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(g : ∀ s, B s → C s) `{!IsHomomorphism g} (f : ∀ s, A s → B s) `{!IsHomomorphism f} {w : SymbolType σ} (α : Operation A w) (β : Operation B w) (γ : Operation C w) (G : OpPreserving g β γ) (F : OpPreserving f α β) : OpPreserving (λ s, g s o f s) α γ. Proof. induction w; simpl in *. - by path_induction. - intro x. by apply (IHw _ (β (f _ x))). Defined. | Lemma | Require Import Import algebra_notations ne_list.notations. | Classes\theory\ua_homomorphism.v | oppreserving_compose | 3,500 |
(g : Homomorphism B C) (f : Homomorphism A B) : Homomorphism A C := BuildHomomorphism (λ s, g s o f s). | Definition | Require Import Import algebra_notations ne_list.notations. | Classes\theory\ua_homomorphism.v | hom_compose | 3,501 |
{w : SymbolType σ} (α : Operation A w) (β : Operation B w) (f : ∀ (s : Sort σ), A s <~> B s) (P : OpPreserving f α β) : transport (λ C : Carriers σ, Operation C w) (path_equiv_family f) α = β. Proof. induction w; simpl in *. - transport_path_forall_hammer. exact (ap10 (transport_idmap_path_universe (f t)) α @ P). - funext y. transport_path_forall_hammer. rewrite transport_forall_constant. rewrite transport_arrow_toconst. rewrite (transport_path_universe_V (f t)). apply IHw. specialize (P ((f t)^-1 y)). by rewrite (eisretr (f t) y) in P. Qed. | Lemma | Require Import Import algebra_notations ne_list.notations. | Classes\theory\ua_homomorphism.v | path_operations_equiv | 3,502 |
(f : ∀ s, A s → B s) `{IsIsomorphism σ A B f} (u : Symbol σ) : transport (λ C : Carriers σ, Operation C (σ u)) (path_equiv_family (equiv_isomorphism f)) u.#A = u.#B. Proof. by apply path_operations_equiv. Defined. | Lemma | Require Import Import algebra_notations ne_list.notations. | Classes\theory\ua_homomorphism.v | path_operations_isomorphism | 3,503 |
(f : ∀ s, A s → B s) `{IsIsomorphism σ A B f} : A = B. Proof. apply (path_algebra _ _ (path_equiv_family (equiv_isomorphism f))). abstract ( funext u; exact (transport_forall_constant _ _ u @ path_operations_isomorphism f u)). Defined. | Theorem | Require Import Import algebra_notations ne_list.notations. | Classes\theory\ua_homomorphism.v | path_isomorphism | 3,504 |
{σ : Signature} (A B : Algebra σ) := BuildIsomorphic | Record | Require Import Import isomorphic_notations. | Classes\theory\ua_isomorphic.v | Isomorphic | 3,505 |
{σ : Signature} (A B : Algebra σ) := { def_iso : ∀ s, A s → B s | { _ : IsHomomorphism def_iso | IsIsomorphism def_iso }}. | Definition | Require Import Import isomorphic_notations. | Classes\theory\ua_isomorphic.v | SigIsomorphic | 3,506 |
{σ : Signature} (A B : Algebra σ) : SigIsomorphic A B <~> A ≅ B. Proof. issig. Defined. | Lemma | Require Import Import isomorphic_notations. | Classes\theory\ua_isomorphic.v | issig_isomorphic | 3,507 |
`{Univalence} {σ} {A B : Algebra σ} (e : A ≅ B) : A = B. Proof. exact (path_isomorphism (def_isomorphic e)). Defined. | Corollary | Require Import Import isomorphic_notations. | Classes\theory\ua_isomorphic.v | id_isomorphic | 3,508 |
{σ} {A B : Algebra σ} (p : A = B) : A ≅ B. Proof. destruct p. exact (BuildIsomorphic (hom_id A)). Defined. | Lemma | Require Import Import isomorphic_notations. | Classes\theory\ua_isomorphic.v | isomorphic_id | 3,509 |
`{Funext} {σ : Signature} {A B : Algebra σ} (F G : A ≅ B) (a : def_isomorphic F = def_isomorphic G) (b : a#(is_homomorphism_isomorphic F) = is_homomorphism_isomorphic G) : F = G. Proof. apply (ap (issig_isomorphic A B)^-1)^-1. srapply path_sigma. - exact a. - apply path_sigma_hprop. refine (ap _ (transport_sigma _ _) @ _). apply b. Defined. | Lemma | Require Import Import isomorphic_notations. | Classes\theory\ua_isomorphic.v | path_isomorphic | 3,510 |
`{Funext} {σ : Signature} {A B : Algebra σ} `{IsHSetAlgebra B} (F G : A ≅ B) (a : def_isomorphic F = def_isomorphic G) : F = G. Proof. apply (path_isomorphic F G a). apply path_ishprop. Defined. | Lemma | Require Import Import isomorphic_notations. | Classes\theory\ua_isomorphic.v | path_hset_isomorphic | 3,511 |
(p : A = B) : def_isomorphic (isomorphic_id p) = transport (λ C, ∀ s, C s → B s) (ap carriers p)^ (hom_id B). Proof. by path_induction. Defined. | Lemma | Require Import Import isomorphic_notations. | Classes\theory\ua_isomorphic.v | path_def_isomorphic_id_transport_dom | 3,512 |
(p : A = B) : def_isomorphic (isomorphic_id p) = transport (λ C, ∀ s, A s → C s) (ap carriers p) (hom_id A). Proof. by path_induction. Defined. | Lemma | Require Import Import isomorphic_notations. | Classes\theory\ua_isomorphic.v | path_def_isomorphic_id_transport_cod | 3,513 |
