _ _) ((pr2 isl) _ _ _ s (pr2 (lg _ _) rab)))). Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | ispartinvbinophrellogeqf | 609 |
Lemma ispartinvbinophrelif {X : setwithbinop} (S : hsubtype X) (R : hrel X) (is : iscomm (@op X)) (isl : β a b c : X, S c β R (op c a) (op c b) β R a b) : ispartinvbinophrel S R. Proof. exists isl. intros a b c s rab. induction (is c a). induction (is c b). apply (isl _ _ _ s rab). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | ispartinvbinophrelif | 610 |
Lemma binophrelandfun {X Y : setwithbinop} (f : binopfun X Y) (R : hrel Y) (is : @isbinophrel Y R) : @isbinophrel X (Ξ» x x', R (f x) (f x')). Proof. set (ish := (pr2 f) : β a0 b0, f (op a0 b0) = op (f a0) (f b0)). split. - intros a b c r. rewrite (ish _ _). rewrite (ish _ _). apply (pr1 is). apply r. - intros a b c r. rewrite (ish _ _). rewrite (ish _ _). apply (pr2 is). apply r. Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | binophrelandfun | 611 |
Lemma ispartbinophrelandfun {X Y : setwithbinop} (f : binopfun X Y) (SX : hsubtype X) (SY : hsubtype Y) (iss : β x : X, (SX x) β (SY (f x))) (R : hrel Y) (is : @ispartbinophrel Y SY R) : @ispartbinophrel X SX (Ξ» x x', R (f x) (f x')). Proof. set (ish := (pr2 f) : β a0 b0, f (op a0 b0) = op (f a0) (f b0)). split. - intros a b c s r. rewrite (ish _ _). rewrite (ish _ _). apply ((pr1 is) _ _ _ (iss _ s) r). - intros a b c s r. rewrite (ish _ _). rewrite (ish _ _). apply ((pr2 is) _ _ _ (iss _ s) r). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | ispartbinophrelandfun | 612 |
Lemma invbinophrelandfun {X Y : setwithbinop} (f : binopfun X Y) (R : hrel Y) (is : @isinvbinophrel Y R) : @isinvbinophrel X (Ξ» x x', R (f x) (f x')). Proof. set (ish := (pr2 f) : β a0 b0, f (op a0 b0) = op (f a0) (f b0)). split. - intros a b c r. rewrite (ish _ _) in r. rewrite (ish _ _) in r. apply ((pr1 is) _ _ _ r). - intros a b c r. rewrite (ish _ _) in r. rewrite (ish _ _) in r. apply ((pr2 is) _ _ _ r). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | invbinophrelandfun | 613 |
Lemma ispartinvbinophrelandfun {X Y : setwithbinop} (f : binopfun X Y) (SX : hsubtype X) (SY : hsubtype Y) (iss : β x : X, (SX x) β (SY (f x))) (R : hrel Y) (is : @ispartinvbinophrel Y SY R) : @ispartinvbinophrel X SX (Ξ» x x', R (f x) (f x')). Proof. set (ish := (pr2 f) : β a0 b0, f (op a0 b0) = op (f a0) (f b0)). split. - intros a b c s r. rewrite (ish _ _) in r. rewrite (ish _ _) in r. apply ((pr1 is) _ _ _ (iss _ s) r). - intros a b c s r. rewrite (ish _ _) in r. rewrite (ish _ _) in r. apply ((pr2 is) _ _ _ (iss _ s) r). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | ispartinvbinophrelandfun | 614 |
Lemma isbinopquotrel {X : setwithbinop} (R : binopeqrel X) {L : hrel X} (is : iscomprelrel R L) (isl : isbinophrel L) : @isbinophrel (setwithbinopquot R) (quotrel is). Proof. unfold isbinophrel. split. - assert (int : β (a b c : setwithbinopquot R), isaprop (quotrel is a b β quotrel is (op c a) (op c b))). { intros a b c. apply impred. intro. apply (pr2 (quotrel is _ _)). } apply (setquotuniv3prop R (Ξ» a b c, make_hProp _ (int a b c))). exact (pr1 isl). - assert (int : β (a b c : setwithbinopquot R), isaprop (quotrel is a b β quotrel is (op a c) (op b c))). { intros a b c. apply impred. intro. apply (pr2 (quotrel is _ _)). } apply (setquotuniv3prop R (Ξ» a b c, make_hProp _ (int a b c))). exact (pr2 isl). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isbinopquotrel | 615 |
