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Lemma iscomm_free_abmonoid (X : hSet) : iscomm (@op (free_abmonoid' X)). Proof. refine (setquotuniv2prop' _ _ _). - intros. apply (isasetmonoid (free_abmonoid' X)). - intros x1 x2. apply (iscompsetquotpr _ (x1 * x2) (x2 * x1)). apply generated_binopeqrel_intro, free_abmonoid_hrel_intro. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
iscomm_free_abmonoid
| 1,000
|
Definition free_abmonoid (X : hSet) : abmonoid := abmonoid_of_monoid (free_abmonoid' X) (iscomm_free_abmonoid X).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abmonoid
| 1,001
|
Definition free_abmonoid_pr (X : hSet) : monoidfun (free_monoid X) (free_abmonoid X) := presented_monoid_pr _.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abmonoid_pr
| 1,002
|
Definition free_abmonoid_unit {X : hSet} (x : X) : free_abmonoid X := presented_monoid_intro x.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abmonoid_unit
| 1,003
|
Definition free_abmonoid_extend {X : hSet} {Y : abmonoid} (f : X β Y) : monoidfun (free_abmonoid X) Y. Proof. apply (presented_monoid_extend f). unfold iscomprelfun. apply free_abmonoid_hrel_univ. - intros. apply (isasetmonoid Y). - intros x y. rewrite !monoidfunmul. apply commax. Defined.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abmonoid_extend
| 1,004
|
Lemma free_abmonoid_extend_homot {X : hSet} {Y : abmonoid} {f f' : X β Y} (h : f ~ f') : free_abmonoid_extend f ~ free_abmonoid_extend f'. Proof. unfold homot. apply setquotunivprop'. - intro x. apply isasetmonoid. - exact (free_monoid_extend_homot h). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abmonoid_extend_homot
| 1,005
|
Lemma free_abmonoid_extend_comp {X : hSet} {Y : abmonoid} (g : monoidfun (free_abmonoid X) Y): free_abmonoid_extend (g β free_abmonoid_unit) ~ g. Proof. apply (presented_monoid_extend_comp g). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abmonoid_extend_comp
| 1,006
|
Definition free_abmonoid_universal_property (X : hSet) (Y : abmonoid) : (X β Y) β monoidfun (free_abmonoid X) Y. Proof. use weq_iso. - apply free_abmonoid_extend. - intro g. exact (g β free_abmonoid_unit). - intro f. apply funextfun. intro x. reflexivity. - intro g. apply monoidfun_paths, funextfun, free_abmonoid_extend_comp. Defined.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abmonoid_universal_property
| 1,007
|
Definition free_abmonoidfun {X Y : hSet} (f : X β Y) : monoidfun (free_abmonoid X) (free_abmonoid Y) := free_abmonoid_extend (free_abmonoid_unit β f).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abmonoidfun
| 1,008
|
Lemma free_abmonoidfun_setquotpr {X Y : hSet} (f : X β Y) (x : free_monoid X) : free_abmonoidfun f (setquotpr _ x) = setquotpr _ (free_monoidfun f x). Proof. refine (setquotunivcomm _ _ _ _ _ @ _). rewrite free_monoid_extend_funcomp. apply free_monoid_extend_comp. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abmonoidfun_setquotpr
| 1,009
|
Lemma free_abmonoid_extend_funcomp {X Y : hSet} {Z : abmonoid} (f : X β Y) (g : Y β Z) : free_abmonoid_extend (g β f) ~ free_abmonoid_extend g β free_abmonoidfun f. Proof. unfold homot. apply setquotunivprop'. - intro. apply isasetmonoid. - intro x. refine (setquotunivcomm _ _ _ _ _ @ _). refine (free_monoid_extend_funcomp f g x @ _). unfold funcomp. rewrite free_abmonoidfun_setquotpr. refine (!setquotunivcomm _ _ _ _ _). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abmonoid_extend_funcomp
| 1,010
|
Proposition free_abmonoid_mor_eq {X : hSet} {Y : abmonoid} {f g : monoidfun (free_abmonoid X) Y} (p : β (x : X), f (free_abmonoid_unit x) = g (free_abmonoid_unit x)) : f = g. Proof. use (invmaponpathsweq (invweq (free_abmonoid_universal_property X Y)) f g). use funextsec. exact p. Qed.