`{Univalence} {σ : Signature} (A : Algebra σ) `{IsHSetAlgebra A} : path_isomorphism (hom_id A) = idpath. Proof. apply path_path_hset_algebra. rewrite path_ap_carriers_path_algebra. apply (paths_ind (λ s, idpath) (λ f _, path_forall A A f = idpath)). - apply path_forall_1. - intros. funext s. symmetry. rewrite (path_ishprop _ (isequiv_idmap (A s))). apply path_universe_1. Qed. | Lemma | Require Import Import isomorphic_notations. | Classes\theory\ua_isomorphic.v | path_path_isomorphism_hom_id_hset | 3,514 |
(@isomorphic_id σ A B) o id_isomorphic == idmap. Proof. intro F. apply path_hset_isomorphic. rewrite path_def_isomorphic_id_transport_cod. funext s x. rewrite !transport_forall_constant. rewrite path_ap_carriers_path_algebra. transport_path_forall_hammer. apply transport_path_universe. Qed. | Lemma | Require Import Import isomorphic_notations. | Classes\theory\ua_isomorphic.v | sect_id_isomorphic | 3,515 |
id_isomorphic o (@isomorphic_id σ A B) == idmap. Proof. intro p. destruct p. apply path_path_isomorphism_hom_id_hset. exact _. Qed. | Lemma | Require Import Import isomorphic_notations. | Classes\theory\ua_isomorphic.v | sect_isomorphic_id | 3,516 |
Carriers σ := λ (s : Sort σ), ∀ (i:I), A i s. | Definition | Require Import Import algebra_notations ne_list.notations. Import prod_algebra_notations. | Classes\theory\ua_prod_algebra.v | carriers_prod_algebra | 3,517 |
(w : SymbolType σ) : (∀ i, Operation (A i) w) → Operation carriers_prod_algebra w := match w return (∀ i, Operation (A i) w) → Operation carriers_prod_algebra w with | [:_:] => idmap | _ ::: g => λ f p, g (λ i, f i (p i)) end. | Fixpoint | Require Import Import algebra_notations ne_list.notations. Import prod_algebra_notations. | Classes\theory\ua_prod_algebra.v | op_prod_algebra | 3,518 |
(u : Symbol σ) : Operation carriers_prod_algebra (σ u) := op_prod_algebra (σ u) (λ (i:I), u. | Definition | Require Import Import algebra_notations ne_list.notations. Import prod_algebra_notations. | Classes\theory\ua_prod_algebra.v | ops_prod_algebra | 3,519 |
Algebra σ := BuildAlgebra carriers_prod_algebra ops_prod_algebra. Global Instance trunc_prod_algebra {n : trunc_index} `{!∀ i, IsTruncAlgebra n (A i)} : IsTruncAlgebra n . Proof. intro s. exact _. Qed. | Definition | Require Import Import algebra_notations ne_list.notations. Import prod_algebra_notations. | Classes\theory\ua_prod_algebra.v | ProdAlgebra | 3,520 |
(i:I) (s : Sort σ) (c : ProdAlgebra I A s) : A i s := c i. | Definition | Require Import Import algebra_notations ne_list.notations. Import prod_algebra_notations. | Classes\theory\ua_prod_algebra.v | def_proj_prod_algebra | 3,521 |
{w : SymbolType σ} (i : I) (v : ∀ i : I, Operation (A i) w) (α : Operation (A i) w) (P : v i = α) : OpPreserving (def_proj_prod_algebra i) (op_prod_algebra I A w v) α. Proof. induction w. - exact P. - intro p. apply (IHw (λ i, v i (p i)) (α (p i))). f_ap. Defined. | Lemma | Require Import Import algebra_notations ne_list.notations. Import prod_algebra_notations. | Classes\theory\ua_prod_algebra.v | oppreserving_proj_prod_algebra | 3,522 |
(i : I) : Homomorphism (ProdAlgebra I A) (A i) := BuildHomomorphism (def_proj_prod_algebra i). | Definition | Require Import Import algebra_notations ne_list.notations. Import prod_algebra_notations. | Classes\theory\ua_prod_algebra.v | hom_proj_prod_algebra | 3,523 |
(f : Homomorphism C (ProdAlgebra I A)) (i:I) : Homomorphism C (A i) := hom_compose (hom_proj_prod_algebra I A i) f. | Definition | Require Import Import algebra_notations ne_list.notations. Import prod_algebra_notations. | Classes\theory\ua_prod_algebra.v | hom_prod_algebra_mapout | 3,524 |
(f : ∀ (i:I) s, C s → A i s) : ∀ (s : Sort σ) , C s → ProdAlgebra I A s := λ (s : Sort σ) (x : C s) (i : I), f i s x. | Definition | Require Import Import algebra_notations ne_list.notations. Import prod_algebra_notations. | Classes\theory\ua_prod_algebra.v | def_prod_algebra_mapin | 3,525 |
{w : SymbolType σ} (f : ∀ (i:I) s, C s → A i s) (α : ∀ (i:I), Operation (A i) w) (β : Operation C w) (P : ∀ (i:I), OpPreserving (f i) β (α i)) : OpPreserving (def_prod_algebra_mapin f) β (op_prod_algebra I A w (λ i, α i)). Proof. induction w. - funext i. apply P. - intro x. apply IHw. intro i. apply P. Defined. | Lemma | Require Import Import algebra_notations ne_list.notations. Import prod_algebra_notations. | Classes\theory\ua_prod_algebra.v | oppreserving_prod_algebra_mapin | 3,526 |
