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style(hott): rename Pointed to pType

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also rename sigma_equiv_sigma_id to sigma_equiv_sigma_right and similarly for pi
This commit is contained in:
Floris van Doorn 2016-02-15 16:05:31 -05:00 committed by Leonardo de Moura
parent 43cf2ad23d
commit bac6d99cc7
23 changed files with 82 additions and 82 deletions
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6
hott/algebra/category/functor/attributes.hlean
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@ -143,15 +143,15 @@ namespace category
calc
fully_faithful F
≃ (Π(c c' : C), is_embedding (to_fun_hom F) × is_surjective (to_fun_hom F))
: pi_equiv_pi_id (λc, pi_equiv_pi_id
: pi_equiv_pi_right (λc, pi_equiv_pi_right
(λc', !is_equiv_equiv_is_embedding_times_is_surjective))
... ≃ (Π(c : C), (Π(c' : C), is_embedding (to_fun_hom F)) ×
(Π(c' : C), is_surjective (to_fun_hom F)))
: pi_equiv_pi_id (λc, !equiv_prod_corec)
: pi_equiv_pi_right (λc, !equiv_prod_corec)
... ≃ (Π(c c' : C), is_embedding (to_fun_hom F)) × full F
: equiv_prod_corec
... ≃ faithful F × full F
: prod_equiv_prod_right (pi_equiv_pi_id (λc, pi_equiv_pi_id
: prod_equiv_prod_right (pi_equiv_pi_right (λc, pi_equiv_pi_right
(λc', !is_embedding_equiv_is_injective)))
-/

2
hott/algebra/category/functor/yoneda.hlean
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@ -142,7 +142,7 @@ namespace yoneda
: is_prop (is_representable F) :=
begin
fapply is_trunc_equiv_closed,
{ exact proof fiber.sigma_char ɏ F qed ⬝e sigma.sigma_equiv_sigma_id (λc, !eq_equiv_iso)},
{ exact proof fiber.sigma_char ɏ F qed ⬝e sigma.sigma_equiv_sigma_right (λc, !eq_equiv_iso)},
{ apply function.is_prop_fiber_of_is_embedding, apply is_embedding_yoneda_embedding}
end
end

4
hott/algebra/homotopy_group.hlean
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@ -31,7 +31,7 @@ namespace eq
local attribute comm_group_homotopy_group [instance]
definition Pointed_homotopy_group [constructor] (n : ℕ) (A : Type*) : Type* :=
definition pType_homotopy_group [constructor] (n : ℕ) (A : Type*) : Type* :=
pointed.Mk (π[n] A)
definition Group_homotopy_group [constructor] (n : ℕ) (A : Type*) : Group :=
@ -43,7 +43,7 @@ namespace eq
definition fundamental_group [constructor] (A : Type*) : Group :=
Group_homotopy_group zero A
notation `πP[`:95 n:0 `] `:0 A:95 := Pointed_homotopy_group n A
notation `πP[`:95 n:0 `] `:0 A:95 := pType_homotopy_group n A
notation `πG[`:95 n:0 ` +1] `:0 A:95 := Group_homotopy_group n A
notation `πaG[`:95 n:0 ` +2] `:0 A:95 := CommGroup_homotopy_group n A

2
hott/function.hlean
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@ -158,7 +158,7 @@ namespace function
end,
apply is_trunc_equiv_closed, exact H2,
fapply is_trunc_equiv_closed,
{apply sigma_equiv_sigma_id, intro g, apply eq_equiv_homotopy},
{apply sigma_equiv_sigma_right, intro g, apply eq_equiv_homotopy},
fapply is_trunc_equiv_closed,
{apply fiber.sigma_char},
fapply is_contr_fiber_of_is_equiv,

2
hott/hit/coeq.hlean
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@ -131,7 +131,7 @@ section
{ intros b b' r, cases r, reflexivity },
rewrite (eq_of_homotopy3 H2) },
apply ap10 H1 },
apply equiv.trans (sigma_equiv_sigma_id H),
apply equiv.trans (sigma_equiv_sigma_right H),
apply equiv.trans !quotient.flattening.flattening_lemma,
fapply quotient.equiv,
{ reflexivity },

