phanerozoic/Coq-UniMath · Datasets at Hugging Face

fact stringlengths 13 15.5k	type stringclasses 9 values	library stringclasses 15 values	imports stringlengths 14 7.64k ⌀	filename stringlengths 12 97	symbolic_name stringlengths 1 78	index_level int64 0 38.7k
Definition isrinvertible {X : UU} (opp : binop X) (x : X) : UU := isweq (λ x0 : X, opp x0 x).	Definition	Algebra	null	Algebra\BinaryOperations.v	isrinvertible	300
Definition isinvertible {X : UU} (opp : binop X) (x : X) : UU := (islinvertible opp x) × (isrinvertible opp x).	Definition	Algebra	null	Algebra\BinaryOperations.v	isinvertible	301
Definition binop_weq_fwd {X Y : UU} (H : X ≃ Y) : binop X → binop Y := λ (opp : binop X) (x y : Y), H (opp (invmap H x) (invmap H y)).	Definition	Algebra	null	Algebra\BinaryOperations.v	binop_weq_fwd	302
Definition binop_weq_bck {X Y : UU} (H : X ≃ Y) : binop Y → binop X := λ (opp : binop Y) (x y : X), invmap H (opp (H x) (H y)).	Definition	Algebra	null	Algebra\BinaryOperations.v	binop_weq_bck	303
Definition isassoc {X : UU} (opp : binop X) : UU := ∏ x x' x'', opp (opp x x') x'' = opp x (opp x' x'').	Definition	Algebra	null	Algebra\BinaryOperations.v	isassoc	304
Lemma isapropisassoc {X : hSet} (opp : binop X) : isaprop (isassoc opp). Proof. apply impred. intro x. apply impred. intro x'. apply impred. intro x''. simpl. apply (setproperty X). Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isapropisassoc	305
Lemma assoc4 {X : UU} (opp : binop X) (isa : isassoc opp) : ∏ w x y z : X, opp (opp (opp w x) y) z = opp (opp w (opp x y)) z. Proof. intros. repeat rewrite isa; exact (idpath _). Qed.	Lemma	Algebra	null	Algebra\BinaryOperations.v	assoc4	306
Definition isrcancellative {X : UU} (opp : binop X) : UU := ∏ x:X, isrcancelable opp x.	Definition	Algebra	null	Algebra\BinaryOperations.v	isrcancellative	307
Definition islcancellative {X : UU} (opp : binop X) : UU := ∏ x:X, islcancelable opp x.	Definition	Algebra	null	Algebra\BinaryOperations.v	islcancellative	308
Definition islunit {X : UU} (opp : binop X) (un0 : X) : UU := ∏ x : X, opp un0 x = x.	Definition	Algebra	null	Algebra\BinaryOperations.v	islunit	309
Lemma isapropislunit {X : hSet} (opp : binop X) (un0 : X) : isaprop (islunit opp un0). Proof. apply impred. intro x. simpl. apply (setproperty X). Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isapropislunit	310
Definition isrunit {X : UU} (opp : binop X) (un0 : X) : UU := ∏ x : X, opp x un0 = x.	Definition	Algebra	null	Algebra\BinaryOperations.v	isrunit	311
Lemma isapropisrunit {X : hSet} (opp : binop X) (un0 : X) : isaprop (isrunit opp un0). Proof. apply impred. intro x. simpl. apply (setproperty X). Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isapropisrunit	312
Definition isunit {X : UU} (opp : binop X) (un0 : X) : UU := (islunit opp un0) × (isrunit opp un0).	Definition	Algebra	null	Algebra\BinaryOperations.v	isunit	313
Definition make_isunit {X : UU} {opp : binop X} {un0 : X} (H1 : islunit opp un0) (H2 : isrunit opp un0) : isunit opp un0 := H1 ,, H2.	Definition	Algebra	null	Algebra\BinaryOperations.v	make_isunit	314
Definition isunital {X : UU} (opp : binop X) : UU := ∑ (un0 : X), isunit opp un0.	Definition	Algebra	null	Algebra\BinaryOperations.v	isunital	315
Definition make_isunital {X : UU} {opp : binop X} (un0 : X) (is : isunit opp un0) : isunital opp := un0 ,, is.	Definition	Algebra	null	Algebra\BinaryOperations.v	make_isunital	316
Lemma isapropisunital {X : hSet} (opp : binop X) : isaprop (isunital opp). Proof. apply (@isapropsubtype X (λ un0, hconj (make_hProp _ (isapropislunit opp un0)) (make_hProp _ (isapropisrunit opp un0)))). intros u1 u2. intros ua1 ua2. apply (!(pr2 ua2 u1) @ (pr1 ua1 u2)). Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isapropisunital	317
Definition ismonoidop {X : UU} (opp : binop X) : UU := (isassoc opp) × (isunital opp).	Definition	Algebra	null	Algebra\BinaryOperations.v	ismonoidop	318
Definition make_ismonoidop {X : UU} {opp : binop X} (H1 : isassoc opp) (H2 : isunital opp) : ismonoidop opp := H1 ,, H2.	Definition	Algebra	null	Algebra\BinaryOperations.v	make_ismonoidop	319
Definition assocax_is {X : UU} {opp : binop X} : ismonoidop opp → isassoc opp := @pr1 _ _.	Definition	Algebra	null	Algebra\BinaryOperations.v	assocax_is	320
Definition unel_is {X : UU} {opp : binop X} (is : ismonoidop opp) : X := pr1 (pr2 is).	Definition	Algebra	null	Algebra\BinaryOperations.v	unel_is	321
Definition lunax_is {X : UU} {opp : binop X} (is : ismonoidop opp) : islunit opp (pr1 (pr2 is)) := pr1 (pr2 (pr2 is)).	Definition	Algebra	null	Algebra\BinaryOperations.v	lunax_is	322
Definition runax_is {X : UU} {opp : binop X} (is : ismonoidop opp) : isrunit opp (pr1 (pr2 is)) := pr2 (pr2 (pr2 is)).	Definition	Algebra	null	Algebra\BinaryOperations.v	runax_is	323
Definition unax_is {X : UU} {opp : binop X} (is : ismonoidop opp) : isunit opp (pr1 (pr2 is)) := lunax_is is ,, runax_is is.	Definition	Algebra	null	Algebra\BinaryOperations.v	unax_is	324
Lemma isapropismonoidop {X : hSet} (opp : binop X) : isaprop (ismonoidop opp). Proof. apply (isofhleveldirprod 1). apply isapropisassoc. apply isapropisunital. Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isapropismonoidop	325
Definition islinvel (x : X) : X → UU := λ x0, x0 * x = u.	Definition	Algebra	null	Algebra\BinaryOperations.v	islinvel	326
Definition isrinvel (x : X) : X → UU := λ x0, x * x0 = u.	Definition	Algebra	null	Algebra\BinaryOperations.v	isrinvel	327
Definition isinvel (x : X) : X → UU := λ x0, (islinvel x x0) × (isrinvel x x0).	Definition	Algebra	null	Algebra\BinaryOperations.v	isinvel	328
Definition haslinv (x : X) : UU := ∑ x0 : X, islinvel x x0.	Definition	Algebra	null	Algebra\BinaryOperations.v	haslinv	329
Definition hasrinv (x : X) : UU := ∑ x0 : X, isrinvel x x0.	Definition	Algebra	null	Algebra\BinaryOperations.v	hasrinv	330
Definition hasinv (x : X) : UU := ∑ x0 : X, isinvel x x0.	Definition	Algebra	null	Algebra\BinaryOperations.v	hasinv	331
Definition haslinv_to_linvel {x : X} : haslinv x → X := pr1.	Definition	Algebra	null	Algebra\BinaryOperations.v	haslinv_to_linvel	332
Definition hasrinv_to_rinvel {x : X} : hasrinv x → X := pr1.	Definition	Algebra	null	Algebra\BinaryOperations.v	hasrinv_to_rinvel	333
Definition hasinv_to_invel {x : X} : hasinv x → X := pr1.	Definition	Algebra	null	Algebra\BinaryOperations.v	hasinv_to_invel	334
Definition merely_haslinv (x : X) : hProp := ∥ haslinv x ∥.	Definition	Algebra	null	Algebra\BinaryOperations.v	merely_haslinv	335
Definition merely_hasrinv (x : X) : hProp := ∥ hasrinv x ∥.	Definition	Algebra	null	Algebra\BinaryOperations.v	merely_hasrinv	336