Definition iscommsetquotfun2 {X: hSet} {R : eqrel X} (f : binop X) (is : iscomprelrelfun2 R R f) (p : iscomm f) : iscomm (setquotfun2 R R f is). Proof. use (setquotuniv2prop _ (Ξ» x y , @eqset (setquotinset _) ((setquotfun2 _ _ _ _) x y) ((setquotfun2 _ _ _ _) y x) )). intros. cbn. now rewrite p. Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | iscommsetquotfun2 | 616 |
Definition isassocsetquotfun2 {X : hSet} {R : eqrel X} (f : binop X) (is : iscomprelrelfun2 R R f) (p : isassoc f) : isassoc (setquotfun2 R R f is). Proof. set (ff := setquotfun2 _ _ _ is). intros ? ? ?. use (setquotuniv3prop _ (Ξ» x y z, @eqset (setquotinset _) (ff (ff z x) y) (ff z (ff x y)))). intros. cbn. now rewrite p. Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | isassocsetquotfun2 | 617 |
Definition setwithbinopdirprod (X Y : setwithbinop) : setwithbinop. Proof. exists (setdirprod X Y). unfold binop. simpl. apply (Ξ» xy xy' : X Γ Y, op (pr1 xy) (pr1 xy') ,, op (pr2 xy) (pr2 xy')). Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | setwithbinopdirprod | 618 |
Definition setwith2binop : UU := β (X : hSet), (binop X) Γ (binop X). | Definition | Algebra | null | Algebra\BinaryOperations.v | setwith2binop | 619 |
Definition make_setwith2binop (X : hSet) (opps : (binop X) Γ (binop X)) : setwith2binop := X ,, opps. | Definition | Algebra | null | Algebra\BinaryOperations.v | make_setwith2binop | 620 |
Definition pr1setwith2binop : setwith2binop β hSet := @pr1 _ (Ξ» X : hSet, (binop X) Γ (binop X)). | Definition | Algebra | null | Algebra\BinaryOperations.v | pr1setwith2binop | 621 |
Definition op1 {X : setwith2binop} : binop X := pr1 (pr2 X). | Definition | Algebra | null | Algebra\BinaryOperations.v | op1 | 622 |
Definition op2 {X : setwith2binop} : binop X := pr2 (pr2 X). | Definition | Algebra | null | Algebra\BinaryOperations.v | op2 | 623 |
Definition setwithbinop1 (X : setwith2binop) : setwithbinop := make_setwithbinop (pr1 X) (@op1 X). | Definition | Algebra | null | Algebra\BinaryOperations.v | setwithbinop1 | 624 |
Definition setwithbinop2 (X : setwith2binop) : setwithbinop := make_setwithbinop (pr1 X) (@op2 X). | Definition | Algebra | null | Algebra\BinaryOperations.v | setwithbinop2 | 625 |
Definition isasettwobinoponhSet (X : hSet) : isaset ((binop X) Γ (binop X)). | Definition | Algebra | null | Algebra\BinaryOperations.v | isasettwobinoponhSet | 626 |
Definition istwobinopfun {X Y : setwith2binop} (f : X β Y) : UU := (β x x' : X, f (op1 x x') = op1 (f x) (f x')) Γ (β x x' : X, f (op2 x x') = op2 (f x) (f x')). | Definition | Algebra | null | Algebra\BinaryOperations.v | istwobinopfun | 627 |
Definition make_istwobinopfun {X Y : setwith2binop} (f : X β Y) (H1 : β x x' : X, f (op1 x x') = op1 (f x) (f x')) (H2 : β x x' : X, f (op2 x x') = op2 (f x) (f x')) : istwobinopfun f := H1 ,, H2. | Definition | Algebra | null | Algebra\BinaryOperations.v | make_istwobinopfun | 628 |
Lemma isapropistwobinopfun {X Y : setwith2binop} (f : X β Y) : isaprop (istwobinopfun f). Proof. apply isofhleveldirprod. - apply impred. intro x. apply impred. intro x'. apply (setproperty Y). - apply impred. intro x. apply impred. intro x'. apply (setproperty Y). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isapropistwobinopfun | 629 |
