feat(pointed): redefine pequiv
Now the underlying pointed function and pointed inverse are the functions which were put in definitionally
This commit is contained in:
Floris van Doorn
parent
66ea4a4725
commit
9265094f96
10 changed files with 167 additions and 106 deletions
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@ -184,9 +184,9 @@ namespace group
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begin
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induction p,
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apply pequiv_eq,
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fapply pmap_eq,
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{ intro g, reflexivity},
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{ apply is_prop.elim}
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fapply phomotopy.mk,
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{ intro g, reflexivity },
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{ apply is_prop.elim }
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end
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definition to_ginv [constructor] (φ : G₁ ≃g G₂) : G₂ →g G₁ :=
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@ -179,7 +179,7 @@ namespace EM
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[is_conn 0 X] [is_trunc 1 X] : EM1 G ≃* X :=
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begin
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apply EM1_pequiv' (pequiv_of_isomorphism e ⬝e* ptrunc_pequiv 0 (Ω X)),
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refine is_equiv.preserve_binary_of_inv_preserve _ mul concat _,
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refine equiv.preserve_binary_of_inv_preserve _ mul concat _,
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intro p q,
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exact to_respect_mul e⁻¹ᵍ (tr p) (tr q)
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end
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@ -75,7 +75,7 @@ We get the long exact sequence of homotopy groups by taking the set-truncation o
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import .chain_complex algebra.homotopy_group eq2
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open eq pointed sigma fiber equiv is_equiv is_trunc nat trunc algebra function sum
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open eq pointed sigma fiber equiv is_equiv is_trunc nat trunc algebra function
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/--------------
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PART 1
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--------------/
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@ -247,7 +247,8 @@ namespace chain_complex
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fiber_sequence_carrier_pequiv n (fiber_sequence_fun (n + 3) x) =
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ap1 (fiber_sequence_fun n) (fiber_sequence_carrier_pequiv (n + 1) x)⁻¹ :=
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begin
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apply homotopy_of_inv_homotopy_pre (fiber_sequence_carrier_pequiv (n + 1)),
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refine @(homotopy_of_inv_homotopy_pre (fiber_sequence_carrier_pequiv (n + 1)))
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!pequiv.to_is_equiv _ _ _,
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apply fiber_sequence_fun_eq_helper n
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end
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@ -239,7 +239,7 @@ namespace chain_complex
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intro y q, esimp at *,
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have H2 : tcc_to_fn X (f m) ((equiv_of_eq (ap (λx, X x) (c m)))⁻¹ᵉ (e⁻¹ y)) = pt,
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begin
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refine _ ⬝ ap e⁻¹ᵉ* q ⬝ (respect_pt (e⁻¹ᵉ*)), apply eq_inv_of_eq, clear q, revert y,
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refine _ ⬝ ap e⁻¹ᵉ* q ⬝ (respect_pt (e⁻¹ᵉ*)), apply @eq_inv_of_eq _ _ e, clear q, revert y,
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apply inv_homotopy_of_homotopy_pre e,
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apply inv_homotopy_of_homotopy_pre, apply p
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end,
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@ -119,13 +119,4 @@ namespace pointed
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abbreviation to_homotopy [coercion] [unfold 5] (p : f ~* g) : Πa, f a = g a :=
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phomotopy.homotopy p
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/- pointed equivalences -/
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structure pequiv (A B : Type*) extends equiv A B, pmap A B
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attribute pequiv._trans_of_to_pmap pequiv._trans_of_to_equiv pequiv.to_pmap pequiv.to_equiv
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[unfold 3]
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attribute pequiv.to_is_equiv [instance]
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attribute pequiv.to_pmap [coercion]
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infix ` ≃* `:25 := pequiv
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end pointed
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@ -308,11 +308,49 @@ namespace equiv
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end equiv
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namespace pointed