Definition merely_hasinv (x : X) : hProp := ∥ hasinv x ∥.	Definition	Algebra	null	Algebra\BinaryOperations.v	merely_hasinv	337
Definition is_inv_inv : ∏ (x x0 : X), (isinvel x x0 → isinvel x0 x) := λ x x0 isinv, pr2 isinv ,, pr1 isinv.	Definition	Algebra	null	Algebra\BinaryOperations.v	is_inv_inv	338
Lemma invop_l : ∏ (x y x' y' : X), (islinvel x x' → islinvel y y' → islinvel (x * y) (y' * x')). Proof. intros x y x' y' xinv yinv. unfold islinvel. pose (assoc := pr1 is). cbn; unfold islinvel. rewrite <- assoc. rewrite (assoc4 opp assoc), xinv. rewrite (runax_is is). exact yinv. Qed.	Lemma	Algebra	null	Algebra\BinaryOperations.v	invop_l	339
Lemma invop_r : ∏ (x y x' y' : X), (isrinvel x x' → isrinvel y y' → isrinvel (x * y) (y' * x')). Proof. intros x y x' y' xinv yinv. pose (assoc := pr1 is). cbn; unfold isrinvel. rewrite <- assoc. rewrite (assoc4 opp assoc), yinv. rewrite (runax_is is). exact xinv. Qed.	Lemma	Algebra	null	Algebra\BinaryOperations.v	invop_r	340
Lemma invop : ∏ (x y x' y' : X), (isinvel x x' → isinvel y y' → isinvel (x * y) (y' * x')). Proof. intros x y x' y' xinv yinv. use make_dirprod. - apply invop_l. + exact (dirprod_pr1 xinv). + exact (dirprod_pr1 yinv). - apply invop_r. + exact (dirprod_pr2 xinv). + exact (dirprod_pr2 yinv). Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	invop	341
Lemma mere_invop : ∏ (x y : X), (merely_hasinv x → merely_hasinv y → merely_hasinv (x * y)). Proof. intros x y. apply hinhfun2. intros xinv yinv. exists ((hasinv_to_invel yinv) * (hasinv_to_invel xinv)). apply invop. - exact (pr2 xinv). - exact (pr2 yinv). Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	mere_invop	342
Lemma linv_eq_rinv (x lx rx : X) (lxlinv : islinvel x lx) (rxrinv : isrinvel x rx) : lx = rx. Proof. intros. refine (!runax_is is _ @ _). refine (!maponpaths (λ z, lx * z) rxrinv @ _). refine (!assocax_is is _ _ _ @ _). refine (maponpaths (λ z, z * rx) lxlinv @ _). apply lunax_is. Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	linv_eq_rinv	343
Definition islinv : UU := ∏ x : X, (inv x) * x = u.	Definition	Algebra	null	Algebra\BinaryOperations.v	islinv	344
Definition isrinv : UU := ∏ x : X, x * (inv x) = u.	Definition	Algebra	null	Algebra\BinaryOperations.v	isrinv	345
Definition isinv : UU := islinv × isrinv.	Definition	Algebra	null	Algebra\BinaryOperations.v	isinv	346
Definition isapropislinvel (x x0 : X) : isaprop (islinvel opp is x x0) := setproperty X _ _.	Definition	Algebra	null	Algebra\BinaryOperations.v	isapropislinvel	347
Definition isapropisrinvel (x x0 : X) : isaprop (isrinvel opp is x x0) := setproperty X _ _.	Definition	Algebra	null	Algebra\BinaryOperations.v	isapropisrinvel	348
Definition isapropisinvel (x x0 : X) : isaprop (isinvel opp is x x0) := isapropdirprod _ _ (isapropislinvel _ _) (isapropisrinvel _ _).	Definition	Algebra	null	Algebra\BinaryOperations.v	isapropisinvel	349
Definition isaprop_haslinv (x : X) (can : islcancelable opp x) : isaprop (hasrinv opp is x). Proof. apply isaproptotal2. - intro; apply isapropislinvel. - intros x' x'' islinvx' islinvx''. apply (Injectivity (λ x0 : X, x * x0)). + apply incl_injectivity; assumption. + exact (islinvx' @ !islinvx''). Defined.	Definition	Algebra	null	Algebra\BinaryOperations.v	isaprop_haslinv	350
Definition isaprop_hasrinv (x : X) (can : isrcancelable opp x) : isaprop (haslinv opp is x). Proof. apply isaproptotal2. - intro; apply isapropisrinvel. - intros x' x'' isrinvx' isrinvx''. apply (Injectivity (λ x0 : X, x0 * x)). + apply incl_injectivity; assumption. + exact (isrinvx' @ !isrinvx''). Defined.	Definition	Algebra	null	Algebra\BinaryOperations.v	isaprop_hasrinv	351
Definition isaprop_hasinv (x : X) (can : iscancelable opp x) : isaprop (hasinv opp is x). Proof. apply isaproptotal2. - intro; apply isapropdirprod. + apply isapropislinvel. + apply isapropisrinvel. - intros x' x'' isinvx' isinvx''. apply (Injectivity (λ x0 : X, x * x0)). + apply incl_injectivity; apply (pr1 can). + exact (pr2 isinvx' @ !pr2 isinvx''). Defined.	Definition	Algebra	null	Algebra\BinaryOperations.v	isaprop_hasinv	352
Definition merely_invertible_elements : hsubtype X := merely_hasinv opp is.	Definition	Algebra	null	Algebra\BinaryOperations.v	merely_invertible_elements	353
Definition invertible_elements (can : ∏ x, iscancelable opp x) : hsubtype X. Proof. intro x. use make_hProp. - exact (hasinv opp is x). - apply isaprop_hasinv, can. Defined.	Definition	Algebra	null	Algebra\BinaryOperations.v	invertible_elements	354
Definition lcanfromlinv (a b c : X) (c' : haslinv opp is c) : c * a = c * b → a = b. Proof. intros e. refine (!lunax_is is a @ _ @ lunax_is is b). refine (!maponpaths (λ z, z * _) (pr2 c') @ _ @ maponpaths (λ z, z * _) (pr2 c')). refine (assocax_is is _ _ _ @ _ @ !assocax_is is _ _ _). apply maponpaths. assumption. Defined.	Definition	Algebra	null	Algebra\BinaryOperations.v	lcanfromlinv	355
Definition rcanfromrinv (a b c : X) (c' : hasrinv opp is c) : a * c = b * c → a = b. Proof. intros e. refine (!runax_is is a @ _ @ runax_is is b). refine (!maponpaths (λ z, _ * z) (pr2 c') @ _ @ maponpaths (λ z, _ * z) (pr2 c')). refine (!assocax_is is _ _ _ @ _ @ assocax_is is _ _ _). apply (maponpaths (λ z, z * _)). assumption. Defined.	Definition	Algebra	null	Algebra\BinaryOperations.v	rcanfromrinv	356
Lemma isapropislinv : isaprop (islinv opp u inv). Proof. intros; apply impred; intro; apply setproperty. Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isapropislinv	357
Lemma isapropisrinv : isaprop (isrinv opp u inv). Proof. intros; apply impred; intro; apply setproperty. Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isapropisrinv	358
Lemma isapropisinv : isaprop (isinv opp u inv). Proof. exact (isofhleveldirprod 1 _ _ isapropislinv isapropisrinv). Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isapropisinv	359
Definition make_isinv {X : UU} {opp : binop X} {un0 : X} {inv0 : X → X} (H1 : islinv opp un0 inv0) (H2 : isrinv opp un0 inv0) : isinv opp un0 inv0 := H1 ,, H2.	Definition	Algebra	null	Algebra\BinaryOperations.v	make_isinv	360
Definition invstruct {X : UU} (opp : binop X) (is : ismonoidop opp) : UU := ∑ (inv0 : X → X), isinv opp (unel_is is) inv0.	Definition	Algebra	null	Algebra\BinaryOperations.v	invstruct	361
Definition make_invstruct {X : UU} {opp : binop X} {is : ismonoidop opp} (inv0 : X → X) (H : isinv opp (unel_is is) inv0) : invstruct opp is := inv0 ,, H.	Definition	Algebra	null	Algebra\BinaryOperations.v	make_invstruct	362
Definition isgrop {X : UU} (opp : binop X) : UU := ∑ (is : ismonoidop opp), invstruct opp is.	Definition	Algebra	null	Algebra\BinaryOperations.v	isgrop	363
