179 lines
9.3 KiB
Text
179 lines
9.3 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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Squareovers
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-/
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import .square
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open eq equiv is_equiv equiv.ops
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namespace eq
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-- we give the argument B explicitly, because Lean would find (λa, B a) by itself, which
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-- makes the type uglier (of course the two terms are definitionally equal)
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inductive squareover {A : Type} (B : A → Type) {a₀₀ : A} {b₀₀ : B a₀₀} :
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Π{a₂₀ a₀₂ a₂₂ : A}
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{p₁₀ : a₀₀ = a₂₀} {p₁₂ : a₀₂ = a₂₂} {p₀₁ : a₀₀ = a₀₂} {p₂₁ : a₂₀ = a₂₂}
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(s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
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{b₂₀ : B a₂₀} {b₀₂ : B a₀₂} {b₂₂ : B a₂₂}
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(q₁₀ : pathover B b₀₀ p₁₀ b₂₀) (q₁₂ : pathover B b₀₂ p₁₂ b₂₂)
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(q₀₁ : pathover B b₀₀ p₀₁ b₀₂) (q₂₁ : pathover B b₂₀ p₂₁ b₂₂),
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Type :=
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idsquareo : squareover B ids idpo idpo idpo idpo
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variables {A A' : Type} {B : A → Type}
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{a a' a'' a₀₀ a₂₀ a₄₀ a₀₂ a₂₂ a₂₄ a₀₄ a₄₂ a₄₄ : A}
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/-a₀₀-/ {p₁₀ : a₀₀ = a₂₀} /-a₂₀-/ {p₃₀ : a₂₀ = a₄₀} /-a₄₀-/
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{p₀₁ : a₀₀ = a₀₂} /-s₁₁-/ {p₂₁ : a₂₀ = a₂₂} /-s₃₁-/ {p₄₁ : a₄₀ = a₄₂}
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/-a₀₂-/ {p₁₂ : a₀₂ = a₂₂} /-a₂₂-/ {p₃₂ : a₂₂ = a₄₂} /-a₄₂-/
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{p₀₃ : a₀₂ = a₀₄} /-s₁₃-/ {p₂₃ : a₂₂ = a₂₄} /-s₃₃-/ {p₄₃ : a₄₂ = a₄₄}
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/-a₀₄-/ {p₁₄ : a₀₄ = a₂₄} /-a₂₄-/ {p₃₄ : a₂₄ = a₄₄} /-a₄₄-/
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{s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁}
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{b₀₀ : B a₀₀} {b₂₀ : B a₂₀} {b₄₀ : B a₄₀}
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{b₀₂ : B a₀₂} {b₂₂ : B a₂₂} {b₄₂ : B a₄₂}
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{b₀₄ : B a₀₄} {b₂₄ : B a₂₄} {b₄₄ : B a₄₄}
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/-b₀₀-/ {q₁₀ : b₀₀ =[p₁₀] b₂₀} /-b₂₀-/ {q₃₀ : b₂₀ =[p₃₀] b₄₀} /-b₄₀-/
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{q₀₁ : b₀₀ =[p₀₁] b₀₂} /-t₁₁-/ {q₂₁ : b₂₀ =[p₂₁] b₂₂} /-t₃₁-/ {q₄₁ : b₄₀ =[p₄₁] b₄₂}
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/-b₀₂-/ {q₁₂ : b₀₂ =[p₁₂] b₂₂} /-b₂₂-/ {q₃₂ : b₂₂ =[p₃₂] b₄₂} /-b₄₂-/
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{q₀₃ : b₀₂ =[p₀₃] b₀₄} /-t₁₃-/ {q₂₃ : b₂₂ =[p₂₃] b₂₄} /-t₃₃-/ {q₄₃ : b₄₂ =[p₄₃] b₄₄}
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/-b₀₄-/ {q₁₄ : b₀₄ =[p₁₄] b₂₄} /-b₂₄-/ {q₃₄ : b₂₄ =[p₃₄] b₄₄} /-b₄₄-/
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definition squareo := @squareover A B a₀₀
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definition idsquareo [reducible] [constructor] (b₀₀ : B a₀₀) := @squareover.idsquareo A B a₀₀ b₀₀
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definition idso [reducible] [constructor] := @squareover.idsquareo A B a₀₀ b₀₀
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definition apds (f : Πa, B a) (s : square p₁₀ p₁₂ p₀₁ p₂₁)
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: squareover B s (apdo f p₁₀) (apdo f p₁₂) (apdo f p₀₁) (apdo f p₂₁) :=
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square.rec_on s idso
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definition vrflo : squareover B vrfl q₁₀ q₁₀ idpo idpo :=
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by induction q₁₀; exact idso
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definition hrflo : squareover B hrfl idpo idpo q₁₀ q₁₀ :=
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by induction q₁₀; exact idso
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definition vdeg_squareover {q₁₀' : b₀₀ =[p₁₀] b₂₀} (r : q₁₀ = q₁₀')
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: squareover B vrfl q₁₀ q₁₀' idpo idpo :=
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by induction r;exact vrflo
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definition hdeg_squareover {q₀₁' : b₀₀ =[p₀₁] b₀₂} (r : q₀₁ = q₀₁')
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: squareover B hrfl idpo idpo q₀₁ q₀₁' :=
