feat(hott): add arity.hlean, about multivariate functions
This commit is contained in:
Floris van Doorn
Leonardo de Moura
parent
71f9a5d1d2
commit
ebba33057c
7 changed files with 196 additions and 147 deletions
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@ -11,91 +11,6 @@ import .basic types.pi .iso
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open function category eq prod equiv is_equiv sigma sigma.ops is_trunc funext iso
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open pi
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section
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variables {A : Type} {B : A → Type} {C : Πa, B a → Type} {D : Πa b, C a b → Type}
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definition homotopy2 [reducible] (f g : Πa b, C a b) : Type :=
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Πa b, f a b = g a b
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definition homotopy3 [reducible] (f g : Πa b c, D a b c) : Type :=
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Πa b c, f a b c = g a b c
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notation f `∼2`:50 g := homotopy2 f g
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notation f `∼3`:50 g := homotopy3 f g
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-- definition apD100 {f g : Πa b, C a b} (p : f = g) : f ∼2 g :=
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-- λa b, eq.rec_on p idp
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definition apD100 {f g : Πa b, C a b} (p : f = g) : f ∼2 g :=
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λa b, apD10 (apD10 p a) b
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definition apD1000 {f g : Πa b c, D a b c} (p : f = g) : f ∼3 g :=
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λa b c, apD100 (apD10 p a) b c
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definition eq_of_homotopy2 {f g : Πa b, C a b} (H : f ∼2 g) : f = g :=
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eq_of_homotopy (λa, eq_of_homotopy (H a))
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definition eq_of_homotopy3 {f g : Πa b c, D a b c} (H : f ∼3 g) : f = g :=
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eq_of_homotopy (λa, eq_of_homotopy2 (H a))
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definition eq_of_homotopy2_id (f : Πa b, C a b)
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: eq_of_homotopy2 (λa b, idpath (f a b)) = idpath f :=
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begin
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apply concat,
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{apply (ap (λx, eq_of_homotopy x)), apply eq_of_homotopy, intro a, apply eq_of_homotopy_id},
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apply eq_of_homotopy_id
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end
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definition eq_of_homotopy3_id (f : Πa b c, D a b c)
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: eq_of_homotopy3 (λa b c, idpath (f a b c)) = idpath f :=
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begin
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apply concat,
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{apply (ap (λx, eq_of_homotopy x)), apply eq_of_homotopy, intro a, apply eq_of_homotopy2_id},
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apply eq_of_homotopy_id
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end
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--TODO: put in namespace funext
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definition is_equiv_apD100 [instance] (f g : Πa b, C a b) : is_equiv (@apD100 A B C f g) :=
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adjointify _
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eq_of_homotopy2
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begin
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intro H, esimp {apD100,eq_of_homotopy2, function.compose},
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apply eq_of_homotopy, intro a,
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apply concat, apply (ap (λx, @apD10 _ (λb : B a, _) _ _ (x a))), apply (retr apD10),
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--TODO: remove implicit argument after #469 is closed
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apply (retr apD10)
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end
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begin
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intro p, cases p, apply eq_of_homotopy2_id
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end
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definition is_equiv_apD1000 [instance] (f g : Πa b c, D a b c) : is_equiv (@apD1000 A B C D f g) :=
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adjointify _
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eq_of_homotopy3
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begin
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intro H, apply eq_of_homotopy, intro a,
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apply concat, {apply (ap (λx, @apD100 _ _ (λ(b : B a)(c : C a b), _) _ _ (x a))), apply (retr apD10)},
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--TODO: remove implicit argument after #469 is closed
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apply (@retr _ _ apD100 !is_equiv_apD100) --is explicit argument needed here?
