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Definition pr1binopfun (X Y : setwithbinop) : binopfun X Y β (X β Y) := @pr1 _ _. | Definition | Algebra | null | Algebra\BinaryOperations.v | pr1binopfun | 500 |
Definition binopfunisbinopfun {X Y : setwithbinop} (f : binopfun X Y) : isbinopfun f := pr2 f. | Definition | Algebra | null | Algebra\BinaryOperations.v | binopfunisbinopfun | 501 |
Lemma isasetbinopfun (X Y : setwithbinop) : isaset (binopfun X Y). Proof. apply (isasetsubset (pr1binopfun X Y)). - change (isofhlevel 2 (X β Y)). apply impred. intro. apply (setproperty Y). - refine (isinclpr1 _ _). intro. apply isapropisbinopfun. Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isasetbinopfun | 502 |
Lemma isbinopfuncomp {X Y Z : setwithbinop} (f : binopfun X Y) (g : binopfun Y Z) : isbinopfun (g β f). Proof. set (axf := binopfunisbinopfun f). set (axg := binopfunisbinopfun g). intros a b. simpl. rewrite (axf a b). rewrite (axg (f a) (f b)). apply idpath. Qed. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isbinopfuncomp | 503 |
Definition binopfuncomp {X Y Z : setwithbinop} (f : binopfun X Y) (g : binopfun Y Z) : binopfun X Z := make_binopfun (g β f) (isbinopfuncomp f g). | Definition | Algebra | null | Algebra\BinaryOperations.v | binopfuncomp | 504 |
Definition binopmono (X Y : setwithbinop) : UU := β (f : incl X Y), isbinopfun (pr1 f). | Definition | Algebra | null | Algebra\BinaryOperations.v | binopmono | 505 |
Definition make_binopmono {X Y : setwithbinop} (f : incl X Y) (is : isbinopfun f) : binopmono X Y := f ,, is. | Definition | Algebra | null | Algebra\BinaryOperations.v | make_binopmono | 506 |
Definition pr1binopmono (X Y : setwithbinop) : binopmono X Y β incl X Y := @pr1 _ _. | Definition | Algebra | null | Algebra\BinaryOperations.v | pr1binopmono | 507 |
Definition binopincltobinopfun (X Y : setwithbinop) : binopmono X Y β binopfun X Y := Ξ» f, make_binopfun (pr1 (pr1 f)) (pr2 f). | Definition | Algebra | null | Algebra\BinaryOperations.v | binopincltobinopfun | 508 |
Definition binopmonocomp {X Y Z : setwithbinop} (f : binopmono X Y) (g : binopmono Y Z) : binopmono X Z := make_binopmono (inclcomp (pr1 f) (pr1 g)) (isbinopfuncomp f g). | Definition | Algebra | null | Algebra\BinaryOperations.v | binopmonocomp | 509 |
Definition binopiso (X Y : setwithbinop) : UU := β (f : X β Y), isbinopfun f. | Definition | Algebra | null | Algebra\BinaryOperations.v | binopiso | 510 |
Definition make_binopiso {X Y : setwithbinop} (f : X β Y) (is : isbinopfun f) : binopiso X Y := f ,, is. | Definition | Algebra | null | Algebra\BinaryOperations.v | make_binopiso | 511 |
Definition pr1binopiso (X Y : setwithbinop) : binopiso X Y β X β Y := @pr1 _ _. | Definition | Algebra | null | Algebra\BinaryOperations.v | pr1binopiso | 512 |
Lemma isasetbinopiso (X Y : setwithbinop) : isaset (binopiso X Y). Proof. use isaset_total2. - use isaset_total2. + use impred_isaset. intros t. use setproperty. + intros x. use isasetaprop. use isapropisweq. - intros w. use isasetaprop. use isapropisbinopfun. Qed. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isasetbinopiso | 513 |
Definition binopisotobinopmono (X Y : setwithbinop) : binopiso X Y β binopmono X Y := Ξ» f, make_binopmono (weqtoincl (pr1 f)) (pr2 f). | Definition | Algebra | null | Algebra\BinaryOperations.v | binopisotobinopmono | 514 |
Definition binopisocomp {X Y Z : setwithbinop} (f : binopiso X Y) (g : binopiso Y Z) : binopiso X Z := make_binopiso (weqcomp (pr1 f) (pr1 g)) (isbinopfuncomp f g). | Definition | Algebra | null | Algebra\BinaryOperations.v | binopisocomp | 515 |
