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feat(hott): make some arguments in init.path implicit and rename apD to apd

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Floris van Doorn 2015-04-24 17:00:32 -04:00 committed by Leonardo de Moura
parent b98c109f73
commit e769fdd9dc
21 changed files with 191 additions and 192 deletions
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4
hott/algebra/precategory/constructions.hlean
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@ -189,7 +189,7 @@ namespace category
end) end)
definition nat_trans_left_inverse : nat_trans_inverse η ∘n η = nat_trans.id := definition nat_trans_left_inverse : nat_trans_inverse η ∘n η = nat_trans.id :=
begin begin
fapply (apD011 nat_trans.mk), fapply (apd011 nat_trans.mk),
apply eq_of_homotopy, intro c, apply left_inverse, 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 eq_of_homotopy, intros, apply eq_of_homotopy, intros, apply eq_of_homotopy, intros,
apply is_hset.elim apply is_hset.elim
@ -197,7 +197,7 @@ namespace category
definition nat_trans_right_inverse : η ∘n nat_trans_inverse η = nat_trans.id := definition nat_trans_right_inverse : η ∘n nat_trans_inverse η = nat_trans.id :=
begin begin
fapply (apD011 nat_trans.mk), fapply (apd011 nat_trans.mk),
apply eq_of_homotopy, intro c, apply right_inverse, 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 eq_of_homotopy, intros, apply eq_of_homotopy, intros, apply eq_of_homotopy, intros,
apply is_hset.elim apply is_hset.elim

12
hott/algebra/precategory/functor.hlean
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@ -50,7 +50,7 @@ namespace functor
{H₂ : Π(a b : C), hom a b → hom (F₂ a) (F₂ b)} (id₁ id₂ comp₁ comp₂) {H₂ : Π(a b : C), hom a b → hom (F₂ a) (F₂ b)} (id₁ id₂ comp₁ comp₂)
(pF : F₁ = F₂) (pH : pF ▹ H₁ = H₂) (pF : F₁ = F₂) (pH : pF ▹ H₁ = H₂)
: functor.mk F₁ H₁ id₁ comp₁ = functor.mk F₂ H₂ id₂ comp₂ := : 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} definition functor_eq' {F₁ F₂ : C ⇒ D}
: Π(p : to_fun_ob F₁ = to_fun_ob F₂), : 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, apply transport_hom,
apply concat, rotate_left 1, apply concat, rotate_left 1,
exact (pi_transport_constant (eq_of_homotopy pF) (H₁ c c') f), exact (pi_transport_constant (eq_of_homotopy pF) (H₁ c c') f),
apply (apD10' f), apply (apd10' f),
apply concat, rotate_left 1, esimp, apply concat, rotate_left 1, esimp,
exact (pi_transport_constant (eq_of_homotopy pF) (H₁ c) c'), exact (pi_transport_constant (eq_of_homotopy pF) (H₁ c) c'),
apply (apD10' c'), apply (apd10' c'),
apply concat, rotate_left 1, esimp, apply concat, rotate_left 1, esimp,
exact (pi_transport_constant (eq_of_homotopy pF) H₁ c), exact (pi_transport_constant (eq_of_homotopy pF) H₁ c),
apply idp 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)) 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 := (id comp) : functor_mk_eq' id id comp comp (idpath F) (idpath H) = idp :=
begin begin
fapply (apD011 (apD01111 functor.mk idp idp)), fapply (apd011 (apd01111 functor.mk idp idp)),
apply is_hset.elim, apply is_hset.elim,
apply is_hset.elim apply is_hset.elim
end end
@ -178,7 +178,7 @@ namespace functor
by (cases F; apply functor_mk_eq'_idp) by (cases F; apply functor_mk_eq'_idp)
definition functor_eq_eta' {F₁ F₂ : C ⇒ D} (p : F₁ = F₂) 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 begin
cases p, cases F₁, cases p, cases F₁,
apply concat, rotate_left 1, apply functor_eq'_idp, 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) -- definition ap010_functor_eq_mk' {F₁ F₂ : C ⇒ D} (p : to_fun_ob F₁ = to_fun_ob F₂)
-- (q : p ▹ F₁ = F₂) (c : C) : -- (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 -- begin
-- cases F₂, revert q, apply (homotopy.rec_on p), clear p, esimp, intros (p, q), -- cases F₂, revert q, apply (homotopy.rec_on p), clear p, esimp, intros (p, q),
-- cases p, clears (e_1, e_2), -- cases p, clears (e_1, e_2),