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end
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begin
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intro p, cases p, apply eq_of_homotopy3_id
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end
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protected definition homotopy2.rec_on {f g : Πa b, C a b} {P : (f ∼2 g) → Type}
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(p : f ∼2 g) (H : Π(q : f = g), P (apD100 q)) : P p :=
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retr apD100 p ▹ H (eq_of_homotopy2 p)
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protected definition homotopy3.rec_on {f g : Πa b c, D a b c} {P : (f ∼3 g) → Type}
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(p : f ∼3 g) (H : Π(q : f = g), P (apD1000 q)) : P p :=
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retr apD1000 p ▹ H (eq_of_homotopy3 p)
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end
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structure functor (C D : Precategory) : Type :=
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(to_fun_ob : C → D)
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(to_fun_hom : Π ⦃a b : C⦄, hom a b → hom (to_fun_ob a) (to_fun_ob b))
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@ -221,20 +136,39 @@ namespace functor
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apply is_trunc_eq, apply is_trunc_succ, apply !homH},
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end
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definition functor_eq_eta' {ob₁ ob₂ : C → D} {hom₁ hom₂ id₁ id₂ comp₁ comp₂}
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(p : functor.mk ob₁ hom₁ id₁ comp₁ = functor.mk ob₂ hom₂ id₂ comp₂)
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: p = p := --functor_eq_mk' _ _ _ _ (ap010 to_fun_ob p) _ :=
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sorry
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--set_option pp.universes true
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-- set_option pp.notation false
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-- set_option pp.implicit true
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definition functor_eq2' {obF obF' : C → D} {homF homF' idF idF' compF compF'}
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(p q : functor.mk obF homF idF compF = functor.mk obF' homF' idF' compF') (r : obF = obF')
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: p = q :=
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-- TODO: REMOVE?
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definition functor_eq2'' {ob₁ ob₂ : C → D} {hom₁ hom₂ id₁ id₂ comp₁ comp₂}
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{pob₁ pob₂ : ob₁ = ob₂} (phom₁ : pob₁ ▹ hom₁ = hom₂) (phom₂ : pob₂ ▹ hom₁ = hom₂)
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(r : pob₁ = pob₂) : functor_eq_mk'' id₁ id₂ comp₁ comp₂ pob₁ phom₁
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= functor_eq_mk'' id₁ id₂ comp₁ comp₂ pob₂ phom₂ :=
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begin
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cases r,
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apply (ap (functor_eq_mk'' id₁ id₂ @comp₁ @comp₂ pob₂)),
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apply is_hprop.elim
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end
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definition functor_eq2' {ob₁ ob₂ : C → D} {hom₁ hom₂ id₁ id₂ comp₁ comp₂} {pob₁ pob₂ : ob₁ ∼ ob₂}
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(phom₁ : Π(a b : C) (f : hom a b), hom_of_eq (pob₁ b) ∘ hom₁ a b f ∘ inv_of_eq (pob₁ a) = hom₂ a b f)
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(phom₂ : Π(a b : C) (f : hom a b), hom_of_eq (pob₂ b) ∘ hom₁ a b f ∘ inv_of_eq (pob₂ a) = hom₂ a b f)
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(r : pob₁ = pob₂) : functor_eq_mk' id₁ id₂ comp₁ comp₂ pob₁ phom₁
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= functor_eq_mk' id₁ id₂ comp₁ comp₂ pob₂ phom₂ :=
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begin
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cases r,
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apply (ap (functor_eq_mk' id₁ id₂ @comp₁ @comp₂ pob₂)),
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apply is_hprop.elim
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end
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definition functor_eq2 {F₁ F₂ : C ⇒ D} (p q : F₁ = F₂) (r : ap010 to_fun_ob p ∼ ap010 to_fun_ob q)
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: p = q :=
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begin
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apply sorry
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end
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-- definition ap010_functor_eq_mk' {F₁ F₂ : C ⇒ D} (p : to_fun_ob F₁ = to_fun_ob F₂)
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@ -6,7 +6,7 @@ Module: algebra.precategory.iso
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Author: Floris van Doorn, Jakob von Raumer
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-/
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import algebra.precategory.basic types.sigma
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import algebra.precategory.basic types.sigma arity
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open eq category prod equiv is_equiv sigma sigma.ops is_trunc
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@ -125,6 +125,7 @@ namespace functor
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functor.mk (functor_uncurry_ob G)
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(functor_uncurry_hom G)
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(functor_uncurry_id G)
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(functor_uncurry_comp G)
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theorem functor_uncurry_functor_curry : functor_uncurry (functor_curry F) = F :=
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functor_eq_mk (λp, ap (to_fun_ob F) !prod.eta)
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159
hott/arity.hlean
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159
hott/arity.hlean
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@ -0,0 +1,159 @@
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/-
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Copyright (c) 2014 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: init.path
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Author: Floris van Doorn
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Theorems about functions with multiple arguments
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-/
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variables {A U V W X Y Z : Type} {B : A → Type} {C : Πa, B a → Type} {D : Πa b, C a b → Type}
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{E : Πa b c, D a b c → Type}
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variables {a a' : A} {u u' : U} {v v' : V} {w w' : W} {x x' : X} {y y' : Y}
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{b : B a} {b' : B a'}
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{c : C a b} {c' : C a' b'}
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{d : D a b c} {d' : D a' b' c'}
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namespace eq
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/-
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Naming convention:
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The theorem which states how to construct an path between two function applications is
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api₀i₁...iₙ.
