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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.
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Lemma
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Algebra
| null |
Algebra\BinaryOperations.v
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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.
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Lemma
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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.
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Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
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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).
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
|
isinvbinophrel
| 603
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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.
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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.
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Lemma
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Algebra
| null |
Algebra\BinaryOperations.v
|
isapropisinvbinophrel
| 605
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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.
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Lemma
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Algebra
| null |
Algebra\BinaryOperations.v
|
isinvbinophrelif
| 606
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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).
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
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ispartinvbinophrel
| 607
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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.
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
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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.
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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.
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Lemma
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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.
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Lemma
|
Algebra
| null |
Algebra\BinaryOperations.v
|
binophrelandfun
| 611
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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.
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Lemma
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Algebra
| null |
Algebra\BinaryOperations.v
|
ispartbinophrelandfun
| 612
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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.
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Lemma
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Algebra
| null |
Algebra\BinaryOperations.v
|
invbinophrelandfun
| 613
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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.
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Lemma
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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.
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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.
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Definition
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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.
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
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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.
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
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setwithbinopdirprod
| 618
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Definition setwith2binop : UU := β (X : hSet), (binop X) Γ (binop X).
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
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setwith2binop
| 619
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Definition make_setwith2binop (X : hSet) (opps : (binop X) Γ (binop X)) : setwith2binop := X ,, opps.
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Definition
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Algebra
| null |
Algebra\BinaryOperations.v
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make_setwith2binop
| 620
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Definition pr1setwith2binop : setwith2binop β hSet := @pr1 _ (Ξ» X : hSet, (binop X) Γ (binop X)).
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Definition
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Algebra
| null |
Algebra\BinaryOperations.v
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pr1setwith2binop
| 621
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Definition op1 {X : setwith2binop} : binop X := pr1 (pr2 X).
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Definition
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Algebra
| null |
Algebra\BinaryOperations.v
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op1
| 622
|
Definition op2 {X : setwith2binop} : binop X := pr2 (pr2 X).
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
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op2
| 623
|
Definition setwithbinop1 (X : setwith2binop) : setwithbinop := make_setwithbinop (pr1 X) (@op1 X).
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
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setwithbinop1
| 624
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Definition setwithbinop2 (X : setwith2binop) : setwithbinop := make_setwithbinop (pr1 X) (@op2 X).
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Definition
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Algebra
| null |
Algebra\BinaryOperations.v
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setwithbinop2
| 625
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Definition isasettwobinoponhSet (X : hSet) : isaset ((binop X) Γ (binop X)).
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
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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')).
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Definition
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Algebra
| null |
Algebra\BinaryOperations.v
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istwobinopfun
| 627
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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.
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
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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.
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Lemma
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Algebra
| null |
Algebra\BinaryOperations.v
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isapropistwobinopfun
| 629
|
Definition twobinopfun (X Y : setwith2binop) : UU := β (f : X β Y), istwobinopfun f.
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
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twobinopfun
| 630
|
Definition make_twobinopfun {X Y : setwith2binop} (f : X β Y) (is : istwobinopfun f) : twobinopfun X Y := f ,, is.
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Definition
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Algebra
| null |
Algebra\BinaryOperations.v
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make_twobinopfun
| 631
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Definition pr1twobinopfun (X Y : setwith2binop) : twobinopfun X Y β (X β Y) := @pr1 _ _.
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Definition
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Algebra
| null |
Algebra\BinaryOperations.v
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pr1twobinopfun
| 632
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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)).
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
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binop1fun
| 633
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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)).
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
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binop2fun
| 634
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Definition twobinopfun_paths {X Y : setwith2binop} (f g : twobinopfun X Y) (e : pr1 f = pr1 g) : f = g.
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Definition
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Algebra
| null |
Algebra\BinaryOperations.v
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twobinopfun_paths
| 635
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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.
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Lemma
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Algebra
| null |
Algebra\BinaryOperations.v
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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.
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Lemma
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Algebra
| null |
Algebra\BinaryOperations.v
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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).
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
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twobinopfuncomp
| 638
|
Definition twobinopmono (X Y : setwith2binop) : UU := β (f : incl X Y), istwobinopfun f.
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
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twobinopmono
| 639
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Definition make_twobinopmono {X Y : setwith2binop} (f : incl X Y) (is : istwobinopfun f) : twobinopmono X Y := f ,, is.
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Definition
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Algebra
| null |
Algebra\BinaryOperations.v
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make_twobinopmono
| 640
|
Definition pr1twobinopmono (X Y : setwith2binop) : twobinopmono X Y β incl X Y := @pr1 _ _.
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Definition
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Algebra
| null |
Algebra\BinaryOperations.v
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pr1twobinopmono
| 641
|
Definition twobinopincltotwobinopfun (X Y : setwith2binop) : twobinopmono X Y β twobinopfun X Y := Ξ» f, make_twobinopfun (pr1 (pr1 f)) (pr2 f).
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Definition
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Algebra
| null |
Algebra\BinaryOperations.v
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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)).
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Definition
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Algebra
| null |
Algebra\BinaryOperations.v
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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)).
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Definition
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Algebra
| null |
Algebra\BinaryOperations.v
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binop2mono
| 644
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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).
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Definition
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Algebra
| null |
Algebra\BinaryOperations.v
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twobinopmonocomp
| 645
|
Definition twobinopiso (X Y : setwith2binop) : UU := β (f : X β Y), istwobinopfun f.
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Definition
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Algebra
| null |
Algebra\BinaryOperations.v
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twobinopiso
| 646
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Definition make_twobinopiso {X Y : setwith2binop} (f : X β Y) (is : istwobinopfun f) : twobinopiso X Y := f ,, is.
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Definition
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Algebra
| null |
Algebra\BinaryOperations.v
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make_twobinopiso
| 647
|
Definition pr1twobinopiso (X Y : setwith2binop) : twobinopiso X Y β X β Y := @pr1 _ _.
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
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pr1twobinopiso
| 648
|
Definition twobinopisototwobinopmono (X Y : setwith2binop) : twobinopiso X Y β twobinopmono X Y := Ξ» f, make_twobinopmono (weqtoincl (pr1 f)) (pr2 f).
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
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twobinopisototwobinopmono
| 649
|
Definition twobinopisototwobinopfun {X Y : setwith2binop} (f : twobinopiso X Y) : twobinopfun X Y := make_twobinopfun f (pr2 f).
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
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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.
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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)).
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Definition
|
Algebra
| null |
Algebra\BinaryOperations.v
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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)).
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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).
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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.
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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).
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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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