|
Proposition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abmonoid_mor_eq
| 1,011
|
Definition presented_abmonoid (X : hSet) (R : hrel (free_abmonoid X)) : abmonoid := abmonoidquot (generated_binopeqrel R).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_abmonoid
| 1,012
|
Definition presented_abmonoid_pr {X : hSet} (R : hrel (free_abmonoid X)) : monoidfun (free_abmonoid X) (presented_abmonoid X R) := monoidquotpr _.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_abmonoid_pr
| 1,013
|
Definition presented_abmonoid_intro {X : hSet} {R : hrel (free_abmonoid X)} : X β presented_abmonoid X R := presented_abmonoid_pr R β free_abmonoid_unit.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_abmonoid_intro
| 1,014
|
Definition presented_abmonoid_extend {X : hSet} {R : hrel (free_abmonoid X)} {Y : abmonoid} (f : X β Y) (H : iscomprelfun R (free_abmonoid_extend f)) : monoidfun (presented_abmonoid X R) Y. Proof. use monoidquotuniv. - exact (free_abmonoid_extend f). - exact (iscomprelfun_generated_binopeqrel _ H). Defined.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_abmonoid_extend
| 1,015
|
Lemma iscomprelfun_presented_abmonoidfun {X : hSet} {R : hrel (free_abmonoid X)} {Y : abmonoid} (g : monoidfun (presented_abmonoid X R) Y) : iscomprelfun R (free_abmonoid_extend (g β presented_abmonoid_intro)). Proof. intros x x' r. rewrite !(free_abmonoid_extend_comp (monoidfuncomp (presented_abmonoid_pr R) g)). apply (maponpaths (pr1 g)). apply iscompsetquotpr. exact (generated_binopeqrel_intro r). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
iscomprelfun_presented_abmonoidfun
| 1,016
|
Lemma presented_abmonoid_extend_comp {X : hSet} {R : hrel (free_abmonoid X)} {Y : abmonoid} (g : monoidfun (presented_abmonoid X R) Y) (H : iscomprelfun R (free_abmonoid_extend (g β presented_abmonoid_intro))) : presented_abmonoid_extend (g β presented_abmonoid_intro) H ~ g. Proof. unfold homot. apply setquotunivprop'. + intro. apply isasetmonoid. + intro x. refine (setquotunivcomm _ _ _ _ _ @ _). exact (free_abmonoid_extend_comp (monoidfuncomp (presented_abmonoid_pr R) g) x). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_abmonoid_extend_comp
| 1,017
|
Definition presented_abmonoid_universal_property {X : hSet} (R : hrel (free_abmonoid X)) (Y : abmonoid) : monoidfun (presented_abmonoid X R) Y β β(f : X β Y), iscomprelfun R (free_abmonoid_extend f). Proof. use weq_iso. - intro g. exact (tpair _ (g β presented_abmonoid_intro) (iscomprelfun_presented_abmonoidfun g)). - intro f. exact (presented_abmonoid_extend (pr1 f) (pr2 f)). - intro g. apply monoidfun_paths, funextfun, presented_abmonoid_extend_comp. - intro f. use total2_paths_f. + apply funextfun. intro x. reflexivity. + apply isapropiscomprelfun. Defined.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_abmonoid_universal_property
| 1,018
|
Definition presented_abmonoidfun {X Y : hSet} {R : hrel (free_abmonoid X)} {S : hrel (free_abmonoid Y)} (f : X β Y) (H : iscomprelrelfun R S (free_abmonoidfun f)) : monoidfun (presented_abmonoid X R) (presented_abmonoid Y S). Proof. apply (presented_abmonoid_extend (presented_abmonoid_intro β f)). intros x x' r. rewrite !(free_abmonoid_extend_funcomp _ _ _). unfold funcomp. rewrite !(free_abmonoid_extend_comp (presented_abmonoid_pr S)). apply iscompsetquotpr. apply generated_binopeqrel_intro. exact (H x x' r). Defined.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_abmonoidfun
| 1,019
|
Definition free_gr_hrel (X : hSet) : hrel (free_monoid (setcoprod X X)) := Ξ» g h, β x, x::coprodcomm X X x::[] = g Γ [] = h.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_gr_hrel
| 1,020
|
