From e769fdd9dc0d1fe3d90fc28fd18e8311845886cf Mon Sep 17 00:00:00 2001 From: Floris van Doorn Date: Fri, 24 Apr 2015 17:00:32 -0400 Subject: [PATCH] feat(hott): make some arguments in init.path implicit and rename apD to apd --- hott/algebra/precategory/constructions.hlean | 4 +- hott/algebra/precategory/functor.hlean | 12 +- hott/algebra/precategory/iso.hlean | 10 +- hott/algebra/precategory/nat_trans.hlean | 2 +- hott/arity.hlean | 56 +++--- hott/hit/circle.hlean | 4 +- hott/hit/coeq.hlean | 2 +- hott/hit/colimit.hlean | 4 +- hott/hit/cylinder.hlean | 2 +- hott/hit/pushout.hlean | 2 +- hott/hit/quotient.hlean | 2 +- hott/hit/suspension.hlean | 2 +- hott/init/axioms/funext_of_ua.hlean | 12 +- hott/init/axioms/funext_varieties.hlean | 16 +- hott/init/equiv.hlean | 10 +- hott/init/hit.hlean | 2 +- hott/init/path.hlean | 199 +++++++++---------- hott/types/eq.hlean | 14 +- hott/types/equiv.hlean | 2 +- hott/types/pi.hlean | 22 +- hott/types/trunc.hlean | 4 +- 21 files changed, 191 insertions(+), 192 deletions(-) diff --git a/hott/algebra/precategory/constructions.hlean b/hott/algebra/precategory/constructions.hlean index 67f76c39a..e0ffa12db 100644 --- a/hott/algebra/precategory/constructions.hlean +++ b/hott/algebra/precategory/constructions.hlean @@ -189,7 +189,7 @@ namespace category end) definition nat_trans_left_inverse : nat_trans_inverse η ∘n η = nat_trans.id := begin - fapply (apD011 nat_trans.mk), + fapply (apd011 nat_trans.mk), apply eq_of_homotopy, intro c, apply left_inverse, apply eq_of_homotopy, intros, apply eq_of_homotopy, intros, apply eq_of_homotopy, intros, apply is_hset.elim @@ -197,7 +197,7 @@ namespace category definition nat_trans_right_inverse : η ∘n nat_trans_inverse η = nat_trans.id := begin - fapply (apD011 nat_trans.mk), + fapply (apd011 nat_trans.mk), apply eq_of_homotopy, intro c, apply right_inverse, apply eq_of_homotopy, intros, apply eq_of_homotopy, intros, apply eq_of_homotopy, intros, apply is_hset.elim diff --git a/hott/algebra/precategory/functor.hlean b/hott/algebra/precategory/functor.hlean index 7413458bf..3b8c06335 100644 --- a/hott/algebra/precategory/functor.hlean +++ b/hott/algebra/precategory/functor.hlean @@ -50,7 +50,7 @@ namespace functor {H₂ : Π(a b : C), hom a b → hom (F₂ a) (F₂ b)} (id₁ id₂ comp₁ comp₂) (pF : F₁ = F₂) (pH : pF ▹ H₁ = H₂) : functor.mk F₁ H₁ id₁ comp₁ = functor.mk F₂ H₂ id₂ comp₂ := - apD01111 functor.mk pF pH !is_hprop.elim !is_hprop.elim + apd01111 functor.mk pF pH !is_hprop.elim !is_hprop.elim definition functor_eq' {F₁ F₂ : C ⇒ D} : Π(p : to_fun_ob F₁ = to_fun_ob F₂), @@ -68,10 +68,10 @@ namespace functor apply concat, rotate_left 1, apply transport_hom, apply concat, rotate_left 1, exact (pi_transport_constant (eq_of_homotopy pF) (H₁ c c') f), - apply (apD10' f), + apply (apd10' f), apply concat, rotate_left 1, esimp, exact (pi_transport_constant (eq_of_homotopy pF) (H₁ c) c'), - apply (apD10' c'), + apply (apd10' c'), apply concat, rotate_left 1, esimp, exact (pi_transport_constant (eq_of_homotopy pF) H₁ c), apply idp @@ -169,7 +169,7 @@ namespace functor definition functor_mk_eq'_idp (F : C → D) (H : Π(a b : C), hom a b → hom (F a) (F b)) (id comp) : functor_mk_eq' id id comp comp (idpath F) (idpath H) = idp := begin - fapply (apD011 (apD01111 functor.mk idp idp)), + fapply (apd011 (apd01111 functor.mk idp idp)), apply is_hset.elim, apply is_hset.elim end @@ -178,7 +178,7 @@ namespace functor by (cases F; apply functor_mk_eq'_idp) definition functor_eq_eta' {F₁ F₂ : C ⇒ D} (p : F₁ = F₂) - : functor_eq' (ap to_fun_ob p) (!transport_compose⁻¹ ⬝ apD to_fun_hom p) = p := + : functor_eq' (ap to_fun_ob p) (!transport_compose⁻¹ ⬝ apd to_fun_hom p) = p := begin cases p, cases F₁, apply concat, rotate_left 1, apply functor_eq'_idp, @@ -203,7 +203,7 @@ namespace functor -- definition ap010_functor_eq_mk' {F₁ F₂ : C ⇒ D} (p : to_fun_ob F₁ = to_fun_ob F₂) -- (q : p ▹ F₁ = F₂) (c : C) : - -- ap to_fun_ob (functor_eq_mk (apD10 p) (λa b f, _)) = p := sorry + -- ap to_fun_ob (functor_eq_mk (apd10 p) (λa b f, _)) = p := sorry -- begin -- cases F₂, revert q, apply (homotopy.rec_on p), clear p, esimp, intros (p, q), -- cases p, clears (e_1, e_2), diff --git a/hott/algebra/precategory/iso.hlean b/hott/algebra/precategory/iso.hlean index 23866a458..c24bb0056 100644 --- a/hott/algebra/precategory/iso.hlean +++ b/hott/algebra/precategory/iso.hlean @@ -131,7 +131,7 @@ namespace iso begin apply is_hprop.mk, intros [H, H'], cases H with [g, li, ri], cases H' with [g', li', ri'], - fapply (apD0111 (@is_iso.mk ob C a b f)), + fapply (apd0111 (@is_iso.mk ob C a b f)), apply left_inverse_eq_right_inverse, apply li, apply ri', @@ -167,7 +167,7 @@ namespace iso protected definition eq_mk' {f f' : a ⟶ b} [H : is_iso f] [H' : is_iso f'] (p : f = f') : iso.mk f = iso.mk f' := - apD011 iso.mk p !is_hprop.elim + apd011 iso.mk p !is_hprop.elim protected definition eq_mk {f f' : a ≅ b} (p : to_hom f = to_hom f') : f = f' := by (cases f; cases f'; apply (iso.eq_mk' p)) @@ -212,16 +212,16 @@ namespace iso open funext variables {X : Type} {x y : X} {F G : X → ob} definition transport_hom_of_eq (p : F = G) (f : hom (F x) (F y)) - : p ▹ f = hom_of_eq (apD10 p y) ∘ f ∘ inv_of_eq (apD10 p x) := + : p ▹ f = hom_of_eq (apd10 p y) ∘ f ∘ inv_of_eq (apd10 p x) := eq.rec_on p !id_leftright⁻¹ definition transport_hom (p : F ∼ G) (f : hom (F x) (F y)) : eq_of_homotopy p ▹ f = hom_of_eq (p y) ∘ f ∘ inv_of_eq (p x) := calc eq_of_homotopy p ▹ f = - hom_of_eq (apD10 (eq_of_homotopy p) y) ∘ f ∘ inv_of_eq (apD10 (eq_of_homotopy p) x) + hom_of_eq (apd10 (eq_of_homotopy p) y) ∘ f ∘ inv_of_eq (apd10 (eq_of_homotopy p) x) : transport_hom_of_eq - ... = hom_of_eq (p y) ∘ f ∘ inv_of_eq (p x) : {retr apD10 p} + ... = hom_of_eq (p y) ∘ f ∘ inv_of_eq (p x) : {retr apd10 p} end structure mono [class] (f : a ⟶ b) := diff --git