4
hott/hit/pointed_pushout.hlean
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@ -3,7 +3,7 @@ Copyright (c) 2016 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Floris van Doorn
Pointed Pushouts
pType Pushouts
-/
import .pushout types.pointed2
@ -17,7 +17,7 @@ namespace pointed
end pointed
open pointed Pointed
open pointed pType
namespace pushout
section

2
hott/hit/pushout.hlean
View file

@ -170,7 +170,7 @@ namespace pushout
{ intros a a' r, cases r, reflexivity },
rewrite (eq_of_homotopy3 H2) },
apply ap10 H1 },
apply equiv.trans (sigma_equiv_sigma_id H),
apply equiv.trans (sigma_equiv_sigma_right H),
apply equiv.trans (quotient.flattening.flattening_lemma R (sum.rec Pinl Pinr) Pglue'),
fapply equiv.MK,
{ intro q, induction q with z z z' fr,

2
hott/homotopy/cofiber.hlean
View file

@ -48,7 +48,7 @@ namespace cofiber
end
end cofiber
-- Pointed version
-- pType version
definition Cofiber {A B : Type*} (f : A →* B) : Type* := Pushout (pconst A Unit) f

6
hott/homotopy/connectedness.hlean
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@ -83,11 +83,11 @@ namespace homotopy
rewrite [-(apd10_ap_precompose_dependent f (eq_of_homotopy r))],
apply equiv.symm,
apply eq_equiv_fn_eq (@apd10 A (λa, P (f a)) (λa, g (f a)) (λa, h (f a))) },
apply equiv.trans (sigma.sigma_equiv_sigma_id e'), clear e',
apply equiv.trans (sigma.sigma_equiv_sigma_right e'), clear e',
apply equiv.trans (equiv.symm (sigma.sigma_equiv_sigma_left
eq_equiv_homotopy)),
apply equiv.symm, apply equiv.trans !fiber_eq_equiv,
apply sigma.sigma_equiv_sigma_id, intro r,
apply sigma.sigma_equiv_sigma_right, intro r,
apply eq_equiv_eq_symm },
apply @is_trunc_equiv_closed _ _ k e, clear e,
apply IH (λb : B, trunctype.mk (g b = h b)
@ -155,7 +155,7 @@ namespace homotopy
-- as special case we get elimination principles for pointed connected types
namespace is_conn
open pointed Pointed unit
open pointed pType unit
section
parameters {n : trunc_index} {A : Type*}
[H : is_conn n .+1 A] (P : A → n -Type)

2
hott/homotopy/red_susp.hlean
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@ -13,7 +13,7 @@ open simple_two_quotient eq unit pointed e_closure
namespace red_susp
section
parameter {A : Pointed}
parameter {A : pType}
inductive red_susp_R : unit → unit → Type :=
| Rmk : Π(a : A), red_susp_R star star

2
hott/homotopy/smash.hlean
View file

@ -8,7 +8,7 @@ The Smash Product of Types
import hit.pushout .wedge .cofiber .susp .sphere
open eq pushout prod pointed Pointed is_trunc
open eq pushout prod pointed pType is_trunc
definition product_of_wedge [constructor] (A B : Type*) : Wedge A B →* A ×* B :=
begin