Lemma free_gr_hrel_in {X : hSet} (x : X β¨Ώ X) : free_gr_hrel X (x::coprodcomm X X x::[]) []. Proof. apply wittohexists with x. split; reflexivity. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_gr_hrel_in
| 1,021
|
Lemma free_gr_hrel_in_rev {X : hSet} (x : X β¨Ώ X) : free_gr_hrel X (coprodcomm X X x::x::[]) []. Proof. pose (H := free_gr_hrel_in (coprodcomm X X x)). rewrite coprodcomm_coprodcomm in H. exact H. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_gr_hrel_in_rev
| 1,022
|
Lemma free_gr_hrel_univ {X : hSet} (P : free_monoid (setcoprod X X) β free_monoid (setcoprod X X) β UU) (HP : β x y, isaprop (P x y)) (Hind : β x, P (x::coprodcomm X X x::[]) []) (x y : free_monoid (setcoprod X X)) : free_gr_hrel X x y β P x y. Proof. apply (@hinhuniv _ (make_hProp (P x y) (HP x y))). intro v. induction v as (x',v), v as (p1,p2), p1, p2. apply Hind. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_gr_hrel_univ
| 1,023
|
Definition fginv_binopfun (X : hSet) : binopfun (free_monoid (setcoprod X X)) (setwithbinop_rev (free_monoid (setcoprod X X))). Proof. refine (binopfuncomp (free_monoidfun _) (reverse_binopfun _)). exact (coprodcomm X X). Defined.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
fginv_binopfun
| 1,024
|
Definition fginv_binopfun_homot {X : hSet} (l : free_monoid (setcoprod X X)) : fginv_binopfun X l = fginv l := idpath _.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
fginv_binopfun_homot
| 1,025
|
Lemma fginv_fginv {X : hSet} (l : free_monoid (setcoprod X X)) : fginv (fginv l) = l. Proof. unfold fginv. rewrite map_reverse, reverse_reverse, <- map_compose, <- map_idfun. apply map_homot. exact coprodcomm_coprodcomm. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
fginv_fginv
| 1,026
|
Lemma generated_free_gr_hrel_in {X : hSet} (l : free_monoid (setcoprod X X)) : generated_binopeqrel (free_gr_hrel X) (l * fginv l) []. Proof. intros R H. revert l. apply list_ind. - apply eqrelrefl. - intros x xs IH. change (R ((free_monoid_unit x * xs) * (fginv xs * @free_monoid_unit (setcoprod X X) (coprodcomm X X x))) []). rewrite assocax, <- (assocax _ _ _ (free_monoid_unit _)). refine (eqreltrans (pr1 R) _ _ _ (binopeqrel_resp_left R _ (binopeqrel_resp_right R _ IH)) _). apply H, free_gr_hrel_in. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
generated_free_gr_hrel_in
| 1,027
|
Lemma generated_free_gr_hrel_in_rev {X : hSet} (l : free_monoid (setcoprod X X)) : generated_binopeqrel (free_gr_hrel X) (fginv l * l) []. Proof. pose (H := generated_free_gr_hrel_in (fginv l)). rewrite fginv_fginv in H. exact H. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
generated_free_gr_hrel_in_rev
| 1,028
|
Definition invstruct_free_gr (X : hSet) : invstruct (@op (free_gr' X)) (pr2 (free_gr' X)). Proof. use make_invstruct. - use setquotfun. + exact fginv. + refine (iscomprelrelfun_generated_binopeqrel_rev (fginv_binopfun X) _). unfold iscomprelrelfun. apply free_gr_hrel_univ. * intros. apply pr2. * intro x. rewrite fginv_binopfun_homot. apply free_gr_hrel_in_rev. - apply make_isinv. + refine (setquotunivprop' _ _ _). * intro. apply isasetmonoid. * intro l. refine (maponpaths (Ξ» x, x * _) (setquotunivcomm _ _ _ _ _) @ _). apply (iscompsetquotpr _ (fginv l * l) []). apply generated_free_gr_hrel_in_rev. + refine (setquotunivprop' _ _ _). * intro. apply isasetmonoid. * intro l. refine (maponpaths (Ξ» x, _ * x) (setquotunivcomm _ _ _ _ _) @ _). apply (iscompsetquotpr _ (l * fginv l) []). apply generated_free_gr_hrel_in. Defined.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
invstruct_free_gr
| 1,029
|
Definition free_gr (X : hSet) : gr := gr_of_monoid (free_gr' X) (invstruct_free_gr X).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_gr
| 1,030
|