a/hott/algebra/precategory/nat_trans.hlean b/hott/algebra/precategory/nat_trans.hlean index c1310ccde..11320a338 100644 --- a/hott/algebra/precategory/nat_trans.hlean +++ b/hott/algebra/precategory/nat_trans.hlean @@ -43,7 +43,7 @@ namespace nat_trans (nat₂ : Π (a b : C) (f : hom a b), G f ∘ η₂ a = η₂ b ∘ F f) (p : η₁ ∼ η₂) : nat_trans.mk η₁ nat₁ = nat_trans.mk η₂ nat₂ := - apD011 nat_trans.mk (eq_of_homotopy p) !is_hprop.elim + apd011 nat_trans.mk (eq_of_homotopy p) !is_hprop.elim definition nat_trans_eq_mk {η₁ η₂ : F ⟹ G} : natural_map η₁ ∼ natural_map η₂ → η₁ = η₂ := nat_trans.rec_on η₁ (λf₁ nat₁, nat_trans.rec_on η₂ (λf₂ nat₂ p, !nat_trans_eq_mk' p)) diff --git a/hott/arity.hlean b/hott/arity.hlean index 954e0f6ee..72f1709f5 100644 --- a/hott/arity.hlean +++ b/hott/arity.hlean @@ -29,10 +29,10 @@ namespace eq transports in the theorem statement). For the fully-dependent versions (except that the conclusion doesn't contain a transport) we write - apDi₀i₁...iₙ. + apdi₀i₁...iₙ. For versions where only some arguments depend on some other arguments, - or for versions with transport in the conclusion (like apD), we don't have a + or for versions with transport in the conclusion (like apd), we don't have a consistent naming scheme (yet). We don't prove each theorem systematically, but prove only the ones which we actually need. @@ -68,25 +68,25 @@ namespace eq definition ap01000 (f : X → Πa b c, D a b c) (Hx : x = x') : f x ∼3 f x' := λa b c, eq.rec_on Hx idp - definition apD011 (f : Πa, B a → Z) (Ha : a = a') (Hb : (Ha ▹ b) = b') + definition apd011 (f : Πa, B a → Z) (Ha : a = a') (Hb : (Ha ▹ b) = b') : f a b = f a' b' := eq.rec_on Hb (eq.rec_on Ha idp) - definition apD0111 (f : Πa b, C a b → Z) (Ha : a = a') (Hb : (Ha ▹ b) = b') - (Hc : apD011 C Ha Hb ▹ c = c') + definition apd0111 (f : Πa b, C a b → Z) (Ha : a = a') (Hb : (Ha ▹ b) = b') + (Hc : apd011 C Ha Hb ▹ c = c') : f a b c = f a' b' c' := eq.rec_on Hc (eq.rec_on Hb (eq.rec_on Ha idp)) - definition apD01111 (f : Πa b c, D a b c → Z) (Ha : a = a') (Hb : (Ha ▹ b) = b') - (Hc : apD011 C Ha Hb ▹ c = c') (Hd : apD0111 D Ha Hb Hc ▹ d = d') + definition apd01111 (f : Πa b c, D a b c → Z) (Ha : a = a') (Hb : (Ha ▹ b) = b') + (Hc : apd011 C Ha Hb ▹ c = c') (Hd : apd0111 D Ha Hb Hc ▹ d = d') : f a b c d = f a' b' c' d' := eq.rec_on Hd (eq.rec_on Hc (eq.rec_on Hb (eq.rec_on Ha idp))) - definition apD100 {f g : Πa b, C a b} (p : f = g) : f ∼2 g := - λa b, apD10 (apD10 p a) b + definition apd100 {f g : Πa b, C a b} (p : f = g) : f ∼2 g := + λa b, apd10 (apd10 p a) b - definition apD1000 {f g : Πa b c, D a b c} (p : f = g) : f ∼3 g := - λa b c, apD100 (apD10 p a) b c + definition apd1000 {f g : Πa b c, D a b c} (p : f = g) : f ∼3 g := + λa b c, apd100 (apd10 p a) b c /- some properties of these variants of ap -/ @@ -128,30 +128,30 @@ end eq open eq is_equiv namespace funext - definition is_equiv_apD100 [instance] (f g : Πa b, C a b) : is_equiv (@apD100 A B C f g) := + definition is_equiv_apd100 [instance] (f g : Πa b, C a b) : is_equiv (@apd100 A B C f g) := adjointify _ eq_of_homotopy2 begin - intro H, esimp [apD100, eq_of_homotopy2, function.compose], + intro H, esimp [apd100, eq_of_homotopy2, function.compose], apply eq_of_homotopy, intro a, - apply concat, apply (ap (λx, apD10 (x a))), apply (retr apD10), - apply (retr apD10) + apply concat, apply (ap (λx, apd10 (x a))), apply (retr apd10), + apply (retr apd10) end begin intro p, cases p, apply eq_of_homotopy2_id end - definition is_equiv_apD1000 [instance] (f g : Πa b c, D a b c) - : is_equiv (@apD1000 A B C D f g) := + definition is_equiv_apd1000 [instance] (f g : Πa b c, D a b c) + : is_equiv (@apd1000 A B C D f g) := adjointify _ eq_of_homotopy3 begin intro H, apply eq_of_homotopy, intro a, apply concat, - {apply (ap (λx, @apD100 _ _ (λ(b : B a)(c : C a b), _) _ _ (x a))), - apply (retr apD10)}, + {apply (ap (λx, @apd100 _ _ (λ(b : B a)(c : C a b), _) _ _ (x a))), + apply (retr apd10)}, --TODO: remove implicit argument after #469 is closed - apply (@retr _ _ apD100 !is_equiv_apD100) --is explicit argument needed here? + apply (@retr _ _ apd100 !is_equiv_apd100) --is explicit argument needed here? end begin intro p, cases p, apply eq_of_homotopy3_id @@ -160,22 +160,22 @@ end funext namespace eq open funext - local attribute funext.is_equiv_apD100 [instance] + local attribute funext.is_equiv_apd100 [instance] protected definition homotopy2.rec_on {f g : Πa b, C a b} {P : (f ∼2 g) → Type} - (p : f ∼2 g) (H : Π(q : f = g), P (apD100 q)) : P p := - retr apD100 p ▹ H (eq_of_homotopy2 p) + (p : f ∼2 g) (H : Π(q : f = g), P (apd100 q)) : P p := + retr apd100 p ▹ H (eq_of_homotopy2 p) protected definition homotopy3.rec_on {f g : Πa b c, D a b c} {P : (f ∼3 g) → Type} - (p : f ∼3 g) (H : Π(q : f = g), P (apD1000 q)) : P p := - retr apD1000 p ▹ H (eq_of_homotopy3 p) + (p : f ∼3 g) (H : Π(q : f = g), P (apd1000 q)) : P p := + retr apd1000 p ▹ H (eq_of_homotopy3 p) - definition apD10_ap (f : X → Πa, B a) (p : x = x') - : apD10 (ap f p) = ap010 f p := + definition apd10_ap (f : X → Πa, B a) (p : x = x') + : apd10 (ap f p) = ap010 f p := eq.rec_on p idp definition eq_of_homotopy_ap010 (f : X → Πa, B a) (p : x = x') : eq_of_homotopy (ap010 f p) = ap f p := - inv_eq_of_eq !apD10_ap⁻¹ + inv_eq_of_eq !apd10_ap⁻¹ definition ap_eq_ap_of_homotopy {f : X → Πa, B a} {p q : x = x'} (H : ap010 f p ∼ ap010 f q) : ap f p = ap f q := diff --git a/hott/hit/circle.hlean b/hott/hit/circle.hlean index 694aa8208..f50595915 100644 --- a/hott/hit/circle.hlean +++ b/hott/hit/circle.hlean @@ -41,12 +41,12 @@ namespace circle definition rec2_seg1 {P : circle → Type} (Pb1 : P base1) (Pb2 : P base2) (Ps1 : seg1 ▹ Pb1 = Pb2) (Ps2 : seg2 ▹ Pb2 = Pb1) - : apD (rec2 Pb1 Pb2 Ps1 Ps2) seg1 = sorry ⬝ Ps1 ⬝ sorry := + : apd (rec2 Pb1 Pb2 Ps1 Ps2) seg1 = sorry ⬝ Ps1 ⬝ sorry := sorry definition rec2_seg2 {P : circle → Type} (Pb1 : P base1) (Pb2 : P base2) (Ps1 : seg1 ▹ Pb1 = Pb2) (Ps2 : seg2 ▹ Pb2 = Pb1) - : apD (rec2 Pb1 Pb2 Ps1 Ps2) seg2 = sorry ⬝ Ps2 ⬝ sorry := + : apd (rec2 Pb1 Pb2 Ps1 Ps2) seg2 = sorry ⬝ Ps2 ⬝ sorry := sorry definition elim2 {P : Type} (Pb1 Pb2 : P) (Ps1 : Pb1 = Pb2) (Ps2 : Pb2 = Pb1) (x : circle) : P := diff --git a/hott/hit/coeq.hlean b/hott/hit/coeq.hlean index 565fbaf0a..2e7ae7a1a 100644 --- a/hott/hit/coeq.hlean +++ b/hott/hit/coeq.hlean @@ -46,7 +46,7 @@ parameters {A B : Type.