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open equiv is_equiv
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definition pequiv_eq {A B : Type*} {p q : A ≃* B} (H : p = q :> (A →* B)) : p = q :=
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open equiv is_equiv pointed prod
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definition pequiv.sigma_char {A B : Type*} :
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(A ≃* B) ≃ Σ(f : A →* B), (Σ(g : B →* A), f ∘* g ~* pid B) × (Σ(h : B →* A), h ∘* f ~* pid A) :=
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begin
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cases p with f Hf, cases q with g Hg, esimp at *,
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exact apd011 pequiv_of_pmap H !is_prop.elimo
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fapply equiv.MK,
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{ intro f, exact ⟨f, (⟨pequiv.to_pinv1 f, pequiv.pright_inv f⟩,
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⟨pequiv.to_pinv2 f, pequiv.pleft_inv f⟩)⟩, },
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{ intro f, exact pequiv.mk' f.1 (pr1 f.2).1 (pr2 f.2).1 (pr1 f.2).2 (pr2 f.2).2 },
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{ intro f, induction f with f v, induction v with hl hr, induction hl, induction hr,
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reflexivity },
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{ intro f, induction f, reflexivity }
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end
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variables {A B : Type*}
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definition is_contr_pright_inv (f : A ≃* B) : is_contr (Σ(g : B →* A), f ∘* g ~* pid B) :=
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begin
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fapply is_trunc_equiv_closed,
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{ exact !fiber.sigma_char ⬝e sigma_equiv_sigma_right (λg, !pmap_eq_equiv) },
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fapply is_contr_fiber_of_is_equiv,
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exact pequiv.to_is_equiv (pequiv_ppcompose_left f)
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end
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definition is_contr_pleft_inv (f : A ≃* B) : is_contr (Σ(h : B →* A), h ∘* f ~* pid A) :=
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begin
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fapply is_trunc_equiv_closed,
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{ exact !fiber.sigma_char ⬝e sigma_equiv_sigma_right (λg, !pmap_eq_equiv) },
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fapply is_contr_fiber_of_is_equiv,
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exact pequiv.to_is_equiv (pequiv_ppcompose_right f)
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end
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definition pequiv_eq_equiv (f g : A ≃* B) : (f = g) ≃ f ~* g :=
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have Π(f : A →* B), is_prop ((Σ(g : B →* A), f ∘* g ~* pid B) × (Σ(h : B →* A), h ∘* f ~* pid A)),
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begin
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intro f, apply is_prop_of_imp_is_contr, intro v,
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let f' := pequiv.sigma_char⁻¹ᵉ ⟨f, v⟩,
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apply is_trunc_prod, exact is_contr_pright_inv f', exact is_contr_pleft_inv f'
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end,
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calc (f = g) ≃ (pequiv.sigma_char f = pequiv.sigma_char g)
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: eq_equiv_fn_eq pequiv.sigma_char f g
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... ≃ (f = g :> (A →* B)) : subtype_eq_equiv
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... ≃ (f ~* g) : pmap_eq_equiv f g
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definition pequiv_eq {f g : A ≃* B} (H : f ~* g) : f = g :=
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(pequiv_eq_equiv f g)⁻¹ᵉ H
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end pointed
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@ -266,9 +266,9 @@ namespace fiber
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lemma pequiv_precompose_ppoint {A A' B : Type*} (f : A →* B) (g : A' ≃* A)
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: ppoint f ∘* fiber.pequiv_precompose f g ~* g ∘* ppoint (f ∘* g) :=
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begin
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induction f with f f₀, induction g with g hg g₀, induction B with B b₀,
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induction A with A a₀', esimp at *, induction g₀, induction f₀,
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reflexivity,
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induction f with f f₀, induction g with g h₁ h₂ p₁ p₂, induction B with B b₀,
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induction g with g g₀, induction A with A a₀', esimp at *, induction g₀, induction f₀,
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reflexivity
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end
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definition pfiber_pequiv_of_square_ppoint {A B C D : Type*} {f : A →* B} {g : C →* D}
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@ -116,18 +116,18 @@ namespace pointed