Lemma isbinopfuninvmap {X Y : setwithbinop} (f : binopiso X Y) : isbinopfun (invmap (pr1 f)). Proof. set (axf := pr2 f). intros a b. apply (invmaponpathsweq (pr1 f)). rewrite (homotweqinvweq (pr1 f) (op a b)). rewrite (axf (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 | isbinopfuninvmap | 516 |
Definition invbinopiso {X Y : setwithbinop} (f : binopiso X Y) : binopiso Y X := make_binopiso (invweq (pr1 f)) (isbinopfuninvmap f). | Definition | Algebra | null | Algebra\BinaryOperations.v | invbinopiso | 517 |
Definition idbinopiso (X : setwithbinop) : binopiso X X. Proof. use make_binopiso. - exact (idweq X). - intros x1 x2. use idpath. Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | idbinopiso | 518 |
Definition setwithbinop_univalence_weq1 (X Y : setwithbinop) : (X = Y) β (X β Y) := total2_paths_equiv _ X Y. | Definition | Algebra | null | Algebra\BinaryOperations.v | setwithbinop_univalence_weq1 | 519 |
Definition setwithbinop_univalence_weq2 (X Y : setwithbinop) : (X β Y) β (binopiso 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 funextfun. intros x1. use funextfun. intros x2. exact (i x1 x2). + intros e. cbn in e. intros x1 x2. induction e. use idpath. + use isapropisbinopfun. + use isasetbinoponhSet. Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | setwithbinop_univalence_weq2 | 520 |
Definition setwithbinop_univalence_map (X Y : setwithbinop) : X = Y β binopiso X Y. Proof. intro e. induction e. exact (idbinopiso X). Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | setwithbinop_univalence_map | 521 |
Lemma setwithbinop_univalence_isweq (X Y : setwithbinop) : isweq (setwithbinop_univalence_map X Y). Proof. use isweqhomot. - exact (weqcomp (setwithbinop_univalence_weq1 X Y) (setwithbinop_univalence_weq2 X Y)). - intros e. induction e. use weqcomp_to_funcomp_app. - use weqproperty. Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | setwithbinop_univalence_isweq | 522 |
Definition setwithbinop_univalence (X Y : setwithbinop) : (X = Y) β (binopiso X Y) := make_weq (setwithbinop_univalence_map X Y) (setwithbinop_univalence_isweq X Y). | Definition | Algebra | null | Algebra\BinaryOperations.v | setwithbinop_univalence | 523 |
Definition hfiberbinop {A B : setwithbinop} (f : binopfun A B) (b1 b2 : B) (hf1 : hfiber (pr1 f) b1) (hf2 : hfiber (pr1 f) b2) : hfiber (pr1 f) (@op B b1 b2) := make_hfiber (pr1 f) (@op A (pr1 hf1) (pr1 hf2)) (hfiberbinop_path f b1 b2 hf1 hf2). | Definition | Algebra | null | Algebra\BinaryOperations.v | hfiberbinop | 524 |
Lemma islcancelablemonob {X Y : setwithbinop} (f : binopmono X Y) (x : X) (is : islcancelable (@op Y) (f x)) : islcancelable (@op X) x. Proof. unfold islcancelable. apply (isincltwooutof3a (Ξ» x0 : X, op x x0) f (pr2 (pr1 f))). assert (h : homot ((Ξ» y0 : Y, op (f x) y0) β f) (f β (Ξ» x0 : X, op x x0))). { intro x0; simpl. apply (!(pr2 f) x x0). } apply (isinclhomot _ _ h). apply (isinclcomp f (make_incl _ is)). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | islcancelablemonob | 525 |
Lemma isrcancelablemonob {X Y : setwithbinop} (f : binopmono X Y) (x : X) (is : isrcancelable (@op Y) (f x)) : isrcancelable (@op X) x. Proof. unfold islcancelable. apply (isincltwooutof3a (Ξ» x0 : X, op x0 x) f (pr2 (pr1 f))). assert (h : homot ((Ξ» y0 : Y, op y0 (f x)) β f) (f β (Ξ» x0 : X, op x0 x))). { intro x0; simpl. apply (!(pr2 f) x0 x). } apply (isinclhomot _ _ h). apply (isinclcomp f (make_incl _ is)). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isrcancelablemonob | 526 |