Definition twobinopfun (X Y : setwith2binop) : UU := β (f : X β Y), istwobinopfun f. | Definition | Algebra | null | Algebra\BinaryOperations.v | twobinopfun | 630 |
Definition make_twobinopfun {X Y : setwith2binop} (f : X β Y) (is : istwobinopfun f) : twobinopfun X Y := f ,, is. | Definition | Algebra | null | Algebra\BinaryOperations.v | make_twobinopfun | 631 |
Definition pr1twobinopfun (X Y : setwith2binop) : twobinopfun X Y β (X β Y) := @pr1 _ _. | Definition | Algebra | null | Algebra\BinaryOperations.v | pr1twobinopfun | 632 |
Definition binop1fun {X Y : setwith2binop} (f : twobinopfun X Y) : binopfun (setwithbinop1 X) (setwithbinop1 Y) := @make_binopfun (setwithbinop1 X) (setwithbinop1 Y) (pr1 f) (pr1 (pr2 f)). | Definition | Algebra | null | Algebra\BinaryOperations.v | binop1fun | 633 |
Definition binop2fun {X Y : setwith2binop} (f : twobinopfun X Y) : binopfun (setwithbinop2 X) (setwithbinop2 Y) := @make_binopfun (setwithbinop2 X) (setwithbinop2 Y) (pr1 f) (pr2 (pr2 f)). | Definition | Algebra | null | Algebra\BinaryOperations.v | binop2fun | 634 |
Definition twobinopfun_paths {X Y : setwith2binop} (f g : twobinopfun X Y) (e : pr1 f = pr1 g) : f = g. | Definition | Algebra | null | Algebra\BinaryOperations.v | twobinopfun_paths | 635 |
Lemma isasettwobinopfun (X Y : setwith2binop) : isaset (twobinopfun X Y). Proof. apply (isasetsubset (pr1twobinopfun X Y)). - change (isofhlevel 2 (X β Y)). apply impred. intro. apply (setproperty Y). - refine (isinclpr1 _ _). intro. apply isapropistwobinopfun. Qed. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isasettwobinopfun | 636 |
Lemma istwobinopfuncomp {X Y Z : setwith2binop} (f : twobinopfun X Y) (g : twobinopfun Y Z) : istwobinopfun (pr1 g β pr1 f). Proof. set (ax1f := pr1 (pr2 f)). set (ax2f := pr2 (pr2 f)). set (ax1g := pr1 (pr2 g)). set (ax2g := pr2 (pr2 g)). split. - intros a b. simpl. rewrite (ax1f a b). rewrite (ax1g (pr1 f a) (pr1 f b)). apply idpath. - intros a b. simpl. rewrite (ax2f a b). rewrite (ax2g (pr1 f a) (pr1 f b)). apply idpath. Qed. | Lemma | Algebra | null | Algebra\BinaryOperations.v | istwobinopfuncomp | 637 |
Definition twobinopfuncomp {X Y Z : setwith2binop} (f : twobinopfun X Y) (g : twobinopfun Y Z) : twobinopfun X Z := make_twobinopfun (g β f) (istwobinopfuncomp f g). | Definition | Algebra | null | Algebra\BinaryOperations.v | twobinopfuncomp | 638 |
Definition twobinopmono (X Y : setwith2binop) : UU := β (f : incl X Y), istwobinopfun f. | Definition | Algebra | null | Algebra\BinaryOperations.v | twobinopmono | 639 |
Definition make_twobinopmono {X Y : setwith2binop} (f : incl X Y) (is : istwobinopfun f) : twobinopmono X Y := f ,, is. | Definition | Algebra | null | Algebra\BinaryOperations.v | make_twobinopmono | 640 |
Definition pr1twobinopmono (X Y : setwith2binop) : twobinopmono X Y β incl X Y := @pr1 _ _. | Definition | Algebra | null | Algebra\BinaryOperations.v | pr1twobinopmono | 641 |
Definition twobinopincltotwobinopfun (X Y : setwith2binop) : twobinopmono X Y β twobinopfun X Y := Ξ» f, make_twobinopfun (pr1 (pr1 f)) (pr2 f). | Definition | Algebra | null | Algebra\BinaryOperations.v | twobinopincltotwobinopfun | 642 |
Definition binop1mono {X Y : setwith2binop} (f : twobinopmono X Y) : binopmono (setwithbinop1 X) (setwithbinop1 Y) := @make_binopmono (setwithbinop1 X) (setwithbinop1 Y) (pr1 f) (pr1 (pr2 f)). | Definition | Algebra | null | Algebra\BinaryOperations.v | binop1mono | 643 |