Definition free_gr_pr (X : hSet) : monoidfun (free_monoid (setcoprod X X)) (free_gr X) := monoidquotpr _.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_gr_pr
| 1,031
|
Definition free_gr_unit {X : hSet} (x : X) : free_gr X. Proof. apply presented_monoid_intro. exact (inl x). Defined.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_gr_unit
| 1,032
|
Definition free_gr_extend {X : hSet} {Y : gr} (f : X β Y) : monoidfun (free_gr X) Y. Proof. use presented_monoid_extend. - exact (sumofmaps f (grinv Y β f)). - unfold iscomprelfun. apply free_gr_hrel_univ. + intros x y. apply (isasetmonoid Y). + intro x. induction x as [x|x]. * apply (grrinvax Y (f x)). * apply (grlinvax Y (f x)). Defined.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_gr_extend
| 1,033
|
Lemma free_gr_extend_homot {X : hSet} {Y : gr} {f f' : X β Y} (h : f ~ f') : free_gr_extend f ~ free_gr_extend f'. Proof. unfold homot. apply setquotunivprop'. - intro x. apply isasetmonoid. - refine (free_monoid_extend_homot _). apply sumofmaps_homot. + exact h. + intro x. exact (maponpaths (grinv Y) (h x)). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_gr_extend_homot
| 1,034
|
Lemma free_gr_extend_comp {X : hSet} {Y : gr} (g : monoidfun (free_gr X) Y): free_gr_extend (g β free_gr_unit) ~ g. Proof. unfold homot. apply setquotunivprop'. - intro. apply (isasetmonoid Y). - intro l. refine (setquotunivcomm _ _ _ _ _ @ _). refine (_ @ (free_monoid_extend_comp (monoidfuncomp (free_gr_pr X) g)) l). apply (maponpaths iterop_list_mon). apply map_homot. intro x. induction x as [x|x]. + reflexivity. + simpl. refine (!monoidfuninvtoinv g _ @ _). reflexivity. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_gr_extend_comp
| 1,035
|
Definition free_gr_universal_property (X : hSet) (Y : gr) : (X β Y) β monoidfun (free_gr X) Y. Proof. use weq_iso. - apply free_gr_extend. - intro g. exact (g β free_gr_unit). - intro f. apply funextfun. intro x. reflexivity. - intro g. apply monoidfun_paths, funextfun, free_gr_extend_comp. Defined.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_gr_universal_property
| 1,036
|
Definition free_grfun {X Y : hSet} (f : X β Y) : monoidfun (free_gr X) (free_gr Y) := free_gr_extend (free_gr_unit β f).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_grfun
| 1,037
|
Lemma sumofmaps_free_gr_unit {X : hSet} : sumofmaps free_gr_unit (grinv (free_gr X) β free_gr_unit) ~ @presented_monoid_intro (setcoprod X X) (free_gr_hrel X). Proof. intro x. induction x as [x|x]; reflexivity. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
sumofmaps_free_gr_unit
| 1,038
|
Lemma free_grfun_setquotpr {X Y : hSet} (f : X β Y) (x : free_monoid (setcoprod X X)) : free_grfun f (setquotpr _ x) = setquotpr _ (@free_monoidfun (setcoprod X X) (setcoprod Y Y) (coprodf f f) x). Proof. refine (setquotunivcomm _ _ _ _ _ @ _). refine (free_monoid_extend_homot _ x @ _). exact (sumofmaps_funcomp f free_gr_unit f (grinv (free_gr Y) β free_gr_unit)). rewrite (@free_monoid_extend_funcomp (setcoprod X X) (setcoprod Y Y)). refine (free_monoid_extend_homot _ _ @ _). exact (sumofmaps_free_gr_unit). apply (@free_monoid_extend_comp (setcoprod _ _)). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_grfun_setquotpr
| 1,039
|
Lemma free_gr_extend_funcomp {X Y : hSet} {Z : gr} (f : X β Y) (g : Y β Z) : free_gr_extend (g β f) ~ free_gr_extend g β free_grfun f. Proof. unfold homot. apply setquotunivprop'. - intro. apply isasetmonoid. - intro x. refine (setquotunivcomm _ _ _ _ _ @ _). refine (free_monoid_extend_homot _ x @ _). exact (sumofmaps_funcomp f g f (grinv Z β g)). refine (@free_monoid_extend_funcomp (setcoprod X X) (setcoprod Y Y) _ _ _ x @ _). unfold funcomp. rewrite free_grfun_setquotpr. refine (!setquotunivcomm _ _ _ _ _). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_gr_extend_funcomp