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/- categorical properties of pointed maps -/
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definition pmap_of_map [constructor] {A B : Type} (f : A → B) (a : A) :
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pointed.MK A a →* pointed.MK B (f a) :=
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pmap.mk f idp
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definition pid [constructor] [refl] (A : Type*) : A →* A :=
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pmap.mk id idp
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definition pcompose [constructor] [trans] (g : B →* C) (f : A →* B) : A →* C :=
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definition pcompose [constructor] [trans] {A B C : Type*} (g : B →* C) (f : A →* B) : A →* C :=
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pmap.mk (λa, g (f a)) (ap g (respect_pt f) ⬝ respect_pt g)
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infixr ` ∘* `:60 := pcompose
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definition pmap_of_map [constructor] {A B : Type} (f : A → B) (a : A) :
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pointed.MK A a →* pointed.MK B (f a) :=
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pmap.mk f idp
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definition respect_pt_pcompose {A B C : Type*} (g : B →* C) (f : A →* B)
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: respect_pt (g ∘* f) = ap g (respect_pt f) ⬝ respect_pt g :=
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idp
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@ -373,6 +373,19 @@ namespace pointed
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p a = q a :=
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ap010 to_homotopy r a
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definition pwhisker_left [constructor] (h : B →* C) (p : f ~* g) : h ∘* f ~* h ∘* g :=
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phomotopy.mk (λa, ap h (p a))
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abstract !con.assoc⁻¹ ⬝ whisker_right _ (!ap_con⁻¹ ⬝ ap02 _ (to_homotopy_pt p)) end
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definition pwhisker_right [constructor] (h : C →* A) (p : f ~* g) : f ∘* h ~* g ∘* h :=
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phomotopy.mk (λa, p (h a))
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abstract !con.assoc⁻¹ ⬝ whisker_right _ (!ap_con_eq_con_ap)⁻¹ ⬝ !con.assoc ⬝
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whisker_left _ (to_homotopy_pt p) end
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definition pconcat2 [constructor] {A B C : Type*} {h i : B →* C} {f g : A →* B}
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(q : h ~* i) (p : f ~* g) : h ∘* f ~* i ∘* g :=
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pwhisker_left _ p ⬝* pwhisker_right _ q
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definition pmap_eq_equiv_internal {A B : Type*} (f g : A →* B) : (f = g) ≃ (f ~* g) :=
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calc (f = g) ≃ pmap.sigma_char f = pmap.sigma_char g
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: eq_equiv_fn_eq pmap.sigma_char f g
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@ -676,9 +689,65 @@ namespace pointed
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/- pointed equivalences -/
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/- constructors / projections + variants -/
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structure pequiv (A B : Type*) :=
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mk' :: (to_pmap : A →* B)
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(to_pinv1 : B →* A)
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(to_pinv2 : B →* A)
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(pright_inv : to_pmap ∘* to_pinv1 ~* pid B)
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(pleft_inv : to_pinv2 ∘* to_pmap ~* pid A)
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attribute pequiv.to_pmap [coercion]
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infix ` ≃* `:25 := pequiv
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definition to_pinv [unfold 3] (f : A ≃* B) : B →* A :=
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pequiv.to_pinv1 f
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definition pleft_inv' (f : A ≃* B) : to_pinv f ∘* f ~* pid A :=
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let g := to_pinv f in
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let h := pequiv.to_pinv2 f in
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calc g ∘* f ~* pid A ∘* (g ∘* f) : by exact !pid_pcompose⁻¹*
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... ~* (h ∘* f) ∘* (g ∘* f) : by exact pwhisker_right _ (pequiv.pleft_inv f)⁻¹*
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... ~* h ∘* (f ∘* g) ∘* f : by exact !passoc ⬝* pwhisker_left _ !passoc⁻¹*
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... ~* h ∘* pid B ∘* f : by exact !pwhisker_left (!pwhisker_right !pequiv.pright_inv)
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... ~* h ∘* f : by exact pwhisker_left _ !pid_pcompose
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... ~* pid A : by exact pequiv.pleft_inv f
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definition equiv_of_pequiv [coercion] [constructor] (f : A ≃* B) : A ≃ B :=
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have is_equiv f, from adjointify f (to_pinv f) (pequiv.pright_inv f) (pleft_inv' f),
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equiv.mk f _