Definition make_isgrop {X : UU} {opp : binop X} (is1 : ismonoidop opp) (is2 : invstruct opp is1) : isgrop opp := is1 ,, is2.	Definition	Algebra	null	Algebra\BinaryOperations.v	make_isgrop	364
Definition pr1isgrop (X : UU) (opp : binop X) : isgrop opp → ismonoidop opp := @pr1 _ _.	Definition	Algebra	null	Algebra\BinaryOperations.v	pr1isgrop	365
Definition grinv_is {X : UU} {opp : binop X} (is : isgrop opp) : X → X := pr1 (pr2 is).	Definition	Algebra	null	Algebra\BinaryOperations.v	grinv_is	366
Definition grlinvax_is {X : UU} {opp : binop X} (is : isgrop opp) : islinv opp (unel_is is) (pr1 (pr2 is)) := pr1 (pr2 (pr2 is)).	Definition	Algebra	null	Algebra\BinaryOperations.v	grlinvax_is	367
Definition grrinvax_is {X : UU} {opp : binop X} (is : isgrop opp) : isrinv opp (unel_is is) (pr1 (pr2 is)) := pr2 (pr2 (pr2 is)).	Definition	Algebra	null	Algebra\BinaryOperations.v	grrinvax_is	368
Lemma isweqrmultingr_is {X : UU} {opp : binop X} (is : isgrop opp) (x0 : X) : isrinvertible opp x0. Proof. induction is as [ is istr ]. set (f := λ x : X, opp x x0). set (g := λ x : X, opp x ((pr1 istr) x0)). induction is as [ assoc isun0 ]. induction istr as [ inv0 axs ]. induction isun0 as [ un0 unaxs ]. simpl in * \|-. assert (egf : ∏ x, g (f x) = x). { intro x. unfold f. unfold g. induction (!assoc x x0 (inv0 x0)). set (e := pr2 axs x0). simpl in e. rewrite e. apply (pr2 unaxs x). } assert (efg : ∏ x, f (g x) = x). { intro x. unfold f. unfold g. induction (!assoc x (inv0 x0) x0). set (e := pr1 axs x0). simpl in e. rewrite e. apply (pr2 unaxs x). } apply (isweq_iso _ _ egf efg). Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isweqrmultingr_is	369
Lemma isweqlmultingr_is {X : UU} {opp : binop X} (is : isgrop opp) (x0 : X) : islinvertible opp x0. Proof. induction is as [ is istr ]. set (f := λ x : X, opp x0 x). set (g := λ x : X, opp ((pr1 istr) x0) x). induction is as [ assoc isun0 ]. induction istr as [ inv0 axs ]. induction isun0 as [ un0 unaxs ]. simpl in * \|-. assert (egf : ∏ x, g (f x) = x). { intro x. unfold f. unfold g. induction (assoc (inv0 x0) x0 x). set (e := pr1 axs x0). simpl in e. rewrite e. apply (pr1 unaxs x). } assert (efg : ∏ x, f (g x) = x). { intro x. unfold f. unfold g. induction (assoc x0 (inv0 x0) x). set (e := pr2 axs x0). simpl in e. rewrite e. apply (pr1 unaxs x). } apply (isweq_iso _ _ egf efg). Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isweqlmultingr_is	370
Lemma isapropinvstruct {X : hSet} {opp : binop X} (is : ismonoidop opp) : isaprop (invstruct opp is). Proof. apply isofhlevelsn. intro is0. set (un0 := pr1 (pr2 is)). assert (int : ∏ (i : X → X), isaprop ((∏ x : X, opp (i x) x = un0) × (∏ x : X, opp x (i x) = un0))). { intro i. apply (isofhleveldirprod 1). - apply impred. intro x. simpl. apply (setproperty X). - apply impred. intro x. simpl. apply (setproperty X). } apply (isapropsubtype (λ i, make_hProp _ (int i))). intros inv1 inv2. simpl. intro ax1. intro ax2. apply funextfun. intro x0. apply (invmaponpathsweq (make_weq _ (isweqrmultingr_is (is ,, is0) x0))). simpl. rewrite (pr1 ax1 x0). rewrite (pr1 ax2 x0). apply idpath. Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isapropinvstruct	371
Lemma isapropisgrop {X : hSet} (opp : binop X) : isaprop (isgrop opp). Proof. apply (isofhleveltotal2 1). - apply isapropismonoidop. - apply isapropinvstruct. Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isapropisgrop	372
Definition allinvvertibleinv {X : hSet} {opp : binop X} (is : ismonoidop opp) (allinv : ∏ x : X, islinvertible opp x) : X → X := λ x : X, invmap (make_weq _ (allinv x)) (unel_is is).	Definition	Algebra	null	Algebra\BinaryOperations.v	allinvvertibleinv	373
Lemma isgropif {X : hSet} {opp : binop X} (is0 : ismonoidop opp) (is : ∏ x : X, merely_hasrinv opp is0 x) : isgrop opp. Proof. exists is0. induction is0 as [ assoc isun0 ]. induction isun0 as [ un0 unaxs0 ]. simpl in is. simpl in unaxs0. simpl in un0. simpl in assoc. simpl in unaxs0. assert (l1 : ∏ x' : X, isincl (λ x0 : X, opp x0 x')). { intro x'. apply (@hinhuniv (∑ (x0 : X), opp x' x0 = un0) (make_hProp _ (isapropisincl (λ x0 : X, opp x0 x')))). - intro int1. simpl. apply isinclbetweensets. + apply (pr2 X). + apply (pr2 X). + intros a b. intro e. rewrite <- (pr2 unaxs0 a). rewrite <- (pr2 unaxs0 b). induction int1 as [ invx' eq ]. rewrite <- eq. induction (assoc a x' invx'). induction (assoc b x' invx'). rewrite e. apply idpath. - apply (is x'). } assert (is' : ∏ x : X, ∃ (x0 : X), opp x0 x = un0). { intro x. apply (λ f , hinhuniv f (is x)). intro s1. induction s1 as [ x' eq ]. apply hinhpr. exists x'. simpl. apply (invmaponpathsincl _ (l1 x')). rewrite (assoc x' x x'). rewrite eq. rewrite (pr1 unaxs0 x'). unfold unel_is. simpl. rewrite (pr2 unaxs0 x'). apply idpath. } assert (l1' : ∏ x' : X, isincl (λ x0 : X, opp x' x0)). { intro x'. apply (@hinhuniv (∑ (x0 : X), opp x0 x' = un0) (make_hProp _ (isapropisincl (λ x0 : X, opp x' x0)))). - intro int1. simpl. apply isinclbetweensets. + apply (pr2 X). + apply (pr2 X). + intros a b. intro e. rewrite <- (pr1 unaxs0 a). rewrite <- (pr1 unaxs0 b). induction int1 as [ invx' eq ]. rewrite <- eq. induction (!assoc invx' x' a). induction (!assoc invx' x' b). rewrite e. apply idpath. - apply (is' x'). } assert (int : ∏ x : X, isaprop (∑ (x0 : X), opp x0 x = un0)%logic). { intro x. apply isapropsubtype. intros x1 x2. intros eq1 eq2. apply (invmaponpathsincl _ (l1 x)). rewrite eq1. rewrite eq2. apply idpath. } simpl. set (linv0 := λ x : X, hinhunivcor1 (make_hProp _ (int x)) (is' x)). simpl in linv0. set (inv0 := λ x : X, pr1 (linv0 x)). exists inv0. simpl. exists (λ x, pr2 (linv0 x)). intro x. apply (invmaponpathsincl _ (l1 x)). rewrite (assoc x (inv0 x) x). change (inv0 x) with (pr1 (linv0 x)). rewrite (pr2 (linv0 x)). unfold unel_is. simpl. rewrite (pr1 unaxs0 x). rewrite (pr2 unaxs0 x). apply idpath. Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isgropif	374
Definition iscomm {X : UU} (opp : binop X) : UU := ∏ x x' : X, opp x x' = opp x' x.	Definition	Algebra	null	Algebra\BinaryOperations.v	iscomm	375
Lemma isapropiscomm {X : hSet} (opp : binop X) : isaprop (iscomm opp). Proof. apply impred. intros x. apply impred. intro x'. simpl. apply (setproperty X). Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isapropiscomm	376
Definition isabmonoidop {X : UU} (opp : binop X) : UU := (ismonoidop opp) × (iscomm opp).	Definition	Algebra	null	Algebra\BinaryOperations.v	isabmonoidop	377
Definition make_isabmonoidop {X : UU} {opp : binop X} (H1 : ismonoidop opp) (H2 : iscomm opp) : isabmonoidop opp := H1 ,, H2.	Definition	Algebra	null	Algebra\BinaryOperations.v	make_isabmonoidop	378
