fix(init.path): rename transport_compose to tr_compose
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5 changed files with 8 additions and 9 deletions
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@ -154,7 +154,7 @@ namespace functor
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by (cases F; apply functor_mk_eq'_idp)
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by (cases F; apply functor_mk_eq'_idp)
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definition functor_eq_eta' {F₁ F₂ : C ⇒ D} (p : F₁ = F₂)
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definition functor_eq_eta' {F₁ F₂ : C ⇒ D} (p : F₁ = F₂)
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: functor_eq' (ap to_fun_ob p) (!transport_compose⁻¹ ⬝ apd to_fun_hom p) = p :=
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: functor_eq' (ap to_fun_ob p) (!tr_compose⁻¹ ⬝ apd to_fun_hom p) = p :=
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begin
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begin
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cases p, cases F₁,
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cases p, cases F₁,
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apply concat, rotate_left 1, apply functor_eq'_idp,
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apply concat, rotate_left 1, apply functor_eq'_idp,
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@ -177,7 +177,7 @@ namespace is_equiv
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is_equiv_rect f P df (f x)
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is_equiv_rect f P df (f x)
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= right_inv f (f x) ▸ df (f⁻¹ (f x)) : by esimp
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= right_inv f (f x) ▸ df (f⁻¹ (f x)) : by esimp
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... = ap f (left_inv f x) ▸ df (f⁻¹ (f x)) : by rewrite -adj
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... = ap f (left_inv f x) ▸ df (f⁻¹ (f x)) : by rewrite -adj
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... = left_inv f x ▸ df (f⁻¹ (f x)) : by rewrite -transport_compose
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... = left_inv f x ▸ df (f⁻¹ (f x)) : by rewrite -tr_compose
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... = df x : by rewrite (apd df (left_inv f x))
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... = df x : by rewrite (apd df (left_inv f x))
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end
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end
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@ -295,7 +295,7 @@ namespace equiv
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equiv_rect f P df (f x)
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equiv_rect f P df (f x)
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= right_inv f (f x) ▸ df (f⁻¹ (f x)) : by esimp
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= right_inv f (f x) ▸ df (f⁻¹ (f x)) : by esimp
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... = ap f (left_inv f x) ▸ df (f⁻¹ (f x)) : by rewrite -adj
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... = ap f (left_inv f x) ▸ df (f⁻¹ (f x)) : by rewrite -adj
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... = left_inv f x ▸ df (f⁻¹ (f x)) : by rewrite -transport_compose
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... = left_inv f x ▸ df (f⁻¹ (f x)) : by rewrite -tr_compose
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... = df x : by rewrite (apd df (left_inv f x))
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... = df x : by rewrite (apd df (left_inv f x))
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@ -509,8 +509,7 @@ namespace eq
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eq.rec_on r !idp_con⁻¹
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eq.rec_on r !idp_con⁻¹
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-- Transporting in a pulled back fibration.
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-- Transporting in a pulled back fibration.
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-- rename: tr_compose
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definition tr_compose (P : B → Type) (f : A → B) (p : x = y) (z : P (f x)) :
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definition transport_compose (P : B → Type) (f : A → B) (p : x = y) (z : P (f x)) :
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transport (P ∘ f) p z = transport P (ap f p) z :=
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transport (P ∘ f) p z = transport P (ap f p) z :=
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eq.rec_on p idp
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eq.rec_on p idp
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@ -526,7 +525,7 @@ namespace eq
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apd10 (ap (λh : A → B, f ∘ h) p) a = ap f (apd10 p a) :=
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apd10 (ap (λh : A → B, f ∘ h) p) a = ap f (apd10 p a) :=
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eq.rec_on p idp
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eq.rec_on p idp
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-- A special case of [transport_compose] which seems to come up a lot.
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-- A special case of [tr_compose] which seems to come up a lot.
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definition tr_eq_cast_ap {P : A → Type} {x y} (p : x = y) (u : P x) : p ▸ u = cast (ap P p) u :=
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definition tr_eq_cast_ap {P : A → Type} {x y} (p : x = y) (u : P x) : p ▸ u = cast (ap P p) u :=
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eq.rec_on p idp
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eq.rec_on p idp
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@ -175,7 +175,7 @@ namespace pi
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(λ(a : A) (b' : B' (f0⁻¹ a)), transport B (right_inv f0 a) ((f1 (f0⁻¹ a))⁻¹ b')))),
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(λ(a : A) (b' : B' (f0⁻¹ a)), transport B (right_inv f0 a) ((f1 (f0⁻¹ a))⁻¹ b')))),
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begin
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begin
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intro h, apply eq_of_homotopy, intro a', esimp,
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intro h, apply eq_of_homotopy, intro a', esimp,
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rewrite [adj f0 a',-transport_compose,fn_tr_eq_tr_fn _ f1,right_inv (f1 _)],
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rewrite [adj f0 a',-tr_compose,fn_tr_eq_tr_fn _ f1,right_inv (f1 _)],
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apply apd
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apply apd
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end,
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end,
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begin
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begin
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@ -216,7 +216,7 @@ namespace sigma
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apply (sigma_eq (left_inv f a)),
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apply (sigma_eq (left_inv f a)),
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apply pathover_of_tr_eq,
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apply pathover_of_tr_eq,
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rewrite [▸*,adj f,-(fn_tr_eq_tr_fn (left_inv f a) (λ a, (g a)⁻¹)),
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rewrite [▸*,adj f,-(fn_tr_eq_tr_fn (left_inv f a) (λ a, (g a)⁻¹)),
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▸*,transport_compose B' f,tr_inv_tr,left_inv]
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▸*,tr_compose B' f,tr_inv_tr,left_inv]
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end
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end
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definition sigma_equiv_sigma_of_is_equiv [H1 : is_equiv f] [H2 : Π a, is_equiv (g a)]
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definition sigma_equiv_sigma_of_is_equiv [H1 : is_equiv f] [H2 : Π a, is_equiv (g a)]
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@ -237,7 +237,7 @@ namespace sigma
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-- definition ap_sigma_functor_eq (p : u.1 = v.1) (q : u.2 =[p] v.2)
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-- definition ap_sigma_functor_eq (p : u.1 = v.1) (q : u.2 =[p] v.2)
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-- : ap (sigma_functor f g) (sigma_eq p q) =
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-- : ap (sigma_functor f g) (sigma_eq p q) =
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-- sigma_eq (ap f p)
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-- sigma_eq (ap f p)
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-- ((transport_compose B' f p (g u.1 u.2))⁻¹ ⬝ (fn_tr_eq_tr_fn p g u.2)⁻¹ ⬝ ap (g v.1) q) :=
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-- ((tr_compose B' f p (g u.1 u.2))⁻¹ ⬝ (fn_tr_eq_tr_fn p g u.2)⁻¹ ⬝ ap (g v.1) q) :=
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-- by induction u; induction v; apply ap_sigma_functor_eq_dpair
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-- by induction u; induction v; apply ap_sigma_functor_eq_dpair
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/- definition 3.11.9(i): Summing up a contractible family of types does nothing. -/
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/- definition 3.11.9(i): Summing up a contractible family of types does nothing. -/
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