12
hott/homotopy/sphere.hlean
View file

@ -92,7 +92,7 @@ namespace sphere
definition base {n : ℕ} : sphere n := north
definition pointed_sphere [instance] [constructor] (n : ℕ) : pointed (sphere n) :=
pointed.mk base
definition Sphere [constructor] (n : ℕ) : Pointed := pointed.mk' (sphere n)
definition Sphere [constructor] (n : ℕ) : pType := pointed.mk' (sphere n)
namespace ops
abbreviation S := sphere
@ -106,7 +106,7 @@ namespace sphere
definition surf {n : ℕ} : Ω[n] S. n :=
nat.rec_on n (proof base qed)
(begin intro m s, refine cast _ (apn m (equator m) s),
exact ap Pointed.carrier !loop_space_succ_eq_in⁻¹ end)
exact ap pType.carrier !loop_space_succ_eq_in⁻¹ end)
definition bool_of_sphere : S 0 → bool :=
@ -126,10 +126,10 @@ namespace sphere
ua sphere_equiv_bool
definition sphere_eq_bool_pointed : S. 0 = Bool :=
Pointed_eq sphere_equiv_bool idp
pType_eq sphere_equiv_bool idp
-- TODO: the commented-out part makes the forward function below "apn _ surf"
definition pmap_sphere (A : Pointed) (n : ℕ) : map₊ (S. n) A ≃ Ω[n] A :=
definition pmap_sphere (A : pType) (n : ℕ) : map₊ (S. n) A ≃ Ω[n] A :=
begin
-- fapply equiv_change_fun,
-- {
@ -143,10 +143,10 @@ namespace sphere
-- { exact sorry}}
end
protected definition elim {n : ℕ} {P : Pointed} (p : Ω[n] P) : map₊ (S. n) P :=
protected definition elim {n : ℕ} {P : pType} (p : Ω[n] P) : map₊ (S. n) P :=
to_inv !pmap_sphere p
-- definition elim_surf {n : ℕ} {P : Pointed} (p : Ω[n] P) : apn n (sphere.elim p) surf = p :=
-- definition elim_surf {n : ℕ} {P : pType} (p : Ω[n] P) : apn n (sphere.elim p) surf = p :=
-- begin
-- induction n with n IH,
-- { esimp [apn,surf,sphere.elim,pmap_sphere], apply sorry},

14
hott/homotopy/susp.hlean
View file

@ -80,12 +80,12 @@ attribute susp.elim_type_on [unfold 2]
namespace susp
open pointed
variables {X Y Z : Pointed}
variables {X Y Z : pType}
definition pointed_susp [instance] [constructor] (X : Type) : pointed (susp X) :=
pointed.mk north
definition Susp [constructor] (X : Type) : Pointed :=
definition Susp [constructor] (X : Type) : pType :=
pointed.mk' (susp X)
definition Susp_functor (f : X →* Y) : Susp X →* Susp Y :=
@ -112,7 +112,7 @@ namespace susp
-- adjunction from Coq-HoTT
definition loop_susp_unit [constructor] (X : Pointed) : X →* Ω(Susp X) :=
definition loop_susp_unit [constructor] (X : pType) : X →* Ω(Susp X) :=
begin
fconstructor,
{ intro x, exact merid x ⬝ (merid pt)⁻¹},
@ -141,7 +141,7 @@ namespace susp
xrewrite [idp_con_idp, -ap_compose (concat idp)]},
end
definition loop_susp_counit [constructor] (X : Pointed) : Susp (Ω X) →* X :=
definition loop_susp_counit [constructor] (X : pType) : Susp (Ω X) →* X :=
begin
fconstructor,
{ intro x, induction x, exact pt, exact pt, exact a},
@ -162,7 +162,7 @@ namespace susp
{ reflexivity}
end
definition loop_susp_counit_unit (X : Pointed)
definition loop_susp_counit_unit (X : pType)
: ap1 (loop_susp_counit X) ∘* loop_susp_unit (Ω X) ~* pid (Ω X) :=
begin
induction X with X x, fconstructor,
@ -176,7 +176,7 @@ namespace susp
xrewrite [ap_con_right_inv (susp.elim x x (λa, a)) (merid idp),idp_con_idp,-ap_compose]}
end
definition loop_susp_unit_counit (X : Pointed)
definition loop_susp_unit_counit (X : pType)
: loop_susp_counit (Susp X) ∘* Susp_functor (loop_susp_unit X) ~* pid (Susp X) :=
begin
induction X with X x, fconstructor,
@ -189,7 +189,7 @@ namespace susp
{ reflexivity}
end
definition susp_adjoint_loop (X Y : Pointed) : map₊ (pointed.mk' (susp X)) Y ≃ map₊ X (Ω Y) :=
definition susp_adjoint_loop (X Y : pType) : map₊ (pointed.mk' (susp X)) Y ≃ map₊ X (Ω Y) :=
begin
fapply equiv.MK,
{ intro f, exact ap1 f ∘* loop_susp_unit X},