| 1,040
|
Definition presented_gr (X : hSet) (R : hrel (free_gr X)) : gr := grquot (generated_binopeqrel R).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_gr
| 1,041
|
Definition presented_gr_pr {X : hSet} (R : hrel (free_gr X)) : monoidfun (free_gr X) (presented_gr X R) := monoidquotpr _.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_gr_pr
| 1,042
|
Definition presented_gr_intro {X : hSet} {R : hrel (free_gr X)} : X β presented_gr X R := presented_gr_pr R β free_gr_unit.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_gr_intro
| 1,043
|
Definition presented_gr_extend {X : hSet} {R : hrel (free_gr X)} {Y : gr} (f : X β Y) (H : iscomprelfun R (free_gr_extend f)) : monoidfun (presented_gr X R) Y. Proof. use monoidquotuniv. - exact (free_gr_extend f). - exact (iscomprelfun_generated_binopeqrel _ H). Defined.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_gr_extend
| 1,044
|
Lemma iscomprelfun_presented_grfun {X : hSet} {R : hrel (free_gr X)} {Y : gr} (g : monoidfun (presented_gr X R) Y) : iscomprelfun R (free_gr_extend (g β presented_gr_intro)). Proof. intros x x' r. rewrite !(free_gr_extend_comp (monoidfuncomp (presented_gr_pr R) g)). apply (maponpaths (pr1 g)). apply iscompsetquotpr. exact (generated_binopeqrel_intro r). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
iscomprelfun_presented_grfun
| 1,045
|
Lemma presented_gr_extend_comp {X : hSet} {R : hrel (free_gr X)} {Y : gr} (g : monoidfun (presented_gr X R) Y) (H : iscomprelfun R (free_gr_extend (g β presented_gr_intro))) : presented_gr_extend (g β presented_gr_intro) H ~ g. Proof. unfold homot. apply setquotunivprop'. + intro. apply isasetmonoid. + intro x. refine (setquotunivcomm _ _ _ _ _ @ _). exact (free_gr_extend_comp (monoidfuncomp (presented_gr_pr R) g) _). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_gr_extend_comp
| 1,046
|
Definition presented_gr_universal_property {X : hSet} (R : hrel (free_gr X)) (Y : gr) : monoidfun (presented_gr X R) Y β β(f : X β Y), iscomprelfun R (free_gr_extend f). Proof. use weq_iso. - intro g. exact (tpair _ (g β presented_gr_intro) (iscomprelfun_presented_grfun g)). - intro f. exact (presented_gr_extend (pr1 f) (pr2 f)). - intro g. apply monoidfun_paths, funextfun, presented_gr_extend_comp. - intro f. use total2_paths_f. + apply funextfun. intro x. reflexivity. + apply isapropiscomprelfun. Defined.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_gr_universal_property
| 1,047
|
Definition presented_grfun {X Y : hSet} {R : hrel (free_gr X)} {S : hrel (free_gr Y)} (f : X β Y) (H : iscomprelrelfun R S (free_grfun f)) : monoidfun (presented_gr X R) (presented_gr Y S). Proof. apply (presented_gr_extend (presented_gr_intro β f)). intros x x' r. rewrite !(free_gr_extend_funcomp _ _ _). unfold funcomp. rewrite !(free_gr_extend_comp (presented_gr_pr S)). apply iscompsetquotpr. apply generated_binopeqrel_intro. exact (H x x' r). Defined.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_grfun
| 1,048
|
Definition free_abgr_hrel (X : hSet) : hrel (free_gr X) := Ξ» g h, β x y, x * y = g Γ y * x = h.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abgr_hrel
| 1,049
|
Lemma free_abgr_hrel_intro {X : hSet} (l1 l2 : free_gr X) : free_abgr_hrel X (l1 * l2) (l2 * l1). Proof. apply wittohexists with l1. split with l2. split; reflexivity. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abgr_hrel_intro
| 1,050
|