Definition pr1isabmonoidop (X : UU) (opp : binop X) : isabmonoidop opp → ismonoidop opp := @pr1 _ _.	Definition	Algebra	null	Algebra\BinaryOperations.v	pr1isabmonoidop	379
Definition commax_is {X : UU} {opp : binop X} (is : isabmonoidop opp) : iscomm opp := pr2 is.	Definition	Algebra	null	Algebra\BinaryOperations.v	commax_is	380
Lemma isapropisabmonoidop {X : hSet} (opp : binop X) : isaprop (isabmonoidop opp). Proof. apply (isofhleveldirprod 1). apply isapropismonoidop. apply isapropiscomm. Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isapropisabmonoidop	381
Lemma abmonoidoprer {X : UU} {opp : binop X} (is : isabmonoidop opp) (a b c d : X) : opp (opp a b) (opp c d) = opp (opp a c) (opp b d). Proof. induction is as [ is comm ]. induction is as [ assoc unital0 ]. simpl in *. induction (assoc (opp a b) c d). induction (assoc (opp a c) b d). induction (!assoc a b c). induction (!assoc a c b). induction (comm b c). apply idpath. Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	abmonoidoprer	382
Lemma weqlcancelablercancelable {X : UU} (opp : binop X) (is : iscomm opp) (x : X) : (islcancelable opp x) ≃ (isrcancelable opp x). Proof. assert (f : (islcancelable opp x) → (isrcancelable opp x)). { unfold islcancelable. unfold isrcancelable. intro isl. apply (λ h, isinclhomot _ _ h isl). intro x0. apply is. } assert (g : (isrcancelable opp x) → (islcancelable opp x)). { unfold islcancelable. unfold isrcancelable. intro isr. apply (λ h, isinclhomot _ _ h isr). intro x0. apply is. } exists f. apply (isweqimplimpl f g (isapropisincl (λ x0 : X, opp x x0)) (isapropisincl (λ x0 : X, opp x0 x))). Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	weqlcancelablercancelable	383
Lemma weqlinvertiblerinvertible {X : UU} (opp : binop X) (is : iscomm opp) (x : X) : (islinvertible opp x) ≃ (isrinvertible opp x). Proof. assert (f : (islinvertible opp x) → (isrinvertible opp x)). { unfold islinvertible. unfold isrinvertible. intro isl. apply (isweqhomot (λ y, opp x y)). - intro z. apply is. - apply isl. } assert (g : (isrinvertible opp x) → (islinvertible opp x)). { unfold islinvertible. unfold isrinvertible. intro isr. apply (λ h, isweqhomot _ _ h isr). intro x0. apply is. } exists f. apply (isweqimplimpl f g (isapropisweq (λ x0 : X, opp x x0)) (isapropisweq (λ x0 : X, opp x0 x))). Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	weqlinvertiblerinvertible	384
Lemma weqlunitrunit {X : hSet} (opp : binop X) (is : iscomm opp) (un0 : X) : (islunit opp un0) ≃ (isrunit opp un0). Proof. assert (f : (islunit opp un0) → (isrunit opp un0)). { unfold islunit. unfold isrunit. intro isl. intro x. induction (is un0 x). apply (isl x). } assert (g : (isrunit opp un0) → (islunit opp un0)). { unfold islunit. unfold isrunit. intro isr. intro x. induction (is x un0). apply (isr x). } exists f. apply (isweqimplimpl f g (isapropislunit opp un0) (isapropisrunit opp un0)). Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	weqlunitrunit	385
Lemma weqlinvrinv {X : hSet} (opp : binop X) (is : iscomm opp) (un0 : X) (inv0 : X → X) : (islinv opp un0 inv0) ≃ (isrinv opp un0 inv0). Proof. assert (f : (islinv opp un0 inv0) → (isrinv opp un0 inv0)). { unfold islinv. unfold isrinv. intro isl. intro x. induction (is (inv0 x) x). apply (isl x). } assert (g : (isrinv opp un0 inv0) → (islinv opp un0 inv0)). { unfold islinv. unfold isrinv. intro isr. intro x. induction (is x (inv0 x)). apply (isr x). } exists f. apply (isweqimplimpl f g (isapropislinv opp un0 inv0) (isapropisrinv opp un0 inv0)). Qed.	Lemma	Algebra	null	Algebra\BinaryOperations.v	weqlinvrinv	386
Definition isabgrop {X : UU} (opp : binop X) : UU := (isgrop opp) × (iscomm opp).	Definition	Algebra	null	Algebra\BinaryOperations.v	isabgrop	387
Definition make_isabgrop {X : UU} {opp : binop X} (H1 : isgrop opp) (H2 : iscomm opp) : isabgrop opp := H1 ,, H2.	Definition	Algebra	null	Algebra\BinaryOperations.v	make_isabgrop	388
Definition pr1isabgrop (X : UU) (opp : binop X) : isabgrop opp → isgrop opp := @pr1 _ _.	Definition	Algebra	null	Algebra\BinaryOperations.v	pr1isabgrop	389
Definition isabgroptoisabmonoidop (X : UU) (opp : binop X) : isabgrop opp → isabmonoidop opp := λ is, pr1 (pr1 is) ,, pr2 is.	Definition	Algebra	null	Algebra\BinaryOperations.v	isabgroptoisabmonoidop	390
Lemma isapropisabgrop {X : hSet} (opp : binop X) : isaprop (isabgrop opp). Proof. apply (isofhleveldirprod 1). apply isapropisgrop. apply isapropiscomm. Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isapropisabgrop	391
Definition isldistr {X : UU} (opp1 opp2 : binop X) : UU := ∏ x x' x'' : X, opp2 x'' (opp1 x x') = opp1 (opp2 x'' x) (opp2 x'' x').	Definition	Algebra	null	Algebra\BinaryOperations.v	isldistr	392
Lemma isapropisldistr {X : hSet} (opp1 opp2 : binop X) : isaprop (isldistr opp1 opp2). Proof. apply impred. intro x. apply impred. intro x'. apply impred. intro x''. simpl. apply (setproperty X). Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isapropisldistr	393
Definition isrdistr {X : UU} (opp1 opp2 : binop X) : UU := ∏ x x' x'' : X, opp2 (opp1 x x') x'' = opp1 (opp2 x x'') (opp2 x' x'').	Definition	Algebra	null	Algebra\BinaryOperations.v	isrdistr	394
Lemma isapropisrdistr {X : hSet} (opp1 opp2 : binop X) : isaprop (isrdistr opp1 opp2). Proof. apply impred. intro x. apply impred. intro x'. apply impred. intro x''. simpl. apply (setproperty X). Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isapropisrdistr	395
Definition isdistr {X : UU} (opp1 opp2 : binop X) : UU := (isldistr opp1 opp2) × (isrdistr opp1 opp2).	Definition	Algebra	null	Algebra\BinaryOperations.v	isdistr	396
Lemma isapropisdistr {X : hSet} (opp1 opp2 : binop X) : isaprop (isdistr opp1 opp2). Proof. apply (isofhleveldirprod 1 _ _ (isapropisldistr _ _) (isapropisrdistr _ _)). Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	isapropisdistr	397
Lemma weqldistrrdistr {X : hSet} (opp1 opp2 : binop X) (is : iscomm opp2) : (isldistr opp1 opp2) ≃ (isrdistr opp1 opp2). Proof. assert (f : (isldistr opp1 opp2) → (isrdistr opp1 opp2)). { unfold isldistr. unfold isrdistr. intro isl. intros x x' x''. induction (is x'' (opp1 x x')). induction (is x'' x). induction (is x'' x'). apply (isl x x' x''). } assert (g : (isrdistr opp1 opp2) → (isldistr opp1 opp2)). { unfold isldistr. unfold isrdistr. intro isr. intros x x' x''. induction (is (opp1 x x') x''). induction (is x x''). induction (is x' x''). apply (isr x x' x''). } exists f. apply (isweqimplimpl f g (isapropisldistr opp1 opp2) (isapropisrdistr opp1 opp2)). Defined.	Lemma	Algebra	null	Algebra\BinaryOperations.v	weqldistrrdistr	398
Definition isabsorb {X : UU} (opp1 opp2 : binop X) : UU := ∏ x y : X, opp1 x (opp2 x y) = x.	Definition	Algebra	null	Algebra\BinaryOperations.v	isabsorb	399