10
hott/algebra/precategory/iso.hlean
View file

@ -131,7 +131,7 @@ namespace iso
begin begin
apply is_hprop.mk, intros [H, H'], apply is_hprop.mk, intros [H, H'],
cases H with [g, li, ri], cases H' with [g', li', ri'], 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 left_inverse_eq_right_inverse,
apply li, apply li,
apply ri', 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') 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' := : 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' := 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)) by (cases f; cases f'; apply (iso.eq_mk' p))
@ -212,16 +212,16 @@ namespace iso
open funext open funext
variables {X : Type} {x y : X} {F G : X → ob} variables {X : Type} {x y : X} {F G : X → ob}
definition transport_hom_of_eq (p : F = G) (f : hom (F x) (F y)) 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 eq.rec_on p !id_leftright⁻¹
definition transport_hom (p : F ∼ G) (f : hom (F x) (F y)) 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) := : eq_of_homotopy p ▹ f = hom_of_eq (p y) ∘ f ∘ inv_of_eq (p x) :=
calc calc
eq_of_homotopy p ▹ f = 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 : 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 end
structure mono [class] (f : a ⟶ b) := structure mono [class] (f : a ⟶ b) :=

2
hott/algebra/precategory/nat_trans.hlean
View file

@ -43,7 +43,7 @@ namespace nat_trans
(nat₂ : Π (a b : C) (f : hom a b), G f ∘ η₂ a = η₂ b ∘ F f) (nat₂ : Π (a b : C) (f : hom a b), G f ∘ η₂ a = η₂ b ∘ F f)
(p : η₁ ∼ η₂) (p : η₁ ∼ η₂)
: nat_trans.mk η₁ nat₁ = nat_trans.mk η₂ nat₂ := : 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 := 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)) nat_trans.rec_on η₁ (λf₁ nat₁, nat_trans.rec_on η₂ (λf₂ nat₂ p, !nat_trans_eq_mk' p))

56
hott/arity.hlean
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@ -29,10 +29,10 @@ namespace eq
transports in the theorem statement). transports in the theorem statement).
For the fully-dependent versions (except that the conclusion doesn't contain a transport) For the fully-dependent versions (except that the conclusion doesn't contain a transport)
we write we write
apDi₀i₁...iₙ. apdi₀i₁...iₙ.
For versions where only some arguments depend on some other arguments, 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). consistent naming scheme (yet).
We don't prove each theorem systematically, but prove only the ones which we actually need. 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' := 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 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' := : f a b = f a' b' :=
eq.rec_on Hb (eq.rec_on Ha idp) 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') definition apd0111 (f : Πa b, C a b → Z) (Ha : a = a') (Hb : (Ha ▹ b) = b')
(Hc : apD011 C Ha Hb ▹ c = c') (Hc : apd011 C Ha Hb ▹ c = c')
: f a b c = f a' b' c' := : f a b c = f a' b' c' :=
eq.rec_on Hc (eq.rec_on Hb (eq.rec_on Ha idp)) 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') 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') (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' := : 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))) 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 := definition apd100 {f g : Πa b, C a b} (p : f = g) : f ∼2 g :=
λa b, apD10 (apD10 p a) b λa b, apd10 (apd10 p a) b
definition apD1000 {f g : Πa b c, D a b c} (p : f = g) : f ∼3 g := 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 λa b c, apd100 (apd10 p a) b c
/- some properties of these variants of ap -/ /- some properties of these variants of ap -/
@ -128,30 +128,30 @@ end eq
open eq is_equiv open eq is_equiv
namespace funext 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 _ adjointify _
eq_of_homotopy2 eq_of_homotopy2
begin 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 eq_of_homotopy, intro a,
apply concat, apply (ap (λx, apD10 (x a))), apply (retr apD10), apply concat, apply (ap (λx, apd10 (x a))), apply (retr apd10),
apply (retr apD10) apply (retr apd10)
end end
begin begin
intro p, cases p, apply eq_of_homotopy2_id intro p, cases p, apply eq_of_homotopy2_id
end end
definition is_equiv_apD1000 [instance] (f g : Πa b c, D a b c) definition is_equiv_apd1000 [instance] (f g : Πa b c, D a b c)
: is_equiv (@apD1000 A B C D f g) := : is_equiv (@apd1000 A B C D f g) :=
adjointify _ adjointify _
eq_of_homotopy3 eq_of_homotopy3
begin begin
intro H, apply eq_of_homotopy, intro a, intro H, apply eq_of_homotopy, intro a,
apply concat, apply concat,
{apply (ap (λx, @apD100 _ _ (λ(b : B a)(c : C a b), _) _ _ (x a))), {apply (ap (λx, @apd100 _ _ (λ(b : B a)(c : C a b), _) _ _ (x a))),
apply (retr apD10)}, apply (retr apd10)},
--TODO: remove implicit argument after #469 is closed --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 end
begin begin
intro p, cases p, apply eq_of_homotopy3_id intro p, cases p, apply eq_of_homotopy3_id
@ -160,22 +160,22 @@ end funext
namespace eq namespace eq
open funext 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} 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 := (p : f ∼2 g) (H : Π(q : f = g), P (apd100 q)) : P p :=
retr apD100 p ▹ H (eq_of_homotopy2 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} 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 := (p : f ∼3 g) (H : Π(q : f = g), P (apd1000 q)) : P p :=
retr apD1000 p ▹ H (eq_of_homotopy3 p) retr apd1000 p ▹ H (eq_of_homotopy3 p)
definition apD10_ap (f : X → Πa, B a) (p : x = x') definition apd10_ap (f : X → Πa, B a) (p : x = x')
: apD10 (ap f p) = ap010 f p := : apd10 (ap f p) = ap010 f p :=
eq.rec_on p idp eq.rec_on p idp
definition eq_of_homotopy_ap010 (f : X → Πa, B a) (p : x = x') definition eq_of_homotopy_ap010 (f : X → Πa, B a) (p : x = x')
: eq_of_homotopy (ap010 f p) = ap f p := : 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) 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 := : ap f p = ap f q :=