Lemma iscancelablemonob {X Y : setwithbinop} (f : binopmono X Y) (x : X) (is : iscancelable (@op Y) (f x)) : iscancelable (@op X) x. Proof. apply (islcancelablemonob f x (pr1 is) ,, isrcancelablemonob f x (pr2 is)). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | iscancelablemonob | 527 |
Lemma islinvertibleisob {X Y : setwithbinop} (f : binopiso X Y) (x : X) (is : islinvertible (@op Y) (f x)) : islinvertible (@op X) x. Proof. unfold islinvertible. apply (twooutof3a (Ξ» x0 : X, op x x0) f). - assert (h : homot ((Ξ» y0 : Y, op (f x) y0) β f) (f β (Ξ» x0 : X, op x x0))). { intro x0; simpl. apply (!(pr2 f) x x0). } apply (isweqhomot _ _ h). apply (pr2 (weqcomp f (make_weq _ is))). - apply (pr2 (pr1 f)). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | islinvertibleisob | 528 |
Lemma isrinvertibleisob {X Y : setwithbinop} (f : binopiso X Y) (x : X) (is : isrinvertible (@op Y) (f x)) : isrinvertible (@op X) x. Proof. unfold islinvertible. apply (twooutof3a (Ξ» x0 : X, op x0 x) f). - assert (h : homot ((Ξ» y0 : Y, op y0 (f x)) β f) (f β (Ξ» x0 : X, op x0 x))). { intro x0; simpl. apply (!(pr2 f) x0 x). } apply (isweqhomot _ _ h). apply (pr2 (weqcomp f (make_weq _ is))). - apply (pr2 (pr1 f)). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isrinvertibleisob | 529 |
Lemma isinvertiblemonob {X Y : setwithbinop} (f : binopiso X Y) (x : X) (is : isinvertible (@op Y) (f x)) : isinvertible (@op X) x. Proof. apply (islinvertibleisob f x (pr1 is) ,, isrinvertibleisob f x (pr2 is)). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isinvertiblemonob | 530 |
Definition islinvertibleisof {X Y : setwithbinop} (f : binopiso X Y) (x : X) (is : islinvertible (@op X) x) : islinvertible (@op Y) (f x). Proof. unfold islinvertible. apply (twooutof3b f). - apply (pr2 (pr1 f)). - assert (h : homot (f β (Ξ» x0 : X, op x x0)) (Ξ» x0 : X, op (f x) (f x0))). { intro x0; simpl. apply (pr2 f x x0). } apply (isweqhomot _ _ h). apply (pr2 (weqcomp (make_weq _ is) f)). Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | islinvertibleisof | 531 |
Definition isrinvertibleisof {X Y : setwithbinop} (f : binopiso X Y) (x : X) (is : isrinvertible (@op X) x) : isrinvertible (@op Y) (f x). Proof. unfold isrinvertible. apply (twooutof3b f). - apply (pr2 (pr1 f)). - assert (h : homot (f β (Ξ» x0 : X, op x0 x)) (Ξ» x0 : X, op (f x0) (f x))). { intro x0; simpl. apply (pr2 f x0 x). } apply (isweqhomot _ _ h). apply (pr2 (weqcomp (make_weq _ is) f)). Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | isrinvertibleisof | 532 |
Lemma isinvertiblemonof {X Y : setwithbinop} (f : binopiso X Y) (x : X) (is : isinvertible (@op X) x) : isinvertible (@op Y) (f x). Proof. apply (islinvertibleisof f x (pr1 is) ,, isrinvertibleisof f x (pr2 is)). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isinvertiblemonof | 533 |
Lemma isassocmonob {X Y : setwithbinop} (f : binopmono X Y) (is : isassoc (@op Y)) : isassoc (@op X). Proof. set (axf := pr2 f). simpl in axf. intros a b c. apply (invmaponpathsincl _ (pr2 (pr1 f))). rewrite (axf (op a b) c). rewrite (axf a b). rewrite (axf a (op b c)). rewrite (axf b c). apply is. Qed. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isassocmonob | 534 |
Lemma iscommmonob {X Y : setwithbinop} (f : binopmono X Y) (is : iscomm (@op Y)) : iscomm (@op X). Proof. set (axf := pr2 f). simpl in axf. intros a b. apply (invmaponpathsincl _ (pr2 (pr1 f))). rewrite (axf a b). rewrite (axf b a). apply is. Qed. | Lemma | Algebra | null | Algebra\BinaryOperations.v | iscommmonob | 535 |
Lemma isassocisof {X Y : setwithbinop} (f : binopiso X Y) (is : isassoc (@op X)) : isassoc (@op Y). Proof. apply (isassocmonob (invbinopiso f) is). Qed. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isassocisof | 536 |