Definition isrinvertible {X : UU} (opp : binop X) (x : X) : UU := isweq (λ x0 : X, opp x0 x).

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Definition isinvertible {X : UU} (opp : binop X) (x : X) : UU := (islinvertible opp x) × (isrinvertible opp x).

Definition

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isinvertible

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Definition binop_weq_fwd {X Y : UU} (H : X ≃ Y) : binop X → binop Y := λ (opp : binop X) (x y : Y), H (opp (invmap H x) (invmap H y)).

Definition

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binop_weq_fwd

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Definition binop_weq_bck {X Y : UU} (H : X ≃ Y) : binop Y → binop X := λ (opp : binop Y) (x y : X), invmap H (opp (H x) (H y)).

Definition

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binop_weq_bck

303

Definition isassoc {X : UU} (opp : binop X) : UU := ∏ x x' x'', opp (opp x x') x'' = opp x (opp x' x'').

Definition

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isassoc

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Lemma isapropisassoc {X : hSet} (opp : binop X) : isaprop (isassoc opp). Proof. apply impred. intro x. apply impred. intro x'. apply impred. intro x''. simpl. apply (setproperty X). Defined.

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isapropisassoc

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Lemma assoc4 {X : UU} (opp : binop X) (isa : isassoc opp) : ∏ w x y z : X, opp (opp (opp w x) y) z = opp (opp w (opp x y)) z. Proof. intros. repeat rewrite isa; exact (idpath _). Qed.

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assoc4

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Definition isrcancellative {X : UU} (opp : binop X) : UU := ∏ x:X, isrcancelable opp x.

Definition

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isrcancellative

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Definition islcancellative {X : UU} (opp : binop X) : UU := ∏ x:X, islcancelable opp x.

Definition

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islcancellative

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Definition islunit {X : UU} (opp : binop X) (un0 : X) : UU := ∏ x : X, opp un0 x = x.

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islunit

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Lemma isapropislunit {X : hSet} (opp : binop X) (un0 : X) : isaprop (islunit opp un0). Proof. apply impred. intro x. simpl. apply (setproperty X). Defined.

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isapropislunit

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Definition isrunit {X : UU} (opp : binop X) (un0 : X) : UU := ∏ x : X, opp x un0 = x.

Definition

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isrunit

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Lemma isapropisrunit {X : hSet} (opp : binop X) (un0 : X) : isaprop (isrunit opp un0). Proof. apply impred. intro x. simpl. apply (setproperty X). Defined.

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isapropisrunit

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Definition isunit {X : UU} (opp : binop X) (un0 : X) : UU := (islunit opp un0) × (isrunit opp un0).

Definition

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isunit

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Definition make_isunit {X : UU} {opp : binop X} {un0 : X} (H1 : islunit opp un0) (H2 : isrunit opp un0) : isunit opp un0 := H1 ,, H2.

Definition

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make_isunit

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Definition isunital {X : UU} (opp : binop X) : UU := ∑ (un0 : X), isunit opp un0.

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isunital

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Definition make_isunital {X : UU} {opp : binop X} (un0 : X) (is : isunit opp un0) : isunital opp := un0 ,, is.

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make_isunital

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Lemma isapropisunital {X : hSet} (opp : binop X) : isaprop (isunital opp). Proof. apply (@isapropsubtype X (λ un0, hconj (make_hProp _ (isapropislunit opp un0)) (make_hProp _ (isapropisrunit opp un0)))). intros u1 u2. intros ua1 ua2. apply (!(pr2 ua2 u1) @ (pr1 ua1 u2)). Defined.

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isapropisunital

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Definition ismonoidop {X : UU} (opp : binop X) : UU := (isassoc opp) × (isunital opp).

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ismonoidop

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Definition make_ismonoidop {X : UU} {opp : binop X} (H1 : isassoc opp) (H2 : isunital opp) : ismonoidop opp := H1 ,, H2.

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make_ismonoidop

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Definition assocax_is {X : UU} {opp : binop X} : ismonoidop opp → isassoc opp := @pr1 _ _.

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assocax_is

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Definition unel_is {X : UU} {opp : binop X} (is : ismonoidop opp) : X := pr1 (pr2 is).

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unel_is

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Definition lunax_is {X : UU} {opp : binop X} (is : ismonoidop opp) : islunit opp (pr1 (pr2 is)) := pr1 (pr2 (pr2 is)).

Definition

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lunax_is

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Definition runax_is {X : UU} {opp : binop X} (is : ismonoidop opp) : isrunit opp (pr1 (pr2 is)) := pr2 (pr2 (pr2 is)).

Definition

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runax_is

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Definition unax_is {X : UU} {opp : binop X} (is : ismonoidop opp) : isunit opp (pr1 (pr2 is)) := lunax_is is ,, runax_is is.

Definition

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unax_is

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Lemma isapropismonoidop {X : hSet} (opp : binop X) : isaprop (ismonoidop opp). Proof. apply (isofhleveldirprod 1). apply isapropisassoc. apply isapropisunital. Defined.

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Algebra

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Algebra\BinaryOperations.v

isapropismonoidop

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Definition islinvel (x : X) : X → UU := λ x0, x0 * x = u.

Definition

Algebra

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Algebra\BinaryOperations.v

islinvel

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Definition isrinvel (x : X) : X → UU := λ x0, x * x0 = u.

Definition

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isrinvel

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Definition isinvel (x : X) : X → UU := λ x0, (islinvel x x0) × (isrinvel x x0).

Definition

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isinvel

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Definition haslinv (x : X) : UU := ∑ x0 : X, islinvel x x0.

Definition

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haslinv

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Definition hasrinv (x : X) : UU := ∑ x0 : X, isrinvel x x0.

Definition

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hasrinv

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Definition hasinv (x : X) : UU := ∑ x0 : X, isinvel x x0.

Definition

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hasinv

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Definition haslinv_to_linvel {x : X} : haslinv x → X := pr1.

Definition

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Algebra\BinaryOperations.v

haslinv_to_linvel

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Definition hasrinv_to_rinvel {x : X} : hasrinv x → X := pr1.

Definition

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Algebra\BinaryOperations.v

hasrinv_to_rinvel

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Definition hasinv_to_invel {x : X} : hasinv x → X := pr1.

Definition

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Algebra\BinaryOperations.v

hasinv_to_invel

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Definition merely_haslinv (x : X) : hProp := ∥ haslinv x ∥.

Definition

Algebra

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merely_haslinv

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Definition merely_hasrinv (x : X) : hProp := ∥ hasrinv x ∥.

Definition

Algebra

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Algebra\BinaryOperations.v

merely_hasrinv

336

Definition merely_hasinv (x : X) : hProp := ∥ hasinv x ∥.

Definition

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merely_hasinv

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Definition is_inv_inv : ∏ (x x0 : X), (isinvel x x0 → isinvel x0 x) := λ x x0 isinv, pr2 isinv ,, pr1 isinv.

Definition

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is_inv_inv

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Lemma invop_l : ∏ (x y x' y' : X), (islinvel x x' → islinvel y y' → islinvel (x * y) (y' * x')). Proof. intros x y x' y' xinv yinv. unfold islinvel. pose (assoc := pr1 is). cbn; unfold islinvel. rewrite <- assoc. rewrite (assoc4 opp assoc), xinv. rewrite (runax_is is). exact yinv. Qed.

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Algebra

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Algebra\BinaryOperations.v

invop_l

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Lemma invop_r : ∏ (x y x' y' : X), (isrinvel x x' → isrinvel y y' → isrinvel (x * y) (y' * x')). Proof. intros x y x' y' xinv yinv. pose (assoc := pr1 is). cbn; unfold isrinvel. rewrite <- assoc. rewrite (assoc4 opp assoc), yinv. rewrite (runax_is is). exact xinv. Qed.

