141 lines
6.9 KiB
Text
141 lines
6.9 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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Theorems about square
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-/
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open eq equiv is_equiv
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namespace cubical
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variables {A : Type} {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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inductive square {A : Type} {a₀₀ : A}
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: Π{a₂₀ a₀₂ a₂₂ : A}, a₀₀ = a₂₀ → a₀₂ = a₂₂ → a₀₀ = a₀₂ → a₂₀ = a₂₂ → Type :=
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ids : square idp idp idp idp
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/- square top bottom left right -/
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variables {s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁} {s₃₁ : square p₃₀ p₃₂ p₂₁ p₄₁}
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{s₁₃ : square p₁₂ p₁₄ p₀₃ p₂₃} {s₃₃ : square p₃₂ p₃₄ p₂₃ p₄₃}
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definition ids [reducible] [constructor] := @square.ids
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definition idsquare [reducible] [constructor] (a : A) := @square.ids A a
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definition hrefl [unfold-c 4] (p : a = a') : square idp idp p p :=
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by cases p; exact ids
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definition vrefl [unfold-c 4] (p : a = a') : square p p idp idp :=
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by cases p; exact ids
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definition hconcat (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) (s₃₁ : square p₃₀ p₃₂ p₂₁ p₄₁)
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: square (p₁₀ ⬝ p₃₀) (p₁₂ ⬝ p₃₂) p₀₁ p₄₁ :=
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by cases s₃₁; exact s₁₁
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definition vconcat (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) (s₁₃ : square p₁₂ p₁₄ p₀₃ p₂₃)
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: square p₁₀ p₁₄ (p₀₁ ⬝ p₀₃) (p₂₁ ⬝ p₂₃) :=
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by cases s₁₃; exact s₁₁
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definition hinverse (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) : square p₁₀⁻¹ p₁₂⁻¹ p₂₁ p₀₁ :=
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by cases s₁₁;exact ids
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definition vinverse (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) : square p₁₂ p₁₀ p₀₁⁻¹ p₂₁⁻¹ :=
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by cases s₁₁;exact ids
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definition transpose (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) : square p₀₁ p₂₁ p₁₀ p₁₂ :=
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by cases s₁₁;exact ids
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definition eq_of_square (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) : p₁₀ ⬝ p₂₁ = p₀₁ ⬝ p₁₂ :=
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by cases s₁₁; apply idp
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definition hdegen_square {p q : a = a'} (r : p = q) : square idp idp p q :=
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by cases r;apply hrefl
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definition vdegen_square {p q : a = a'} (r : p = q) : square p q idp idp :=
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by cases r;apply vrefl
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definition square_of_eq (r : p₁₀ ⬝ p₂₁ = p₀₁ ⬝ p₁₂) : square p₁₀ p₁₂ p₀₁ p₂₁ :=
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by cases p₁₂; (esimp [concat] at r); cases r; cases p₂₁; cases p₁₀; exact ids
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definition aps {B : Type} (f : A → B) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
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: square (ap f p₁₀) (ap f p₁₂) (ap f p₀₁) (ap f p₂₁) :=
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by cases s₁₁;exact ids
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definition square_of_homotopy {B : Type} {f g : A → B} (H : f ∼ g) (p : a = a')
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: square (H a) (H a') (ap f p) (ap g p) :=
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by cases p; apply vrefl
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definition square_of_homotopy' {B : Type} {f g : A → B} (H : f ∼ g) (p : a = a')
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: square (ap f p) (ap g p) (H a) (H a') :=
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by cases p; apply hrefl
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definition square_equiv_eq (t : a₀₀ = a₀₂) (b : a₂₀ = a₂₂) (l : a₀₀ = a₂₀) (r : a₀₂ = a₂₂)
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: square t b l r ≃ t ⬝ r = l ⬝ b :=
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begin
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fapply equiv.MK,