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attribute pointed._trans_of_equiv_of_pequiv pequiv._trans_of_to_pmap [unfold 3]
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definition pequiv.to_is_equiv [instance] [constructor] (f : A ≃* B) :
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is_equiv (pointed._trans_of_equiv_of_pequiv f) :=
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adjointify f (to_pinv f) (pequiv.pright_inv f) (pleft_inv' f)
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definition pequiv.to_is_equiv' [instance] [constructor] (f : A ≃* B) :
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is_equiv (pequiv._trans_of_to_pmap f) :=
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pequiv.to_is_equiv f
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protected definition pequiv.MK2 [constructor] (f : A →* B) (g : B →* A)
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(gf : g ∘* f ~* !pid) (fg : f ∘* g ~* !pid) : A ≃* B :=
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pequiv.mk' f g g fg gf
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definition pinv [constructor] (f : A →* B) (H : is_equiv f) : B →* A :=
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pmap.mk f⁻¹ᶠ (ap f⁻¹ᶠ (respect_pt f)⁻¹ ⬝ (left_inv f pt))
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definition pequiv_of_pmap [constructor] (f : A →* B) (H : is_equiv f) : A ≃* B :=
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pequiv.mk f _ (respect_pt f)
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pequiv.mk' f (pinv f H) (pinv f H)
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abstract begin
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fapply phomotopy.mk, exact right_inv f,
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induction f with f f₀, induction B with B b₀, esimp at *, induction f₀, esimp,
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exact adj f pt ⬝ ap02 f !idp_con⁻¹
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end end
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abstract begin
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fapply phomotopy.mk, exact left_inv f,
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induction f with f f₀, induction B with B b₀, esimp at *, induction f₀, esimp,
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exact !idp_con⁻¹ ⬝ !idp_con⁻¹
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end end
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definition pequiv.mk [constructor] (f : A → B) (H : is_equiv f) (p : f pt = pt) : A ≃* B :=
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pequiv_of_pmap (pmap.mk f p) H
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definition pequiv_of_equiv [constructor] (f : A ≃ B) (H : f pt = pt) : A ≃* B :=
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pequiv.mk f _ H
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@ -687,15 +756,30 @@ namespace pointed
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(gf : Πa, g (f a) = a) (fg : Πb, f (g b) = b) : A ≃* B :=
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pequiv.mk f (adjointify f g fg gf) (respect_pt f)
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definition equiv_of_pequiv [constructor] (f : A ≃* B) : A ≃ B :=
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equiv.mk f _
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/- categorical properties of pointed equivalences -/
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definition to_pinv [constructor] (f : A ≃* B) : B →* A :=
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pmap.mk f⁻¹ ((ap f⁻¹ (respect_pt f))⁻¹ ⬝ left_inv f pt)
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protected definition pequiv.refl [refl] [constructor] (A : Type*) : A ≃* A :=
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pequiv.mk' (pid A) (pid A) (pid A) !pid_pcompose !pcompose_pid
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protected definition pequiv.rfl [constructor] : A ≃* A :=
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pequiv.refl A
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protected definition pequiv.symm [symm] [constructor] (f : A ≃* B) : B ≃* A :=
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pequiv.mk' (pequiv.to_pinv1 f) f f (pleft_inv' f) (pequiv.pright_inv f)
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protected definition pequiv.trans [trans] [constructor] (f : A ≃* B) (g : B ≃* C) : A ≃* C :=
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pequiv_of_pmap (g ∘* f) !is_equiv_compose
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definition pequiv_compose {A B C : Type*} (g : B ≃* C) (f : A ≃* B) : A ≃* C :=
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pequiv_of_pmap (g ∘* f) (is_equiv_compose g f)
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postfix `⁻¹ᵉ*`:(max + 1) := pequiv.symm
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infix ` ⬝e* `:75 := pequiv.trans
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infixr ` ∘*ᵉ `:60 := pequiv_compose
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definition to_pmap_pequiv_of_pmap {A B : Type*} (f : A →* B) (H : is_equiv f)
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: pequiv.to_pmap (pequiv_of_pmap f H) = f :=
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by cases f; reflexivity
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by reflexivity
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/-
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A version of pequiv.MK with stronger conditions.
