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