Definition binop2mono {X Y : setwith2binop} (f : twobinopmono X Y) : binopmono (setwithbinop2 X) (setwithbinop2 Y) := @make_binopmono (setwithbinop2 X) (setwithbinop2 Y) (pr1 f) (pr2 (pr2 f)). | Definition | Algebra | null | Algebra\BinaryOperations.v | binop2mono | 644 |
Definition twobinopmonocomp {X Y Z : setwith2binop} (f : twobinopmono X Y) (g : twobinopmono Y Z) : twobinopmono X Z := make_twobinopmono (inclcomp (pr1 f) (pr1 g)) (istwobinopfuncomp f g). | Definition | Algebra | null | Algebra\BinaryOperations.v | twobinopmonocomp | 645 |
Definition twobinopiso (X Y : setwith2binop) : UU := β (f : X β Y), istwobinopfun f. | Definition | Algebra | null | Algebra\BinaryOperations.v | twobinopiso | 646 |
Definition make_twobinopiso {X Y : setwith2binop} (f : X β Y) (is : istwobinopfun f) : twobinopiso X Y := f ,, is. | Definition | Algebra | null | Algebra\BinaryOperations.v | make_twobinopiso | 647 |
Definition pr1twobinopiso (X Y : setwith2binop) : twobinopiso X Y β X β Y := @pr1 _ _. | Definition | Algebra | null | Algebra\BinaryOperations.v | pr1twobinopiso | 648 |
Definition twobinopisototwobinopmono (X Y : setwith2binop) : twobinopiso X Y β twobinopmono X Y := Ξ» f, make_twobinopmono (weqtoincl (pr1 f)) (pr2 f). | Definition | Algebra | null | Algebra\BinaryOperations.v | twobinopisototwobinopmono | 649 |
Definition twobinopisototwobinopfun {X Y : setwith2binop} (f : twobinopiso X Y) : twobinopfun X Y := make_twobinopfun f (pr2 f). | Definition | Algebra | null | Algebra\BinaryOperations.v | twobinopisototwobinopfun | 650 |
Lemma twobinopiso_paths {X Y : setwith2binop} (f g : twobinopiso X Y) (e : pr1 f = pr1 g) : f = g. Proof. use total2_paths_f. - exact e. - use proofirrelevance. use isapropistwobinopfun. Qed. | Lemma | Algebra | null | Algebra\BinaryOperations.v | twobinopiso_paths | 651 |
Definition binop1iso {X Y : setwith2binop} (f : twobinopiso X Y) : binopiso (setwithbinop1 X) (setwithbinop1 Y) := @make_binopiso (setwithbinop1 X) (setwithbinop1 Y) (pr1 f) (pr1 (pr2 f)). | Definition | Algebra | null | Algebra\BinaryOperations.v | binop1iso | 652 |
Definition binop2iso {X Y : setwith2binop} (f : twobinopiso X Y) : binopiso (setwithbinop2 X) (setwithbinop2 Y) := @make_binopiso (setwithbinop2 X) (setwithbinop2 Y) (pr1 f) (pr2 (pr2 f)). | Definition | Algebra | null | Algebra\BinaryOperations.v | binop2iso | 653 |
Definition twobinopisocomp {X Y Z : setwith2binop} (f : twobinopiso X Y) (g : twobinopiso Y Z) : twobinopiso X Z := make_twobinopiso (weqcomp (pr1 f) (pr1 g)) (istwobinopfuncomp f g). | Definition | Algebra | null | Algebra\BinaryOperations.v | twobinopisocomp | 654 |
Lemma istwobinopfuninvmap {X Y : setwith2binop} (f : twobinopiso X Y) : istwobinopfun (invmap (pr1 f)). Proof. set (ax1f := pr1 (pr2 f)). set (ax2f := pr2 (pr2 f)). split. - intros a b. apply (invmaponpathsweq (pr1 f)). rewrite (homotweqinvweq (pr1 f) (op1 a b)). rewrite (ax1f (invmap (pr1 f) a) (invmap (pr1 f) b)). rewrite (homotweqinvweq (pr1 f) a). rewrite (homotweqinvweq (pr1 f) b). apply idpath. - intros a b. apply (invmaponpathsweq (pr1 f)). rewrite (homotweqinvweq (pr1 f) (op2 a b)). rewrite (ax2f (invmap (pr1 f) a) (invmap (pr1 f) b)). rewrite (homotweqinvweq (pr1 f) a). rewrite (homotweqinvweq (pr1 f) b). apply idpath. Qed. | Lemma | Algebra | null | Algebra\BinaryOperations.v | istwobinopfuninvmap | 655 |
