48 lines
1.3 KiB
Text
48 lines
1.3 KiB
Text
/-
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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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Module: types.prod
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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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open eq equiv is_equiv is_trunc 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 eta (u : A × B) : (pr₁ u , pr₂ u) = u :=
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by cases u; apply idp
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definition pair_eq (pa : a = a') (pb : b = b') : (a , b) = (a' , b') :=
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by cases pa; cases pb; apply idp
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definition prod_eq (H₁ : pr₁ u = pr₁ v) (H₂ : pr₂ u = pr₂ v) : u = v :=
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begin
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cases u with [a₁, b₁],
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cases v with [a₂, b₂],
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apply (transport _ (eta (a₁, b₁))),
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apply (transport _ (eta (a₂, b₂))),
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apply (pair_eq H₁ H₂),
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end
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/- Symmetry -/
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definition is_equiv_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 prod_comm_equiv (A B : Type) : A × B ≃ B × A :=
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equiv.mk flip _
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-- is_trunc_prod is defined in sigma
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end prod
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