(f : ∀ i, Homomorphism C (A i)) : Homomorphism C (ProdAlgebra I A) := BuildHomomorphism (def_prod_algebra_mapin f). | Definition | Require Import Import algebra_notations ne_list.notations. Import prod_algebra_notations. | Classes\theory\ua_prod_algebra.v | hom_prod_algebra_mapin | 3,527 |
`{!∀ i, IsHSetAlgebra (A i)} : (∀ (i:I), Homomorphism C (A i)) <~> Homomorphism C (ProdAlgebra I A). Proof. apply (equiv_adjointify hom_prod_algebra_mapin hom_prod_algebra_mapout). - intro f. by apply path_hset_homomorphism. - intro f. funext i. by apply path_hset_homomorphism. Defined. | Lemma | Require Import Import algebra_notations ne_list.notations. Import prod_algebra_notations. | Classes\theory\ua_prod_algebra.v | ump_prod_algebra | 3,528 |
(b:Bool) : Algebra σ := if b then B else A. Global Instance trunc_bin_prod_algebras {n : trunc_index} `{!IsTruncAlgebra n A} `{!IsTruncAlgebra n B} : ∀ (b:Bool), IsTruncAlgebra n ( b). Proof. intros []; exact _. Qed. | Definition | Require Import Import algebra_notations ne_list.notations. Import prod_algebra_notations. | Classes\theory\ua_prod_algebra.v | bin_prod_algebras | 3,529 |
Algebra σ := ProdAlgebra Bool bin_prod_algebras. | Definition | Require Import Import algebra_notations ne_list.notations. Import prod_algebra_notations. | Classes\theory\ua_prod_algebra.v | BinProdAlgebra | 3,530 |
Homomorphism BinProdAlgebra A := hom_proj_prod_algebra Bool bin_prod_algebras false. | Definition | Require Import Import algebra_notations ne_list.notations. Import prod_algebra_notations. | Classes\theory\ua_prod_algebra.v | fst_prod_algebra | 3,531 |
Homomorphism BinProdAlgebra B := hom_proj_prod_algebra Bool bin_prod_algebras true. | Definition | Require Import Import algebra_notations ne_list.notations. Import prod_algebra_notations. | Classes\theory\ua_prod_algebra.v | snd_prod_algebra | 3,532 |
Homomorphism C A * Homomorphism C B <~> Homomorphism C (A × B). Proof. set (k := λ (b:Bool), Homomorphism C (bin_prod_algebras A B b)). exact (equiv_compose (ump_prod_algebra Bool (bin_prod_algebras A B) C) (equiv_bool_forall_prod k)^-1). Defined. | Lemma | Require Import Import algebra_notations ne_list.notations. Import prod_algebra_notations. | Classes\theory\ua_prod_algebra.v | ump_bin_prod_algebra | 3,533 |
Carriers σ := λ s, Quotient (Φ s). | Definition | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | carriers_quotient_algebra | 3,534 |
{w : list (Sort σ)} : ∀ (P : FamilyProd carriers_quotient_algebra w → Type) `{!∀ a, IsHProp (P a)} (dclass : ∀ x, P (map_family_prod (λ s, class_of (Φ s)) x)) (a : FamilyProd carriers_quotient_algebra w), P a := match w with | nil => λ P _ dclass 'tt, dclass tt | s :: w' => λ P _ dclass a, Quotient_ind_hprop (Φ s) (λ a, ∀ b, P (a,b)) (λ a, (λ c, P (class_of (Φ s) a, c)) (λ c, dclass (a, c))) (fst a) (snd a) end. | Fixpoint | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | quotient_ind_prop_family_prod | 3,535 |
{w : SymbolType σ} (f : Operation A w) (g : Operation carriers_quotient_algebra w) := ∀ (a : FamilyProd A (dom_symboltype w)), ap_operation g (map_family_prod (λ s, class_of (Φ s)) a) = class_of (Φ (cod_symboltype w)) (ap_operation f a). | Definition | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | ComputeOpQuotient | 3,536 |
(q : ∀ (w : SymbolType σ), QuotOp w) (s : Sort σ) (w : SymbolType σ) (f : Operation A (s ::: w)) (P : OpCompatible A Φ f) (x y : A s) (C : Φ s x y) : op_qalg_cons q f P x = op_qalg_cons q f P y. Proof. apply (@path_forall_ap_operation _ σ). apply quotient_ind_prop_family_prod; try exact _. intro a. destruct (q _ _ (op_compatible_cons Φ s w f x P)) as [g1 P1]. destruct (q _ _ (op_compatible_cons Φ s w f y P)) as [g2 P2]. refine ((P1 a) @ _ @ (P2 a)^). apply qglue. exact (P (x,a) (y,a) (C, reflexive_for_all_2_family_prod A Φ a)). Defined. | Lemma | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | op_quotient_algebra_well_def | 3,537 |
{w : SymbolType σ} : QuotOp w. Proof. refine ( match w return QuotOp w with | [:s:] => λ (f : A s) P, (class_of (Φ s) f; λ a, idpath) | s ::: w' => λ (f : A s → Operation A w') P, (Quotient_rec (Φ s) _ (λ (x : A s), op_qalg_cons f P x) (op_quotient_algebra_well_def s w' f P) ; _) end ). intros [x a]. apply ( w' (f x) (op_compatible_cons Φ s w' f x P)). Defined. | Fixpoint | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | op_quotient_algebra | 3,538 |
