lean2/hott/algebra/e_closure.hlean

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/-
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Floris van Doorn
The "equivalence closure" of a type-valued relation.
Given a binary type-valued relation (fibration), we add reflexivity, symmetry and transitivity terms
-/
import .relation eq2 arity
open eq
inductive e_closure {A : Type} (R : A → A → Type) : A → A → Type :=
| of_rel : Π{a a'} (r : R a a'), e_closure R a a'
| refl : Πa, e_closure R a a
| symm : Π{a a'} (r : e_closure R a a'), e_closure R a' a
| trans : Π{a a' a''} (r : e_closure R a a') (r' : e_closure R a' a''), e_closure R a a''
namespace e_closure
infix `⬝r`:75 := e_closure.trans
postfix `⁻¹ʳ`:(max+10) := e_closure.symm
notation `[`:max a `]`:0 := e_closure.of_rel a
abbreviation rfl {A : Type} {R : A → A → Type} {a : A} := refl R a
end e_closure
namespace relation
section
parameters {A : Type}
(R : A → A → Type)
local abbreviation T := e_closure R
variables ⦃a a' : A⦄ {s : R a a'} {r : T a a}
parameter {R}
protected definition e_closure.elim [unfold 8] {B : Type} {f : A → B}
(e : Π⦃a a' : A⦄, R a a' → f a = f a') (t : T a a') : f a = f a' :=
begin
induction t,
exact e r,
reflexivity,
exact v_0⁻¹,
exact v_0 ⬝ v_1
end
definition ap_e_closure_elim_h [unfold 12] {B C : Type} {f : A → B} {g : B → C}
(e : Π⦃a a' : A⦄, R a a' → f a = f a')
{e' : Π⦃a a' : A⦄, R a a' → g (f a) = g (f a')}
(p : Π⦃a a' : A⦄ (s : R a a'), ap g (e s) = e' s) (t : T a a')
: ap g (e_closure.elim e t) = e_closure.elim e' t :=
begin
induction t,
apply p,
reflexivity,
exact ap_inv g (e_closure.elim e r) ⬝ inverse2 v_0,
exact ap_con g (e_closure.elim e r) (e_closure.elim e r') ⬝ (v_0 ◾ v_1)
end
definition ap_e_closure_elim {B C : Type} {f : A → B} (g : B → C)
(e : Π⦃a a' : A⦄, R a a' → f a = f a') (t : T a a')
: ap g (e_closure.elim e t) = e_closure.elim (λa a' r, ap g (e r)) t :=
ap_e_closure_elim_h e (λa a' s, idp) t
definition ap_e_closure_elim_h_eq {B C : Type} {f : A → B} {g : B → C}
(e : Π⦃a a' : A⦄, R a a' → f a = f a')
{e' : Π⦃a a' : A⦄, R a a' → g (f a) = g (f a')}
(p : Π⦃a a' : A⦄ (s : R a a'), ap g (e s) = e' s) (t : T a a')
: ap_e_closure_elim_h e p t =
ap_e_closure_elim g e t ⬝ ap (λx, e_closure.elim x t) (eq_of_homotopy3 p) :=
begin
fapply homotopy3.rec_on p,
intro q, esimp at q, induction q,
esimp, rewrite eq_of_homotopy3_id
end
theorem ap_ap_e_closure_elim_h {B C D : Type} {f : A → B}
{g : B → C} (h : C → D)
(e : Π⦃a a' : A⦄, R a a' → f a = f a')
{e' : Π⦃a a' : A⦄, R a a' → g (f a) = g (f a')}
(p : Π⦃a a' : A⦄ (s : R a a'), ap g (e s) = e' s) (t : T a a')
: square (ap (ap h) (ap_e_closure_elim_h e p t))
(ap_e_closure_elim_h e (λa a' s, ap_compose h g (e s)) t)
(ap_compose h g (e_closure.elim e t))⁻¹
(ap_e_closure_elim_h e' (λa a' s, (ap (ap h) (p s))⁻¹) t) :=
begin
induction t,
{ esimp,
apply square_of_eq, exact !con.right_inv ⬝ !con.left_inv⁻¹},
{ apply ids},
{ rewrite [▸*,ap_con (ap h)],
refine (transpose !ap_compose_inv)⁻¹ᵛ ⬝h _,
rewrite [con_inv,inv_inv,-inv2_inv],
exact !ap_inv2 ⬝v square_inv2 v_0},
{ rewrite [▸*,ap_con (ap h)],
refine (transpose !ap_compose_con)⁻¹ᵛ ⬝h _,
rewrite [con_inv,inv_inv,con2_inv],
refine !ap_con2 ⬝v square_con2 v_0 v_1},
end
theorem ap_ap_e_closure_elim {B C D : Type} {f : A → B}
(g : B → C) (h : C → D)
(e : Π⦃a a' : A⦄, R a a' → f a = f a') (t : T a a')
: square (ap (ap h) (ap_e_closure_elim g e t))
(ap_e_closure_elim_h e (λa a' s, ap_compose h g (e s)) t)
(ap_compose h g (e_closure.elim e t))⁻¹
(ap_e_closure_elim h (λa a' r, ap g (e r)) t) :=
!ap_ap_e_closure_elim_h
open e_closure
definition is_equivalence_e_closure : is_equivalence T :=
begin
constructor,
intro a, exact rfl,
intro a a' t, exact t⁻¹ʳ,
intro a a' a'' t t', exact t ⬝r t',
end
end
end relation