4
hott/hit/circle.hlean
View file

@ -41,12 +41,12 @@ namespace circle
definition rec2_seg1 {P : circle → Type} (Pb1 : P base1) (Pb2 : P base2) definition rec2_seg1 {P : circle → Type} (Pb1 : P base1) (Pb2 : P base2)
(Ps1 : seg1 ▹ Pb1 = Pb2) (Ps2 : seg2 ▹ Pb2 = Pb1) (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 sorry
definition rec2_seg2 {P : circle → Type} (Pb1 : P base1) (Pb2 : P base2) definition rec2_seg2 {P : circle → Type} (Pb1 : P base1) (Pb2 : P base2)
(Ps1 : seg1 ▹ Pb1 = Pb2) (Ps2 : seg2 ▹ Pb2 = Pb1) (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 sorry
definition elim2 {P : Type} (Pb1 Pb2 : P) (Ps1 : Pb1 = Pb2) (Ps2 : Pb2 = Pb1) (x : circle) : P := definition elim2 {P : Type} (Pb1 Pb2 : P) (Ps1 : Pb1 = Pb2) (Ps2 : Pb2 = Pb1) (x : circle) : P :=

2
hott/hit/coeq.hlean
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@ -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)) 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)) (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 sorry
protected definition elim {P : Type} (P_i : B → P) protected definition elim {P : Type} (P_i : B → P)

4
hott/hit/colimit.hlean
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@ -52,7 +52,7 @@ section
definition rec_cglue [reducible] {P : colimit → Type} definition rec_cglue [reducible] {P : colimit → Type}
(Pincl : Π⦃i : I⦄ (x : A i), P (ι x)) (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) (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 sorry
protected definition elim {P : Type} (Pincl : Π⦃i : I⦄ (x : A i), P) 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)) 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) (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 sorry
definition elim_glue {P : Type} (Pincl : Π⦃n : ℕ⦄ (a : A n), P) definition elim_glue {P : Type} (Pincl : Π⦃n : ℕ⦄ (a : A n), P)

2
hott/hit/cylinder.hlean
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@ -54,7 +54,7 @@ parameters {A B : Type.{u}} (f : A → B)
definition rec_seg {P : cylinder → Type} definition rec_seg {P : cylinder → Type}
(Pbase : Π(b : B), P (base b)) (Ptop : Π(a : A), P (top a)) (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) (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 sorry
protected definition elim {P : Type} (Pbase : B → P) (Ptop : A → P) protected definition elim {P : Type} (Pbase : B → P) (Ptop : A → P)