Definition invtwobinopiso {X Y : setwith2binop} (f : twobinopiso X Y) : twobinopiso Y X := make_twobinopiso (invweq (pr1 f)) (istwobinopfuninvmap f). | Definition | Algebra | null | Algebra\BinaryOperations.v | invtwobinopiso | 656 |
Definition idtwobinopiso (X : setwith2binop) : twobinopiso X X. Proof. use make_twobinopiso. - use (idweq X). - use make_istwobinopfun. + intros x x'. use idpath. + intros x x'. use idpath. Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | idtwobinopiso | 657 |
Definition setwith2binop_univalence_weq1 (X Y : setwith2binop) : (X = Y) β (X β Y) := total2_paths_equiv _ X Y. | Definition | Algebra | null | Algebra\BinaryOperations.v | setwith2binop_univalence_weq1 | 658 |
Definition setwith2binop_univalence_weq2 (X Y : setwith2binop) : (X β Y) β (twobinopiso X Y). Proof. use weqbandf. - use hSet_univalence. - intros e. use invweq. induction X as [X Xop]. induction Y as [Y Yop]. cbn in e. induction e. use weqimplimpl. + intros i. use dirprod_paths. * use funextfun. intros x1. use funextfun. intros x2. exact ((dirprod_pr1 i) x1 x2). * use funextfun. intros x1. use funextfun. intros x2. exact ((dirprod_pr2 i) x1 x2). + intros e. cbn in e. use make_istwobinopfun. * intros x1 x2. induction e. use idpath. * intros x1 x2. induction e. use idpath. + use isapropistwobinopfun. + use isasettwobinoponhSet. Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | setwith2binop_univalence_weq2 | 659 |
Definition setwith2binop_univalence_map (X Y : setwith2binop) : X = Y β twobinopiso X Y. Proof. intro e. induction e. exact (idtwobinopiso X). Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | setwith2binop_univalence_map | 660 |
Lemma setwith2binop_univalence_isweq (X Y : setwith2binop) : isweq (setwith2binop_univalence_map X Y). Proof. use isweqhomot. - exact (weqcomp (setwith2binop_univalence_weq1 X Y) (setwith2binop_univalence_weq2 X Y)). - intros e. induction e. use weqcomp_to_funcomp_app. - use weqproperty. Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | setwith2binop_univalence_isweq | 661 |
Definition setwith2binop_univalence (X Y : setwith2binop) : (X = Y) β (twobinopiso X Y) := make_weq (setwith2binop_univalence_map X Y) (setwith2binop_univalence_isweq X Y). | Definition | Algebra | null | Algebra\BinaryOperations.v | setwith2binop_univalence | 662 |
Lemma isldistrmonob {X Y : setwith2binop} (f : twobinopmono X Y) (is : isldistr (@op1 Y) (@op2 Y)) : isldistr (@op1 X) (@op2 X). Proof. set (ax1f := pr1 (pr2 f)). set (ax2f := pr2 (pr2 f)). intros a b c. apply (invmaponpathsincl _ (pr2 (pr1 f))). change ((pr1 f) (op2 c (op1 a b)) = (pr1 f) (op1 (op2 c a) (op2 c b))). rewrite (ax2f c (op1 a b)). rewrite (ax1f a b). rewrite (ax1f (op2 c a) (op2 c b)). rewrite (ax2f c a). rewrite (ax2f c b). apply is. Qed. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isldistrmonob | 663 |
Lemma isrdistrmonob {X Y : setwith2binop} (f : twobinopmono X Y) (is : isrdistr (@op1 Y) (@op2 Y)) : isrdistr (@op1 X) (@op2 X). Proof. set (ax1f := pr1 (pr2 f)). set (ax2f := pr2 (pr2 f)). intros a b c. apply (invmaponpathsincl _ (pr2 (pr1 f))). change ((pr1 f) (op2 (op1 a b) c) = (pr1 f) (op1 (op2 a c) (op2 b c))). rewrite (ax2f (op1 a b) c). rewrite (ax1f a b). rewrite (ax1f (op2 a c) (op2 b c)). rewrite (ax2f a c). rewrite (ax2f b c). apply is. Qed. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isrdistrmonob | 664 |