4
hott/homotopy/wedge.hlean
View file

@ -3,11 +3,11 @@ Copyright (c) 2016 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Ulrik Buchholtz
The Wedge Sum of Two Pointed Types
The Wedge Sum of Two pType Types
-/
import hit.pointed_pushout .connectedness
open eq pushout pointed Pointed unit
open eq pushout pointed pType unit
definition Wedge (A B : Type*) : Type* := Pushout (pconst Unit A) (pconst Unit B)

2
hott/types/arrow.hlean
View file

@ -49,7 +49,7 @@ namespace pi
variables {B}
definition arrow_equiv_arrow_right' [constructor] (f1 : A → (B ≃ B')) : (A → B) ≃ (A → B') :=
pi_equiv_pi_id f1
pi_equiv_pi_right f1
/- Equivalence if one of the types is contractible -/

4
hott/types/equiv.hlean
View file

@ -32,7 +32,7 @@ namespace is_equiv
definition is_contr_right_inverse : is_contr (Σ(g : B → A), f ∘ g ~ id) :=
begin
fapply is_trunc_equiv_closed,
{apply sigma_equiv_sigma_id, intro g, apply eq_equiv_homotopy},
{apply sigma_equiv_sigma_right, intro g, apply eq_equiv_homotopy},
fapply is_trunc_equiv_closed,
{apply fiber.sigma_char},
fapply is_contr_fiber_of_is_equiv,
@ -45,7 +45,7 @@ namespace is_equiv
fapply is_trunc_equiv_closed,
{apply equiv.symm, apply sigma_pi_equiv_pi_sigma},
fapply is_trunc_equiv_closed,
{apply pi_equiv_pi_id, intro a,
{apply pi_equiv_pi_right, intro a,
apply (fiber_eq_equiv (fiber.mk (u.1 (f a)) (u.2 (f a))) (fiber.mk a idp))},
end

16
hott/types/fiber.hlean
View file

@ -34,7 +34,7 @@ namespace fiber
apply eq_equiv_fn_eq_of_equiv, apply fiber.sigma_char,
apply equiv.trans,
apply sigma_eq_equiv,
apply sigma_equiv_sigma_id,
apply sigma_equiv_sigma_right,
intro p,
apply pathover_eq_equiv_Fl,
end
@ -48,13 +48,13 @@ namespace fiber
calc
fiber pr1 a ≃ Σu, u.1 = a : fiber.sigma_char
... ≃ Σa' (b : B a'), a' = a : sigma_assoc_equiv
... ≃ Σa' (p : a' = a), B a' : sigma_equiv_sigma_id (λa', !comm_equiv_nondep)
... ≃ Σa' (p : a' = a), B a' : sigma_equiv_sigma_right (λa', !comm_equiv_nondep)
... ≃ Σu, B u.1 : sigma_assoc_equiv
... ≃ B a : !sigma_equiv_of_is_contr_left
definition sigma_fiber_equiv (f : A → B) : (Σb, fiber f b) ≃ A :=
calc
(Σb, fiber f b) ≃ Σb a, f a = b : sigma_equiv_sigma_id (λb, !fiber.sigma_char)
(Σb, fiber f b) ≃ Σb a, f a = b : sigma_equiv_sigma_right (λb, !fiber.sigma_char)
... ≃ Σa b, f a = b : sigma_comm_equiv
... ≃ A : sigma_equiv_of_is_contr_right
@ -77,7 +77,7 @@ namespace fiber
calc
fiber (g ∘ f) (g b) ≃ Σa : A, g (f a) = g b : fiber.sigma_char
... ≃ Σa : A, f a = b : begin
apply sigma_equiv_sigma_id, intro a,
apply sigma_equiv_sigma_right, intro a,
apply equiv.symm, apply eq_equiv_fn_eq
end
... ≃ fiber f b : fiber.sigma_char
@ -142,14 +142,14 @@ namespace fiber
... ≃ Σ(x : A), Σ(p : P x), Σ(H : x = a), f x p =[H] q
:
begin
apply sigma_equiv_sigma_id, intro x,
apply sigma_equiv_sigma_id, intro p,
apply sigma_equiv_sigma_right, intro x,
apply sigma_equiv_sigma_right, intro p,
apply sigma_eq_equiv
end
... ≃ Σ(x : A), Σ(H : x = a), Σ(p : P x), f x p =[H] q
:
begin
apply sigma_equiv_sigma_id, intro x,
apply sigma_equiv_sigma_right, intro x,
apply sigma_comm_equiv
end
... ≃ Σ(w : Σx, x = a), Σ(p : P w.1), f w.1 p =[w.2] q
@ -161,7 +161,7 @@ namespace fiber
... ≃ Σ(p : P a), f a p = q
:
begin
apply sigma_equiv_sigma_id, intro p,
apply sigma_equiv_sigma_right, intro p,
apply pathover_idp
end
... ≃ fiber (f a) q