Lemma iscommisof {X Y : setwithbinop} (f : binopiso X Y) (is : iscomm (@op X)) : iscomm (@op Y). Proof. apply (iscommmonob (invbinopiso f) is). Qed. | Lemma | Algebra | null | Algebra\BinaryOperations.v | iscommisof | 537 |
Lemma isunitisof {X Y : setwithbinop} (f : binopiso X Y) (unx : X) (is : isunit (@op X) unx) : isunit (@op Y) (f unx). Proof. set (axf := pr2 f). split. - intro a. change (f unx) with (pr1 f unx). apply (invmaponpathsweq (pr1 (invbinopiso f))). rewrite (pr2 (invbinopiso f) (pr1 f unx) a). simpl. rewrite (homotinvweqweq (pr1 f) unx). apply (pr1 is). - intro a. change (f unx) with (pr1 f unx). apply (invmaponpathsweq (pr1 (invbinopiso f))). rewrite (pr2 (invbinopiso f) a (pr1 f unx)). simpl. rewrite (homotinvweqweq (pr1 f) unx). apply (pr2 is). Qed. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isunitisof | 538 |
Definition isunitalisof {X Y : setwithbinop} (f : binopiso X Y) (is : isunital (@op X)) : isunital (@op Y) := make_isunital (f (pr1 is)) (isunitisof f (pr1 is) (pr2 is)). | Definition | Algebra | null | Algebra\BinaryOperations.v | isunitalisof | 539 |
Lemma isunitisob {X Y : setwithbinop} (f : binopiso X Y) (uny : Y) (is : isunit (@op Y) uny) : isunit (@op X) ((invmap f) uny). Proof. set (int := isunitisof (invbinopiso f)). simpl. simpl in int. apply int. apply is. Qed. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isunitisob | 540 |
Definition isunitalisob {X Y : setwithbinop} (f : binopiso X Y) (is : isunital (@op Y)) : isunital (@op X) := make_isunital ((invmap f) (pr1 is)) (isunitisob f (pr1 is) (pr2 is)). | Definition | Algebra | null | Algebra\BinaryOperations.v | isunitalisob | 541 |
Definition ismonoidopisof {X Y : setwithbinop} (f : binopiso X Y) (is : ismonoidop (@op X)) : ismonoidop (@op Y) := isassocisof f (pr1 is) ,, isunitalisof f (pr2 is). | Definition | Algebra | null | Algebra\BinaryOperations.v | ismonoidopisof | 542 |
Definition ismonoidopisob {X Y : setwithbinop} (f : binopiso X Y) (is : ismonoidop (@op Y)) : ismonoidop (@op X) := isassocisob f (pr1 is) ,, isunitalisob f (pr2 is). | Definition | Algebra | null | Algebra\BinaryOperations.v | ismonoidopisob | 543 |
Lemma isinvisof {X Y : setwithbinop} (f : binopiso X Y) (unx : X) (invx : X β X) (is : isinv (@op X) unx invx) : isinv (@op Y) (pr1 f unx) ((pr1 f) β invx β invmap (pr1 f)). Proof. set (axf := pr2 f). set (axinvf := pr2 (invbinopiso f)). simpl in axf, axinvf. split. - intro a. apply (invmaponpathsweq (pr1 (invbinopiso f))). simpl. rewrite (axinvf ((pr1 f) (invx (invmap (pr1 f) a))) a). rewrite (homotinvweqweq (pr1 f) unx). rewrite (homotinvweqweq (pr1 f) (invx (invmap (pr1 f) a))). apply (pr1 is). - intro a. apply (invmaponpathsweq (pr1 (invbinopiso f))). simpl. rewrite (axinvf a ((pr1 f) (invx (invmap (pr1 f) a)))). rewrite (homotinvweqweq (pr1 f) unx). rewrite (homotinvweqweq (pr1 f) (invx (invmap (pr1 f) a))). apply (pr2 is). Qed. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isinvisof | 544 |
Definition isgropisof {X Y : setwithbinop} (f : binopiso X Y) (is : isgrop (@op X)) : isgrop (@op Y) := ismonoidopisof f is ,, (f β grinv_is is β invmap f) ,, isinvisof f (unel_is is) (grinv_is is) (pr2 (pr2 is)). | Definition | Algebra | null | Algebra\BinaryOperations.v | isgropisof | 545 |
Lemma isinvisob {X Y : setwithbinop} (f : binopiso X Y) (uny : Y) (invy : Y β Y) (is : isinv (@op Y) uny invy) : isinv (@op X) (invmap (pr1 f) uny) (invmap f β invy β f). Proof. apply (isinvisof (invbinopiso f) uny invy is). Qed. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isinvisob | 546 |