Lemma

Algebra

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Algebra\BinaryOperations.v

invop_r

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Lemma invop : ∏ (x y x' y' : X), (isinvel x x' → isinvel y y' → isinvel (x * y) (y' * x')). Proof. intros x y x' y' xinv yinv. use make_dirprod. - apply invop_l. + exact (dirprod_pr1 xinv). + exact (dirprod_pr1 yinv). - apply invop_r. + exact (dirprod_pr2 xinv). + exact (dirprod_pr2 yinv). Defined.

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Algebra

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Algebra\BinaryOperations.v

invop

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Lemma mere_invop : ∏ (x y : X), (merely_hasinv x → merely_hasinv y → merely_hasinv (x * y)). Proof. intros x y. apply hinhfun2. intros xinv yinv. exists ((hasinv_to_invel yinv) * (hasinv_to_invel xinv)). apply invop. - exact (pr2 xinv). - exact (pr2 yinv). Defined.

Lemma

Algebra

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Algebra\BinaryOperations.v

mere_invop

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Lemma linv_eq_rinv (x lx rx : X) (lxlinv : islinvel x lx) (rxrinv : isrinvel x rx) : lx = rx. Proof. intros. refine (!runax_is is _ @ _). refine (!maponpaths (λ z, lx * z) rxrinv @ _). refine (!assocax_is is _ _ _ @ _). refine (maponpaths (λ z, z * rx) lxlinv @ _). apply lunax_is. Defined.

Lemma

Algebra

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Algebra\BinaryOperations.v

linv_eq_rinv

343

Definition islinv : UU := ∏ x : X, (inv x) * x = u.

Definition

Algebra

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Algebra\BinaryOperations.v

islinv

344

Definition isrinv : UU := ∏ x : X, x * (inv x) = u.

Definition

Algebra

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Algebra\BinaryOperations.v

isrinv

345

Definition isinv : UU := islinv × isrinv.

Definition

Algebra

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Algebra\BinaryOperations.v

isinv

346

Definition isapropislinvel (x x0 : X) : isaprop (islinvel opp is x x0) := setproperty X _ _.

Definition

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Algebra\BinaryOperations.v

isapropislinvel

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Definition isapropisrinvel (x x0 : X) : isaprop (isrinvel opp is x x0) := setproperty X _ _.

Definition

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isapropisrinvel

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Definition isapropisinvel (x x0 : X) : isaprop (isinvel opp is x x0) := isapropdirprod _ _ (isapropislinvel _ _) (isapropisrinvel _ _).

Definition

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isapropisinvel

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Definition isaprop_haslinv (x : X) (can : islcancelable opp x) : isaprop (hasrinv opp is x). Proof. apply isaproptotal2. - intro; apply isapropislinvel. - intros x' x'' islinvx' islinvx''. apply (Injectivity (λ x0 : X, x * x0)). + apply incl_injectivity; assumption. + exact (islinvx' @ !islinvx''). Defined.

Definition

Algebra

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Algebra\BinaryOperations.v

isaprop_haslinv

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Definition isaprop_hasrinv (x : X) (can : isrcancelable opp x) : isaprop (haslinv opp is x). Proof. apply isaproptotal2. - intro; apply isapropisrinvel. - intros x' x'' isrinvx' isrinvx''. apply (Injectivity (λ x0 : X, x0 * x)). + apply incl_injectivity; assumption. + exact (isrinvx' @ !isrinvx''). Defined.

Definition

Algebra

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Algebra\BinaryOperations.v

isaprop_hasrinv

351

Definition isaprop_hasinv (x : X) (can : iscancelable opp x) : isaprop (hasinv opp is x). Proof. apply isaproptotal2. - intro; apply isapropdirprod. + apply isapropislinvel. + apply isapropisrinvel. - intros x' x'' isinvx' isinvx''. apply (Injectivity (λ x0 : X, x * x0)). + apply incl_injectivity; apply (pr1 can). + exact (pr2 isinvx' @ !pr2 isinvx''). Defined.

Definition

Algebra

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Algebra\BinaryOperations.v

isaprop_hasinv

352

Definition merely_invertible_elements : hsubtype X := merely_hasinv opp is.

Definition

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merely_invertible_elements

353

Definition invertible_elements (can : ∏ x, iscancelable opp x) : hsubtype X. Proof. intro x. use make_hProp. - exact (hasinv opp is x). - apply isaprop_hasinv, can. Defined.

Definition

Algebra

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Algebra\BinaryOperations.v

invertible_elements

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Definition lcanfromlinv (a b c : X) (c' : haslinv opp is c) : c * a = c * b → a = b. Proof. intros e. refine (!lunax_is is a @ _ @ lunax_is is b). refine (!maponpaths (λ z, z * _) (pr2 c') @ _ @ maponpaths (λ z, z * _) (pr2 c')). refine (assocax_is is _ _ _ @ _ @ !assocax_is is _ _ _). apply maponpaths. assumption. Defined.

Definition

Algebra

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Algebra\BinaryOperations.v

lcanfromlinv

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Definition rcanfromrinv (a b c : X) (c' : hasrinv opp is c) : a * c = b * c → a = b. Proof. intros e. refine (!runax_is is a @ _ @ runax_is is b). refine (!maponpaths (λ z, _ * z) (pr2 c') @ _ @ maponpaths (λ z, _ * z) (pr2 c')). refine (!assocax_is is _ _ _ @ _ @ assocax_is is _ _ _). apply (maponpaths (λ z, z * _)). assumption. Defined.

Definition

Algebra

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rcanfromrinv

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Lemma isapropislinv : isaprop (islinv opp u inv). Proof. intros; apply impred; intro; apply setproperty. Defined.

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Algebra

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isapropislinv

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Lemma isapropisrinv : isaprop (isrinv opp u inv). Proof. intros; apply impred; intro; apply setproperty. Defined.

Lemma

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isapropisrinv

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Lemma isapropisinv : isaprop (isinv opp u inv). Proof. exact (isofhleveldirprod 1 _ _ isapropislinv isapropisrinv). Defined.

Lemma

Algebra

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isapropisinv

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Definition make_isinv {X : UU} {opp : binop X} {un0 : X} {inv0 : X → X} (H1 : islinv opp un0 inv0) (H2 : isrinv opp un0 inv0) : isinv opp un0 inv0 := H1 ,, H2.

Definition

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make_isinv

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Definition invstruct {X : UU} (opp : binop X) (is : ismonoidop opp) : UU := ∑ (inv0 : X → X), isinv opp (unel_is is) inv0.

Definition

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invstruct

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Definition make_invstruct {X : UU} {opp : binop X} {is : ismonoidop opp} (inv0 : X → X) (H : isinv opp (unel_is is) inv0) : invstruct opp is := inv0 ,, H.

Definition

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make_invstruct

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Definition isgrop {X : UU} (opp : binop X) : UU := ∑ (is : ismonoidop opp), invstruct opp is.

Definition

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isgrop

363

Definition make_isgrop {X : UU} {opp : binop X} (is1 : ismonoidop opp) (is2 : invstruct opp is1) : isgrop opp := is1 ,, is2.

Definition

Algebra

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make_isgrop

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Definition pr1isgrop (X : UU) (opp : binop X) : isgrop opp → ismonoidop opp := @pr1 _ _.

Definition

Algebra

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Algebra\BinaryOperations.v

pr1isgrop

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Definition grinv_is {X : UU} {opp : binop X} (is : isgrop opp) : X → X := pr1 (pr2 is).

Definition

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grinv_is

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Definition grlinvax_is {X : UU} {opp : binop X} (is : isgrop opp) : islinv opp (unel_is is) (pr1 (pr2 is)) := pr1 (pr2 (pr2 is)).

Definition

Algebra

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grlinvax_is

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Definition grrinvax_is {X : UU} {opp : binop X} (is : isgrop opp) : isrinv opp (unel_is is) (pr1 (pr2 is)) := pr2 (pr2 (pr2 is)).

