21 lines
758 B
Text
21 lines
758 B
Text
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import hott data.sigma
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open path sigma
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variables {A : Type} {B : A → Type} {C : Πa, B a → Type} {D : Πa b, C a b → Type}
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{a a' a'' : A} {b b₁ b₂ : B a} {b' : B a'} {b'' : B a''} {u v w : Σa, B a}
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definition path_sigma_dpair (p : a ≈ a') (q : p ▹ b ≈ b') : dpair a b ≈ dpair a' b' :=
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path.rec_on p (λb b' q, path.rec_on q idp) b b' q
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definition path_sigma (p : dpr1 u ≈ dpr1 v) (q : p ▹ dpr2 u ≈ dpr2 v) : u ≈ v :=
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destruct u
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(λu1 u2, destruct v (λ v1 v2, path_sigma_dpair))
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p q
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definition path_path_sigma_lemma' {p1 : a ≈ a'} {p2 : p1 ▹ b ≈ b'} {q2 : p1 ▹ b ≈ b'}
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(s : idp ▹ p2 ≈ q2) : path_sigma p1 p2 ≈ path_sigma p1 q2 :=
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begin
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apply (path.rec_on s),
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apply idp,
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end
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