Definition isgropisob {X Y : setwithbinop} (f : binopiso X Y) (is : isgrop (@op Y)) : isgrop (@op X) := ismonoidopisob f is ,, invmap f β grinv_is is β f ,, isinvisob f (unel_is is) (grinv_is is) (pr2 (pr2 is)). | Definition | Algebra | null | Algebra\BinaryOperations.v | isgropisob | 547 |
Definition isabmonoidopisof {X Y : setwithbinop} (f : binopiso X Y) (is : isabmonoidop (@op X)) : isabmonoidop (@op Y) := ismonoidopisof f is ,, iscommisof f (commax_is is). | Definition | Algebra | null | Algebra\BinaryOperations.v | isabmonoidopisof | 548 |
Definition isabmonoidopisob {X Y : setwithbinop} (f : binopiso X Y) (is : isabmonoidop (@op Y)) : isabmonoidop (@op X) := ismonoidopisob f is ,, iscommisob f (commax_is is). | Definition | Algebra | null | Algebra\BinaryOperations.v | isabmonoidopisob | 549 |
Definition isabgropisof {X Y : setwithbinop} (f : binopiso X Y) (is : isabgrop (@op X)) : isabgrop (@op Y) := isgropisof f is ,, iscommisof f (commax_is is). | Definition | Algebra | null | Algebra\BinaryOperations.v | isabgropisof | 550 |
Definition isabgropisob {X Y : setwithbinop} (f : binopiso X Y) (is : isabgrop (@op Y)) : isabgrop (@op X) := isgropisob f is ,, iscommisob f (commax_is is). | Definition | Algebra | null | Algebra\BinaryOperations.v | isabgropisob | 551 |
Definition issubsetwithbinop {X : hSet} (opp : binop X) (A : hsubtype X) : UU := β a a' : A, A (opp (pr1 a) (pr1 a')). | Definition | Algebra | null | Algebra\BinaryOperations.v | issubsetwithbinop | 552 |
Lemma isapropissubsetwithbinop {X : hSet} (opp : binop X) (A : hsubtype X) : isaprop (issubsetwithbinop opp A). Proof. apply impred. intro a. apply impred. intros a'. apply (pr2 (A (opp (pr1 a) (pr1 a')))). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isapropissubsetwithbinop | 553 |
Definition subsetswithbinop (X : setwithbinop) : UU := β (A : hsubtype X), issubsetwithbinop (@op X) A. | Definition | Algebra | null | Algebra\BinaryOperations.v | subsetswithbinop | 554 |
Definition make_subsetswithbinop {X : setwithbinop} (t : hsubtype X) (H : issubsetwithbinop op t) : subsetswithbinop X := t ,, H. | Definition | Algebra | null | Algebra\BinaryOperations.v | make_subsetswithbinop | 555 |
Definition subsetswithbinopconstr {X : setwithbinop} : β (t : hsubtype X), (Ξ» A : hsubtype X, issubsetwithbinop op A) t β β A : hsubtype X, issubsetwithbinop op A := @make_subsetswithbinop X. | Definition | Algebra | null | Algebra\BinaryOperations.v | subsetswithbinopconstr | 556 |
Definition pr1subsetswithbinop (X : setwithbinop) : subsetswithbinop X β hsubtype X := @pr1 _ (Ξ» A : hsubtype X, issubsetwithbinop (@op X) A). | Definition | Algebra | null | Algebra\BinaryOperations.v | pr1subsetswithbinop | 557 |
Definition pr2subsetswithbinop {X : setwithbinop} (Y : subsetswithbinop X) : issubsetwithbinop (@op X) (pr1subsetswithbinop X Y) := pr2 Y. | Definition | Algebra | null | Algebra\BinaryOperations.v | pr2subsetswithbinop | 558 |
Definition totalsubsetwithbinop (X : setwithbinop) : subsetswithbinop X. Proof. exists (Ξ» x : X, htrue). intros x x'. apply tt. Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | totalsubsetwithbinop | 559 |
Definition carrierofasubsetwithbinop {X : setwithbinop} (A : subsetswithbinop X) : setwithbinop. Proof. set (aset := (make_hSet (carrier A) (isasetsubset (pr1carrier A) (setproperty X) (isinclpr1carrier A))) : hSet). exists aset. set (subopp := (Ξ» a a' : A, make_carrier A (op (pr1carrier _ a) (pr1carrier _ a')) (pr2 A a a')) : (A β A β A)). simpl. unfold binop. apply subopp. Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | carrierofasubsetwithbinop | 560 |