Definition

Algebra

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Algebra\BinaryOperations.v

grrinvax_is

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Lemma isweqrmultingr_is {X : UU} {opp : binop X} (is : isgrop opp) (x0 : X) : isrinvertible opp x0. Proof. induction is as [ is istr ]. set (f := λ x : X, opp x x0). set (g := λ x : X, opp x ((pr1 istr) x0)). induction is as [ assoc isun0 ]. induction istr as [ inv0 axs ]. induction isun0 as [ un0 unaxs ]. simpl in * |-. assert (egf : ∏ x, g (f x) = x). { intro x. unfold f. unfold g. induction (!assoc x x0 (inv0 x0)). set (e := pr2 axs x0). simpl in e. rewrite e. apply (pr2 unaxs x). } assert (efg : ∏ x, f (g x) = x). { intro x. unfold f. unfold g. induction (!assoc x (inv0 x0) x0). set (e := pr1 axs x0). simpl in e. rewrite e. apply (pr2 unaxs x). } apply (isweq_iso _ _ egf efg). Defined.

Lemma

Algebra

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Algebra\BinaryOperations.v

isweqrmultingr_is

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Lemma isweqlmultingr_is {X : UU} {opp : binop X} (is : isgrop opp) (x0 : X) : islinvertible opp x0. Proof. induction is as [ is istr ]. set (f := λ x : X, opp x0 x). set (g := λ x : X, opp ((pr1 istr) x0) x). induction is as [ assoc isun0 ]. induction istr as [ inv0 axs ]. induction isun0 as [ un0 unaxs ]. simpl in * |-. assert (egf : ∏ x, g (f x) = x). { intro x. unfold f. unfold g. induction (assoc (inv0 x0) x0 x). set (e := pr1 axs x0). simpl in e. rewrite e. apply (pr1 unaxs x). } assert (efg : ∏ x, f (g x) = x). { intro x. unfold f. unfold g. induction (assoc x0 (inv0 x0) x). set (e := pr2 axs x0). simpl in e. rewrite e. apply (pr1 unaxs x). } apply (isweq_iso _ _ egf efg). Defined.

Lemma

Algebra

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Algebra\BinaryOperations.v

isweqlmultingr_is

370

Lemma isapropinvstruct {X : hSet} {opp : binop X} (is : ismonoidop opp) : isaprop (invstruct opp is). Proof. apply isofhlevelsn. intro is0. set (un0 := pr1 (pr2 is)). assert (int : ∏ (i : X → X), isaprop ((∏ x : X, opp (i x) x = un0) × (∏ x : X, opp x (i x) = un0))). { intro i. apply (isofhleveldirprod 1). - apply impred. intro x. simpl. apply (setproperty X). - apply impred. intro x. simpl. apply (setproperty X). } apply (isapropsubtype (λ i, make_hProp _ (int i))). intros inv1 inv2. simpl. intro ax1. intro ax2. apply funextfun. intro x0. apply (invmaponpathsweq (make_weq _ (isweqrmultingr_is (is ,, is0) x0))). simpl. rewrite (pr1 ax1 x0). rewrite (pr1 ax2 x0). apply idpath. Defined.

Lemma

Algebra

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Algebra\BinaryOperations.v

isapropinvstruct

371

Lemma isapropisgrop {X : hSet} (opp : binop X) : isaprop (isgrop opp). Proof. apply (isofhleveltotal2 1). - apply isapropismonoidop. - apply isapropinvstruct. Defined.

Lemma

Algebra

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Algebra\BinaryOperations.v

isapropisgrop

372

Definition allinvvertibleinv {X : hSet} {opp : binop X} (is : ismonoidop opp) (allinv : ∏ x : X, islinvertible opp x) : X → X := λ x : X, invmap (make_weq _ (allinv x)) (unel_is is).

Definition

Algebra

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Algebra\BinaryOperations.v

allinvvertibleinv

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Lemma isgropif {X : hSet} {opp : binop X} (is0 : ismonoidop opp) (is : ∏ x : X, merely_hasrinv opp is0 x) : isgrop opp. Proof. exists is0. induction is0 as [ assoc isun0 ]. induction isun0 as [ un0 unaxs0 ]. simpl in is. simpl in unaxs0. simpl in un0. simpl in assoc. simpl in unaxs0. assert (l1 : ∏ x' : X, isincl (λ x0 : X, opp x0 x')). { intro x'. apply (@hinhuniv (∑ (x0 : X), opp x' x0 = un0) (make_hProp _ (isapropisincl (λ x0 : X, opp x0 x')))). - intro int1. simpl. apply isinclbetweensets. + apply (pr2 X). + apply (pr2 X). + intros a b. intro e. rewrite <- (pr2 unaxs0 a). rewrite <- (pr2 unaxs0 b). induction int1 as [ invx' eq ]. rewrite <- eq. induction (assoc a x' invx'). induction (assoc b x' invx'). rewrite e. apply idpath. - apply (is x'). } assert (is' : ∏ x : X, ∃ (x0 : X), opp x0 x = un0). { intro x. apply (λ f , hinhuniv f (is x)). intro s1. induction s1 as [ x' eq ]. apply hinhpr. exists x'. simpl. apply (invmaponpathsincl _ (l1 x')). rewrite (assoc x' x x'). rewrite eq. rewrite (pr1 unaxs0 x'). unfold unel_is. simpl. rewrite (pr2 unaxs0 x'). apply idpath. } assert (l1' : ∏ x' : X, isincl (λ x0 : X, opp x' x0)). { intro x'. apply (@hinhuniv (∑ (x0 : X), opp x0 x' = un0) (make_hProp _ (isapropisincl (λ x0 : X, opp x' x0)))). - intro int1. simpl. apply isinclbetweensets. + apply (pr2 X). + apply (pr2 X). + intros a b. intro e. rewrite <- (pr1 unaxs0 a). rewrite <- (pr1 unaxs0 b). induction int1 as [ invx' eq ]. rewrite <- eq. induction (!assoc invx' x' a). induction (!assoc invx' x' b). rewrite e. apply idpath. - apply (is' x'). } assert (int : ∏ x : X, isaprop (∑ (x0 : X), opp x0 x = un0)%logic). { intro x. apply isapropsubtype. intros x1 x2. intros eq1 eq2. apply (invmaponpathsincl _ (l1 x)). rewrite eq1. rewrite eq2. apply idpath. } simpl. set (linv0 := λ x : X, hinhunivcor1 (make_hProp _ (int x)) (is' x)). simpl in linv0. set (inv0 := λ x : X, pr1 (linv0 x)). exists inv0. simpl. exists (λ x, pr2 (linv0 x)). intro x. apply (invmaponpathsincl _ (l1 x)). rewrite (assoc x (inv0 x) x). change (inv0 x) with (pr1 (linv0 x)). rewrite (pr2 (linv0 x)). unfold unel_is. simpl. rewrite (pr1 unaxs0 x). rewrite (pr2 unaxs0 x). apply idpath. Defined.

Lemma

Algebra

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Algebra\BinaryOperations.v

isgropif

374

Definition iscomm {X : UU} (opp : binop X) : UU := ∏ x x' : X, opp x x' = opp x' x.

Definition

Algebra

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Algebra\BinaryOperations.v

iscomm

375

Lemma isapropiscomm {X : hSet} (opp : binop X) : isaprop (iscomm opp). Proof. apply impred. intros x. apply impred. intro x'. simpl. apply (setproperty X). Defined.

Lemma

Algebra

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Algebra\BinaryOperations.v

isapropiscomm

376

Definition isabmonoidop {X : UU} (opp : binop X) : UU := (ismonoidop opp) × (iscomm opp).

Definition

Algebra

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Algebra\BinaryOperations.v

isabmonoidop

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Definition make_isabmonoidop {X : UU} {opp : binop X} (H1 : ismonoidop opp) (H2 : iscomm opp) : isabmonoidop opp := H1 ,, H2.

Definition

Algebra

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Algebra\BinaryOperations.v

make_isabmonoidop

378

Definition pr1isabmonoidop (X : UU) (opp : binop X) : isabmonoidop opp → ismonoidop opp := @pr1 _ _.

Definition

Algebra

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Algebra\BinaryOperations.v

pr1isabmonoidop

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Definition commax_is {X : UU} {opp : binop X} (is : isabmonoidop opp) : iscomm opp := pr2 is.

Definition

Algebra

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Algebra\BinaryOperations.v

commax_is

380

Lemma isapropisabmonoidop {X : hSet} (opp : binop X) : isaprop (isabmonoidop opp). Proof. apply (isofhleveldirprod 1). apply isapropismonoidop. apply isapropiscomm. Defined.