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Here i₀, ... iₙ are digits, n is the arity of the function(s),
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and iⱼ specifies the dimension of the path between the jᵗʰ argument
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(i₀ specifies the dimension of the path between the functions).
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A value iⱼ ≡ 0 means that the jᵗʰ arguments are definitionally equal
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The functions are non-dependent, except when the theorem name contains trailing zeroes
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(where the function is dependent only in the arguments where it doesn't result in any
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transports in the theorem statement).
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For the fully-dependent versions (except that the conclusion doesn't contain a transport)
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we write
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apDi₀i₁...iₙ.
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For versions where only some arguments depend on some other arguments,
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or for versions with transport in the conclusion (like apD), we don't have a
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consistent naming scheme (yet).
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We don't prove each theorem systematically, but prove only the ones which we actually need.
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-/
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definition homotopy2 [reducible] (f g : Πa b, C a b) : Type :=
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Πa b, f a b = g a b
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definition homotopy3 [reducible] (f g : Πa b c, D a b c) : Type :=
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Πa b c, f a b c = g a b c
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definition homotopy4 [reducible] (f g : Πa b c d, E a b c d) : Type :=
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Πa b c d, f a b c d = g a b c d
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notation f `∼2`:50 g := homotopy2 f g
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notation f `∼3`:50 g := homotopy3 f g
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definition ap011 (f : U → V → W) (Hu : u = u') (Hv : v = v') : f u v = f u' v' :=
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eq.rec_on Hu (ap (f u) Hv)
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definition ap0111 (f : U → V → W → X) (Hu : u = u') (Hv : v = v') (Hw : w = w')
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: f u v w = f u' v' w' :=
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eq.rec_on Hu (ap011 (f u) Hv Hw)
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definition ap01111 (f : U → V → W → X → Y) (Hu : u = u') (Hv : v = v') (Hw : w = w') (Hx : x = x')
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: f u v w x = f u' v' w' x' :=
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eq.rec_on Hu (ap0111 (f u) Hv Hw Hx)
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definition ap010 (f : X → Πa, B a) (Hx : x = x') : f x ∼ f x' :=
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λa, eq.rec_on Hx idp
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definition ap0100 (f : X → Πa b, C a b) (Hx : x = x') : f x ∼2 f x' :=
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λa b, eq.rec_on Hx idp
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definition ap01000 (f : X → Πa b c, D a b c) (Hx : x = x') : f x ∼3 f x' :=
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λa b c, eq.rec_on Hx idp
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definition apD011 (f : Πa, B a → Z) (Ha : a = a') (Hb : (Ha ▹ b) = b')
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: f a b = f a' b' :=
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eq.rec_on Hb (eq.rec_on Ha idp)
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definition apD0111 (f : Πa b, C a b → Z) (Ha : a = a') (Hb : (Ha ▹ b) = b')
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(Hc : apD011 C Ha Hb ▹ c = c')
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: f a b c = f a' b' c' :=
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eq.rec_on Hc (eq.rec_on Hb (eq.rec_on Ha idp))
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definition apD01111 (f : Πa b c, D a b c → Z) (Ha : a = a') (Hb : (Ha ▹ b) = b')
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(Hc : apD011 C Ha Hb ▹ c = c') (Hd : apD0111 D Ha Hb Hc ▹ d = d')
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: f a b c d = f a' b' c' d' :=
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eq.rec_on Hd (eq.rec_on Hc (eq.rec_on Hb (eq.rec_on Ha idp)))
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definition apD100 {f g : Πa b, C a b} (p : f = g) : f ∼2 g :=
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λa b, apD10 (apD10 p a) b
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definition apD1000 {f g : Πa b c, D a b c} (p : f = g) : f ∼3 g :=
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λa b c, apD100 (apD10 p a) b c
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/- the following theorems are function extentionality for functions with multiple arguments -/
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definition eq_of_homotopy2 {f g : Πa b, C a b} (H : f ∼2 g) : f = g :=
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eq_of_homotopy (λa, eq_of_homotopy (H a))
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definition eq_of_homotopy3 {f g : Πa b c, D a b c} (H : f ∼3 g) : f = g :=
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eq_of_homotopy (λa, eq_of_homotopy2 (H a))