{u}} (f g : A → B) definition rec_cp {P : coeq → Type} (P_i : Π(x : B), P (coeq_i x)) (Pcp : Π(x : A), cp x ▹ P_i (f x) = P_i (g x)) - (x : A) : apD (rec P_i Pcp) (cp x) = sorry ⬝ Pcp x ⬝ sorry := + (x : A) : apd (rec P_i Pcp) (cp x) = sorry ⬝ Pcp x ⬝ sorry := sorry protected definition elim {P : Type} (P_i : B → P) diff --git a/hott/hit/colimit.hlean b/hott/hit/colimit.hlean index 3ee947f81..a67ab835d 100644 --- a/hott/hit/colimit.hlean +++ b/hott/hit/colimit.hlean @@ -52,7 +52,7 @@ section definition rec_cglue [reducible] {P : colimit → Type} (Pincl : Π⦃i : I⦄ (x : A i), P (ι x)) (Pglue : Π(j : J) (x : A (dom j)), cglue j x ▹ Pincl (f j x) = Pincl x) - {j : J} (x : A (dom j)) : apD (rec Pincl Pglue) (cglue j x) = Pglue j x := + {j : J} (x : A (dom j)) : apd (rec Pincl Pglue) (cglue j x) = Pglue j x := sorry protected definition elim {P : Type} (Pincl : Π⦃i : I⦄ (x : A i), P) @@ -137,7 +137,7 @@ section definition rec_glue {P : seq_colim → Type} (Pincl : Π⦃n : ℕ⦄ (a : A n), P (sι a)) (Pglue : Π⦃n : ℕ⦄ (a : A n), glue a ▹ Pincl (f a) = Pincl a) {n : ℕ} (a : A n) - : apD (rec Pincl Pglue) (glue a) = sorry ⬝ Pglue a ⬝ sorry := + : apd (rec Pincl Pglue) (glue a) = sorry ⬝ Pglue a ⬝ sorry := sorry definition elim_glue {P : Type} (Pincl : Π⦃n : ℕ⦄ (a : A n), P) diff --git a/hott/hit/cylinder.hlean b/hott/hit/cylinder.hlean index 105732db2..f4f000959 100644 --- a/hott/hit/cylinder.hlean +++ b/hott/hit/cylinder.hlean @@ -54,7 +54,7 @@ parameters {A B : Type.{u}} (f : A → B) definition rec_seg {P : cylinder → Type} (Pbase : Π(b : B), P (base b)) (Ptop : Π(a : A), P (top a)) (Pseg : Π(a : A), seg a ▹ Pbase (f a) = Ptop a) - (a : A) : apD (rec Pbase Ptop Pseg) (seg a) = sorry ⬝ Pseg a ⬝ sorry := + (a : A) : apd (rec Pbase Ptop Pseg) (seg a) = sorry ⬝ Pseg a ⬝ sorry := sorry protected definition elim {P : Type} (Pbase : B → P) (Ptop : A → P) diff --git a/hott/hit/pushout.hlean b/hott/hit/pushout.hlean index 1f0ffeaae..8c586ff05 100644 --- a/hott/hit/pushout.hlean +++ b/hott/hit/pushout.hlean @@ -63,7 +63,7 @@ parameters {TL BL TR : Type} (f : TL → BL) (g : TL → TR) definition rec_glue {P : pushout → Type} (Pinl : Π(x : BL), P (inl x)) (Pinr : Π(x : TR), P (inr x)) (Pglue : Π(x : TL), glue x ▹ Pinl (f x) = Pinr (g x)) - (x : TL) : apD (rec Pinl Pinr Pglue) (glue x) = Pglue x := + (x : TL) : apd (rec Pinl Pinr Pglue) (glue x) = Pglue x := sorry protected definition elim {P : Type} (Pinl : BL → P) (Pinr : TR → P) diff --git a/hott/hit/quotient.hlean b/hott/hit/quotient.hlean index ec071db54..3c1fe1901 100644 --- a/hott/hit/quotient.hlean +++ b/hott/hit/quotient.hlean @@ -46,7 +46,7 @@ parameters {A : Type} (R : A → A → hprop) definition rec_eq_of_rel {P : quotient → Type} [Pt : Πaa, is_hset (P aa)] (Pc : Π(a : A), P (class_of a)) (Pp : Π⦃a a' : A⦄ (H : R a a'), eq_of_rel H ▹ Pc a = Pc a') - {a a' : A} (H : R a a') : apD (rec Pc Pp) (eq_of_rel H) = sorry ⬝ Pp H ⬝ sorry := + {a a' : A} (H : R a a') : apd (rec Pc Pp) (eq_of_rel H) = sorry ⬝ Pp H ⬝ sorry := sorry protected definition elim {P : Type} [Pt : is_hset P] (Pc : A → P) diff --git a/hott/hit/suspension.hlean b/hott/hit/suspension.hlean index 228999e50..40c8c8e11 100644 --- a/hott/hit/suspension.hlean +++ b/hott/hit/suspension.hlean @@ -41,7 +41,7 @@ namespace suspension definition rec_merid {P : suspension A → Type} (PN : P !north) (PS : P !south) (Pm : Π(a : A), merid a ▹ PN = PS) (a : A) - : apD (rec PN PS Pm) (merid a) = sorry ⬝ Pm a ⬝ sorry := + : apd (rec PN PS Pm) (merid a) = sorry ⬝ Pm a ⬝ sorry := sorry protected definition elim {P : Type} (PN : P) (PS : P) (Pm : A → PN = PS) diff --git a/hott/init/axioms/funext_of_ua.hlean b/hott/init/axioms/funext_of_ua.hlean index ab14fca08..0685535f0 100644 --- a/hott/init/axioms/funext_of_ua.hlean +++ b/hott/init/axioms/funext_of_ua.hlean @@ -138,25 +138,25 @@ definition funext_of_ua : funext := variables {A : Type} {P : A → Type} {f g : Π x, P x} namespace funext - definition is_equiv_apD [instance] (f g : Π x, P x) : is_equiv (@apD10 A P f g) := + definition is_equiv_apd [instance] (f g : Π x, P x) : is_equiv (@apd10 A P f g) := funext_of_ua f g end funext open funext definition eq_equiv_homotopy : (f = g) ≃ (f ∼ g) := -equiv.mk apD10 _ +equiv.mk apd10 _ definition eq_of_homotopy [reducible] : f ∼ g → f = g := -(@apD10 A P f g)⁻¹ +(@apd10 A P f g)⁻¹ --TODO: rename to eq_of_homotopy_idp definition eq_of_homotopy_id (f : Π x, P x) : eq_of_homotopy (λx : A, idpath (f x)) = idpath f := -is_equiv.sect apD10 idp +is_equiv.sect apd10 idp definition naive_funext_of_ua : naive_funext := λ A P f g h, eq_of_homotopy h protected definition homotopy.rec_on {Q : (f ∼ g) → Type} (p : f ∼ g) - (H : Π(q : f = g), Q (apD10 q)) : Q p := -retr apD10 p ▹ H (eq_of_homotopy p) + (H : Π(q : f = g), Q (apd10 q)) : Q p := +retr apd10 p ▹ H (eq_of_homotopy p) diff --git a/hott/init/axioms/funext_varieties.hlean b/hott/init/axioms/funext_varieties.hlean index c1a8e1faa..c3b442536 100644 --- a/hott/init/axioms/funext_varieties.hlean +++ b/hott/init/axioms/funext_varieties.hlean @@ -14,7 +14,7 @@ import ..path ..trunc ..equiv open eq is_trunc sigma function /- In init.axioms.funext, we defined function extensionality to be the assertion - that the map apD10 is an equivalence. We now prove that this follows + that the map apd10 is an equivalence. We now prove that this follows from a couple of weaker-looking forms of function extensionality. We do require eta conversion, which Coq 8.4+ has judgmentally. @@ -22,7 +22,7 @@ open eq is_trunc sigma function by Peter Lumsdaine and Michael Shulman. -/ definition funext.