(u : Symbol σ) : Operation carriers_quotient_algebra (σ u) := (op_quotient_algebra u. | Definition | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | ops_quotient_algebra | 3,539 |
Algebra σ := BuildAlgebra carriers_quotient_algebra ops_quotient_algebra. Global Instance hset_quotient_algebra : IsHSetAlgebra . Proof. intro s. exact _. Qed. | Definition | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | QuotientAlgebra | 3,540 |
(u : Symbol σ) : ComputeOpQuotient u.#A u.#QuotientAlgebra. Proof. apply op_quotient_algebra. Defined. | Lemma | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | compute_op_quotient | 3,541 |
`{Funext} (p : Φ = Ψ) : A/Φ = A/Ψ. Proof. by destruct p, (path_ishprop CΦ CΨ). Defined. | Lemma | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | path_quotient_algebra | 3,542 |
`{Univalence} (R : ∀ s x y, Φ s x y <-> Ψ s x y) : A/Φ = A/Ψ. Proof. apply path_quotient_algebra. funext s x y. refine (path_universe_uncurried _). apply equiv_iff_hprop; apply R. Defined. | Lemma | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | path_quotient_algebra_iff | 3,543 |
∀ (s : Sort σ), A s → (A/Φ) s := λ s x, class_of (Φ s) x. | Definition | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | def_hom_quotient | 3,544 |
`{Funext} (w : SymbolType σ) (g : Operation (A/Φ) w) (α : Operation A w) (G : ComputeOpQuotient A Φ α g) : OpPreserving def_hom_quotient α g. Proof. unfold ComputeOpQuotient in G. induction w; cbn in *. - by destruct (G tt)^. - intro x. apply IHw. intro a. apply (G (x,a)). Defined. | Lemma | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | oppreserving_quotient | 3,545 |
Homomorphism A (A/Φ) := BuildHomomorphism def_hom_quotient. Global Instance surjection_quotient : ∀ s, IsSurjection ( s). Proof. intro s. apply issurj_class_of. Qed. | Definition | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | hom_quotient | 3,546 |
∀ (s : Sort σ), (A/Φ) s → B s := λ s, (equiv_quotient_ump (Φ s) (Build_HSet (B s)))^-1 (f s; R s). | Definition | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | def_hom_quotient_algebra_mapout | 3,547 |
{w : SymbolType σ} (g : Operation (A/Φ) w) (α : Operation A w) (β : Operation B w) (G : ComputeOpQuotient A Φ α g) (P : OpPreserving f α β) : OpPreserving def_hom_quotient_algebra_mapout g β. Proof. unfold ComputeOpQuotient in G. induction w; cbn in *. - destruct (G tt)^. apply P. - refine (Quotient_ind_hprop (Φ t) _ _). intro x. apply (IHw (g (class_of (Φ t) x)) (α x) (β (f t x))). + intro a. apply (G (x,a)). + apply P. Defined. | Lemma | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | oppreserving_quotient_algebra_mapout | 3,548 |
Homomorphism (A/Φ) B := BuildHomomorphism def_hom_quotient_algebra_mapout. | Definition | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | hom_quotient_algebra_mapout | 3,549 |
∀ (s : Sort σ) (x : A s), hom_quotient_algebra_mapout s (class_of (Φ s) x) = f s x. Proof. reflexivity. Defined. | Lemma | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | compute_quotient_algebra_mapout | 3,550 |
(g : Homomorphism (A/Φ) B) : Homomorphism A B := hom_compose g (hom_quotient Φ). | Definition | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | hom_quotient_algebra_mapin | 3,551 |
{f : Homomorphism A B | ∀ s (x y : A s), Φ s x y → f s x = f s y} → Homomorphism (A/Φ) B. Proof. intros [f P]. exists (hom_quotient_algebra_mapout f P). exact _. Defined. | Lemma | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | ump_quotient_algebra_lr | 3,552 |
Homomorphism (A/Φ) B → {f : Homomorphism A B | ∀ s (x y : A s), Φ s x y → f s x = f s y}. Proof. intro g. exists (hom_quotient_algebra_mapin g). intros s x y E. exact (transport (λ z, g s (class_of (Φ s) x) = g s z) (qglue E) idpath). Defined. | Lemma | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | ump_quotient_algebra_rl | 3,553 |
{f : Homomorphism A B | ∀ s (x y : A s), Φ s x y → f s x = f s y} <~> Homomorphism (A/Φ) B. Proof. apply (equiv_adjointify ump_quotient_algebra_lr ump_quotient_algebra_rl). - intro G. apply path_hset_homomorphism. funext s. exact (eissect (equiv_quotient_ump (Φ s) _) (G s)). - intro F. apply path_sigma_hprop. by apply path_hset_homomorphism. Defined. | Lemma | Require Import Basics.Notations. Require Import Import algebra_notations ne_list.notations. Import quotient_algebra_notations. | Classes\theory\ua_quotient_algebra.v | ump_quotient_algebra | 3,554 |