2
hott/hit/pushout.hlean
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@ -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)) 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)) (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 sorry
protected definition elim {P : Type} (Pinl : BL → P) (Pinr : TR → P) protected definition elim {P : Type} (Pinl : BL → P) (Pinr : TR → P)

2
hott/hit/quotient.hlean
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@ -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)] 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') (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 sorry
protected definition elim {P : Type} [Pt : is_hset P] (Pc : A → P) protected definition elim {P : Type} [Pt : is_hset P] (Pc : A → P)

2
hott/hit/suspension.hlean
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@ -41,7 +41,7 @@ namespace suspension
definition rec_merid {P : suspension A → Type} (PN : P !north) (PS : P !south) definition rec_merid {P : suspension A → Type} (PN : P !north) (PS : P !south)
(Pm : Π(a : A), merid a ▹ PN = PS) (a : A) (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 sorry
protected definition elim {P : Type} (PN : P) (PS : P) (Pm : A → PN = PS) protected definition elim {P : Type} (PN : P) (PS : P) (Pm : A → PN = PS)

12
hott/init/axioms/funext_of_ua.hlean
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@ -138,25 +138,25 @@ definition funext_of_ua : funext :=
variables {A : Type} {P : A → Type} {f g : Π x, P x} variables {A : Type} {P : A → Type} {f g : Π x, P x}
namespace funext 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 funext_of_ua f g
end funext end funext
open funext open funext
definition eq_equiv_homotopy : (f = g) ≃ (f ∼ g) := definition eq_equiv_homotopy : (f = g) ≃ (f ∼ g) :=
equiv.mk apD10 _ equiv.mk apd10 _
definition eq_of_homotopy [reducible] : f ∼ g → f = g := 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 --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 := 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 := definition naive_funext_of_ua : naive_funext :=
λ A P f g h, eq_of_homotopy h λ A P f g h, eq_of_homotopy h
protected definition homotopy.rec_on {Q : (f ∼ g) → Type} (p : f ∼ g) protected definition homotopy.rec_on {Q : (f ∼ g) → Type} (p : f ∼ g)
(H : Π(q : f = g), Q (apD10 q)) : Q p := (H : Π(q : f = g), Q (apd10 q)) : Q p :=
retr apD10 p ▹ H (eq_of_homotopy p) retr apd10 p ▹ H (eq_of_homotopy p)

16
hott/init/axioms/funext_varieties.hlean
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@ -14,7 +14,7 @@ import ..path ..trunc ..equiv
open eq is_trunc sigma function open eq is_trunc sigma function
/- In init.axioms.funext, we defined function extensionality to be the assertion /- 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 from a couple of weaker-looking forms of function extensionality. We do
require eta conversion, which Coq 8.4+ has judgmentally. 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. -/ by Peter Lumsdaine and Michael Shulman. -/
definition funext.{l k} := 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. -- Naive funext is the simple assertion that pointwise equal functions are equal.
-- TODO think about universe levels -- 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 let sim2path := homotopy_ind f eq_to_f idp in
assert t1 : sim2path f (homotopy.refl f) = idp, assert t1 : sim2path f (homotopy.refl f) = idp,
proof homotopy_ind_comp f eq_to_f idp qed, proof homotopy_ind_comp f eq_to_f idp qed,
assert t2 : apD10 (sim2path f (homotopy.refl f)) = (homotopy.refl f), assert t2 : apd10 (sim2path f (homotopy.refl f)) = (homotopy.refl f),
proof ap apD10 t1 qed, proof ap apd10 t1 qed,
have sect : apD10 ∘ (sim2path g) ∼ id, have sect : apd10 ∘ (sim2path g) ∼ id,
proof (homotopy_ind f (λ g' x, apD10 (sim2path g' x) = x) t2) g qed, proof (homotopy_ind f (λ g' x, apd10 (sim2path g' x) = x) t2) g qed,
have retr : (sim2path g) ∘ apD10 ∼ id, have retr : (sim2path g) ∘ apd10 ∼ id,
from (λ h, eq.rec_on h (homotopy_ind_comp f _ idp)), 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 := definition funext_from_naive_funext : naive_funext -> funext :=
compose funext_of_weak_funext weak_funext_of_naive_funext compose funext_of_weak_funext weak_funext_of_naive_funext