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by induction r; exact hrflo
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-- relating squareovers to squares
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definition square_of_squareover (t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁) :
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square (!con_tr ⬝ ap (λa, p₂₁ ▸ a) (tr_eq_of_pathover q₁₀))
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(tr_eq_of_pathover q₁₂)
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(ap (λq, q ▸ b₀₀) (eq_of_square s₁₁) ⬝ !con_tr ⬝ ap (λa, p₁₂ ▸ a) (tr_eq_of_pathover q₀₁))
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(tr_eq_of_pathover q₂₁) :=
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by induction t₁₁; esimp; constructor
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/-
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definition squareover_of_square
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(q : square (!con_tr ⬝ ap (λa, p₂₁ ▸ a) (tr_eq_of_pathover q₁₀))
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(tr_eq_of_pathover q₁₂)
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(ap (λq, q ▸ b₀₀) (eq_of_square s₁₁) ⬝ !con_tr ⬝ ap (λa, p₁₂ ▸ a) (tr_eq_of_pathover q₀₁))
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(tr_eq_of_pathover q₂₁))
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: squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁ :=
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sorry
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-/
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definition square_of_squareover_ids {b₀₀ b₀₂ b₂₀ b₂₂ : B a}
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(t : b₀₀ = b₂₀) (b : b₀₂ = b₂₂) (l : b₀₀ = b₀₂) (r : b₂₀ = b₂₂)
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(so : squareover B ids (pathover_idp_of_eq t)
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(pathover_idp_of_eq b)
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(pathover_idp_of_eq l)
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(pathover_idp_of_eq r)) : square t b l r :=
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begin
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let H := square_of_squareover so, -- use apply ... in
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rewrite [▸* at H,+idp_con at H,+ap_id at H,↑pathover_idp_of_eq at H],
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rewrite [+to_right_inv !(pathover_equiv_tr_eq (refl a)) at H],
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exact H
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end
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definition squareover_ids_of_square {b₀₀ b₀₂ b₂₀ b₂₂ : B a}
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(t : b₀₀ = b₂₀) (b : b₀₂ = b₂₂) (l : b₀₀ = b₀₂) (r : b₂₀ = b₂₂) (q : square t b l r)
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: squareover B ids (pathover_idp_of_eq t)
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(pathover_idp_of_eq b)
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(pathover_idp_of_eq l)
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(pathover_idp_of_eq r) :=
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square.rec_on q idso
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-- relating pathovers to squareovers
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definition pathover_of_squareover (t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁)
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: q₁₀ ⬝o q₂₁ =[eq_of_square s₁₁] q₀₁ ⬝o q₁₂ :=
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by induction t₁₁; constructor
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definition squareover_of_pathover {s : p₁₀ ⬝ p₂₁ = p₀₁ ⬝ p₁₂}
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(r : q₁₀ ⬝o q₂₁ =[s] q₀₁ ⬝o q₁₂) : squareover B (square_of_eq s) q₁₀ q₁₂ q₀₁ q₂₁ :=
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by induction q₁₂; esimp [concato] at r;induction r;induction q₂₁;induction q₁₀;constructor
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definition pathover_top_of_squareover (t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁)
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: q₁₀ =[eq_top_of_square s₁₁] q₀₁ ⬝o q₁₂ ⬝o q₂₁⁻¹ᵒ :=
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by induction t₁₁; constructor
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definition squareover_of_pathover_top {s : p₁₀ = p₀₁ ⬝ p₁₂ ⬝ p₂₁⁻¹}
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(r : q₁₀ =[s] q₀₁ ⬝o q₁₂ ⬝o q₂₁⁻¹ᵒ)
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: squareover B (square_of_eq_top s) q₁₀ q₁₂ q₀₁ q₂₁ :=
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by induction q₂₁; induction q₁₂; esimp at r;induction r;induction q₁₀;constructor