(s : Sort σ) (x : (A&&P) s) (y : (A&&P) s) := Φ s (i s x) (i s y). Global Instance equiv_rel_trace_congruence (s : Sort σ) : EquivRel ( s). Proof. unfold . constructor. - intros [y Y]. reflexivity. - intros [y1 Y1] [y2 Y2] S. by symmetry. - intros [y1 Y1] [y2 Y2] [y3 Y3] S T. by transitivity y2. Qed. | Definition | Require Import Import | Classes\theory\ua_second_isomorphism.v | cong_trace | 3,555 |
{w : SymbolType σ} (a b : FamilyProd (A&&P) (dom_symboltype w)) (R : for_all_2_family_prod (A&&P) (A&&P) cong_trace a b) : for_all_2_family_prod A A Φ (map_family_prod i a) (map_family_prod i b). Proof with try assumption. induction w... destruct a as [x a], b as [y b], R as [C R]. split... apply IHw... Qed. | Lemma | Require Import Import | Classes\theory\ua_second_isomorphism.v | for_all_2_family_prod_trace_congruence | 3,556 |
(s : Sort σ) (x : (A/Φ) s) : HProp := hexists (λ (y : (A&&P) s), in_class (Φ s) x (i s y)). | Definition | Require Import Import | Classes\theory\ua_second_isomorphism.v | is_subalgebra_class | 3,557 |
{w : SymbolType σ} (γ : Operation (A/Φ) w) (α : Operation A w) (Q : ComputeOpQuotient A Φ α γ) (C : ClosedUnderOp A P α) : ClosedUnderOp (A/Φ) is_subalgebra_class γ. Proof. induction w. - specialize (Q tt). apply tr. exists (α; C). cbn in Q. destruct Q^. exact (EquivRel_Reflexive α). - refine (Quotient_ind_hprop (Φ t) _ _). intro x. refine (Trunc_rec _). intros [y R]. apply (IHw (γ (class_of (Φ t) x)) (α (i t y))). + intro a. destruct (qglue R)^. apply (Q (i t y,a)). + apply C. exact y.2. Qed. | Lemma | Require Import Import | Classes\theory\ua_second_isomorphism.v | op_closed_subalgebra_is_subalgebra_class | 3,558 |
IsClosedUnderOps (A/Φ) is_subalgebra_class. Proof. intro u. eapply op_closed_subalgebra_is_subalgebra_class. - apply compute_op_quotient. - apply is_closed_under_ops_subalgebra_predicate. exact _. Qed. | Definition | Require Import Import | Classes\theory\ua_second_isomorphism.v | is_closed_under_ops_is_subalgebra_class | 3,559 |
(s : Sort σ) : ((A&&P) / Ψ) s → ((A/Φ) && Q) s := Quotient_rec (Ψ s) _ (λ (x : (A&&P) s), (class_of (Φ s) (i s x); tr (x; EquivRel_Reflexive x))) (λ (x y : (A&&P) s) (T : Ψ s x y), path_sigma_hprop (class_of (Φ s) (i s x); _) (class_of (Φ s) (i s y); _) (@qglue _ (Φ s) _ _ T)). | Definition | Require Import Import | Classes\theory\ua_second_isomorphism.v | def_second_isomorphism | 3,560 |
{w : SymbolType σ} (α : Operation A w) (γ : Operation (A/Φ) w) (ζ : Operation ((A&&P) / Ψ) w) (CA : ClosedUnderOp (A/Φ) Q γ) (CB : ClosedUnderOp A P α) (QA : ComputeOpQuotient A Φ α γ) (QB : ComputeOpQuotient (A&&P) Ψ (op_subalgebra A P α CB) ζ) : OpPreserving def_second_isomorphism ζ (op_subalgebra (A/Φ) Q γ CA). Proof. unfold ComputeOpQuotient in *. induction w; cbn in *. - apply path_sigma_hprop. cbn. destruct (QB tt)^, (QA tt)^. by apply qglue. - refine (Quotient_ind_hprop (Ψ t) _ _). intro x. apply (IHw (α (i t x)) (γ (class_of (Φ t) (i t x))) (ζ (class_of (Ψ t) x)) (CA (class_of (Φ t) (i t x)) (tr (x; _))) (CB (i t x) x.2)). + intro a. exact (QA (i t x, a)). + intro a. exact (QB (x, a)). Defined. | Lemma | Require Import Import | Classes\theory\ua_second_isomorphism.v | oppreserving_second_isomorphism | 3,561 |
Homomorphism ((A&&P) / Ψ) ((A/Φ) && Q) := BuildHomomorphism def_second_isomorphism. Global Instance embedding_second_isomorphism (s : Sort σ) : IsEmbedding ( s). Proof. apply isembedding_isinj_hset. refine (Quotient_ind_hprop (Ψ s) _ _). intro x. refine (Quotient_ind_hprop (Ψ s) _ _). intros y p. apply qglue. exact (related_quotient_paths (Φ s) (i s x) (i s y) (p..1)). Qed. | Definition | Require Import Import | Classes\theory\ua_second_isomorphism.v | hom_second_isomorphism | 3,562 |
IsIsomorphism hom_second_isomorphism. Proof. intro s. apply isequiv_surj_emb; exact _. Qed. | Theorem | Require Import Import | Classes\theory\ua_second_isomorphism.v | is_isomorphism_second_isomorphism | 3,563 |
(A&&P) / Ψ ≅ (A/Φ) && Q. Proof. exact (BuildIsomorphic def_second_isomorphism). Defined. | Theorem | Require Import Import | Classes\theory\ua_second_isomorphism.v | isomorphic_second_isomorphism | 3,564 |
(A&&P) / Ψ = (A/Φ) && Q. Proof. exact (id_isomorphic isomorphic_second_isomorphism). Defined. | Corollary | Require Import Import | Classes\theory\ua_second_isomorphism.v | id_second_isomorphism | 3,565 |