Definition isdistrmonob {X Y : setwith2binop} (f : twobinopmono X Y) (is : isdistr (@op1 Y) (@op2 Y)) : isdistr (@op1 X) (@op2 X) := isldistrmonob f (pr1 is) ,, isrdistrmonob f (pr2 is). | Definition | Algebra | null | Algebra\BinaryOperations.v | isdistrmonob | 665 |
Lemma isldistrisof {X Y : setwith2binop} (f : twobinopiso X Y) (is : isldistr (@op1 X) (@op2 X)) : isldistr (@op1 Y) (@op2 Y). Proof. apply (isldistrisob (invtwobinopiso f) is). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isldistrisof | 666 |
Lemma isrdistrisof {X Y : setwith2binop} (f : twobinopiso X Y) (is : isrdistr (@op1 X) (@op2 X)) : isrdistr (@op1 Y) (@op2 Y). Proof. apply (isrdistrisob (invtwobinopiso f) is). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isrdistrisof | 667 |
Lemma isdistrisof {X Y : setwith2binop} (f : twobinopiso X Y) (is : isdistr (@op1 X) (@op2 X)) : isdistr (@op1 Y) (@op2 Y). Proof. apply (isdistrisob (invtwobinopiso f) is). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isdistrisof | 668 |
Definition isrigopsisof {X Y : setwith2binop} (f : twobinopiso X Y) (is : isrigops (@op1 X) (@op2 X)) : isrigops (@op1 Y) (@op2 Y). Proof. split. - exists (isabmonoidopisof (binop1iso f) (rigop1axs_is is) ,, ismonoidopisof (binop2iso f) (rigop2axs_is is)). simpl. change (unel_is (ismonoidopisof (binop1iso f) (rigop1axs_is is))) with ((pr1 f) (rigunel1_is is)). split. + intro y. rewrite <- (homotweqinvweq f y). rewrite <- ((pr2 (pr2 f)) _ _). apply (maponpaths (pr1 f)). apply (rigmult0x_is is). + intro y. rewrite <- (homotweqinvweq f y). rewrite <- ((pr2 (pr2 f)) _ _). apply (maponpaths (pr1 f)). apply (rigmultx0_is is). - apply (isdistrisof f). apply (rigdistraxs_is is). Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | isrigopsisof | 669 |
Definition isrigopsisob {X Y : setwith2binop} (f : twobinopiso X Y) (is : isrigops (@op1 Y) (@op2 Y)) : isrigops (@op1 X) (@op2 X). Proof. apply (isrigopsisof (invtwobinopiso f) is). Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | isrigopsisob | 670 |
Definition isringopsisof {X Y : setwith2binop} (f : twobinopiso X Y) (is : isringops (@op1 X) (@op2 X)) : isringops (@op1 Y) (@op2 Y) := (isabgropisof (binop1iso f) (ringop1axs_is is) ,, ismonoidopisof (binop2iso f) (ringop2axs_is is)) ,, isdistrisof f (pr2 is). | Definition | Algebra | null | Algebra\BinaryOperations.v | isringopsisof | 671 |
Definition isringopsisob {X Y : setwith2binop} (f : twobinopiso X Y) (is : isringops (@op1 Y) (@op2 Y)) : isringops (@op1 X) (@op2 X) := (isabgropisob (binop1iso f) (ringop1axs_is is) ,, ismonoidopisob (binop2iso f) (ringop2axs_is is)) ,, isdistrisob f (pr2 is). | Definition | Algebra | null | Algebra\BinaryOperations.v | isringopsisob | 672 |
Definition iscommringopsisof {X Y : setwith2binop} (f : twobinopiso X Y) (is : iscommringops (@op1 X) (@op2 X)) : iscommringops (@op1 Y) (@op2 Y) := isringopsisof f is ,, iscommisof (binop2iso f) (pr2 is). | Definition | Algebra | null | Algebra\BinaryOperations.v | iscommringopsisof | 673 |
Definition iscommringopsisob {X Y : setwith2binop} (f : twobinopiso X Y) (is : iscommringops (@op1 Y) (@op2 Y)) : iscommringops (@op1 X) (@op2 X) := isringopsisob f is ,, iscommisob (binop2iso f) (pr2 is). | Definition | Algebra | null | Algebra\BinaryOperations.v | iscommringopsisob | 674 |