10
hott/init/equiv.hlean
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@ -78,7 +78,7 @@ namespace is_equiv
... = ap f (ap invf ff'a) ⬝ retrf'a : by rewrite ap_compose, ... = ap f (ap invf ff'a) ⬝ retrf'a : by rewrite ap_compose,
have eq2 : _ = _, have eq2 : _ = _,
from calc retrf'a 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 ⬝ (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 ap_con_eq_con_ap
... = ((ap f (ap invf ff'a))⁻¹ ⬝ Hty (invf (f a))) ⬝ ap f' secta : by rewrite con.assoc ... = ((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, ... = Hty (invf (f' a)) ⬝ ((ap f' (ap invf ff'a))⁻¹ ⬝ ap f' secta) : by rewrite con.assoc,
have eq3 : _ = _, have eq3 : _ = _,
from calc (Hty (invf (f' a)))⁻¹ ⬝ retrf'a 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) ⬝ ap f' secta : by rewrite ap_inv
... = ap f' ((ap invf ff'a)⁻¹ ⬝ secta) : by rewrite ap_con, ... = ap f' ((ap invf ff'a)⁻¹ ⬝ secta) : by rewrite ap_con,
eq3) in eq3) in
@ -119,7 +119,7 @@ namespace is_equiv
from !con_idp ⬝ eq1, from !con_idp ⬝ eq1,
have eq3 : idp = _, have eq3 : idp = _,
from calc 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 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 ap_inv
... = ((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 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))) ⬝ 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, ... = 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), 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) eq4)
definition adjointify : is_equiv f := 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 (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 (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 ... = 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 end

2
hott/init/hit.hlean
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@ -82,5 +82,5 @@ namespace type_quotient
constant rec_eq_of_rel {A : Type} {R : A → A → Type} {P : type_quotient R → Type} 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') (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 end type_quotient

