48 lines
1.4 KiB
Text
48 lines
1.4 KiB
Text
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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Leonardo de Moura
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import logic.classes.inhabited logic.connectives.eq
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inductive prod (A B : Type) : Type :=
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| pair : A → B → prod A B
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precedence `×`:30
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infixr × := prod
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-- notation for n-ary tuples
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notation `(` h `,` t:(foldl `,` (e r, pair r e) h) `)` := t
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namespace prod
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section
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parameters {A B : Type}
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abbreviation pr1 (p : prod A B) := prod_rec (λ x y, x) p
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abbreviation pr2 (p : prod A B) := prod_rec (λ x y, y) p
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theorem pr1_pair (a : A) (b : B) : pr1 (a, b) = a := refl a
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theorem pr2_pair (a : A) (b : B) : pr2 (a, b) = b := refl b
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-- TODO: remove prefix when we can protect it
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theorem pair_destruct {P : A × B → Prop} (p : A × B) (H : ∀a b, P (a, b)) : P p :=
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prod_rec H p
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theorem prod_ext (p : prod A B) : pair (pr1 p) (pr2 p) = p :=
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pair_destruct p (λx y, refl (x, y))
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theorem pair_eq {p1 p2 : prod A B} (H1 : pr1 p1 = pr1 p2) (H2 : pr2 p1 = pr2 p2) : p1 = p2 :=
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calc p1 = pair (pr1 p1) (pr2 p1) : symm (prod_ext p1)
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... = pair (pr1 p2) (pr2 p1) : {H1}
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... = pair (pr1 p2) (pr2 p2) : {H2}
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... = p2 : prod_ext p2
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theorem prod_inhabited (H1 : inhabited A) (H2 : inhabited B) : inhabited (prod A B) :=
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inhabited_elim H1 (λa, inhabited_elim H2 (λb, inhabited_intro (pair a b)))
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end
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instance prod_inhabited
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end prod
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