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definition eq_of_homotopy2_id (f : Πa b, C a b)
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: eq_of_homotopy2 (λa b, idpath (f a b)) = idpath f :=
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begin
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apply concat,
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{apply (ap (λx, eq_of_homotopy x)), apply eq_of_homotopy, intro a, apply eq_of_homotopy_id},
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apply eq_of_homotopy_id
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end
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definition eq_of_homotopy3_id (f : Πa b c, D a b c)
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: eq_of_homotopy3 (λa b c, idpath (f a b c)) = idpath f :=
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begin
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apply concat,
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{apply (ap (λx, eq_of_homotopy x)), apply eq_of_homotopy, intro a, apply eq_of_homotopy2_id},
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apply eq_of_homotopy_id
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end
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end eq
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open is_equiv eq
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namespace funext
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definition is_equiv_apD100 [instance] (f g : Πa b, C a b) : is_equiv (@apD100 A B C f g) :=
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adjointify _
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eq_of_homotopy2
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begin
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intro H, esimp {apD100,eq_of_homotopy2, function.compose},
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apply eq_of_homotopy, intro a,
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apply concat, apply (ap (λx, @apD10 _ (λb : B a, _) _ _ (x a))), apply (retr apD10),
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--TODO: remove implicit argument after #469 is closed
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apply (retr apD10)
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end
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begin
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intro p, cases p, apply eq_of_homotopy2_id
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end
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definition is_equiv_apD1000 [instance] (f g : Πa b c, D a b c)
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: is_equiv (@apD1000 A B C D f g) :=
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adjointify _
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eq_of_homotopy3
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begin
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intro H, apply eq_of_homotopy, intro a,
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apply concat,
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{apply (ap (λx, @apD100 _ _ (λ(b : B a)(c : C a b), _) _ _ (x a))),
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apply (retr apD10)},
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--TODO: remove implicit argument after #469 is closed
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apply (@retr _ _ apD100 !is_equiv_apD100) --is explicit argument needed here?
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end
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begin
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intro p, cases p, apply eq_of_homotopy3_id
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end
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end funext
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namespace eq
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open funext
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local attribute funext.is_equiv_apD100 [instance]
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protected definition homotopy2.rec_on {f g : Πa b, C a b} {P : (f ∼2 g) → Type}
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(p : f ∼2 g) (H : Π(q : f = g), P (apD100 q)) : P p :=
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retr apD100 p ▹ H (eq_of_homotopy2 p)
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protected definition homotopy3.rec_on {f g : Πa b c, D a b c} {P : (f ∼3 g) → Type}
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(p : f ∼3 g) (H : Π(q : f = g), P (apD1000 q)) : P p :=
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retr apD1000 p ▹ H (eq_of_homotopy3 p)
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end eq
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@ -135,28 +135,28 @@ theorem weak_funext_of_ua : weak_funext :=
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definition funext_of_ua : funext :=
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funext_of_weak_funext (@weak_funext_of_ua)
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variables {A : Type} {P : A → Type} {f g : Π x, P x}
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namespace funext
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definition is_equiv_apD [instance] {A : Type} {P : A → Type} (f g : Π x, P x)
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: is_equiv (@apD10 A P f g) :=
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definition is_equiv_apD [instance] (f g : Π x, P x) : is_equiv (@apD10 A P f g) :=
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funext_of_ua f g
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end funext
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open funext
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definition eq_equiv_homotopy {A : Type} {P : A → Type} {f g : Π x, P x} : (f = g) ≃ (f ∼ g) :=