199
hott/init/path.hlean
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@ -112,69 +112,69 @@ namespace eq
/- Theorems for moving things around in equations -/ /- 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 := 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 := 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 := 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 := 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 := 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 := 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 := 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 := 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 := 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 := 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 := 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 := 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 := 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 := 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 := 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 := 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 -/ /- Transport -/
@ -184,13 +184,16 @@ namespace eq
-- This idiom makes the operation right associative. -- This idiom makes the operation right associative.
notation p `▹`:65 x:64 := transport _ p x 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 := definition tr_inv [reducible] (P : A → Type) {x y : A} (p : x = y) (u : P y) : P x :=
p⁻¹ ▹ u p⁻¹ ▹ u
definition ap ⦃A B : Type⦄ (f : A → B) {x y:A} (p : x = y) : f x = f y := definition ap ⦃A B : Type⦄ (f : A → B) {x y:A} (p : x = y) : f x = f y :=
eq.rec_on p idp eq.rec_on p idp
definition ap01 [reducible] := ap abbreviation ap01 [parsing-only] := ap
definition homotopy [reducible] (f g : Πx, P x) : Type := definition homotopy [reducible] (f g : Πx, P x) : Type :=
Πx : A, f x = g x Πx : A, f x = g x
@ -212,19 +215,19 @@ namespace eq
calc_trans trans calc_trans trans
end homotopy 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 λx, eq.rec_on H idp
--the next theorem is useful if you want to write "apply (apD10' 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 := definition apd10' {f g : Πx, P x} (a : A) (H : f = g) : f a = g a :=
eq.rec_on H idp 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 := 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) 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 eq.rec_on p idp
/- calc enviroment -/ /- calc enviroment -/
@ -236,19 +239,19 @@ namespace eq
/- More theorems for moving things around in equations -/ /- 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 := u = p⁻¹ ▹ v → p ▹ u = v :=
eq.rec_on p (take v, id) 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 := u = p ▹ v → p⁻¹ ▹ u = v :=
eq.rec_on p (take u, id) u 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 := p ▹ u = v → u = p⁻¹ ▹ v :=
eq.rec_on p (take v, id) 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 := p⁻¹ ▹ u = v → u = p ▹ v :=
eq.rec_on p (take u, id) u eq.rec_on p (take u, id) u
@ -258,9 +261,8 @@ namespace eq
-- functorial. -- functorial.
-- Functions take identity paths to identity paths -- 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 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
definition apD_idp (x : A) (f : Π x : A, P x) : apD f idp = idp :> (f x = f x) := idp
-- Functions commute with concatenation. -- Functions commute with concatenation.
definition ap_con (f : A → B) {x y z : A} (p : x = y) (q : y = z) : 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 eq.rec_on p idp
-- The action of constant maps. -- The action of constant maps.
definition ap_constant (p : x = y) (z : B) : definition ap_constant (p : x = y) (z : B) : ap (λu, z) p = idp :=
ap (λu, z) p = idp :=
eq.rec_on p idp eq.rec_on p idp
-- Naturality of [ap]. -- 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) : 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) := ap f q ⬝ p y = p x ⬝ ap g q :=
eq.rec_on q (idp_con _ ⬝ (con_idp _)⁻¹) eq.rec_on q (!idp_con ⬝ !con_idp⁻¹)
-- Naturality of [ap] at identity. -- Naturality of [ap] at identity.
definition ap_con_eq_con {f : A → A} (p : Πx, f x = x) {x y : A} (q : x = y) : 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 := ap f q ⬝ p y = p x ⬝ q :=
eq.rec_on q (idp_con _ ⬝ (con_idp _)⁻¹) 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) : 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) := p x ⬝ ap f q = q ⬝ p y :=
eq.rec_on q (con_idp _ ⬝ (idp_con _)⁻¹) eq.rec_on q (!con_idp ⬝ !idp_con⁻¹)
-- Naturality with other paths hanging around. -- Naturality with other paths hanging around.
@ -330,14 +331,14 @@ namespace eq
-- TODO: try this using the simplifier, and compare proofs -- 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) 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) : {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 eq.rec_on s (eq.rec_on q
(calc (calc
(ap f idp) ⬝ (p x ⬝ idp) = idp ⬝ p x : idp (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)) ... = (p x) ⬝ (ap g idp ⬝ idp) : idp))
-- This also works: -- 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) 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) : {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) definition ap_con_con_eq_con_con {f : A → A} (p : f ∼ id) {x y : A} (q : x = y)
{z : A} (s : y = z) : {z : A} (s : y = z) :
(ap f q) ⬝ (p y ⬝ s) = (p x) ⬝ (q ⬝ s) := ap f q ⬝ (p y ⬝ s) = p x ⬝ (q ⬝ s) :=
eq.rec_on s (eq.rec_on q (idp_con _ ▹ idp)) 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) definition con_con_ap_eq_con_con {g : A → A} (p : id ∼ g) {x y : A} (q : x = y)
{w : A} (r : w = x) : {w : A} (r : w = x) :
@ -373,27 +374,27 @@ namespace eq
apply (idp_con (p x) ▹ idp) apply (idp_con (p x) ▹ idp)
end end
/- Action of [apD10] and [ap10] on paths -/ /- Action of [apd10] and [ap10] on paths -/