{w : SymbolType σ} : Operation A w → Type := match w with | [:s:] => P s | s ::: w' => λ (α : A s → Operation A w'), ∀ (x : A s), P s x → (α x) end. Global Instance trunc_closed_under_op {n} `{∀ s x, IsTrunc n (P s x)} {w : SymbolType σ} (α : Operation A w) : IsTrunc n ( α). Proof. induction w; cbn; exact _. Qed. | Fixpoint | Require Import Import algebra_notations ne_list.notations. Import subalgebra_notations. | Classes\theory\ua_subalgebra.v | ClosedUnderOp | 3,566 |
Type := ∀ (u : Symbol σ), ClosedUnderOp u.#A. Global Instance trunc_is_closed_under_ops {n} `{∀ s x, IsTrunc n (P s x)} : IsTrunc n . Proof. apply istrunc_forall. Qed. | Definition | Require Import Import algebra_notations ne_list.notations. Import subalgebra_notations. | Classes\theory\ua_subalgebra.v | IsClosedUnderOps | 3,567 |
Carriers σ := λ (s : Sort σ), {x | P s x}. | Definition | Require Import Import algebra_notations ne_list.notations. Import subalgebra_notations. | Classes\theory\ua_subalgebra.v | carriers_subalgebra | 3,568 |
{w : SymbolType σ} : ∀ (α : Operation A w), ClosedUnderOp A P α → Operation carriers_subalgebra w := match w with | [:t:] => λ α c, (α; c) | s ::: w' => λ α c x, (α x. | Fixpoint | Require Import Import algebra_notations ne_list.notations. Import subalgebra_notations. | Classes\theory\ua_subalgebra.v | op_subalgebra | 3,569 |
(u : Symbol σ) : Operation carriers_subalgebra (σ u) := op_subalgebra u. | Definition | Require Import Import algebra_notations ne_list.notations. Import subalgebra_notations. | Classes\theory\ua_subalgebra.v | ops_subalgebra | 3,570 |
Algebra σ := BuildAlgebra carriers_subalgebra ops_subalgebra. Global Instance trunc_subalgebra {n : trunc_index} `{!IsTruncAlgebra n.+1 A} : IsTruncAlgebra n.+1 . Proof. pose proof (hprop_subalgebra_predicate A P). intro s. apply @istrunc_sigma. - exact _. - intro. induction n; exact _. Qed. | Definition | Require Import Import algebra_notations ne_list.notations. Import subalgebra_notations. | Classes\theory\ua_subalgebra.v | Subalgebra | 3,571 |
(s : Sort σ) : (A&&P) s → A s := pr1. | Definition | Require Import Import algebra_notations ne_list.notations. Import subalgebra_notations. | Classes\theory\ua_subalgebra.v | def_inc_subalgebra | 3,572 |
{w : SymbolType σ} (α : Operation A w) (C : ClosedUnderOp A P α) : OpPreserving def_inc_subalgebra (op_subalgebra A P α C) α. Proof. induction w. - reflexivity. - intros x. apply IHw. Defined. | Lemma | Require Import Import algebra_notations ne_list.notations. Import subalgebra_notations. | Classes\theory\ua_subalgebra.v | oppreserving_inc_subalgebra | 3,573 |
Homomorphism (A&&P) A := BuildHomomorphism def_inc_subalgebra. | Definition | Require Import Import algebra_notations ne_list.notations. Import subalgebra_notations. | Classes\theory\ua_subalgebra.v | hom_inc_subalgebra | 3,574 |
(improper : ∀ s (x : A s), P s x) : IsIsomorphism hom_inc_subalgebra. Proof. intro s. refine (isequiv_adjointify _ (λ x, (x; improper s x)) _ _). - intro x. reflexivity. - intro x. by apply path_sigma_hprop. Qed. | Lemma | Require Import Import algebra_notations ne_list.notations. Import subalgebra_notations. | Classes\theory\ua_subalgebra.v | is_isomorphism_inc_improper_subalgebra | 3,575 |
`{Funext} (p : P = Q) : A&&P = A&&Q. Proof. by destruct p, (path_ishprop CP CQ). Defined. | Lemma | Require Import Import algebra_notations ne_list.notations. Import subalgebra_notations. | Classes\theory\ua_subalgebra.v | path_subalgebra | 3,576 |
`{Univalence} (R : ∀ s x, P s x <-> Q s x) : A&&P = A&&Q. Proof. apply path_subalgebra. funext s x. apply (@path_universe _ _ _ (fst (R s x))). apply (equiv_equiv_iff_hprop _ _ (R s x)). Defined. | Lemma | Require Import Import algebra_notations ne_list.notations. Import subalgebra_notations. | Classes\theory\ua_subalgebra.v | path_subalgebra_iff | 3,577 |
(_ : ∀ s x y, Ψ s x y → Φ s x y) (s : Sort σ) (a b : (A/Ψ) s) := ∀ (x y : A s), in_class (Ψ s) a x → in_class (Ψ s) b y → Φ s x y. Global Instance equivalence_relation_quotient (s : Sort σ) : EquivRel ( subrel s). Proof. constructor. - refine (Quotient_ind_hprop (Ψ s) _ _). intros x y z P Q. apply subrel. by transitivity x. - refine (Quotient_ind_hprop (Ψ s) _ _). intro x1. refine (Quotient_ind_hprop (Ψ s) _ _). intros x2 C y1 y2 P Q. symmetry. by apply C. - refine (Quotient_ind_hprop (Ψ s) _ _). intro x1. refine (Quotient_ind_hprop (Ψ s) _ _). intro x2. refine (Quotient_ind_hprop (Ψ s) _ _). intros x3 C D y1 y2 P Q. transitivity x2. + exact (C y1 x2 P (EquivRel_Reflexive x2)). + exact (D x2 y2 (EquivRel_Reflexive x2) Q). Defined. | Definition | Require Import Import algebra_notations quotient_algebra_notations isomorphic_notations. | Classes\theory\ua_third_isomorphism.v | cong_quotient | 3,578 |