Definition issubsetwith2binop {X : setwith2binop} (A : hsubtype X) : UU := (β a a' : A, A (op1 (pr1 a) (pr1 a'))) Γ (β a a' : A, A (op2 (pr1 a) (pr1 a'))). | Definition | Algebra | null | Algebra\BinaryOperations.v | issubsetwith2binop | 675 |
Lemma isapropissubsetwith2binop {X : setwith2binop} (A : hsubtype X) : isaprop (issubsetwith2binop A). Proof. apply (isofhleveldirprod 1). - apply impred. intro a. apply impred. intros a'. apply (pr2 (A (op1 (pr1 a) (pr1 a')))). - apply impred. intro a. apply impred. intros a'. apply (pr2 (A (op2 (pr1 a) (pr1 a')))). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isapropissubsetwith2binop | 676 |
Definition subsetswith2binop (X : setwith2binop) : UU := β (A : hsubtype X), issubsetwith2binop A. | Definition | Algebra | null | Algebra\BinaryOperations.v | subsetswith2binop | 677 |
Definition make_subsetswith2binop {X : setwith2binop} (t : hsubtype X) (H : issubsetwith2binop t) : subsetswith2binop X := t ,, H. | Definition | Algebra | null | Algebra\BinaryOperations.v | make_subsetswith2binop | 678 |
Definition subsetswith2binopconstr {X : setwith2binop} : β (t : hsubtype X), (Ξ» A : hsubtype X, issubsetwith2binop A) t β β A : hsubtype X, issubsetwith2binop A := @make_subsetswith2binop X. | Definition | Algebra | null | Algebra\BinaryOperations.v | subsetswith2binopconstr | 679 |
Definition pr1subsetswith2binop (X : setwith2binop) : subsetswith2binop X β hsubtype X := @pr1 _ (Ξ» A : hsubtype X, issubsetwith2binop A). | Definition | Algebra | null | Algebra\BinaryOperations.v | pr1subsetswith2binop | 680 |
Definition totalsubsetwith2binop (X : setwith2binop) : subsetswith2binop X. Proof. exists (Ξ» x : X, htrue). split. - intros x x'. apply tt. - intros. apply tt. Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | totalsubsetwith2binop | 681 |
Definition carrierofsubsetwith2binop {X : setwith2binop} (A : subsetswith2binop X) : setwith2binop. Proof. set (aset := (make_hSet (carrier A) (isasetsubset (pr1carrier A) (setproperty X) (isinclpr1carrier A))) : hSet). exists aset. set (subopp1 := (Ξ» a a' : A, make_carrier A (op1 (pr1carrier _ a) (pr1carrier _ a')) (pr1 (pr2 A) a a')) : (A β A β A)). set (subopp2 := (Ξ» a a' : A, make_carrier A (op2 (pr1carrier _ a) (pr1carrier _ a')) (pr2 (pr2 A) a a')) : (A β A β A)). simpl. exact (subopp1 ,, subopp2). Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | carrierofsubsetwith2binop | 682 |
Definition is2binophrel {X : setwith2binop} (R : hrel X) : UU := (@isbinophrel (setwithbinop1 X) R) Γ (@isbinophrel (setwithbinop2 X) R). | Definition | Algebra | null | Algebra\BinaryOperations.v | is2binophrel | 683 |
Lemma isapropis2binophrel {X : setwith2binop} (R : hrel X) : isaprop (is2binophrel R). Proof. apply (isofhleveldirprod 1). - apply isapropisbinophrel. - apply isapropisbinophrel. Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isapropis2binophrel | 684 |
Lemma iscomp2binoptransrel {X : setwith2binop} (R : hrel X) (is : istrans R) (isb : is2binophrel R) : (iscomprelrelfun2 R R (@op1 X)) Γ (iscomprelrelfun2 R R (@op2 X)). Proof. split. - apply (@iscompbinoptransrel (setwithbinop1 X) R is (pr1 isb)). - apply (@iscompbinoptransrel (setwithbinop2 X) R is (pr2 isb)). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | iscomp2binoptransrel | 685 |