-- Application of paths between functions preserves the groupoid structure -- 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) : 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 := apd10 (h ⬝ h') x = apd10 h x ⬝ apd10 h' x :=
eq.rec_on h (take h', eq.rec_on h' idp) h' 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) : definition apd10_inv {f g : Πx : A, P x} (h : f = g) (x : A) :
apD10 h⁻¹ x = (apD10 h x)⁻¹ := apd10 h⁻¹ x = (apd10 h x)⁻¹ :=
eq.rec_on h idp eq.rec_on h idp
definition ap10_idp {f : A → B} (x : A) : ap10 (refl f) x = idp := 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) : 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 := 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] -- [ap10] also behaves nicely on paths produced by [ap]
definition ap_ap10 (f g : A → B) (h : B → C) (p : f = g) (a : A) : 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 -/ /- Transport and the groupoid structure of paths -/
definition tr_idp (P : A → Type) {x : A} (u : P x) : definition tr_idp (P : A → Type) {x : A} (u : P x) : idp ▹ u = u := idp
idp ▹ u = u := idp
definition tr_con (P : A → Type) {x y z : A} (p : x = y) (q : y = z) (u : P x) : 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 := p ⬝ q ▹ u = q ▹ p ▹ u :=
@ -432,12 +432,12 @@ namespace eq
eq.rec_on p idp eq.rec_on p idp
-- Dependent transport in a doubly dependent type. -- 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) := {a a' : A} (p : a = a') (b : P a) (z : Q a b) : Q a' (p ▹ b) :=
eq.rec_on p z 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 -- Transporting along higher-dimensional paths
definition transport2 (P : A → Type) {x y : A} {p q : x = y} (r : p = q) (z : P x) : 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) 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) : (s : z = w) :
ap (transport P p) s ⬝ transport2 P r w = transport2 P r z ⬝ ap (transport P q) s := 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) : 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) : 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 := 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. -- Transporting in a pulled back fibration.
-- rename: tr_compose -- rename: tr_compose
@ -506,33 +506,31 @@ namespace eq
ap (λh, h ∘ f) p = transport (λh : B → C, g ∘ f = h ∘ f) p idp := ap (λh, h ∘ f) p = transport (λh : B → C, g ∘ f = h ∘ f) p idp :=
eq.rec_on p idp eq.rec_on p idp
definition apD10_ap_precompose (f : A → B) (g g' : B → C) (p : g = g') (a : 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) := apd10 (ap (λh : B → C, h ∘ f) p) a = apd10 p (f a) :=
eq.rec_on p idp eq.rec_on p idp
definition apD10_ap_postcompose (f : B → C) (g g' : A → B) (p : g = g') (a : 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) := apd10 (ap (λh : A → B, f ∘ h) p) a = ap f (apd10 p a) :=
eq.rec_on p idp eq.rec_on p idp
-- A special case of [transport_compose] which seems to come up a lot. -- 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) : definition tr_eq_cast_ap (P : A → Type) {x y} (p : x = y) (u : P x) : p ▹ u = cast (ap P p) u :=
transport P p u = transport id (ap P p) u :=
eq.rec_on p idp 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]. -- In a constant fibration, [apd] reduces to [ap], modulo [transport_const].
definition apD_eq_tr_constant_con_ap (f : A → B) (p: x = y) : definition apd_eq_tr_constant_con_ap (f : A → B) (p : x = y) :
apD f p = tr_constant p (f x) ⬝ ap f p := apd f p = tr_constant p (f x) ⬝ ap f p :=
eq.rec_on p idp eq.rec_on p idp
/- The 2-dimensional groupoid structure -/ /- The 2-dimensional groupoid structure -/
-- Horizontal composition of 2-dimensional paths. -- Horizontal composition of 2-dimensional paths.
definition concat2 {p p' : x = y} {q q' : y = z} (h : p = p') (h' : q = q') : definition concat2 {p p' : x = y} {q q' : y = z} (h : p = p') (h' : q = q') : p ⬝ q = p' ⬝ q' :=
p ⬝ q = p' ⬝ q' :=
eq.rec_on h (eq.rec_on h' idp) eq.rec_on h (eq.rec_on h' idp)
infixl `◾`:75 := concat2 infixl `◾`:75 := concat2
@ -552,10 +550,10 @@ namespace eq
-- Unwhiskering, a.k.a. cancelling -- 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 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 λs, !con_inv_cancel_right⁻¹ ⬝ whisker_right s r⁻¹ ⬝ !con_inv_cancel_right
-- Whiskering and identity paths. -- Whiskering and identity paths.
@ -590,9 +588,10 @@ namespace eq
(a ◾ c) ⬝ (b ◾ d) = (a ⬝ b) ◾ (c ⬝ d) := (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))) 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') := (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. -- 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) := 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) 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 (!whisker_right_idp_right ◾ !whisker_left_idp_left)⁻¹
⬝ (!con_idp ◾ !con_idp) ⬝ (!con_idp ◾ !con_idp)
⬝ (!idp_con ◾ !idp_con) ⬝ (!idp_con ◾ !idp_con)
@ -619,7 +618,7 @@ namespace eq
⬝ (!whisker_left_idp_left ◾ !whisker_right_idp_right) ⬝ (!whisker_left_idp_left ◾ !whisker_right_idp_right)
-- The action of functions on 2-dimensional paths -- 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 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'') : 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 := ⬝ (ap_con f p' q')⁻¹ :=
eq.rec_on r (eq.rec_on s (eq.rec_on q (eq.rec_on p idp))) 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) : 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 := apd f p = transport2 P r (f x) ⬝ apd f q :=
eq.rec_on r (idp_con _)⁻¹ eq.rec_on r !idp_con⁻¹
-- And now for a lemma whose statement is much longer than its proof. -- 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) : {p1 p2 p3 : x = y} (r1 : p1 = p2) (r2 : p2 = p3) :
apD02 f (r1 ⬝ r2) = apD02 f r1 apd02 f (r1 ⬝ r2) = apd02 f r1
⬝ whisker_left (transport2 P r1 (f x)) (apD02 f r2) ⬝ whisker_left (transport2 P r1 (f x)) (apd02 f r2)
⬝ con.assoc' _ _ _ ⬝ 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)) eq.rec_on r2 (eq.rec_on r1 (eq.rec_on p1 idp))
end eq end eq