Definition isbinophrel {X : setwithbinop} (R : hrel X) : UU := (β a b c : X, R a b β R (op c a) (op c b)) Γ (β a b c : X, R a b β R (op a c) (op b c)). | Definition | Algebra | null | Algebra\BinaryOperations.v | isbinophrel | 561 |
Definition make_isbinophrel {X : setwithbinop} {R : hrel X} (H1 : β a b c : X, R a b β R (op c a) (op c b)) (H2 : β a b c : X, R a b β R (op a c) (op b c)) : isbinophrel R := H1 ,, H2. | Definition | Algebra | null | Algebra\BinaryOperations.v | make_isbinophrel | 562 |
Definition isbinophrellogeqf {X : setwithbinop} {L R : hrel X} (lg : hrellogeq L R) (isl : isbinophrel L) : isbinophrel 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 | isbinophrellogeqf | 563 |
Lemma isapropisbinophrel {X : setwithbinop} (R : hrel X) : isaprop (isbinophrel 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 | isapropisbinophrel | 564 |
Lemma isbinophrelif {X : setwithbinop} (R : hrel X) (is : iscomm (@op X)) (isl : β a b c : X, R a b β R (op c a) (op c b)) : isbinophrel 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 | isbinophrelif | 565 |
Lemma iscompbinoptransrel {X : setwithbinop} (R : hrel X) (ist : istrans R) (isb : isbinophrel R) : iscomprelrelfun2 R R (@op X). Proof. intros a b c d. intros rab rcd. set (racbc := pr2 isb a b c rab). set (rbcbd := pr1 isb c d b rcd). apply (ist _ _ _ racbc rbcbd). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | iscompbinoptransrel | 566 |
Lemma isbinopreflrel {X : setwithbinop} (R : hrel X) (isr : isrefl R) (isb : iscomprelrelfun2 R R (@op X)) : isbinophrel R. Proof. split. - intros a b c rab. apply (isb c c a b (isr c) rab). - intros a b c rab. apply (isb a b c c rab (isr c)). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isbinopreflrel | 567 |
Definition binophrel (X : setwithbinop) : UU := β (R : hrel X), isbinophrel R. | Definition | Algebra | null | Algebra\BinaryOperations.v | binophrel | 568 |
Definition make_binophrel {X : setwithbinop} (t : hrel X) (H : isbinophrel t) : binophrel X := t ,, H. | Definition | Algebra | null | Algebra\BinaryOperations.v | make_binophrel | 569 |
Definition pr1binophrel (X : setwithbinop) : binophrel X β hrel X := @pr1 _ (Ξ» R : hrel X, isbinophrel R). | Definition | Algebra | null | Algebra\BinaryOperations.v | pr1binophrel | 570 |
Definition binophrel_resp_left {X : setwithbinop} (R : binophrel X) {a b : X} (c : X) (r : R a b) : R (op c a) (op c b) := pr1 (pr2 R) a b c r. | Definition | Algebra | null | Algebra\BinaryOperations.v | binophrel_resp_left | 571 |
Definition binophrel_resp_right {X : setwithbinop} (R : binophrel X) {a b : X} (c : X) (r : R a b) : R (op a c) (op b c) := pr2 (pr2 R) a b c r. | Definition | Algebra | null | Algebra\BinaryOperations.v | binophrel_resp_right | 572 |
Definition binoppo (X : setwithbinop) : UU := β (R : po X), isbinophrel R. | Definition | Algebra | null | Algebra\BinaryOperations.v | binoppo | 573 |
Definition make_binoppo {X : setwithbinop} (t : po X) (H : isbinophrel t) : binoppo X := t ,, H. | Definition | Algebra | null | Algebra\BinaryOperations.v | make_binoppo | 574 |
Definition pr1binoppo (X : setwithbinop) : binoppo X β po X := @pr1 _ (Ξ» R : po X, isbinophrel R). | Definition | Algebra | null | Algebra\BinaryOperations.v | pr1binoppo | 575 |
Definition binopeqrel (X : setwithbinop) : UU := β (R : eqrel X), isbinophrel R. | Definition | Algebra | null | Algebra\BinaryOperations.v | binopeqrel | 576 |