Definition twobinophrel (X : setwith2binop) : UU := β (R : hrel X), is2binophrel R. | Definition | Algebra | null | Algebra\BinaryOperations.v | twobinophrel | 686 |
Definition make_twobinophrel {X : setwith2binop} (t : hrel X) (H : is2binophrel t) : twobinophrel X := t ,, H. | Definition | Algebra | null | Algebra\BinaryOperations.v | make_twobinophrel | 687 |
Definition pr1twobinophrel (X : setwith2binop) : twobinophrel X β hrel X := @pr1 _ (Ξ» R : hrel X, is2binophrel R). | Definition | Algebra | null | Algebra\BinaryOperations.v | pr1twobinophrel | 688 |
Definition twobinoppo (X : setwith2binop) : UU := β (R : po X), is2binophrel R. | Definition | Algebra | null | Algebra\BinaryOperations.v | twobinoppo | 689 |
Definition make_twobinoppo {X : setwith2binop} (t : po X) (H : is2binophrel t) : twobinoppo X := t ,, H. | Definition | Algebra | null | Algebra\BinaryOperations.v | make_twobinoppo | 690 |
Definition pr1twobinoppo (X : setwith2binop) : twobinoppo X β po X := @pr1 _ (Ξ» R : po X, is2binophrel R). | Definition | Algebra | null | Algebra\BinaryOperations.v | pr1twobinoppo | 691 |
Definition twobinopeqrel (X : setwith2binop) : UU := β (R : eqrel X), is2binophrel R. | Definition | Algebra | null | Algebra\BinaryOperations.v | twobinopeqrel | 692 |
Definition make_twobinopeqrel {X : setwith2binop} (t : eqrel X) (H : is2binophrel t) : twobinopeqrel X := t ,, H. | Definition | Algebra | null | Algebra\BinaryOperations.v | make_twobinopeqrel | 693 |
Definition pr1twobinopeqrel (X : setwith2binop) : twobinopeqrel X β eqrel X := @pr1 _ (Ξ» R : eqrel X, is2binophrel R). | Definition | Algebra | null | Algebra\BinaryOperations.v | pr1twobinopeqrel | 694 |
Definition setwith2binopquot {X : setwith2binop} (R : twobinopeqrel X) : setwith2binop. Proof. exists (setquotinset R). set (qt := setquot R). set (qtset := setquotinset R). assert (iscomp1 : iscomprelrelfun2 R R (@op1 X)) by apply (pr1 (iscomp2binoptransrel (pr1 R) (eqreltrans _) (pr2 R))). set (qtop1 := setquotfun2 R R (@op1 X) iscomp1). assert (iscomp2 : iscomprelrelfun2 R R (@op2 X)) by apply (pr2 (iscomp2binoptransrel (pr1 R) (eqreltrans _) (pr2 R))). set (qtop2 := setquotfun2 R R (@op2 X) iscomp2). simpl. exact (qtop1 ,, qtop2). Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | setwith2binopquot | 695 |
Definition setwith2binopdirprod (X Y : setwith2binop) : setwith2binop. Proof. exists (setdirprod X Y). simpl. exact ( (Ξ» xy xy', (op1 (pr1 xy) (pr1 xy')) ,, (op1 (pr2 xy) (pr2 xy'))) ,, (Ξ» xy xy', (op2 (pr1 xy) (pr1 xy')) ,, (op2 (pr2 xy) (pr2 xy'))) ). Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | setwith2binopdirprod | 696 |
Lemma infinitary_op_to_binop {X : hSet} (op : β I : UU, (I β X) β X) : binop X. Proof. intros x y; exact (op _ (bool_rect (Ξ» _, X) x y)). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | infinitary_op_to_binop | 697 |
Definition isnonzeroCR (X : rig) (R : tightap X) := R 1%rig 0%rig. | Definition | Algebra | Require Import UniMath.MoreFoundations.Tactics. | Algebra\ConstructiveStructures.v | isnonzeroCR | 698 |
Definition isConstrDivRig (X : rig) (R : tightap X) := isnonzeroCR X R Γ (β x : X, R x 0%rig β multinvpair X x). | Definition | Algebra | Require Import UniMath.MoreFoundations.Tactics. | Algebra\ConstructiveStructures.v | isConstrDivRig | 699 |
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