{w : list (Sort σ)} (a b : FamilyProd A w) : for_all_2_family_prod (A/Ψ) (A/Ψ) (cong_quotient subrel) (map_family_prod (λ s, class_of (Ψ s)) a) (map_family_prod (λ s, class_of (Ψ s)) b) → for_all_2_family_prod A A Φ a b. Proof. intro F. induction w; cbn in *. - constructor. - destruct a as [x a], b as [y b], F as [Q F]. split. + apply Q; simpl; reflexivity. + by apply IHw. Qed. | Lemma | Require Import Import algebra_notations quotient_algebra_notations isomorphic_notations. | Classes\theory\ua_third_isomorphism.v | for_all_relation_quotient | 3,579 |
(s : Sort σ) (x y : A s) (P : Ψ s x y) : class_of (Φ s) x = class_of (Φ s) y. Proof. apply qglue. exact (subrel s x y P). Defined. | Lemma | Require Import Import algebra_notations quotient_algebra_notations isomorphic_notations. | Classes\theory\ua_third_isomorphism.v | third_surjecton_well_def | 3,580 |
(s : Sort σ) : (A/Ψ) s → (A/Φ) s := Quotient_rec (Ψ s) _ (class_of (Φ s)) (third_surjecton_well_def s). | Definition | Require Import Import algebra_notations quotient_algebra_notations isomorphic_notations. | Classes\theory\ua_third_isomorphism.v | def_third_surjection | 3,581 |
{w : SymbolType σ} (f : Operation A w) : ∀ (α : Operation (A/Φ) w) (Qα : ComputeOpQuotient A Φ f α) (β : Operation (A/Ψ) w) (Qβ : ComputeOpQuotient A Ψ f β), OpPreserving def_third_surjection β α. Proof. induction w. - refine (Quotient_ind_hprop (Φ t) _ _). intros α Qα. refine (Quotient_ind_hprop (Ψ t) _ _). intros β Qβ. apply qglue. transitivity f. + apply subrel. apply (related_quotient_paths (Ψ t)). exact (Qβ tt). + apply (related_quotient_paths (Φ t)). symmetry. exact (Qα tt). - intros α Qα β Qβ. refine (Quotient_ind_hprop (Ψ t) _ _). intro x. exact (IHw (f x) (α (class_of (Φ t) x)) (λ a, Qα (x,a)) (β (class_of (Ψ t) x)) (λ a, Qβ (x,a))). Defined. | Lemma | Require Import Import algebra_notations quotient_algebra_notations isomorphic_notations. | Classes\theory\ua_third_isomorphism.v | oppreserving_third_surjection | 3,582 |
Homomorphism (A/Ψ) (A/Φ) := BuildHomomorphism def_third_surjection. Global Instance surjection_third_surjection (s : Sort σ) : IsSurjection ( s). Proof. apply BuildIsSurjection. refine (Quotient_ind_hprop (Φ s) _ _). intro x. apply tr. by exists (class_of (Ψ s) x). Qed. | Definition | Require Import Import algebra_notations quotient_algebra_notations isomorphic_notations. | Classes\theory\ua_third_isomorphism.v | hom_third_surjection | 3,583 |
A/Ψ / cong_ker hom_third_surjection = A/Ψ / Θ. Proof. apply path_quotient_algebra_iff. intros s x y. split; generalize dependent y; generalize dependent x; refine (Quotient_ind_hprop (Ψ s) _ _); intro x; refine (Quotient_ind_hprop (Ψ s) _ _); intro y. - intros K x' y' Cx Cy. apply subrel in Cx. apply subrel in Cy. apply (related_quotient_paths (Φ s)) in K. transitivity x. + by symmetry. + by transitivity y. - intro T. apply qglue. exact (T x y (EquivRel_Reflexive x) (EquivRel_Reflexive y)). Defined. | Lemma | Require Import Import algebra_notations quotient_algebra_notations isomorphic_notations. | Classes\theory\ua_third_isomorphism.v | path_quotient_algebras_third_surjection | 3,584 |
Homomorphism (A/Ψ/Θ) (A/Φ) := transport (λ X, Homomorphism X (A/Φ)) path_quotient_algebras_third_surjection (hom_first_isomorphism_surjection hom_third_surjection). | Definition | Require Import Import algebra_notations quotient_algebra_notations isomorphic_notations. | Classes\theory\ua_third_isomorphism.v | hom_third_isomorphism | 3,585 |
IsIsomorphism hom_third_isomorphism. Proof. unfold hom_third_isomorphism. destruct path_quotient_algebras_third_surjection. exact _. Qed. | Theorem | Require Import Import algebra_notations quotient_algebra_notations isomorphic_notations. | Classes\theory\ua_third_isomorphism.v | is_isomorphism_third_isomorphism | 3,586 |
A/Ψ/Θ ≅ A/Φ. Proof. exact (BuildIsomorphic hom_third_isomorphism). Defined. | Corollary | Require Import Import algebra_notations quotient_algebra_notations isomorphic_notations. | Classes\theory\ua_third_isomorphism.v | isomorphic_third_isomorphism | 3,587 |
A/Ψ/Θ = A/Φ. Proof. exact (id_isomorphic isomorphic_third_isomorphism). Defined. | Corollary | Require Import Import algebra_notations quotient_algebra_notations isomorphic_notations. | Classes\theory\ua_third_isomorphism.v | id_third_isomorphism | 3,588 |