2
hott/types/pi.hlean
View file

@ -214,7 +214,7 @@ namespace pi
: (Πa, B a) ≃ (Πa', B' a') :=
pi_equiv_pi_of_is_equiv (to_fun f0) (λa', to_fun (f1 a'))
definition pi_equiv_pi_id [constructor] {P Q : A → Type} (g : Πa, P a ≃ Q a)
definition pi_equiv_pi_right [constructor] {P Q : A → Type} (g : Πa, P a ≃ Q a)
: (Πa, P a) ≃ (Πa, Q a) :=
pi_equiv_pi equiv.refl g

26
hott/types/pointed.hlean
View file

@ -12,23 +12,23 @@ open is_trunc eq prod sigma nat equiv option is_equiv bool unit algebra equiv.op
structure pointed [class] (A : Type) :=
(point : A)
structure Pointed :=
structure pType :=
(carrier : Type)
(Point : carrier)
open Pointed
open pType
notation `Type*` := Pointed
notation `Type*` := pType
namespace pointed
attribute Pointed.carrier [coercion]
attribute pType.carrier [coercion]
variables {A B : Type}
definition pt [unfold 2] [H : pointed A] := point A
protected definition Mk [constructor] {A : Type} (a : A) := Pointed.mk A a
protected definition MK [constructor] (A : Type) (a : A) := Pointed.mk A a
protected definition Mk [constructor] {A : Type} (a : A) := pType.mk A a
protected definition MK [constructor] (A : Type) (a : A) := pType.mk A a
protected definition mk' [constructor] (A : Type) [H : pointed A] : Type* :=
Pointed.mk A (point A)
pType.mk A (point A)
definition pointed_carrier [instance] [constructor] (A : Type*) : pointed A :=
pointed.mk (Point A)
@ -89,15 +89,15 @@ namespace pointed
Ω[n] (pointed.mk' A)
open equiv.ops
definition Pointed_eq {A B : Type*} (f : A ≃ B) (p : f pt = pt) : A = B :=
definition pType_eq {A B : Type*} (f : A ≃ B) (p : f pt = pt) : A = B :=
begin
cases A with A a, cases B with B b, esimp at *,
fapply apd011 @Pointed.mk,
fapply apd011 @pType.mk,
{ apply ua f},
{ rewrite [cast_ua,p]},
end
protected definition Pointed.sigma_char.{u} : Pointed.{u} ≃ Σ(X : Type.{u}), X :=
protected definition pType.sigma_char.{u} : pType.{u} ≃ Σ(X : Type.{u}), X :=
begin
fapply equiv.MK,
{ intro x, induction x with X x, exact ⟨X, x⟩},
@ -149,7 +149,7 @@ namespace pointed
definition loop_space_loop_irrel (p : point A = point A) : Ω(pointed.Mk p) = Ω[2] A :=
begin
intros, fapply Pointed_eq,
intros, fapply pType_eq,
{ esimp, transitivity _,
apply eq_equiv_fn_eq_of_equiv (equiv_eq_closed_right _ p⁻¹),
esimp, apply eq_equiv_eq_closed, apply con.right_inv, apply con.right_inv},
@ -157,7 +157,7 @@ namespace pointed
end
definition iterated_loop_space_loop_irrel (n : ℕ) (p : point A = point A)
: Ω[succ n](pointed.Mk p) = Ω[succ (succ n)] A :> Pointed :=
: Ω[succ n](pointed.Mk p) = Ω[succ (succ n)] A :> pType :=
calc
Ω[succ n](pointed.Mk p) = Ω[n](Ω (pointed.Mk p)) : loop_space_succ_eq_in
... = Ω[n] (Ω[2] A) : loop_space_loop_irrel
@ -297,7 +297,7 @@ namespace pointed
end
definition pcast [constructor] {A B : Type*} (p : A = B) : A →* B :=
proof pmap.mk (cast (ap Pointed.carrier p)) (by induction p; reflexivity) qed
proof pmap.mk (cast (ap pType.carrier p)) (by induction p; reflexivity) qed
definition pinverse [constructor] {X : Type*} : Ω X →* Ω X :=
pmap.mk eq.inverse idp