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definition eq_equiv_homotopy : (f = g) ≃ (f ∼ g) :=
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equiv.mk apD10 _
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definition eq_of_homotopy {A : Type} {P : A → Type} {f g : Π x, P x} : f ∼ g → f = g :=
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definition eq_of_homotopy [reducible] : f ∼ g → f = g :=
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(@apD10 A P f g)⁻¹
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--rename to eq_of_homotopy_idp
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definition eq_of_homotopy_id {A : Type} {P : A → Type} (f : Π x, P x)
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: eq_of_homotopy (λx : A, idpath (f x)) = idpath f :=
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--TODO: rename to eq_of_homotopy_idp
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definition eq_of_homotopy_id (f : Π x, P x) : eq_of_homotopy (λx : A, idpath (f x)) = idpath f :=
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is_equiv.sect apD10 idp
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definition naive_funext_of_ua : naive_funext :=
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λ A P f g h, eq_of_homotopy h
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protected definition homotopy.rec_on {A : Type} {B : A → Type} {f g : Πa, B a} {P : (f ∼ g) → Type}
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(p : f ∼ g) (H : Π(q : f = g), P (apD10 q)) : P p :=
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protected definition homotopy.rec_on {Q : (f ∼ g) → Type} (p : f ∼ g)
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(H : Π(q : f = g), Q (apD10 q)) : Q p :=
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retr apD10 p ▹ H (eq_of_homotopy p)
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@ -3,7 +3,7 @@ Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: init.path
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Author: Jeremy Avigad, Jakob von Raumer, Floris van Doorn
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Authors: Jeremy Avigad, Jakob von Raumer, Floris van Doorn
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Ported from Coq HoTT
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-/
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@ -186,7 +186,7 @@ namespace eq
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definition ap ⦃A B : Type⦄ (f : A → B) {x y:A} (p : x = y) : f x = f y :=
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eq.rec_on p idp
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definition ap01 := ap
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definition ap01 [reducible] := ap
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definition homotopy [reducible] (f g : Πx, P x) : Type :=
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Πx : A, f x = g x
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@ -590,8 +590,6 @@ namespace eq
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(whisker_right a q) ⬝ (whisker_left p' b) = (whisker_left p b) ⬝ (whisker_right a q') :=
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eq.rec_on b (eq.rec_on a (idp_con _)⁻¹)
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-- Structure corresponding to the coherence equations of a bicategory.
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||||
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-- The "pentagonator": the 3-cell witnessing the associativity pentagon.
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@ -630,7 +628,6 @@ namespace eq
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⬝ (ap02 f r ◾ ap02 f s)
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⬝ (ap_con f p' q')⁻¹ :=
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eq.rec_on r (eq.rec_on s (eq.rec_on q (eq.rec_on p idp)))
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-- eq.rec_on r (eq.rec_on s (eq.rec_on p (eq.rec_on q idp)))
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||||
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||||
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 :=
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||||
|
|
@ -645,45 +642,3 @@ namespace eq
|
|||
⬝ (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
|
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|
||||
namespace eq
|
||||
section
|
||||
variables {A B C D E : Type} {a a' : A} {b b' : B} {c c' : C} {d d' : D}
|
||||
|
||||
definition ap011 (f : A → B → C) (Ha : a = a') (Hb : b = b') : f a b = f a' b' :=
|
||||
eq.rec_on Ha (eq.rec_on Hb idp)
|
||||
|
||||
definition ap0111 (f : A → B → C → D) (Ha : a = a') (Hb : b = b') (Hc : c = c')
|
||||
: f a b c = f a' b' c' :=
|
||||
eq.rec_on Ha (ap011 (f a) Hb Hc)
|
||||
|
||||
definition ap01111 (f : A → B → C → D → E) (Ha : a = a') (Hb : b = b') (Hc : c = c') (Hd : d = d')
|
||||
: f a b c d = f a' b' c' d' :=
|
||||
eq.rec_on Ha (ap0111 (f a) Hb Hc Hd)
|
||||
|
||||
definition ap010 {C : B → Type} (f : A → Π(b : B), C b) (Ha : a = a') (b : B) : f a b = f a' b :=
|
||||
eq.rec_on Ha idp
|
||||
end
|
||||
section
|
||||
variables {A : Type} {B : A → Type} {C : Πa, B a → Type} {D : Πa b, C a b → Type}
|
||||
{E : Πa b c, D a b c → Type} {F : Type}
|
||||
variables {a a' : A}
|
||||
{b : B a} {b' : B a'}
|
||||
{c : C a b} {c' : C a' b'}
|
||||
{d : D a b c} {d' : D a' b' c'}
|
||||
|
||||
definition apD011 (f : Πa, B a → F) (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 → F) (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 → F) (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)))
|
||||
end
|
||||
end eq
|
||||
|
|
|
|||
|
|
@ -9,7 +9,7 @@ Ported from Coq HoTT
|
|||
Theorems about the types equiv and is_equiv
|
||||
-/
|
||||
|
||||
import types.fiber types.arrow
|
||||
import types.fiber types.arrow arity
|
||||
|
||||
open eq is_trunc sigma sigma.ops arrow pi
|
||||
|
||||
|
|
|
|||
Loading…
Reference in a new issue