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@ -704,47 +788,14 @@ namespace pointed
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This is not the case when using `pequiv.MK` (if g is a pointed map),
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that will only give an ordinary homotopy.
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-/
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protected definition pequiv.MK2 [constructor] (f : A →* B) (g : B →* A)
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(gf : g ∘* f ~* !pid) (fg : f ∘* g ~* !pid) : A ≃* B :=
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pequiv.MK f g gf fg
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definition to_pmap_pequiv_MK2 [constructor] (f : A →* B) (g : B →* A)
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(gf : g ∘* f ~* !pid) (fg : f ∘* g ~* !pid) : pequiv.MK2 f g gf fg ~* f :=
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phomotopy.mk (λb, idp) !idp_con
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by reflexivity
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definition to_pinv_pequiv_MK2 [constructor] (f : A →* B) (g : B →* A)
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(gf : g ∘* f ~* !pid) (fg : f ∘* g ~* !pid) : to_pinv (pequiv.MK2 f g gf fg) ~* g :=
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phomotopy.mk (λb, idp)
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abstract [irreducible] begin
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esimp,
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note H := to_homotopy_pt gf, note H2 := to_homotopy_pt fg,
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note H3 := eq_top_of_square (natural_square (to_homotopy fg) (respect_pt f)),
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rewrite [▸* at *, H, H3, H2, ap_id, - +con.assoc, ap_compose' f g, con_inv,
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- ap_inv, - +ap_con g],
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apply whisker_right, apply ap02 g,
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rewrite [ap_con, - + con.assoc, +ap_inv, +inv_con_cancel_right, con.left_inv],
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end end
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/- categorical properties of pointed equivalences -/
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protected definition pequiv.refl [refl] [constructor] (A : Type*) : A ≃* A :=
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pequiv_of_pmap !pid !is_equiv_id
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protected definition pequiv.rfl [constructor] : A ≃* A :=
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pequiv.refl A
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protected definition pequiv.symm [symm] [constructor] (f : A ≃* B) : B ≃* A :=
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pequiv_of_pmap (to_pinv f) !is_equiv_inv