Definition make_binopeqrel {X : setwithbinop} (t : eqrel X) (H : isbinophrel t) : binopeqrel X := t ,, H. | Definition | Algebra | null | Algebra\BinaryOperations.v | make_binopeqrel | 577 |
Definition pr1binopeqrel (X : setwithbinop) : binopeqrel X β eqrel X := @pr1 _ (Ξ» R : eqrel X, isbinophrel R). | Definition | Algebra | null | Algebra\BinaryOperations.v | pr1binopeqrel | 578 |
Definition binopeqrel_resp_left {X : setwithbinop} (R : binopeqrel X) {a b : X} (c : X) (r : R a b) : R (op c a) (op c b) := pr1 (pr2 R) a b c r. | Definition | Algebra | null | Algebra\BinaryOperations.v | binopeqrel_resp_left | 579 |
Definition binopeqrel_resp_right {X : setwithbinop} (R : binopeqrel X) {a b : X} (c : X) (r : R a b) : R (op a c) (op b c) := pr2 (pr2 R) a b c r. | Definition | Algebra | null | Algebra\BinaryOperations.v | binopeqrel_resp_right | 580 |
Definition setwithbinopquot {X : setwithbinop} (R : binopeqrel X) : setwithbinop. Proof. exists (setquotinset R). set (qt := setquot R). set (qtset := setquotinset R). assert (iscomp : iscomprelrelfun2 R R op) by apply (iscompbinoptransrel R (eqreltrans R) (pr2 R)). set (qtmlt := setquotfun2 R R op iscomp). simpl. unfold binop. apply qtmlt. Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | setwithbinopquot | 581 |
Definition ispartbinophrel {X : setwithbinop} (S : hsubtype X) (R : hrel X) : UU := (β a b c : X, S c β R a b β R (op c a) (op c b)) Γ (β a b c : X, S c β R a b β R (op a c) (op b c)). | Definition | Algebra | null | Algebra\BinaryOperations.v | ispartbinophrel | 582 |
Lemma isaprop_ispartbinophrel {X : setwithbinop} (S : hsubtype X) (R : hrel X) : isaprop (ispartbinophrel S R). Proof. apply isapropdirprod ; apply impred_isaprop ; intros a ; apply impred_isaprop ; intros b ; apply impred_isaprop ; intros c ; apply isapropimpl, isapropimpl, propproperty. Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isaprop_ispartbinophrel | 583 |
Definition isbinoptoispartbinop {X : setwithbinop} (S : hsubtype X) (L : hrel X) (d2 : isbinophrel L) : ispartbinophrel S L. Proof. unfold isbinophrel in d2. unfold ispartbinophrel. split. - intros a b c is. apply (pr1 d2 a b c). - intros a b c is. apply (pr2 d2 a b c). Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | isbinoptoispartbinop | 584 |
Definition ispartbinophrellogeqf {X : setwithbinop} (S : hsubtype X) {L R : hrel X} (lg : hrellogeq L R) (isl : ispartbinophrel S L) : ispartbinophrel S R. Proof. split. - intros a b c is rab. apply ((pr1 (lg _ _) ((pr1 isl) _ _ _ is (pr2 (lg _ _) rab)))). - intros a b c is rab. apply ((pr1 (lg _ _) ((pr2 isl) _ _ _ is (pr2 (lg _ _) rab)))). Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | ispartbinophrellogeqf | 585 |
Lemma ispartbinophrelif {X : setwithbinop} (S : hsubtype X) (R : hrel X) (is : iscomm (@op X)) (isl : β a b c : X, S c β R a b β R (op c a) (op c b)) : ispartbinophrel 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 | ispartbinophrelif | 586 |
Definition generated_binophrel_hrel {X : setwithbinop} (R : hrel X) : hrel X := Ξ» x x', β(R' : binophrel X), (β xβ xβ, R xβ xβ β R' xβ xβ) β R' x x'. | Definition | Algebra | null | Algebra\BinaryOperations.v | generated_binophrel_hrel | 587 |
Lemma isbinophrel_generated_binophrel {X : setwithbinop} (R : hrel X) : isbinophrel (generated_binophrel_hrel R). Proof. split. - intros a b c H R' H2. apply binophrel_resp_left. exact (H R' H2). - intros a b c H R' H2. apply binophrel_resp_right. exact (H R' H2). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isbinophrel_generated_binophrel | 588 |