{l k} := - Π ⦃A : Type.{l}⦄ {P : A → Type.{k}} (f g : Π x, P x), is_equiv (@apD10 A P f g) + Π ⦃A : Type.{l}⦄ {P : A → Type.{k}} (f g : Π x, P x), is_equiv (@apd10 A P f g) -- Naive funext is the simple assertion that pointwise equal functions are equal. -- TODO think about universe levels @@ -98,13 +98,13 @@ theorem funext_of_weak_funext (wf : weak_funext.{l k}) : funext.{l k} := let sim2path := homotopy_ind f eq_to_f idp in assert t1 : sim2path f (homotopy.refl f) = idp, proof homotopy_ind_comp f eq_to_f idp qed, - assert t2 : apD10 (sim2path f (homotopy.refl f)) = (homotopy.refl f), - proof ap apD10 t1 qed, - have sect : apD10 ∘ (sim2path g) ∼ id, - proof (homotopy_ind f (λ g' x, apD10 (sim2path g' x) = x) t2) g qed, - have retr : (sim2path g) ∘ apD10 ∼ id, + assert t2 : apd10 (sim2path f (homotopy.refl f)) = (homotopy.refl f), + proof ap apd10 t1 qed, + have sect : apd10 ∘ (sim2path g) ∼ id, + proof (homotopy_ind f (λ g' x, apd10 (sim2path g' x) = x) t2) g qed, + have retr : (sim2path g) ∘ apd10 ∼ id, from (λ h, eq.rec_on h (homotopy_ind_comp f _ idp)), - is_equiv.adjointify apD10 (sim2path g) sect retr + is_equiv.adjointify apd10 (sim2path g) sect retr definition funext_from_naive_funext : naive_funext -> funext := compose funext_of_weak_funext weak_funext_of_naive_funext diff --git a/hott/init/equiv.hlean b/hott/init/equiv.hlean index 0e653471e..b21894be8 100644 --- a/hott/init/equiv.hlean +++ b/hott/init/equiv.hlean @@ -78,7 +78,7 @@ namespace is_equiv ... = ap f (ap invf ff'a) ⬝ retrf'a : by rewrite ap_compose, have eq2 : _ = _, from calc retrf'a - = (ap f (ap invf ff'a))⁻¹ ⬝ (ap f secta ⬝ ff'a) : eq_inv_con_of_con_eq _ _ _ eq1⁻¹ + = (ap f (ap invf ff'a))⁻¹ ⬝ (ap f secta ⬝ ff'a) : eq_inv_con_of_con_eq eq1⁻¹ ... = (ap f (ap invf ff'a))⁻¹ ⬝ (ap f secta ⬝ Hty a) : ap_inv invf ff'a ... = (ap f (ap invf ff'a))⁻¹ ⬝ (Hty (invf (f a)) ⬝ ap f' secta) : by rewrite ap_con_eq_con_ap ... = ((ap f (ap invf ff'a))⁻¹ ⬝ Hty (invf (f a))) ⬝ ap f' secta : by rewrite con.assoc @@ -88,7 +88,7 @@ namespace is_equiv ... = Hty (invf (f' a)) ⬝ ((ap f' (ap invf ff'a))⁻¹ ⬝ ap f' secta) : by rewrite con.assoc, have eq3 : _ = _, from calc (Hty (invf (f' a)))⁻¹ ⬝ retrf'a - = (ap f' (ap invf ff'a))⁻¹ ⬝ ap f' secta : inv_con_eq_of_eq_con _ _ _ eq2 + = (ap f' (ap invf ff'a))⁻¹ ⬝ ap f' secta : inv_con_eq_of_eq_con eq2 ... = (ap f' (ap invf ff'a)⁻¹) ⬝ ap f' secta : by rewrite ap_inv ... = ap f' ((ap invf ff'a)⁻¹ ⬝ secta) : by rewrite ap_con, eq3) in @@ -119,7 +119,7 @@ namespace is_equiv from !con_idp ⬝ eq1, have eq3 : idp = _, from calc idp - = (ap f (sec a))⁻¹ ⬝ ((ret fgfa)⁻¹ ⬝ (fgretrfa ⬝ ap f (sec a))) : eq_inv_con_of_con_eq _ _ _ eq2 + = (ap f (sec a))⁻¹ ⬝ ((ret fgfa)⁻¹ ⬝ (fgretrfa ⬝ ap f (sec a))) : eq_inv_con_of_con_eq eq2 ... = ((ap f (sec a))⁻¹ ⬝ (ret fgfa)⁻¹) ⬝ (fgretrfa ⬝ ap f (sec a)) : by rewrite con.assoc' ... = (ap f (sec a)⁻¹ ⬝ (ret fgfa)⁻¹) ⬝ (fgretrfa ⬝ ap f (sec a)) : by rewrite ap_inv ... = ((ap f (sec a)⁻¹ ⬝ (ret fgfa)⁻¹) ⬝ fgretrfa) ⬝ ap f (sec a) : by rewrite con.assoc' @@ -130,7 +130,7 @@ namespace is_equiv ... = retrfa⁻¹ ⬝ (ap f (ap g (ap f (sec a)⁻¹) ⬝ ap g (ret (f a))) ⬝ ap f (sec a)) : by rewrite con.assoc' ... = retrfa⁻¹ ⬝ ap f ((ap g (ap f (sec a)⁻¹) ⬝ ap g (ret (f a))) ⬝ sec a) : by rewrite -ap_con, have eq4 : ret (f a) = ap f ((ap g (ap f (sec a)⁻¹) ⬝ ap g (ret (f a))) ⬝ sec a), - from eq_of_idp_eq_inv_con _ _ eq3, + from eq_of_idp_eq_inv_con eq3, eq4) definition adjointify : is_equiv f := @@ -188,7 +188,7 @@ namespace is_equiv = transport P (retr f (f x)) (df (f⁻¹ (f x))) : by esimp ... = transport P (eq.ap f (sect f x)) (df (f⁻¹ (f x))) : by rewrite adj ... = transport (P ∘ f) (sect f x) (df (f⁻¹ (f x))) : by rewrite -transport_compose - ... = df x : by rewrite (apD df (sect f x)) + ... = df x : by rewrite (apd df (sect f x)) end diff --git a/hott/init/hit.hlean b/hott/init/hit.hlean index 5013afa37..8c46bbc89 100644 --- a/hott/init/hit.hlean +++ b/hott/init/hit.hlean @@ -82,5 +82,5 @@ namespace type_quotient constant rec_eq_of_rel {A : Type} {R : A → A → Type} {P : type_quotient R → Type} (Pc : Π(a : A), P (class_of R a)) (Pp : Π⦃a a' : A⦄ (H : R a a'), eq_of_rel H ▹ Pc a = Pc a') - {a a' : A} (H : R a a') : apD (type_quotient.rec Pc Pp) (eq_of_rel H) = Pp H + {a a' : A} (H : R a a') : apd (type_quotient.rec Pc Pp) (eq_of_rel H) = Pp H end type_quotient diff --git a/hott/init/path.hlean b/hott/init/path.hlean index e3afe84a4..5724c0006 100644 --- a/hott/init/path.hlean +++ b/hott/init/path.hlean @@ -112,69 +112,69 @@ namespace eq /- Theorems for moving things around in equations -/ - definition con_eq_of_eq_inv_con (p : x = z) (q : y = z) (r : y = x) : + definition con_eq_of_eq_inv_con {p : x = z} {q : y = z} {r : y = x} : p = r⁻¹ ⬝ q → r ⬝ p = q := - eq.rec_on r (take p h, idp_con _ ⬝ h ⬝ idp_con _) p + eq.rec_on r (take p h, !idp_con ⬝ h ⬝ !idp_con) p - definition con_eq_of_eq_con_inv (p : x = z) (q : y = z) (r : y = x) : + definition con_eq_of_eq_con_inv {p : x = z} {q : y = z} {r : y = x} : r = q ⬝ p⁻¹ → r ⬝ p = q := - eq.rec_on p (take q h, (con_idp _ ⬝ h ⬝ con_idp _)) q + eq.rec_on p (take q h, (!con_idp ⬝ h ⬝ !con_idp)) q - definition inv_con_eq_of_eq_con (p : x = z) (q : y = z) (r : x = y) : + definition inv_con_eq_of_eq_con {p : x = z} {q : y = z} {r : x = y} : p = r ⬝ q → r⁻¹ ⬝ p = q := - eq.rec_on r (take q h, idp_con _ ⬝ h ⬝ idp_con _) q + eq.rec_on r (take q h, !idp_con ⬝ h ⬝ !idp_con) q - definition con_inv_eq_of_eq_con (p : z = x) (q : y = z) (r : y = x) : + definition con_inv_eq_of_eq_con {p : z = x} {q : y = z} {r : y = x} : r = q ⬝ p → r ⬝ p⁻¹ = q := - eq.rec_on p (take r h, con_idp _ ⬝ h ⬝ con_idp _) r + eq.rec_on p (take r h, !con_idp ⬝ h ⬝ !con_idp) r - definition eq_con_of_inv_con_eq (p : x = z) (q : y = z) (r : y = x) : + definition eq_con_of_inv_con_eq {p : x = z} {q : y = z} {r : y = x} : r⁻¹ ⬝ q = p → q = r ⬝ p := - eq.rec_on r (take p h, (idp_con _)⁻¹ ⬝ h ⬝ (idp_con _)⁻¹) p + eq.rec_on r (take p h, !idp_con⁻¹ ⬝ h ⬝ !idp_con⁻¹) p - definition eq_con_of_con_inv_eq (p : x = z) (q : y = z) (r : y = x) : + definition eq_con_of_con_inv_eq {p : x = z} {q : y = z} {r : y = x} : q ⬝ p⁻¹ = r → q = r ⬝ p := - eq.rec_on p (take q h, (con_idp _)⁻¹ ⬝ h ⬝ (con_idp _)⁻¹) q + eq.rec_on p (take q h, !con_idp⁻¹ ⬝ h ⬝ !con_idp⁻¹) q - definition eq_inv_con_of_con_eq (p : x = z) (q : y = z) (r : x = y) : + definition eq_inv_con_of_con_eq {p : x = z} {q : y = z} {r : x = y} : r ⬝ q = p → q = r⁻¹ ⬝ p := - eq.rec_on r (take q h, (idp_con _)⁻¹ ⬝ h ⬝ (idp_con _)⁻¹) q + eq.rec_on r (take q h, !idp_con⁻¹ ⬝ h ⬝ !idp_con⁻¹) q - definition eq_con_inv_of_con_eq (p : z = x) (q : y = z) (r : y = x) : + definition eq_con_inv_of_con_eq {p : z = x} {q : y = z} {r : y = x} : q ⬝ p = r → q = r ⬝ p⁻¹ := - eq.rec_on p (take r h, (con_idp _)⁻¹ ⬝ h ⬝ (con_idp _)⁻¹) r + eq.rec_on p (take r h, !con_idp⁻¹ ⬝ h ⬝ !con_idp⁻¹) r - definition eq_of_con_inv_eq_idp (p q : x = y) : + definition eq_of_con_inv_eq_idp {p q : x = y} : p ⬝ q⁻¹ = idp → p = q := - eq.rec_on q (take p h, (con_idp _)⁻¹ ⬝ h) p + eq.rec_on q (take p h, !con_idp⁻¹ ⬝ h) p - definition eq_of_inv_con_eq_idp (p q : x = y) : + definition eq_of_inv_con_eq_idp {p q : x = y} : q⁻¹ ⬝ p = idp → p = q := - eq.rec_on q (take p h, (idp_con _)⁻¹ ⬝ h) p + eq.rec_on q (take p h, !idp_con⁻¹ ⬝ h) p - definition eq_inv_of_con_eq_idp' (p : x = y) (q : y = x) : + definition eq_inv_of_con_eq_idp' {p : x = y} {q : y = x} : p ⬝ q = idp → p = q⁻¹ := - eq.rec_on q (take p h, (con_idp _)⁻¹ ⬝ h) p + eq.rec_on q (take p h, !con_idp⁻¹ ⬝ h) p - definition eq_inv_of_con_eq_idp (p : x = y) (q : y = x) : + definition eq_inv_of_con_eq_idp {p : x = y} {q : y = x} : q ⬝ p = idp → p = q⁻¹ := - eq.rec_on q (take p h, (idp_con _)⁻¹ ⬝ h) p + eq.rec_on q (take p h, !idp_con⁻¹ ⬝ h) p - definition eq_of_idp_eq_inv_con (p q : x = y) : + definition eq_of_idp_eq_inv_con {p q : x = y} : idp = p⁻¹ ⬝ q → p = q := - eq.rec_on p (take q h, h ⬝ (idp_con _)) q + eq.rec_on p (take q h, h ⬝ !idp_con) q - definition eq_of_idp_eq_con_inv (p q : x = y) : + definition eq_of_idp_eq_con_inv {p q : x = y} : idp = q ⬝ p⁻¹ → p = q := - eq.rec_on p (take q h, h ⬝ (con_idp _)) q + eq.rec_on p (take q h, h ⬝ !con_idp) q - definition inv_eq_of_idp_eq_con (p : x = y) (q : y = x) : + definition inv_eq_of_idp_eq_con {p : x = y} {q : y = x} : idp = q ⬝ p → p⁻¹ = q := - eq.rec_on p (take q h, h ⬝ (con_idp _)) q + eq.rec_on p (take q h, h ⬝ !con_idp) q - definition inv_eq_of_idp_eq_con' (p : x = y) (q : y = x) : + definition inv_eq_of_idp_eq_con' {p : x = y} {q : y = x} : idp = p ⬝ q → p⁻¹ = q := - eq.rec_on p (take q h, h ⬝ (idp_con _)) q + eq.rec_on p (take q h, h ⬝ !idp_con) q /- Transport -/ @@ -184,13 +184,16 @@ namespace eq -- This idiom makes the operation right associative. notation p `▹`:65 x:64 := transport _ p x + definition cast [reducible] {A B : Type} (p : A = B) (a : A) : B := + p ▹ a + definition tr_inv [reducible] (P : A → Type) {x y : A} (p : x = y) (u : P y) : P x := p⁻¹ ▹ u definition ap ⦃A B : Type⦄ (f : A → B) {x y:A} (p : x = y) : f x = f y := eq.rec_on p idp - definition ap01 [reducible] := ap + abbreviation ap01 [parsing-only] := ap definition homotopy [reducible] (f g : Πx, P x) : Type := Πx : A, f x = g x @@ -212,19 +215,19 @@ namespace eq calc_trans trans end homotopy - definition apD10 {f g : Πx, P x} (H : f = g) : f ∼ g := + definition apd10 {f g : Πx, P x} (H : f = g) : f ∼ g := λx, eq.rec_on H idp - --the next theorem is useful if you want to write "apply (apD10' a)" - definition apD10' {f g : Πx, P x} (a : A) (H : f = g) : f a = g a := + --the next theorem is useful if you want to write "apply (apd10' a)" + definition apd10' {f g : Πx, P x} (a : A) (H : f = g) : f a = g a := eq.rec_on H idp - definition ap10 {f g : A → B} (H : f = g) : f ∼ g := apD10 H + definition ap10 [reducible] {f g : A → B} (H : f = g) : f ∼ g := apd10 H definition ap11 {f g : A → B} (H : f = g) {x y : A} (p : x = y) : f x = g y := eq.rec_on H (eq.rec_on p idp) - definition apD (f : Πa:A, P a) {x y : A} (p : x = y) : p ▹ (f x) = f y := + definition apd (f : Πa:A, P a) {x y : A} (p : x = y) : p ▹ (f x) = f y := eq.rec_on p idp /- calc enviroment -/ @@ -236,19 +239,19 @@ namespace eq /- More theorems for moving things around in equations -/ - definition tr_eq_of_eq_inv_tr (P : A → Type) {x y : A} (p : x = y) (u : P x) (v : P y) : + definition tr_eq_of_eq_inv_tr (P : A → Type) {x y : A} {p : x = y} {u : P x} {v : P y} : u = p⁻¹ ▹ v → p ▹ u = v := eq.rec_on p (take v, id) v - definition inv_tr_eq_of_eq_tr (P : A → Type) {x y : A} (p : y = x) (u : P x) (v : P y) : + definition inv_tr_eq_of_eq_tr (P : A → Type) {x y : A} {p : y = x} {u : P x} {v : P y} : u = p ▹ v → p⁻¹ ▹ u = v := eq.rec_on p (take u, id) u - definition eq_inv_tr_of_tr_eq (P : A → Type) {x y : A} (p : x = y) (u : P x) (v : P y) : + definition eq_inv_tr_of_tr_eq (P : A → Type) {x y : A} {p : x = y} {u : P x} {v : P y} : p ▹ u = v → u = p⁻¹ ▹ v := eq.rec_on p (take v, id) v - definition eq_tr_of_inv_tr_eq (P : A → Type) {x y : A} (p : y = x) (u : P x) (v : P y) : + definition eq_tr_of_inv_tr_eq (P : A → Type) {x y : A} {p : y = x} {u : P x} {v : P y} : p⁻¹ ▹ u = v → u = p ▹ v := eq.rec_on p (take u, id) u @@ -258,9 +261,8 @@ namespace eq -- functorial. -- Functions take identity paths to identity paths - definition ap_idp (x : A) (f : A → B) : (ap f idp) = idp :> (f x = f x) := idp - - definition apD_idp (x : A) (f : Π x : A, P x) : apD f idp = idp :> (f x = f x) := idp + definition ap_idp (x : A) (f : A → B) : ap f idp = idp :> (f