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protected definition pequiv.trans [trans] [constructor] (f : A ≃* B) (g : B ≃* C) : A ≃* C :=
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pequiv_of_pmap (g ∘* f) !is_equiv_compose
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definition pequiv_compose {A B C : Type*} (g : B ≃* C) (f : A ≃* B) : A ≃* C :=
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pequiv_of_pmap (g ∘* f) (is_equiv_compose g f)
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infixr ` ∘*ᵉ `:60 := pequiv_compose
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postfix `⁻¹ᵉ*`:(max + 1) := pequiv.symm
|
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infix ` ⬝e* `:75 := pequiv.trans
|
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by reflexivity
|
||||
|
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/- more on pointed equivalences -/
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||||
|
|
@ -754,7 +805,7 @@ namespace pointed
|
|||
|
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definition to_pmap_pequiv_trans {A B C : Type*} (f : A ≃* B) (g : B ≃* C)
|
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: pequiv.to_pmap (f ⬝e* g) = g ∘* f :=
|
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!to_pmap_pequiv_of_pmap
|
||||
by reflexivity
|
||||
|
||||
definition to_fun_pequiv_trans {X Y Z : Type*} (f : X ≃* Y) (g :Y ≃* Z) : f ⬝e* g ~ g ∘ f :=
|
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λx, idp
|
||||
|
|
@ -796,8 +847,8 @@ namespace pointed
|
|||
{a₁ a₂ : A} (p : a₁ = a₂) : pequiv_of_eq (ap C p) ∘* f a₁ ~* f a₂ ∘* pequiv_of_eq (ap B p) :=
|
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pcast_commute f p
|
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|
||||
definition pequiv.eta_expand [constructor] {A B : Type*} (f : A ≃* B) : A ≃* B :=
|
||||
pequiv.mk f _ (pequiv.resp_pt f)
|
||||
-- definition pequiv.eta_expand [constructor] {A B : Type*} (f : A ≃* B) : A ≃* B :=
|
||||
-- pequiv.mk' f (to_pinv f) (pequiv.to_pinv2 f) (pright_inv f) _
|
||||
|
||||
/-
|
||||
the theorem pequiv_eq, which gives a condition for two pointed equivalences are equal
|
||||
|
|
@ -805,35 +856,11 @@ namespace pointed
|
|||
-/
|
||||
|
||||
/- computation rules of pointed homotopies, possibly combined with pointed equivalences -/
|
||||
definition pwhisker_left [constructor] (h : B →* C) (p : f ~* g) : h ∘* f ~* h ∘* g :=
|
||||
phomotopy.mk (λa, ap h (p a))
|
||||
abstract !con.assoc⁻¹ ⬝ whisker_right _ (!ap_con⁻¹ ⬝ ap02 _ (to_homotopy_pt p)) end
|
||||
|
||||
definition pwhisker_right [constructor] (h : C →* A) (p : f ~* g) : f ∘* h ~* g ∘* h :=
|
||||
phomotopy.mk (λa, p (h a))
|
||||
abstract !con.assoc⁻¹ ⬝ whisker_right _ (!ap_con_eq_con_ap)⁻¹ ⬝ !con.assoc ⬝
|
||||
whisker_left _ (to_homotopy_pt p) end
|
||||
|
||||
definition pconcat2 [constructor] {A B C : Type*} {h i : B →* C} {f g : A →* B}
|
||||
(q : h ~* i) (p : f ~* g) : h ∘* f ~* i ∘* g :=
|
||||
pwhisker_left _ p ⬝* pwhisker_right _ q