6
hott/types/pointed2.hlean
View file

@ -19,7 +19,7 @@ structure pequiv (A B : Type*) extends equiv A B, pmap A B
section
universe variable u
structure ptrunctype (n : trunc_index) extends trunctype.{u} n, Pointed.{u}
structure ptrunctype (n : trunc_index) extends trunctype.{u} n, pType.{u}
end
namespace pointed
@ -45,7 +45,7 @@ namespace pointed
pmap.mk f⁻¹ (ap f⁻¹ (respect_pt f)⁻¹ ⬝ !left_inv)
definition pua {A B : Type*} (f : A ≃* B) : A = B :=
Pointed_eq (equiv_of_pequiv f) !respect_pt
pType_eq (equiv_of_pequiv f) !respect_pt
protected definition pequiv.refl [refl] [constructor] (A : Type*) : A ≃* A :=
pequiv_of_pmap !pid !is_equiv_id
@ -76,7 +76,7 @@ namespace pointed
pequiv_of_eq p ⬝e* q
definition eq_of_pequiv {A B : Type*} (p : A ≃* B) : A = B :=
Pointed_eq (equiv_of_pequiv p) !respect_pt
pType_eq (equiv_of_pequiv p) !respect_pt
definition peap {A B : Type*} (F : Type* → Type*) (p : A ≃* B) : F A ≃* F B :=
pequiv_of_pmap (pcast (ap F (eq_of_pequiv p))) begin cases eq_of_pequiv p, apply is_equiv_id end

8
hott/types/sigma.hlean
View file

@ -242,7 +242,7 @@ namespace sigma
(Σa, B a) ≃ (Σa', B' a') :=
sigma_equiv_sigma_of_is_equiv (to_fun Hf) (λ a, to_fun (Hg a))
definition sigma_equiv_sigma_id [constructor] {B' : A → Type} (Hg : Π a, B a ≃ B' a)
definition sigma_equiv_sigma_right [constructor] {B' : A → Type} (Hg : Π a, B a ≃ B' a)
: (Σa, B a) ≃ Σa, B' a :=
sigma_equiv_sigma equiv.refl Hg
@ -332,9 +332,9 @@ namespace sigma
: (Σ(v : Σa, B a), C v.1) ≃ (Σ(u : Σa, C a), B u.1) :=
calc (Σ(v : Σa, B a), C v.1)
≃ (Σa (b : B a), C a) : !sigma_assoc_equiv⁻¹ᵉ
... ≃ (Σa, B a × C a) : sigma_equiv_sigma_id (λa, !equiv_prod)
... ≃ (Σa, C a × B a) : sigma_equiv_sigma_id (λa, !prod_comm_equiv)
... ≃ (Σa (c : C a), B a) : sigma_equiv_sigma_id (λa, !equiv_prod)
... ≃ (Σa, B a × C a) : sigma_equiv_sigma_right (λa, !equiv_prod)
... ≃ (Σa, C a × B a) : sigma_equiv_sigma_right (λa, !prod_comm_equiv)
... ≃ (Σa (c : C a), B a) : sigma_equiv_sigma_right (λa, !equiv_prod)
... ≃ (Σ(u : Σa, C a), B u.1) : sigma_assoc_equiv
/- Interaction with other type constructors -/

