48 lines
1.3 KiB
Text
48 lines
1.3 KiB
Text
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/-
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Copyright (c) 2014 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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Ported from Coq HoTT
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Theorems about products
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-/
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import ..trunc data.prod
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open path equiv is_equiv truncation prod
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variables {A A' B B' C D : Type}
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{a a' a'' : A} {b b₁ b₂ b' b'' : B} {u v w : A × B}
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namespace prod
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-- prod.eta is already used for the eta rule for strict equality
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protected definition peta (u : A × B) : (pr₁ u , pr₂ u) ≈ u :=
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destruct u (λu1 u2, idp)
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definition pair_path (pa : a ≈ a') (pb : b ≈ b') : (a , b) ≈ (a' , b') :=
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path.rec_on pa (path.rec_on pb idp)
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protected definition path : (pr₁ u ≈ pr₁ v) → (pr₂ u ≈ pr₂ v) → u ≈ v :=
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begin
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apply (prod.rec_on u), intros (a₁, b₁),
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apply (prod.rec_on v), intros (a₂, b₂, H₁, H₂),
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apply (transport _ (peta (a₁, b₁))),
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apply (transport _ (peta (a₂, b₂))),
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apply (pair_path H₁ H₂),
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end
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/- Symmetry -/
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definition isequiv_flip [instance] (A B : Type) : is_equiv (@flip A B) :=
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adjointify flip
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flip
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(λu, destruct u (λb a, idp))
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(λu, destruct u (λa b, idp))
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definition symm_equiv (A B : Type) : A × B ≃ B × A :=
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equiv.mk flip _
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-- trunc_prod is defined in sigma
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end prod
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