Lemma free_abgr_hrel_univ {X : hSet} (P : free_gr X β free_gr X β UU) (HP : β (x y : free_gr X), isaprop (P x y)) (Hind : β x y, P (x * y) (y * x)) (x y : free_gr X) : free_abgr_hrel X x y β P x y. Proof. apply (@hinhuniv _ (make_hProp (P x y) (HP x y))). intro v. induction v as (x',v). induction v as (y',v). induction v as (p1,p2). induction p1. induction p2. apply Hind. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abgr_hrel_univ
| 1,051
|
Definition free_abgr' (X : hSet) : gr := presented_gr X (free_abgr_hrel X).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abgr'
| 1,052
|
Lemma iscomm_free_abgr (X : hSet) : iscomm (@op (free_abgr' X)). Proof. refine (setquotuniv2prop' _ _ _). - intros. apply (isasetmonoid (free_abgr' X)). - intros x1 x2. apply (iscompsetquotpr _ (x1 * x2) (x2 * x1)). apply generated_binopeqrel_intro, free_abgr_hrel_intro. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
iscomm_free_abgr
| 1,053
|
Definition free_abgr (X : hSet) : abgr := abgr_of_gr (free_abgr' X) (iscomm_free_abgr X).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abgr
| 1,054
|
Definition free_abgr_pr (X : hSet) : monoidfun (free_gr X) (free_abgr X) := monoidquotpr _.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abgr_pr
| 1,055
|
Definition free_abgr_unit {X : hSet} (x : X) : free_abgr X := presented_gr_intro x.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abgr_unit
| 1,056
|
Definition free_abgr_extend {X : hSet} {Y : abgr} (f : X β Y) : monoidfun (free_abgr X) Y. Proof. apply (presented_gr_extend f). unfold iscomprelfun. apply free_abgr_hrel_univ. - intros. apply (isasetmonoid Y). - intros x y. rewrite !monoidfunmul. apply (commax Y). Defined.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abgr_extend
| 1,057
|
Lemma free_abgr_extend_homot {X : hSet} {Y : abgr} {f f' : X β Y} (h : f ~ f') : free_abgr_extend f ~ free_abgr_extend f'. Proof. unfold homot. apply setquotunivprop'. - intro x. apply isasetmonoid. - exact (free_gr_extend_homot h). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abgr_extend_homot
| 1,058
|
Lemma free_abgr_extend_comp {X : hSet} {Y : abgr} (g : monoidfun (free_abgr X) Y): free_abgr_extend (g β free_abgr_unit) ~ g. Proof. exact (presented_gr_extend_comp g _). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abgr_extend_comp
| 1,059
|
Definition free_abgr_universal_property (X : hSet) (Y : abgr) : (X β Y) β monoidfun (free_abgr X) Y. Proof. use weq_iso. - apply free_abgr_extend. - intro g. exact (g β free_abgr_unit). - intro f. apply funextfun. intro x. reflexivity. - intro g. apply monoidfun_paths, funextfun, free_abgr_extend_comp. Defined.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abgr_universal_property
| 1,060
|
Definition free_abgrfun {X Y : hSet} (f : X β Y) : monoidfun (free_abgr X) (free_abgr Y) := free_abgr_extend (free_abgr_unit β f).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abgrfun
| 1,061
|
Lemma free_abgrfun_setquotpr {X Y : hSet} (f : X β Y) (x : free_gr X) : free_abgrfun f (setquotpr _ x) = setquotpr _ (free_grfun f x). Proof. refine (setquotunivcomm _ _ _ _ _ @ _). rewrite free_gr_extend_funcomp. exact (free_gr_extend_comp _ (free_grfun f x)). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abgrfun_setquotpr
| 1,062
|
Lemma free_abgr_extend_funcomp {X Y : hSet} {Z : abgr} (f : X β Y) (g : Y β Z) : free_abgr_extend (g β f) ~ free_abgr_extend g β free_abgrfun f. Proof. unfold homot. apply setquotunivprop'. - intro. apply isasetmonoid. - intro x. refine (setquotunivcomm _ _ _ _ _ @ _). refine (free_gr_extend_funcomp _ _ x @ _). unfold funcomp. rewrite free_abgrfun_setquotpr. refine (!setquotunivcomm _ _ _ _ _). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
free_abgr_extend_funcomp
| 1,063
|
Definition presented_abgr (X : hSet) (R : hrel (free_abgr X)) : abgr := abgrquot (generated_binopeqrel R).