14
hott/types/eq.hlean
View file

@ -96,7 +96,7 @@ namespace eq
definition transport_eq_FlFr_D {B : A → Type} {f g : Πa, B a} definition transport_eq_FlFr_D {B : A → Type} {f g : Πa, B a}
(p : a1 = a2) (q : f a1 = g a1) (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 begin
apply (eq.rec_on p), apply (eq.rec_on p),
apply inverse, apply inverse,
@ -201,7 +201,7 @@ namespace eq
equiv.mk _ !is_equiv_whisker_right equiv.mk _ !is_equiv_whisker_right
definition is_equiv_con_eq_of_eq_inv_con (p : a1 = a3) (q : a2 = a3) (r : a2 = a1) 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 begin
cases r, cases r,
apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right), 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 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) 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 begin
cases p, cases p,
apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right) 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 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) 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 begin
cases r, cases r,
apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right) 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 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) 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 begin
cases p, cases p,
apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right) 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 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) 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 begin
cases r, cases r,
apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right) 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 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) 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 begin
cases p, cases p,
apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right) apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right)

2
hott/types/equiv.hlean
View file

@ -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') 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' := : 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' := 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)) by (cases f; cases f'; apply (equiv.eq_mk' p))

22
hott/types/pi.hlean
View file

@ -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]. /- 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. -/ /- Now we show how these things compute. -/
definition apD10_eq_of_homotopy (h : f ∼ g) : apD10 (eq_of_homotopy h) ∼ h := definition apd10_eq_of_homotopy (h : f ∼ g) : apd10 (eq_of_homotopy h) ∼ h :=
apD10 (retr apD10 h) apd10 (retr apd10 h)
definition eq_of_homotopy_eta (p : f = g) : eq_of_homotopy (apD10 p) = p := definition eq_of_homotopy_eta (p : f = g) : eq_of_homotopy (apd10 p) = p :=
sect apD10 p sect apd10 p
definition eq_of_homotopy_idp (f : Πa, B a) : eq_of_homotopy (λx : A, refl (f x)) = refl f := definition eq_of_homotopy_idp (f : Πa, B a) : eq_of_homotopy (λx : A, refl (f x)) = refl f :=
!eq_of_homotopy_eta !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. -/ /- 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) := 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) definition is_equiv_eq_of_homotopy [instance] (f g : Πx, B x)
: is_equiv (@eq_of_homotopy _ _ f g) := : 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) := definition homotopy_equiv_eq (f g : Πx, B x) : (f ∼ g) ≃ (f = g) :=
equiv.mk _ !is_equiv_eq_of_homotopy 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) 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) : (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) eq.rec_on p (λx, idp)
/- A special case of [transport_pi] where the type [B] does not depend on [A], /- 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') 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')))) := : ap (pi_functor f0 f1) (eq_of_homotopy h) = eq_of_homotopy (λa':A', (ap (f1 a') (h (f0 a')))) :=
begin 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, cases p,
apply concat, apply concat,
exact (ap (ap (pi_functor f0 f1)) (eq_of_homotopy_idp g)), 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 (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, 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 (tr_inv (λx, sect f0 a' ▹ x = h a') (retr (f1 _) _)), unfold function.id,
apply apD apply apd
end, end,
begin begin
intro h, intro h,
apply eq_of_homotopy, intro a, apply eq_of_homotopy, intro a,
apply (tr_inv (λx, retr f0 a ▹ x = h a) (sect (f1 _) _)), unfold function.id, apply (tr_inv (λx, retr f0 a ▹ x = h a) (sect (f1 _) _)), unfold function.id,
apply apD apply apd
end end
end end

4
hott/types/trunc.hlean
View file

@ -80,7 +80,7 @@ namespace is_trunc
(imp : Π{a b : A}, R a b → a = b) : is_hset A := (imp : Π{a b : A}, R a b → a = b) : is_hset A :=
is_hset_of_axiom_K is_hset_of_axiom_K
(λa p, (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) have H3 : Π(r : R a a), transport (λx, a = x) p (imp r)
= imp (transport (λx, R a x) p r), from = imp (transport (λx, R a x) p r), from
to_fun (equiv.symm !heq_pi) H2, 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 (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 (transport (λx, R a x) p (refl a)) : H3
... = imp (refl a) : is_hprop.elim, ... = 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) 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) (mere : Π(a b : A), is_hprop (R a b)) (refl : Π(a : A), R a a)