Lemma

Algebra

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Algebra\BinaryOperations.v

isapropisabmonoidop

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Lemma abmonoidoprer {X : UU} {opp : binop X} (is : isabmonoidop opp) (a b c d : X) : opp (opp a b) (opp c d) = opp (opp a c) (opp b d). Proof. induction is as [ is comm ]. induction is as [ assoc unital0 ]. simpl in *. induction (assoc (opp a b) c d). induction (assoc (opp a c) b d). induction (!assoc a b c). induction (!assoc a c b). induction (comm b c). apply idpath. Defined.

Lemma

Algebra

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Algebra\BinaryOperations.v

abmonoidoprer

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Lemma weqlcancelablercancelable {X : UU} (opp : binop X) (is : iscomm opp) (x : X) : (islcancelable opp x) ≃ (isrcancelable opp x). Proof. assert (f : (islcancelable opp x) → (isrcancelable opp x)). { unfold islcancelable. unfold isrcancelable. intro isl. apply (λ h, isinclhomot _ _ h isl). intro x0. apply is. } assert (g : (isrcancelable opp x) → (islcancelable opp x)). { unfold islcancelable. unfold isrcancelable. intro isr. apply (λ h, isinclhomot _ _ h isr). intro x0. apply is. } exists f. apply (isweqimplimpl f g (isapropisincl (λ x0 : X, opp x x0)) (isapropisincl (λ x0 : X, opp x0 x))). Defined.

Lemma

Algebra

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Algebra\BinaryOperations.v

weqlcancelablercancelable

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Lemma weqlinvertiblerinvertible {X : UU} (opp : binop X) (is : iscomm opp) (x : X) : (islinvertible opp x) ≃ (isrinvertible opp x). Proof. assert (f : (islinvertible opp x) → (isrinvertible opp x)). { unfold islinvertible. unfold isrinvertible. intro isl. apply (isweqhomot (λ y, opp x y)). - intro z. apply is. - apply isl. } assert (g : (isrinvertible opp x) → (islinvertible opp x)). { unfold islinvertible. unfold isrinvertible. intro isr. apply (λ h, isweqhomot _ _ h isr). intro x0. apply is. } exists f. apply (isweqimplimpl f g (isapropisweq (λ x0 : X, opp x x0)) (isapropisweq (λ x0 : X, opp x0 x))). Defined.

Lemma

Algebra

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Algebra\BinaryOperations.v

weqlinvertiblerinvertible

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Lemma weqlunitrunit {X : hSet} (opp : binop X) (is : iscomm opp) (un0 : X) : (islunit opp un0) ≃ (isrunit opp un0). Proof. assert (f : (islunit opp un0) → (isrunit opp un0)). { unfold islunit. unfold isrunit. intro isl. intro x. induction (is un0 x). apply (isl x). } assert (g : (isrunit opp un0) → (islunit opp un0)). { unfold islunit. unfold isrunit. intro isr. intro x. induction (is x un0). apply (isr x). } exists f. apply (isweqimplimpl f g (isapropislunit opp un0) (isapropisrunit opp un0)). Defined.

Lemma

Algebra

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Algebra\BinaryOperations.v

weqlunitrunit

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Lemma weqlinvrinv {X : hSet} (opp : binop X) (is : iscomm opp) (un0 : X) (inv0 : X → X) : (islinv opp un0 inv0) ≃ (isrinv opp un0 inv0). Proof. assert (f : (islinv opp un0 inv0) → (isrinv opp un0 inv0)). { unfold islinv. unfold isrinv. intro isl. intro x. induction (is (inv0 x) x). apply (isl x). } assert (g : (isrinv opp un0 inv0) → (islinv opp un0 inv0)). { unfold islinv. unfold isrinv. intro isr. intro x. induction (is x (inv0 x)). apply (isr x). } exists f. apply (isweqimplimpl f g (isapropislinv opp un0 inv0) (isapropisrinv opp un0 inv0)). Qed.

Lemma

Algebra

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Algebra\BinaryOperations.v

weqlinvrinv

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Definition isabgrop {X : UU} (opp : binop X) : UU := (isgrop opp) × (iscomm opp).

Definition

Algebra

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Algebra\BinaryOperations.v

isabgrop

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Definition make_isabgrop {X : UU} {opp : binop X} (H1 : isgrop opp) (H2 : iscomm opp) : isabgrop opp := H1 ,, H2.

Definition

Algebra

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Algebra\BinaryOperations.v

make_isabgrop

388

Definition pr1isabgrop (X : UU) (opp : binop X) : isabgrop opp → isgrop opp := @pr1 _ _.

Definition

Algebra

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Algebra\BinaryOperations.v

pr1isabgrop

389

Definition isabgroptoisabmonoidop (X : UU) (opp : binop X) : isabgrop opp → isabmonoidop opp := λ is, pr1 (pr1 is) ,, pr2 is.

Definition

Algebra

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Algebra\BinaryOperations.v

isabgroptoisabmonoidop

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Lemma isapropisabgrop {X : hSet} (opp : binop X) : isaprop (isabgrop opp). Proof. apply (isofhleveldirprod 1). apply isapropisgrop. apply isapropiscomm. Defined.

Lemma

Algebra

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Algebra\BinaryOperations.v

isapropisabgrop

391

Definition isldistr {X : UU} (opp1 opp2 : binop X) : UU := ∏ x x' x'' : X, opp2 x'' (opp1 x x') = opp1 (opp2 x'' x) (opp2 x'' x').

Definition

Algebra

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Algebra\BinaryOperations.v

isldistr

392

Lemma isapropisldistr {X : hSet} (opp1 opp2 : binop X) : isaprop (isldistr opp1 opp2). Proof. apply impred. intro x. apply impred. intro x'. apply impred. intro x''. simpl. apply (setproperty X). Defined.

Lemma

Algebra

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Algebra\BinaryOperations.v

isapropisldistr

393

Definition isrdistr {X : UU} (opp1 opp2 : binop X) : UU := ∏ x x' x'' : X, opp2 (opp1 x x') x'' = opp1 (opp2 x x'') (opp2 x' x'').

Definition

Algebra

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Algebra\BinaryOperations.v

isrdistr

394

Lemma isapropisrdistr {X : hSet} (opp1 opp2 : binop X) : isaprop (isrdistr opp1 opp2). Proof. apply impred. intro x. apply impred. intro x'. apply impred. intro x''. simpl. apply (setproperty X). Defined.

Lemma

Algebra

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Algebra\BinaryOperations.v

isapropisrdistr

395

Definition isdistr {X : UU} (opp1 opp2 : binop X) : UU := (isldistr opp1 opp2) × (isrdistr opp1 opp2).

Definition

Algebra

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Algebra\BinaryOperations.v

isdistr

396

Lemma isapropisdistr {X : hSet} (opp1 opp2 : binop X) : isaprop (isdistr opp1 opp2). Proof. apply (isofhleveldirprod 1 _ _ (isapropisldistr _ _) (isapropisrdistr _ _)). Defined.

Lemma

Algebra

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Algebra\BinaryOperations.v

isapropisdistr

397

Lemma weqldistrrdistr {X : hSet} (opp1 opp2 : binop X) (is : iscomm opp2) : (isldistr opp1 opp2) ≃ (isrdistr opp1 opp2). Proof. assert (f : (isldistr opp1 opp2) → (isrdistr opp1 opp2)). { unfold isldistr. unfold isrdistr. intro isl. intros x x' x''. induction (is x'' (opp1 x x')). induction (is x'' x). induction (is x'' x'). apply (isl x x' x''). } assert (g : (isrdistr opp1 opp2) → (isldistr opp1 opp2)). { unfold isldistr. unfold isrdistr. intro isr. intros x x' x''. induction (is (opp1 x x') x''). induction (is x x''). induction (is x' x''). apply (isr x x' x''). } exists f. apply (isweqimplimpl f g (isapropisldistr opp1 opp2) (isapropisrdistr opp1 opp2)). Defined.

Lemma

Algebra

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Algebra\BinaryOperations.v

weqldistrrdistr

398

Definition isabsorb {X : UU} (opp1 opp2 : binop X) : UU := ∏ x y : X, opp1 x (opp2 x y) = x.

Definition

Algebra

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Algebra\BinaryOperations.v

isabsorb

399