18
hott/types/univ.hlean
View file

@ -103,26 +103,26 @@ namespace univ
end
definition is_object_classifier (f : A → B)
: pullback_square (pointed_fiber f) (fiber f) f Pointed.carrier :=
: pullback_square (pointed_fiber f) (fiber f) f pType.carrier :=
pullback_square.mk
(λa, idp)
(is_equiv_of_equiv_of_homotopy
(calc
A ≃ Σb, fiber f b : sigma_fiber_equiv
... ≃ Σb (v : ΣX, X = fiber f b), v.1 : sigma_equiv_sigma_id
... ≃ Σb (v : ΣX, X = fiber f b), v.1 : sigma_equiv_sigma_right
(λb, !sigma_equiv_of_is_contr_left)
... ≃ Σb X (p : X = fiber f b), X : sigma_equiv_sigma_id
... ≃ Σb X (p : X = fiber f b), X : sigma_equiv_sigma_right
(λb, !sigma_assoc_equiv)
... ≃ Σb X (x : X), X = fiber f b : sigma_equiv_sigma_id
(λb, sigma_equiv_sigma_id
... ≃ Σb X (x : X), X = fiber f b : sigma_equiv_sigma_right
(λb, sigma_equiv_sigma_right
(λX, !comm_equiv_nondep))
... ≃ Σb (v : ΣX, X), v.1 = fiber f b : sigma_equiv_sigma_id
... ≃ Σb (v : ΣX, X), v.1 = fiber f b : sigma_equiv_sigma_right
(λb, !sigma_assoc_equiv⁻¹)
... ≃ Σb (Y : Type*), Y = fiber f b : sigma_equiv_sigma_id
(λb, sigma_equiv_sigma (Pointed.sigma_char)⁻¹
... ≃ Σb (Y : Type*), Y = fiber f b : sigma_equiv_sigma_right
(λb, sigma_equiv_sigma (pType.sigma_char)⁻¹
(λv, sigma.rec_on v (λx y, equiv.refl)))
... ≃ Σ(Y : Type*) b, Y = fiber f b : sigma_comm_equiv
... ≃ pullback Pointed.carrier (fiber f) : !pullback.sigma_char⁻¹ᵉ
... ≃ pullback pType.carrier (fiber f) : !pullback.sigma_char⁻¹ᵉ
)
proof λb, idp qed)

4
src/emacs/lean-syntax.el
View file

@ -175,8 +175,8 @@
(,(rx word-start (group "example") ".") (1 'font-lock-keyword-face))
(,(rx (or "∎")) . 'font-lock-keyword-face)
;; Types
(,(rx word-start (or "Prop" "Type" "Type'" "Type₊" "Type₀" "Type₁" "Type₂" "Type₃" "Type*" "Set") symbol-end) . 'font-lock-type-face)
(,(rx word-start (group (or "Prop" "Type" "Set")) ".") (1 'font-lock-type-face))
(,(rx word-start (or "Prop" "Type" "Type'" "Type₊" "Type₀" "Type₁" "Type₂" "Type₃" "Type*" "pType" "Set") symbol-end) . 'font-lock-type-face)
(,(rx word-start (group (or "Prop" "Type" "Set" "pType")) ".") (1 'font-lock-type-face))
;; String
("\"[^\"]*\"" . 'font-lock-string-face)
;; ;; Constants