Coeq@{i j u} {B : Type@{i}} {A : Type@{j}} (f g : B -> A) : Type@{u} := GraphQuotient@{i j u} (fun a b => {x : B & (f x = a) * (g x = b)}). | Definition | Require Import Basics. Require Import Types.Paths Types.Arrow Types.Sigma Types.Forall Types.Universe Types.Prod. Require Import Colimits.GraphQuotient. | Colimits\Coeq.v | Coeq@ | 3,589 |
{B A f g} (a : A) : @Coeq B A f g := gq a. | Definition | Require Import Basics. Require Import Types.Paths Types.Arrow Types.Sigma Types.Forall Types.Universe Types.Prod. Require Import Colimits.GraphQuotient. | Colimits\Coeq.v | coeq | 3,590 |
{B A f g} b : @coeq B A f g (f b) = coeq (g b) := gqglue (b; (idpath,idpath)). | Definition | Require Import Basics. Require Import Types.Paths Types.Arrow Types.Sigma Types.Forall Types.Universe Types.Prod. Require Import Colimits.GraphQuotient. | Colimits\Coeq.v | cglue | 3,591 |
{B A f g} (P : @Coeq B A f g -> Type) (coeq' : forall a, P (coeq a)) (cglue' : forall b, (cglue b) # (coeq' (f b)) = coeq' (g b)) : forall w, P w. Proof. rapply GraphQuotient_ind. intros a b [x [[] []]]. exact (cglue' x). Defined. | Definition | Require Import Basics. Require Import Types.Paths Types.Arrow Types.Sigma Types.Forall Types.Universe Types.Prod. Require Import Colimits.GraphQuotient. | Colimits\Coeq.v | Coeq_ind | 3,592 |
{B A f g} (P : @Coeq B A f g -> Type) (coeq' : forall a, P (coeq a)) (cglue' : forall b, (cglue b) # (coeq' (f b)) = coeq' (g b)) (b:B) : apD (Coeq_ind P coeq' cglue') (cglue b) = cglue' b. Proof. rapply GraphQuotient_ind_beta_gqglue. Defined. | Lemma | Require Import Basics. Require Import Types.Paths Types.Arrow Types.Sigma Types.Forall Types.Universe Types.Prod. Require Import Colimits.GraphQuotient. | Colimits\Coeq.v | Coeq_ind_beta_cglue | 3,593 |
{B A f g} (P : Type) (coeq' : A -> P) (cglue' : forall b, coeq' (f b) = coeq' (g b)) : @Coeq B A f g -> P. Proof. rapply GraphQuotient_rec. intros a b [x [[] []]]. exact (cglue' x). Defined. | Definition | Require Import Basics. Require Import Types.Paths Types.Arrow Types.Sigma Types.Forall Types.Universe Types.Prod. Require Import Colimits.GraphQuotient. | Colimits\Coeq.v | Coeq_rec | 3,594 |
{B A f g} (P : Type) (coeq' : A -> P) (cglue' : forall b:B, coeq' (f b) = coeq' (g b)) (b:B) : ap (Coeq_rec P coeq' cglue') (cglue b) = cglue' b. Proof. rapply GraphQuotient_rec_beta_gqglue. Defined. | Definition | Require Import Basics. Require Import Types.Paths Types.Arrow Types.Sigma Types.Forall Types.Universe Types.Prod. Require Import Colimits.GraphQuotient. | Colimits\Coeq.v | Coeq_rec_beta_cglue | 3,595 |
{B A f g} (P : @Coeq B A f g -> Type) `{forall x, IsHProp (P x)} (i : forall a, P (coeq a)) : forall x, P x. Proof. snrapply Coeq_ind. 1: exact i. intros b. rapply path_ishprop. Defined. | Definition | Require Import Basics. Require Import Types.Paths Types.Arrow Types.Sigma Types.Forall Types.Universe Types.Prod. Require Import Colimits.GraphQuotient. | Colimits\Coeq.v | Coeq_ind_hprop | 3,596 |
{B A f g} {P : @Coeq B A f g -> Type} (h : forall w : Coeq f g, P w) : h == Coeq_ind P (h o coeq) (fun b => apD h (cglue b)). Proof. unfold pointwise_paths. nrapply (Coeq_ind _ (fun _ => 1)). intros b. lhs nrapply transport_paths_FlFr_D. lhs nrapply (whiskerL _ (Coeq_ind_beta_cglue _ _ _ _)). lhs nrapply (whiskerR (concat_p1 _)). nrapply concat_Vp. Defined. | Definition | Require Import Basics. Require Import Types.Paths Types.Arrow Types.Sigma Types.Forall Types.Universe Types.Prod. Require Import Colimits.GraphQuotient. | Colimits\Coeq.v | Coeq_ind_eta_homotopic | 3,597 |
{B A f g} {P : Type} (h : @Coeq B A f g -> P) : h == Coeq_rec P (h o coeq) (fun b => ap h (cglue b)). Proof. unfold pointwise_paths. nrapply (Coeq_ind _ (fun _ => 1)). intros b. apply transport_paths_FlFr', equiv_p1_1q. symmetry; nrapply Coeq_rec_beta_cglue. Defined. | Definition | Require Import Basics. Require Import Types.Paths Types.Arrow Types.Sigma Types.Forall Types.Universe Types.Prod. Require Import Colimits.GraphQuotient. | Colimits\Coeq.v | Coeq_rec_eta_homotopic | 3,598 |
`{Funext} {B A f g} {P : @Coeq B A f g -> Type} (h : forall w : Coeq f g, P w) : h = Coeq_ind P (h o coeq) (fun b => apD h (cglue b)) := path_forall _ _ (Coeq_ind_eta_homotopic h). | Definition | Require Import Basics. Require Import Types.Paths Types.Arrow Types.Sigma Types.Forall Types.Universe Types.Prod. Require Import Colimits.GraphQuotient. | Colimits\Coeq.v | Coeq_ind_eta | 3,599 |