x = f x) := idp + definition apd_idp (x : A) (f : Πx, P x) : apd f idp = idp :> (f x = f x) := idp -- Functions commute with concatenation. definition ap_con (f : A → B) {x y z : A} (p : x = y) (q : y = z) : @@ -297,23 +299,22 @@ namespace eq eq.rec_on p idp -- The action of constant maps. - definition ap_constant (p : x = y) (z : B) : - ap (λu, z) p = idp := + definition ap_constant (p : x = y) (z : B) : ap (λu, z) p = idp := eq.rec_on p idp -- Naturality of [ap]. - definition ap_con_eq_con_ap {f g : A → B} (p : Π x, f x = g x) {x y : A} (q : x = y) : - (ap f q) ⬝ (p y) = (p x) ⬝ (ap g q) := - eq.rec_on q (idp_con _ ⬝ (con_idp _)⁻¹) + definition ap_con_eq_con_ap {f g : A → B} (p : f ∼ g) {x y : A} (q : x = y) : + ap f q ⬝ p y = p x ⬝ ap g q := + eq.rec_on q (!idp_con ⬝ !con_idp⁻¹) -- Naturality of [ap] at identity. definition ap_con_eq_con {f : A → A} (p : Πx, f x = x) {x y : A} (q : x = y) : - (ap f q) ⬝ (p y) = (p x) ⬝ q := - eq.rec_on q (idp_con _ ⬝ (con_idp _)⁻¹) + ap f q ⬝ p y = p x ⬝ q := + eq.rec_on q (!idp_con ⬝ !con_idp⁻¹) definition con_ap_eq_con {f : A → A} (p : Πx, x = f x) {x y : A} (q : x = y) : - (p x) ⬝ (ap f q) = q ⬝ (p y) := - eq.rec_on q (con_idp _ ⬝ (idp_con _)⁻¹) + p x ⬝ ap f q = q ⬝ p y := + eq.rec_on q (!con_idp ⬝ !idp_con⁻¹) -- Naturality with other paths hanging around. @@ -330,14 +331,14 @@ namespace eq -- TODO: try this using the simplifier, and compare proofs definition ap_con_con_eq_con_ap_con {f g : A → B} (p : f ∼ g) {x y : A} (q : x = y) {z : B} (s : g y = z) : - (ap f q) ⬝ (p y ⬝ s) = (p x) ⬝ (ap g q ⬝ s) := + ap f q ⬝ (p y ⬝ s) = p x ⬝ (ap g q ⬝ s) := eq.rec_on s (eq.rec_on q (calc (ap f idp) ⬝ (p x ⬝ idp) = idp ⬝ p x : idp - ... = p x : idp_con _ + ... = p x : !idp_con ... = (p x) ⬝ (ap g idp ⬝ idp) : idp)) -- This also works: - -- eq.rec_on s (eq.rec_on q (idp_con _ ▹ idp)) + -- eq.rec_on s (eq.rec_on q (!idp_con ▹ idp)) definition con_ap_con_con_eq_con_con_con {f : A → A} (p : f ∼ id) {x y : A} (q : x = y) {w z : A} (r : w = f x) (s : y = z) : @@ -356,8 +357,8 @@ namespace eq definition ap_con_con_eq_con_con {f : A → A} (p : f ∼ id) {x y : A} (q : x = y) {z : A} (s : y = z) : - (ap f q) ⬝ (p y ⬝ s) = (p x) ⬝ (q ⬝ s) := - eq.rec_on s (eq.rec_on q (idp_con _ ▹ idp)) + ap f q ⬝ (p y ⬝ s) = p x ⬝ (q ⬝ s) := + eq.rec_on s (eq.rec_on q (!idp_con ▹ idp)) definition con_con_ap_eq_con_con {g : A → A} (p : id ∼ g) {x y : A} (q : x = y) {w : A} (r : w = x) : @@ -373,27 +374,27 @@ namespace eq apply (idp_con (p x) ▹ idp) end - /- Action of [apD10] and [ap10] on paths -/ + /- Action of [apd10] and [ap10] on paths -/ -- Application of paths between functions preserves the groupoid structure - definition apD10_idp (f : Πx, P x) (x : A) : apD10 (refl f) x = idp := idp + definition apd10_idp (f : Πx, P x) (x : A) : apd10 (refl f) x = idp := idp - definition apD10_con {f f' f'' : Πx, P x} (h : f = f') (h' : f' = f'') (x : A) : - apD10 (h ⬝ h') x = apD10 h x ⬝ apD10 h' x := + definition apd10_con {f f' f'' : Πx, P x} (h : f = f') (h' : f' = f'') (x : A) : + apd10 (h ⬝ h') x = apd10 h x ⬝ apd10 h' x := eq.rec_on h (take h', eq.rec_on h' idp) h' - definition apD10_inv {f g : Πx : A, P x} (h : f = g) (x : A) : - apD10 h⁻¹ x = (apD10 h x)⁻¹ := + definition apd10_inv {f g : Πx : A, P x} (h : f = g) (x : A) : + apd10 h⁻¹ x = (apd10 h x)⁻¹ := eq.rec_on h idp definition ap10_idp {f : A → B} (x : A) : ap10 (refl f) x = idp := idp definition ap10_con {f f' f'' : A → B} (h : f = f') (h' : f' = f'') (x : A) : - ap10 (h ⬝ h') x = ap10 h x ⬝ ap10 h' x := apD10_con h h' x + ap10 (h ⬝ h') x = ap10 h x ⬝ ap10 h' x := apd10_con h h' x definition ap10_inv {f g : A → B} (h : f = g) (x : A) : ap10 h⁻¹ x = (ap10 h x)⁻¹ := - apD10_inv h x + apd10_inv h x -- [ap10] also behaves nicely on paths produced by [ap] definition ap_ap10 (f g : A → B) (h : B → C) (p : f = g) (a : A) : @@ -403,8 +404,7 @@ namespace eq /- Transport and the groupoid structure of paths -/ - definition tr_idp (P : A → Type) {x : A} (u : P x) : - idp ▹ u = u := idp + definition tr_idp (P : A → Type) {x : A} (u : P x) : idp ▹ u = u := idp definition tr_con (P : A → Type) {x y z : A} (p : x = y) (q : y = z) (u : P x) : p ⬝ q ▹ u = q ▹ p ▹ u := @@ -432,12 +432,12 @@ namespace eq eq.rec_on p idp -- Dependent transport in a doubly dependent type. - -- should P, Q and y all be explicit here? - definition transportD (P : A → Type) (Q : Π a : A, P a → Type) + definition transportD {P : A → Type} (Q : Π a : A, P a → Type) {a a' : A} (p : a = a') (b : P a) (z : Q a b) : Q a' (p ▹ b) := eq.rec_on p z - -- In Coq the variables B, C and y are explicit, but in Lean we can probably have them implicit using the following notation - notation p `▹D`:65 x:64 := transportD _ _ p _ x + + -- In Coq the variables P, Q and b are explicit, but in Lean we can probably have them implicit using the following notation + notation p `▹D`:65 x:64 := transportD _ p _ x -- Transporting along higher-dimensional paths definition transport2 (P : A → Type) {x y : A} {p q : x = y} (r : p = q) (z : P x) : @@ -470,7 +470,7 @@ namespace eq definition ap_tr_con_tr2 (P : A → Type) {x y : A} {p q : x = y} {z w : P x} (r : p = q) (s : z = w) : ap (transport P p) s ⬝ transport2 P r w = transport2 P r z ⬝ ap (transport P q) s := - eq.rec_on r (con_idp _ ⬝ (idp_con _)⁻¹) + eq.rec_on r (!con_idp ⬝ !idp_con⁻¹) definition fn_tr_eq_tr_fn {P Q : A → Type} {x y : A} (p : x = y) (f : Πx, P x → Q x) (z : P x) : @@ -494,7 +494,7 @@ namespace eq definition tr2_constant {p q : x = y} (r : p = q) (z : B) : tr_constant p z = transport2 (λu, B) r z ⬝ tr_constant q z := - eq.rec_on r (idp_con _)⁻¹ + eq.rec_on r !idp_con⁻¹ -- Transporting in a pulled back fibration. -- rename: tr_compose @@ -506,33 +506,31 @@ namespace eq ap (λh, h ∘ f) p = transport (λh : B → C, g ∘ f = h ∘ f) p idp := eq.rec_on p idp - definition apD10_ap_precompose (f : A → B) (g