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_abgr
| 1,064
|
Definition presented_abgr_pr {X : hSet} (R : hrel (free_abgr X)) : monoidfun (free_abgr X) (presented_abgr X R) := monoidquotpr _.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_abgr_pr
| 1,065
|
Definition presented_abgr_intro {X : hSet} {R : hrel (free_abgr X)} : X β presented_abgr X R := presented_abgr_pr R β free_abgr_unit.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_abgr_intro
| 1,066
|
Definition presented_abgr_extend {X : hSet} {R : hrel (free_abgr X)} {Y : abgr} (f : X β Y) (H : iscomprelfun R (free_abgr_extend f)) : monoidfun (presented_abgr X R) Y. Proof. use monoidquotuniv. - exact (free_abgr_extend f). - exact (iscomprelfun_generated_binopeqrel _ H). Defined.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_abgr_extend
| 1,067
|
Lemma iscomprelfun_presented_abgrfun {X : hSet} {R : hrel (free_abgr X)} {Y : abgr} (g : monoidfun (presented_abgr X R) Y) : iscomprelfun R (free_abgr_extend (g β presented_abgr_intro)). Proof. intros x x' r. rewrite !(free_abgr_extend_comp (monoidfuncomp (presented_abgr_pr R) g)). apply (maponpaths (pr1 g)). apply iscompsetquotpr. exact (generated_binopeqrel_intro r). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
iscomprelfun_presented_abgrfun
| 1,068
|
Lemma presented_abgr_extend_comp {X : hSet} {R : hrel (free_abgr X)} {Y : abgr} (g : monoidfun (presented_abgr X R) Y) (H : iscomprelfun R (free_abgr_extend (g β presented_abgr_intro))) : presented_abgr_extend (g β presented_abgr_intro) H ~ g. Proof. unfold homot. apply setquotunivprop'. + intro. apply isasetmonoid. + intro x. refine (setquotunivcomm _ _ _ _ _ @ _). exact (free_abgr_extend_comp (monoidfuncomp (presented_abgr_pr R) g) x). Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_abgr_extend_comp
| 1,069
|
Definition presented_abgr_universal_property {X : hSet} (R : hrel (free_abgr X)) (Y : abgr) : monoidfun (presented_abgr X R) Y β β(f : X β Y), iscomprelfun R (free_abgr_extend f). Proof. use weq_iso. - intro g. exact (tpair _ (g β presented_abgr_intro) (iscomprelfun_presented_abgrfun g)). - intro f. exact (presented_abgr_extend (pr1 f) (pr2 f)). - intro g. apply monoidfun_paths, funextfun, presented_abgr_extend_comp. - intro f. use total2_paths_f. + apply funextfun. intro x. reflexivity. + apply isapropiscomprelfun. Defined.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_abgr_universal_property
| 1,070
|
Definition presented_abgrfun {X Y : hSet} {R : hrel (free_abgr X)} {S : hrel (free_abgr Y)} (f : X β Y) (H : iscomprelrelfun R S (free_abgrfun f)) : monoidfun (presented_abgr X R) (presented_abgr Y S). Proof. apply (presented_abgr_extend (presented_abgr_intro β f)). intros x x' r. rewrite !(free_abgr_extend_funcomp _ _ _). unfold funcomp. rewrite !(free_abgr_extend_comp (presented_abgr_pr S)). apply iscompsetquotpr. apply generated_binopeqrel_intro. exact (H x x' r). Defined.
|
Definition
|
Algebra
|
Require Import UniMath.MoreFoundations.Subtypes. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Combinatorics.Lists.
|
Algebra\Free_Monoids_and_Groups.v
|
presented_abgrfun
| 1,071
|
Definition action_op G (X:hSet) : Type := β (g:G) (x:X), X.
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
action_op
| 1,072
|
Definition ActionStructure : Type := β (act_mult : action_op G X) (act_unit : β x, act_mult (unel G) x = x), β g h x, act_mult (op g h) x = act_mult g (act_mult h x).
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
ActionStructure
| 1,073
|
Definition make act_mult act_unit act_assoc : ActionStructure := act_mult,, act_unit,, act_assoc.
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
make
| 1,074
|
Definition act_mult (x:ActionStructure) := pr1 x.
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
act_mult
| 1,075
|
Definition act_unit (x:ActionStructure) := pr12 x.
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
act_unit
| 1,076
|
Definition act_assoc (x:ActionStructure) := pr22 x.
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
act_assoc
| 1,077
|
Lemma isaset_ActionStructure (G:gr) (X:hSet) : isaset (ActionStructure G X). Proof. intros. apply isaset_total2. { apply (impred 2); intro g. apply impred; intro x. apply setproperty. } intro op. apply isaset_total2. { apply (impred 2); intro x. apply hlevelntosn. apply setproperty. } intro un. apply (impred 2); intro g. apply (impred 2); intro h. apply (impred 2); intro x. apply hlevelntosn. apply setproperty. Qed.
|
Lemma
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
isaset_ActionStructure
| 1,078
|
Definition Action (G:gr) := total2 (ActionStructure G).
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
Action
| 1,079
|
Definition makeAction {G:gr} (X:hSet) (ac:ActionStructure G X) := X,,ac : Action G.
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
makeAction
| 1,080
|
Definition ac_set {G:gr} (X:Action G) := pr1 X.
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
ac_set
| 1,081
|
Definition ac_type {G:gr} (X:Action G) := pr1hSet (ac_set X).
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
ac_type
| 1,082
|
Definition ac_str {G:gr} (X:Action G) := pr2 X : ActionStructure G (ac_set X).
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
ac_str
| 1,083
|
Definition ac_mult {G:gr} (X:Action G) := act_mult (pr2 X).
