fact
stringlengths
13
15.5k
type
stringclasses
9 values
library
stringclasses
15 values
imports
stringlengths
14
7.64k
βŒ€
filename
stringlengths
12
97
symbolic_name
stringlengths
1
78
index_level
int64
0
38.7k
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 := (∏ a b c : X, R (op c a) (op c b) β†’ R a b) Γ— (∏ 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 : ∏ 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 := (∏ a b c : X, S c β†’ R (op c a) (op c b) β†’ R a b) Γ— (∏ 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 : ∏ 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 (λ x x', R (f x) (f x')). Proof. set (ish := (pr2 f) : ∏ 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 : ∏ 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) : ∏ 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 (λ x x', R (f x) (f x')). Proof. set (ish := (pr2 f) : ∏ 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 : ∏ 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) : ∏ 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 : ∏ (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 : ∏ (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 := βˆ‘ (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 := (∏ x x' : X, f (op1 x x') = op1 (f x) (f x')) Γ— (∏ 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 : ∏ x x' : X, f (op1 x x') = op1 (f x) (f x')) (H2 : ∏ 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 := βˆ‘ (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 ∘ 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 ∘ f) (istwobinopfuncomp f g).
Definition
Algebra
null
Algebra\BinaryOperations.v
twobinopfuncomp
638
Definition twobinopmono (X Y : setwith2binop) : UU := βˆ‘ (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 := βˆ‘ (f : X ≃ Y), istwobinopfun f.
Definition
Algebra
null
Algebra\BinaryOperations.v
twobinopiso
646
Definition make_twobinopiso {X Y : setwith2binop} (f : X ≃ 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 ≃ 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) ≃ (X ╝ Y) := total2_paths_equiv _ X Y.
Definition
Algebra
null
Algebra\BinaryOperations.v
setwith2binop_univalence_weq1
658
Definition setwith2binop_univalence_weq2 (X Y : setwith2binop) : (X ╝ Y) ≃ (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) ≃ (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 := (∏ a a' : A, A (op1 (pr1 a) (pr1 a'))) Γ— (∏ 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 := βˆ‘ (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} : ∏ (t : hsubtype X), (Ξ» A : hsubtype X, issubsetwith2binop A) t β†’ βˆ‘ 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 := βˆ‘ (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 := βˆ‘ (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 := βˆ‘ (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 : ∏ 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— (∏ x : X, R x 0%rig β†’ multinvpair X x).
Definition
Algebra
Require Import UniMath.MoreFoundations.Tactics.
Algebra\ConstructiveStructures.v
isConstrDivRig
699