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definition squareover_of_eq_top (r : change_path (eq_top_of_square s₁₁) q₁₀ = q₀₁ ⬝o q₁₂ ⬝o q₂₁⁻¹ᵒ)
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: squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁ :=
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begin
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induction s₁₁, revert q₁₂ q₁₀ r,
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eapply idp_rec_on q₂₁, clear q₂₁,
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intro q₁₂,
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eapply idp_rec_on q₁₂, clear q₁₂,
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esimp, intros,
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induction r, eapply idp_rec_on q₁₀,
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constructor
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end
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definition eq_top_of_squareover (r : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁)
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: change_path (eq_top_of_square s₁₁) q₁₀ = q₀₁ ⬝o q₁₂ ⬝o q₂₁⁻¹ᵒ :=
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by induction r; reflexivity
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/-
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definition squareover_equiv_pathover (q₁₀ : b₀₀ =[p₁₀] b₂₀) (q₁₂ : b₀₂ =[p₁₂] b₂₂)
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(q₀₁ : b₀₀ =[p₀₁] b₀₂) (q₂₁ : b₂₀ =[p₂₁] b₂₂)
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: squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁ ≃ q₁₀ ⬝o q₂₁ =[eq_of_square s₁₁] q₀₁ ⬝o q₁₂ :=
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begin
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fapply equiv.MK,
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{ exact pathover_of_squareover},
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{ intro r, rewrite [-to_left_inv !square_equiv_eq s₁₁], apply squareover_of_pathover, exact r},
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{ intro r, }, --need characterization of squareover lying over ids.
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{ intro s, induction s, apply idp},
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end
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-/
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definition eq_of_vdeg_squareover {q₁₀' : b₀₀ =[p₁₀] b₂₀}
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(p : squareover B vrfl q₁₀ q₁₀' idpo idpo) : q₁₀ = q₁₀' :=
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begin
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let H := square_of_squareover p, -- use apply ... in
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induction p₁₀, -- if needed we can remove this induction and use con_tr_idp in types/eq2
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rewrite [▸* at H,idp_con at H,+ap_id at H],
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let H' := eq_of_vdeg_square H,
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exact eq_of_fn_eq_fn !pathover_equiv_tr_eq H'
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end
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-- definition vdeg_tr_squareover {q₁₂ : p₀₁ ▸ b₀₀ =[p₁₂] p₂₁ ▸ b₂₀} (r : q₁₀ =[_] q₁₂)
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-- : squareover B s₁₁ q₁₀ q₁₂ !pathover_tr !pathover_tr :=
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-- by induction p;exact vrflo
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/- charcaterization of pathovers in pathovers -/
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-- in this version the fibration (B) of the pathover does not depend on the variable a
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definition pathover_pathover {a' a₂' : A'} {p : a' = a₂'} {f g : A' → A}
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{b : Πa, B (f a)} {b₂ : Πa, B (g a)} {q : Π(a' : A'), f a' = g a'}
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(r : pathover B (b a') (q a') (b₂ a'))
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(r₂ : pathover B (b a₂') (q a₂') (b₂ a₂'))
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(s : squareover B (natural_square_tr q p) r r₂
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(pathover_ap B f (apdo b p)) (pathover_ap B g (apdo b₂ p)))
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: pathover (λa, pathover B (b a) (q a) (b₂ a)) r p r₂ :=
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begin
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induction p, esimp at s, apply pathover_idp_of_eq, apply eq_of_vdeg_squareover, exact s
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end
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end eq
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