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
ac_mult
| 1,084
|
Definition ac_assoc {G:gr} (X:Action G) := act_assoc _ _ (pr2 X) : β g h x, (op g h)*x = g*(h*x).
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
ac_assoc
| 1,085
|
Definition right_mult {G:gr} {X:Action G} (x:X) := Ξ» g, g*x.
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
right_mult
| 1,086
|
Definition left_mult {G:gr} {X:Action G} (g:G) := Ξ» x:X, g*x.
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
left_mult
| 1,087
|
Definition is_equivariant {G:gr} {X Y:Action G} (f:X->Y) : hProp := (β g x, f (g*x) = g*(f x))%logic.
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
is_equivariant
| 1,088
|
Definition is_equivariant_isaprop {G:gr} {X Y:Action G} (f:X->Y) : isaprop (is_equivariant f).
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
is_equivariant_isaprop
| 1,089
|
Definition is_equivariant_identity {G:gr} {X Y:Action G} (p:ac_set X = ac_set Y) : p # ac_str X = ac_str Y β is_equivariant (cast (maponpaths pr1hSet p)). Proof. revert X Y p; intros [X [Xm [Xu Xa]]] [Y [Ym [Yu Ya]]] ? . simpl in p. destruct p; simpl. unfold transportf; simpl. simple refine (make_weq _ _). { intros p g x. simpl in x. simpl. exact (eqtohomot (eqtohomot (maponpaths act_mult p) g) x). } use isweq_iso. { unfold cast; simpl. intro i. assert (p:Xm=Ym). { apply funextsec; intro g. apply funextsec; intro x; simpl in x. exact (i g x). } destruct p. clear i. assert (p:Xu=Yu). { apply funextsec; intro x; simpl in x. apply setproperty. } destruct p. assert (p:Xa=Ya). { apply funextsec; intro g. apply funextsec; intro h. apply funextsec; intro x. apply setproperty. } destruct p. apply idpath. } { intro p. apply isaset_ActionStructure. } { intro is. apply proofirrelevance. apply impred; intros g. apply impred; intros x. apply setproperty. } Defined.
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
is_equivariant_identity
| 1,090
|
Definition is_equivariant_comp {G:gr} {X Y Z:Action G} (p:X->Y) (i:is_equivariant p) (q:Y->Z) (j:is_equivariant q) : is_equivariant (funcomp p q). Proof. intros. intros g x. exact (maponpaths q (i g x) @ j g (p x)). Defined.
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
is_equivariant_comp
| 1,091
|
Definition ActionMap {G:gr} (X Y:Action G) := total2 (@is_equivariant _ X Y).
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
ActionMap
| 1,092
|
Definition underlyingFunction {G:gr} {X Y:Action G} (f:ActionMap X Y) := pr1 f.
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
underlyingFunction
| 1,093
|
Definition equivariance {G:gr} {X Y:Action G} (f:ActionMap X Y) : is_equivariant f := pr2 f.
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
equivariance
| 1,094
|
Definition composeActionMap {G:gr} (X Y Z:Action G) (p:ActionMap X Y) (q:ActionMap Y Z) : ActionMap X Z. Proof. revert p q; intros [p i] [q j]. exists (funcomp p q). apply is_equivariant_comp. assumption. assumption. Defined.
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
composeActionMap
| 1,095
|
Definition ActionIso {G:gr} (X Y:Action G) : Type. Proof. exact (β f:(ac_set X) β (ac_set Y), is_equivariant f). Defined.
|
Definition
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
ActionIso
| 1,096
|
Lemma ActionIso_isaset {G:gr} (X Y:Action G) : isaset (ActionIso X Y). Proof. apply (isofhlevelsninclb _ pr1). { apply isinclpr1; intro f. apply propproperty. } apply isofhlevelsnweqtohlevelsn. apply setproperty. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
ActionIso_isaset
| 1,097
|
Lemma underlyingIso_incl {G:gr} {X Y:Action G} : isincl (underlyingIso : ActionIso X Y β X β Y). Proof. intros. apply isinclpr1; intro f. apply propproperty. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
underlyingIso_incl
| 1,098
|
Lemma underlyingIso_injectivity {G:gr} {X Y:Action G} (e f:ActionIso X Y) : (e = f) β (underlyingIso e = underlyingIso f). Proof. intros. apply weqonpathsincl. apply underlyingIso_incl. Defined.
|
Lemma
|
Algebra
|
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.OrderTheory.OrderedSets.OrderedSets. Import UniMath.MoreFoundations.PartA.
|
Algebra\GroupAction.v
|
underlyingIso_injectivity
| 1,099
|
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