|
||||
|
||||
definition pleft_inv (f : A ≃* B) : f⁻¹ᵉ* ∘* f ~* pid A :=
|
||||
phomotopy.mk (left_inv f)
|
||||
abstract begin
|
||||
esimp, symmetry, apply con_inv_cancel_left
|
||||
end end
|
||||
pleft_inv' f
|
||||
|
||||
definition pright_inv (f : A ≃* B) : f ∘* f⁻¹ᵉ* ~* pid B :=
|
||||
phomotopy.mk (right_inv f)
|
||||
abstract begin
|
||||
induction f with f H p, esimp,
|
||||
rewrite [ap_con, +ap_inv, -adj f, -ap_compose],
|
||||
note q := natural_square (right_inv f) p,
|
||||
rewrite [ap_id at q],
|
||||
apply eq_bot_of_square,
|
||||
exact q
|
||||
end end
|
||||
pequiv.pright_inv f
|
||||
|
||||
definition pcancel_left (f : B ≃* C) {g h : A →* B} (p : f ∘* g ~* f ∘* h) : g ~* h :=
|
||||
begin
|
||||
|
|
@ -1003,11 +1030,11 @@ namespace pointed
|
|||
|
||||
definition to_pmap_loopn_pequiv_loopn [constructor] (n : ℕ) (f : A ≃* B)
|
||||
: loopn_pequiv_loopn n f ~* apn n f :=
|
||||
!to_pmap_pequiv_MK2
|
||||
by reflexivity
|
||||
|
||||
definition to_pinv_loopn_pequiv_loopn [constructor] (n : ℕ) (f : A ≃* B)
|
||||
: (loopn_pequiv_loopn n f)⁻¹ᵉ* ~* apn n f⁻¹ᵉ* :=
|
||||
!to_pinv_pequiv_MK2
|
||||
by reflexivity
|
||||
|
||||
definition loopn_pequiv_loopn_con (n : ℕ) (f : A ≃* B) (p q : Ω[n+1] A)
|
||||
: loopn_pequiv_loopn (n+1) f (p ⬝ q) =
|
||||
|
|
@ -1030,9 +1057,7 @@ namespace pointed
|
|||
|
||||
definition apn_pinv (n : ℕ) {A B : Type*} (f : A ≃* B) :
|
||||
Ω→[n] f⁻¹ᵉ* ~* (loopn_pequiv_loopn n f)⁻¹ᵉ* :=
|
||||
begin
|
||||
refine !to_pinv_pequiv_MK2⁻¹*
|
||||
end
|
||||
by reflexivity
|
||||
|
||||
definition pmap_functor [constructor] {A A' B B' : Type*} (f : A' →* A) (g : B →* B') :
|
||||
ppmap A B →* ppmap A' B' :=
|
||||
|
|
@ -1061,8 +1086,7 @@ namespace pointed
|
|||
begin
|
||||
fapply phomotopy.mk,
|
||||
{ reflexivity},
|
||||
{ esimp [pequiv.trans, pequiv.symm],
|
||||
exact !con.right_inv⁻¹ ⬝ ((!idp_con⁻¹ ⬝ !ap_id⁻¹) ◾ (!ap_id⁻¹⁻² ⬝ !idp_con⁻¹)), }
|
||||
{ symmetry, exact (!ap_id ⬝ !idp_con) ◾ (!idp_con ⬝ !ap_id) ⬝ !con.right_inv }
|
||||
end
|
||||
|
||||
/- properties of iterated loop space -/
|
||||
|
|
|
|||
|
|
@ -502,6 +502,10 @@ namespace sigma
|
|||
(u.1 = v.1) ≃ (u = v) :=
|
||||
equiv.mk !subtype_eq _
|
||||
|
||||
definition subtype_eq_equiv [constructor] [H : Πa, is_prop (B a)] (u v : {a | B a}) :
|
||||
(u = v) ≃ (u.1 = v.1) :=
|
||||
(equiv_subtype u v)⁻¹ᵉ
|
||||
|
||||
definition subtype_eq_inv {A : Type} {B : A → Type} [H : Πa, is_prop (B a)] (u v : Σa, B a)
|
||||
: u = v → u.1 = v.1 :=
|
||||
subtype_eq⁻¹ᶠ
|
||||
|
|
|
|||
|
|
@ -567,6 +567,9 @@ namespace trunc
|
|||
: (tr a = tr a' :> trunc n.+1 A) ≃ trunc n (a = a') :=
|
||||
!trunc_eq_equiv
|
||||
|
||||
definition trunc_eq {n : ℕ₋₂} {a a' : A} (p : trunc n (a = a')) :tr a = tr a' :> trunc n.+1 A :=
|
||||
!tr_eq_tr_equiv⁻¹ᵉ p
|
||||
|
||||
definition code_mul {n : ℕ₋₂} {aa₁ aa₂ aa₃ : trunc n.+1 A}
|
||||
(g : trunc.code n aa₁ aa₂) (h : trunc.code n aa₂ aa₃) : trunc.code n aa₁ aa₃ :=
|
||||
begin
|
||||
|
|
|
|||
Loading…
Reference in a new issue