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{ exact eq_of_square},
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{ exact square_of_eq},
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{ intro s, cases b, esimp [concat] at s, cases s, cases r, cases t, apply idp},
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{ intro s, cases s, apply idp},
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end
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definition rec_on_b [recursor] {a₀₀ : A}
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{P : Π{a₂₀ a₁₂ : A} {t : a₀₀ = a₂₀} {l : a₀₀ = a₁₂} {r : a₂₀ = a₁₂}, square t idp l r → Type}
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{a₂₀ a₁₂ : A} {t : a₀₀ = a₂₀} {l : a₀₀ = a₁₂} {r : a₂₀ = a₁₂}
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(s : square t idp l r) (H : P ids) : P s :=
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have H2 : P (square_of_eq (eq_of_square s)),
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from eq.rec_on (eq_of_square s : t ⬝ r = l) (by cases r; cases t; exact H),
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left_inv (to_fun !square_equiv_eq) s ▸ H2
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definition rec_on_r [recursor] {a₀₀ : A}
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{P : Π{a₀₂ a₂₁ : A} {t : a₀₀ = a₂₁} {b : a₀₂ = a₂₁} {l : a₀₀ = a₀₂}, square t b l idp → Type}
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{a₀₂ a₂₁ : A} {t : a₀₀ = a₂₁} {b : a₀₂ = a₂₁} {l : a₀₀ = a₀₂}
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(s : square t b l idp) (H : P ids) : P s :=
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let p : l ⬝ b = t := (eq_of_square s)⁻¹ in
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have H2 : P (square_of_eq (eq_of_square s)⁻¹⁻¹),
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from @eq.rec_on _ _ (λx p, P (square_of_eq p⁻¹)) _ p (by cases b; cases l; exact H),
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left_inv (to_fun !square_equiv_eq) s ▸ !inv_inv ▸ H2
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definition rec_on_l [recursor] {a₀₁ : A}
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{P : Π {a₂₀ a₂₂ : A} {t : a₀₁ = a₂₀} {b : a₀₁ = a₂₂} {r : a₂₀ = a₂₂},
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square t b idp r → Type}
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{a₂₀ a₂₂ : A} {t : a₀₁ = a₂₀} {b : a₀₁ = a₂₂} {r : a₂₀ = a₂₂}
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(s : square t b idp r) (H : P ids) : P s :=
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let p : t ⬝ r = b := eq_of_square s ⬝ !idp_con in
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have H2 : P (square_of_eq (p ⬝ !idp_con⁻¹)),
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from eq.rec_on p (by cases r; cases t; exact H),
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left_inv (to_fun !square_equiv_eq) s ▸ !con_inv_cancel_right ▸ H2
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definition rec_on_t [recursor] {a₁₀ : A}
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{P : Π {a₀₂ a₂₂ : A} {b : a₀₂ = a₂₂} {l : a₁₀ = a₀₂} {r : a₁₀ = a₂₂}, square idp b l r → Type}
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{a₀₂ a₂₂ : A} {b : a₀₂ = a₂₂} {l : a₁₀ = a₀₂} {r : a₁₀ = a₂₂}
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(s : square idp b l r) (H : P ids) : P s :=
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let p : l ⬝ b = r := (eq_of_square s)⁻¹ ⬝ !idp_con in
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assert H2 : P (square_of_eq ((p ⬝ !idp_con⁻¹)⁻¹)),
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from eq.rec_on p (by cases b; cases l; exact H),
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assert H3 : P (square_of_eq ((eq_of_square s)⁻¹⁻¹)),
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from eq.rec_on !con_inv_cancel_right H2,
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assert H4 : P (square_of_eq (eq_of_square s)),
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from eq.rec_on !inv_inv H3,
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proof
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left_inv (to_fun !square_equiv_eq) s ▸ H4
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qed
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definition rec_on_tb [recursor] {a : A}
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{P : Π{b : A} {l : a = b} {r : a = b}, square idp idp l r → Type}
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{b : A} {l : a = b} {r : a = b}
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(s : square idp idp l r) (H : P ids) : P s :=
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have H2 : P (square_of_eq (eq_of_square s)),
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from eq.rec_on (eq_of_square s : idp ⬝ r = l) (by cases r; exact H),
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left_inv (to_fun !square_equiv_eq) s ▸ H2
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--we can also do the other recursors (lr, tl, tr, bl, br, tbl, tbr, tlr, blr), but let's postpone this until they are needed
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end cubical
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