g' : B → C) (p : g = g') (a : A) : - apD10 (ap (λh : B → C, h ∘ f) p) a = apD10 p (f a) := + definition apd10_ap_precompose (f : A → B) (g g' : B → C) (p : g = g') (a : A) : + apd10 (ap (λh : B → C, h ∘ f) p) a = apd10 p (f a) := eq.rec_on p idp - definition apD10_ap_postcompose (f : B → C) (g g' : A → B) (p : g = g') (a : A) : - apD10 (ap (λh : A → B, f ∘ h) p) a = ap f (apD10 p a) := + definition apd10_ap_postcompose (f : B → C) (g g' : A → B) (p : g = g') (a : A) : + apd10 (ap (λh : A → B, f ∘ h) p) a = ap f (apd10 p a) := eq.rec_on p idp -- A special case of [transport_compose] which seems to come up a lot. - definition tr_eq_tr_id_ap (P : A → Type) x y (p : x = y) (u : P x) : - transport P p u = transport id (ap P p) u := + definition tr_eq_cast_ap (P : A → Type) {x y} (p : x = y) (u : P x) : p ▹ u = cast (ap P p) u := eq.rec_on p idp - /- The behavior of [ap] and [apD] -/ + /- The behavior of [ap] and [apd] -/ - -- In a constant fibration, [apD] reduces to [ap], modulo [transport_const]. - definition apD_eq_tr_constant_con_ap (f : A → B) (p: x = y) : - apD f p = tr_constant p (f x) ⬝ ap f p := + -- In a constant fibration, [apd] reduces to [ap], modulo [transport_const]. + definition apd_eq_tr_constant_con_ap (f : A → B) (p : x = y) : + apd f p = tr_constant p (f x) ⬝ ap f p := eq.rec_on p idp /- The 2-dimensional groupoid structure -/ -- Horizontal composition of 2-dimensional paths. - definition concat2 {p p' : x = y} {q q' : y = z} (h : p = p') (h' : q = q') : - p ⬝ q = p' ⬝ q' := + definition concat2 {p p' : x = y} {q q' : y = z} (h : p = p') (h' : q = q') : p ⬝ q = p' ⬝ q' := eq.rec_on h (eq.rec_on h' idp) infixl `◾`:75 := concat2 @@ -552,10 +550,10 @@ namespace eq -- Unwhiskering, a.k.a. cancelling - definition cancel_left {x y z : A} (p : x = y) (q r : y = z) : (p ⬝ q = p ⬝ r) → (q = r) := + definition cancel_left {x y z : A} {p : x = y} {q r : y = z} : (p ⬝ q = p ⬝ r) → (q = r) := λs, !inv_con_cancel_left⁻¹ ⬝ whisker_left p⁻¹ s ⬝ !inv_con_cancel_left - definition cancel_right {x y z : A} (p q : x = y) (r : y = z) : (p ⬝ r = q ⬝ r) → (p = q) := + definition cancel_right {x y z : A} {p q : x = y} {r : y = z} : (p ⬝ r = q ⬝ r) → (p = q) := λs, !con_inv_cancel_right⁻¹ ⬝ whisker_right s r⁻¹ ⬝ !con_inv_cancel_right -- Whiskering and identity paths. @@ -590,9 +588,10 @@ namespace eq (a ◾ c) ⬝ (b ◾ d) = (a ⬝ b) ◾ (c ⬝ d) := eq.rec_on d (eq.rec_on c (eq.rec_on b (eq.rec_on a idp))) - definition whisker_right_con_whisker_left {x y z : A} (p p' : x = y) (q q' : y = z) (a : p = p') (b : q = q') : + definition whisker_right_con_whisker_left {x y z : A} {p p' : x = y} {q q' : y = z} + (a : p = p') (b : q = q') : (whisker_right a q) ⬝ (whisker_left p' b) = (whisker_left p b) ⬝ (whisker_right a q') := - eq.rec_on b (eq.rec_on a (idp_con _)⁻¹) + eq.rec_on b (eq.rec_on a !idp_con⁻¹) -- Structure corresponding to the coherence equations of a bicategory. @@ -609,7 +608,7 @@ namespace eq con.assoc' p idp q ⬝ whisker_right (con_idp p) q = whisker_left p (idp_con q) := eq.rec_on q (eq.rec_on p idp) - definition eckmann_hilton {x:A} (p q : idp = idp :> (x = x)) : p ⬝ q = q ⬝ p := + definition eckmann_hilton {x:A} (p q : idp = idp :> x = x) : p ⬝ q = q ⬝ p := (!whisker_right_idp_right ◾ !whisker_left_idp_left)⁻¹ ⬝ (!con_idp ◾ !con_idp) ⬝ (!idp_con ◾ !idp_con) @@ -619,7 +618,7 @@ namespace eq ⬝ (!whisker_left_idp_left ◾ !whisker_right_idp_right) -- The action of functions on 2-dimensional paths - definition ap02 (f:A → B) {x y : A} {p q : x = y} (r : p = q) : ap f p = ap f q := + definition ap02 (f : A → B) {x y : A} {p q : x = y} (r : p = q) : ap f p = ap f q := eq.rec_on r idp definition ap02_con (f : A → B) {x y : A} {p p' p'' : x = y} (r : p = p') (r' : p' = p'') : @@ -633,16 +632,16 @@ namespace eq ⬝ (ap_con f p' q')⁻¹ := eq.rec_on r (eq.rec_on s (eq.rec_on q (eq.rec_on p idp))) - definition apD02 {p q : x = y} (f : Π x, P x) (r : p = q) : - apD f p = transport2 P r (f x) ⬝ apD f q := - eq.rec_on r (idp_con _)⁻¹ + definition apd02 {p q : x = y} (f : Π x, P x) (r : p = q) : + apd f p = transport2 P r (f x) ⬝ apd f q := + eq.rec_on r !idp_con⁻¹ -- And now for a lemma whose statement is much longer than its proof. - definition apD02_con (P : A → Type) (f : Π x:A, P x) {x y : A} + definition apd02_con (P : A → Type) (f : Π x:A, P x) {x y : A} {p1 p2 p3 : x = y} (r1 : p1 = p2) (r2 : p2 = p3) : - apD02 f (r1 ⬝ r2) = apD02 f r1 - ⬝ whisker_left (transport2 P r1 (f x)) (apD02 f r2) + apd02 f (r1 ⬝ r2) = apd02 f r1 + ⬝ whisker_left (transport2 P r1 (f x)) (apd02 f r2) ⬝ con.assoc' _ _ _ - ⬝ (whisker_right (tr2_con P r1 r2 (f x))⁻¹ (apD f p3)) := + ⬝ (whisker_right (tr2_con P r1 r2 (f x))⁻¹ (apd f p3)) := eq.rec_on r2 (eq.rec_on r1 (eq.rec_on p1 idp)) end eq diff --git a/hott/types/eq.hlean b/hott/types/eq.hlean index 85a33b36a..ce45af006 100644 --- a/hott/types/eq.hlean +++ b/hott/types/eq.hlean @@ -96,7 +96,7 @@ namespace eq definition transport_eq_FlFr_D {B : A → Type} {f g : Πa, B a} (p : a1 = a2) (q : f a1 = g a1) - : transport (λx, f x = g x) p q = (apD f p)⁻¹ ⬝ ap (transport B p) q ⬝ (apD g p) := + : transport (λx, f x = g x) p q = (apd f p)⁻¹ ⬝ ap (transport B p) q ⬝ (apd g p) := begin apply (eq.rec_on p), apply inverse, @@ -201,7 +201,7 @@ namespace eq equiv.mk _ !is_equiv_whisker_right definition is_equiv_con_eq_of_eq_inv_con (p : a1 = a3) (q : a2 = a3) (r : a2 = a1) - : is_equiv (con_eq_of_eq_inv_con p q r) := + : is_equiv (con_eq_of_eq_inv_con : p = r⁻¹ ⬝ q → r ⬝ p = q) := begin cases r, apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right), @@ -212,7 +212,7 @@ namespace eq equiv.mk _ !is_equiv_con_eq_of_eq_inv_con definition is_equiv_con_eq_of_eq_con_inv (p : a1 = a3) (q : a2 = a3) (r : a2 = a1) - : is_equiv (con_eq_of_eq_con_inv p q r) := + : is_equiv (con_eq_of_eq_con_inv : r = q ⬝ p⁻¹ → r ⬝ p = q) := begin cases p, apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right) @@ -223,7 +223,7 @@ namespace eq equiv.mk _ !is_equiv_con_eq_of_eq_con_inv definition is_equiv_inv_con_eq_of_eq_con (p : a1 = a3) (q : a2 = a3) (r : a1 = a2) - : is_equiv (inv_con_eq_of_eq_con p q r) := + : is_equiv (inv_con_eq_of_eq_con : p = r ⬝ q → r⁻¹ ⬝ p = q) := begin cases r, apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right) @@ -234,7 +234,7 @@ namespace eq equiv.mk _ !is_equiv_inv_con_eq_of_eq_con definition is_equiv_con_inv_eq_of_eq_con (p : a3 = a1) (q : a2 = a3) (r : a2 = a1) - : is_equiv (con_inv_eq_of_eq_con p q r) := + : is_equiv (con_inv_eq_of_eq_con : r = q ⬝ p → r ⬝ p⁻¹ = q) := begin cases p, apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right) @@ -245,7 +245,7 @@ namespace eq equiv.mk _ !is_equiv_con_inv_eq_of_eq_con definition is_equiv_eq_con_of_inv_con_eq (p : a1 = a3) (q : a2 = a3) (r : a2 = a1) - : is_equiv (eq_con_of_inv_con_eq p q r) := + : is_equiv (eq_con_of_inv_con_eq : r⁻¹ ⬝ q = p → q = r ⬝ p) := begin cases r, apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right) @@ -256,7 +256,7 @@ namespace eq equiv.mk _ !is_equiv_eq_con_of_inv_con_eq definition is_equiv_eq_con_of_con_inv_eq (p : a1 = a3) (q : a2 = a3) (r : a2 = a1) - : is_equiv (eq_con_of_con_inv_eq p q r) := + : is_equiv (eq_con_of_con_inv_eq : q ⬝ p⁻¹ = r → q = r ⬝ p) := begin cases p, apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right) diff --git a/hott/types/equiv.hlean b/hott/types/equiv.hlean index 5c850e995..3322cdd58 100644 --- a/hott/types/equiv.hlean +++ b/hott/types/equiv.hlean @@ -91,7 +91,7 @@ namespace equiv protected definition eq_mk' {f f' : A → B} [H : is_equiv f] [H' : is_equiv f'] (p : f = f') : equiv.mk f H = equiv.mk f' H' := - apD011 equiv.mk p !is_hprop.elim + apd011 equiv.mk p !is_hprop.elim protected definition eq_mk {f f' : A ≃ B} (p : to_fun f = to_fun f') : f = f' := by (cases f; cases f'; apply (equiv.eq_mk' p)) diff --git a/hott/types/pi.hlean b/hott/types/pi.hlean index 040d60f4e..0db46e619 100644 --- a/hott/types/pi.hlean +++ b/hott/types/pi.hlean @@ -22,15 +22,15 @@ namespace pi /- Paths [p : f ≈ g] in a function type [Πx:X, P x] are equivalent to functions taking values in path types, [H : Πx:X, f x ≈ g x], or concisely, [H : f ∼ g]. - This equivalence, however, is just the combination of [apD10] and function extensionality [funext], and as such, [path_forall], et seq. are given in axioms.funext and path: -/ + This equivalence, however, is just the combination of [apd10] and function extensionality [funext], and as such, [path_forall], et seq. are given in axioms.funext and path: -/ /- Now we show how these things compute. -/ - definition apD10_eq_of_homotopy (h : f ∼ g) : apD10 (eq_of_homotopy h) ∼ h := - apD10 (retr apD10 h) + definition apd10_eq_of_homotopy (h : f ∼ g) : apd10 (eq_of_homotopy h) ∼ h := + apd10 (retr apd10 h) - definition eq_of_homotopy_eta (p : f = g) : eq_of_homotopy (apD10 p) = p := - sect apD10 p + definition eq_of_homotopy_eta (p : f = g) : eq_of_homotopy (apd10 p) = p := + sect apd10 p definition eq_of_homotopy_idp (f : Πa, B a) : eq_of_homotopy (λx : A, refl (f x)) = refl f := !eq_of_homotopy_eta @@ -38,11 +38,11 @@ namespace pi /- The identification of the path space of a dependent function space, up to equivalence, is of course just funext. -/ definition eq_equiv_homotopy (f g : Πx, B x) : (f = g) ≃ (f ∼ g) := - equiv.mk _ !is_equiv_apD + equiv.mk _ !is_equiv_apd definition is_equiv_eq_of_homotopy [instance] (f g : Πx, B x) : is_equiv (@eq_of_homotopy _ _ f g) := - is_equiv_inv apD10 + is_equiv_inv apd10 definition homotopy_equiv_eq (f g : Πx, B x) : (f ∼ g) ≃ (f = g) := equiv.mk _ !is_equiv_eq_of_homotopy @@ -52,7 +52,7 @@ namespace pi definition pi_transport (p : a = a') (f : Π(b : B a), C a b) : (transport (λa, Π(b : B a), C a b) p f) - ∼ (λb, transport (C a') !tr_inv_tr (transportD _ _ p _ (f (p⁻¹ ▹ b)))) := + ∼ (λb, transport (C a') !tr_inv_tr (transportD _ p _ (f (p⁻¹ ▹ b)))) := eq.rec_on p (λx, idp) /- A special case of [transport_pi] where the type [B] does not depend on [A], @@ -107,7 +107,7 @@ namespace pi definition ap_pi_functor {g g' : Π(a:A), B a} (h : g ∼ g') : ap (pi_functor f0 f1) (eq_of_homotopy h) = eq_of_homotopy (λa':A', (ap (f1 a') (h (f0 a')))) := begin - apply (equiv_rect (@apD10 A B g g')), intro p, clear h, + apply (equiv_rect (@apd10 A B g g')), intro p, clear h, cases p, apply concat, exact (ap (ap (pi_functor f0 f1)) (eq_of_homotopy_idp g)), @@ -130,13 +130,13 @@ namespace pi apply (transport (λx, f1 a' x = h a') (transport_compose B f0 (sect f0 a') _)), apply (tr_inv (λx, x = h a') (fn_tr_eq_tr_fn _ f1 _)), unfold function.compose, apply (tr_inv (λx, sect f0 a' ▹ x = h a') (retr (f1 _) _)), unfold function.id, - apply apD + apply apd end, begin intro h, apply eq_of_homotopy, intro a, apply (tr_inv (λx, retr f0 a ▹ x = h a) (sect (f1 _) _)), unfold function.id, - apply apD + apply apd end end diff --git a/hott/types/trunc.hlean b/hott/types/trunc.hlean index 5a741e827..479851b0c 100644 --- a/hott/types/trunc.hlean +++ b/hott/types/trunc.hlean @@ -80,7 +80,7 @@ namespace is_trunc (imp : Π{a b : A}, R a b → a = b) : is_hset A := is_hset_of_axiom_K (λa p, - have H2 : transport (λx, R a x → a = x) p (@imp a a) = @imp a a, from !apD, + have H2 : transport (λx, R a x → a = x) p (@imp a a) = @imp a a, from !apd, have H3 : Π(r : R a a), transport (λx, a = x) p (imp r) = imp (transport (λx, R a x) p r), from to_fun (equiv.symm !heq_pi) H2, @@ -89,7 +89,7 @@ namespace is_trunc imp (refl a) ⬝ p = transport (λx, a = x) p (imp (refl a)) : transport_eq_r ... = imp (transport (λx, R a x) p (refl a)) : H3 ... = imp (refl a) : is_hprop.elim, - cancel_left (imp (refl a)) _ _ H4) + cancel_left H4) definition relation_equiv_eq {A : Type} (R : A → A → Type) (mere : Π(a b : A), is_hprop (R a b)) (refl : Π(a : A), R a a)