Definition generated_binophrel {X : setwithbinop} (R : hrel X) : binophrel X := make_binophrel (generated_binophrel_hrel R) (isbinophrel_generated_binophrel R). | Definition | Algebra | null | Algebra\BinaryOperations.v | generated_binophrel | 589 |
Lemma generated_binophrel_intro {X : setwithbinop} {R : hrel X} {x x' : X} (r : R x x') : generated_binophrel R x x'. Proof. intros R' H. exact (H x x' r). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | generated_binophrel_intro | 590 |
Definition generated_binopeqrel_hrel {X : setwithbinop} (R : hrel X) : hrel X := Ξ» x x', β(R' : binopeqrel X), (β xβ xβ, R xβ xβ β R' xβ xβ) β R' x x'. | Definition | Algebra | null | Algebra\BinaryOperations.v | generated_binopeqrel_hrel | 591 |
Lemma isbinophrel_generated_binopeqrel {X : setwithbinop} (R : hrel X) : isbinophrel (generated_binopeqrel_hrel R). Proof. split. - intros a b c H R' H2. apply binopeqrel_resp_left. exact (H R' H2). - intros a b c H R' H2. apply binopeqrel_resp_right. exact (H R' H2). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | isbinophrel_generated_binopeqrel | 592 |
Lemma iseqrel_generated_binopeqrel {X : setwithbinop} (R : hrel X) : iseqrel (generated_binopeqrel_hrel R). Proof. use iseqrelconstr. - intros x1 x2 x3 H1 H2 R' HR. eapply eqreltrans. + exact (H1 R' HR). + exact (H2 R' HR). - intros x R' HR. apply eqrelrefl. - intros x1 x2 H R' HR. apply eqrelsymm. exact (H R' HR). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | iseqrel_generated_binopeqrel | 593 |
Definition generated_binopeqrel {X : setwithbinop} (R : hrel X) : binopeqrel X := make_binopeqrel (make_eqrel (generated_binopeqrel_hrel R) (iseqrel_generated_binopeqrel R)) (isbinophrel_generated_binopeqrel R). | Definition | Algebra | null | Algebra\BinaryOperations.v | generated_binopeqrel | 594 |
Lemma generated_binopeqrel_intro {X : setwithbinop} {R : hrel X} {x x' : X} (r : R x x') : generated_binopeqrel R x x'. Proof. intros R' H. exact (H x x' r). Defined. | Lemma | Algebra | null | Algebra\BinaryOperations.v | generated_binopeqrel_intro | 595 |
Definition pullback_binopeqrel {X Y : setwithbinop} (f : binopfun X Y) (R : binopeqrel Y) : binopeqrel X. Proof. use make_binopeqrel. - use make_eqrel. + intros x x'. exact (R (f x) (f x')). + apply iseqrelconstr. * intros x1 x2 x3 r1 r2. exact (eqreltrans R _ _ _ r1 r2). * intro x. exact (eqrelrefl R _). * intros x x' r. exact (eqrelsymm R _ _ r). - apply make_isbinophrel; simpl; intros x1 x2 x3 r; rewrite !binopfunisbinopfun. + exact (binopeqrel_resp_left R _ r). + exact (binopeqrel_resp_right R _ r). Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | pullback_binopeqrel | 596 |
Definition pullback_binopeqrel_rev {X Y : setwithbinop} (f : binopfun X (setwithbinop_rev Y)) (R : binopeqrel Y) : binopeqrel X. Proof. apply (pullback_binopeqrel f). use make_binopeqrel. - exact R. - apply make_isbinophrel; intros x1 x2 x3 r; apply (pr2 R); exact r. Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | pullback_binopeqrel_rev | 597 |
Definition binopeqrel_eq (X : setwithbinop) : binopeqrel X. Proof. use make_binopeqrel. - use make_eqrel. + intros x x'. exact (make_hProp (x = x') (pr2 (pr1 X) _ _)). + apply iseqrelconstr. * intros x1 x2 x3 r1 r2. exact (r1 @ r2). * intro x. reflexivity. * intros x x' r. exact (!r). - apply make_isbinophrel; simpl; intros x1 x2 x3 r; rewrite r; reflexivity. Defined. | Definition | Algebra | null | Algebra\BinaryOperations.v | binopeqrel_eq | 598 |
Definition binopeqrel_of_binopfun {X Y : setwithbinop} (f : binopfun X Y) : binopeqrel X := pullback_binopeqrel f (binopeqrel_eq Y). | Definition | Algebra | null | Algebra\BinaryOperations.v | binopeqrel_of_binopfun | 599 |