52 lines
2.8 KiB
Text
52 lines
2.8 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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Coherence conditions for operations on squares
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-/
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import .square
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open equiv
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namespace eq
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variables {A B C : Type} {a a' a'' a₀₀ a₂₀ a₄₀ a₀₂ a₂₂ a₂₄ a₀₄ a₄₂ a₄₄ a₁ a₂ a₃ a₄ : A}
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{f : A → B} {b : B} {c : C}
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/-a₀₀-/ {p₁₀ p₁₀' : a₀₀ = a₂₀} /-a₂₀-/ {p₃₀ : a₂₀ = a₄₀} /-a₄₀-/
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{p₀₁ p₀₁' : a₀₀ = a₀₂} /-s₁₁-/ {p₂₁ p₂₁' : a₂₀ = a₂₂} /-s₃₁-/ {p₄₁ : a₄₀ = a₄₂}
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/-a₀₂-/ {p₁₂ 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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theorem whisker_bl_whisker_tl_eq (p : a = a')
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: whisker_bl p (whisker_tl p ids) = con.right_inv p ⬝ph vrfl :=
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by induction p; reflexivity
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theorem ap_is_constant_natural_square {g : B → C} {f : A → B} (H : Πa, g (f a) = c) (p : a = a') :
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(ap_is_constant H p)⁻¹ ⬝ph natural_square H p ⬝hp ap_constant p c =
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whisker_bl (H a') (whisker_tl (H a) ids) :=
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begin induction p, esimp, rewrite inv_inv, rewrite whisker_bl_whisker_tl_eq end
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definition inv_ph_eq_of_eq_ph {p : a₀₀ = a₀₂} {r : p₀₁ = p} {s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁}
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{s₁₁' : square p₁₀ p₁₂ p p₂₁} (t : s₁₁ = r ⬝ph s₁₁') : r⁻¹ ⬝ph s₁₁ = s₁₁' :=
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by induction r; exact t
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-- the following is used for torus.elim_surf
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theorem whisker_square_aps_eq
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{q₁₀ : f a₀₀ = f a₂₀} {q₀₁ : f a₀₀ = f a₀₂} {q₂₁ : f a₂₀ = f a₂₂} {q₁₂ : f a₀₂ = f a₂₂}
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{r₁₀ : ap f p₁₀ = q₁₀} {r₀₁ : ap f p₀₁ = q₀₁} {r₂₁ : ap f p₂₁ = q₂₁} {r₁₂ : ap f p₁₂ = q₁₂}
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{s₁₁ : p₁₀ ⬝ p₂₁ = p₀₁ ⬝ p₁₂} {t₁₁ : square q₁₀ q₁₂ q₀₁ q₂₁}
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(u : square (ap02 f s₁₁) (eq_of_square t₁₁)
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(ap_con f p₁₀ p₂₁ ⬝ (r₁₀ ◾ r₂₁)) (ap_con f p₀₁ p₁₂ ⬝ (r₀₁ ◾ r₁₂)))
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: whisker_square r₁₀ r₁₂ r₀₁ r₂₁ (aps f (square_of_eq s₁₁)) = t₁₁ :=
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begin
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induction r₁₀, induction r₀₁, induction r₁₂, induction r₂₁,
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induction p₁₂, induction p₁₀, induction p₂₁, esimp at *, induction s₁₁, esimp at *,
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esimp [square_of_eq],
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apply eq_of_fn_eq_fn !square_equiv_eq, esimp,
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exact (eq_bot_of_square u)⁻¹
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end
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end eq
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