43 lines
No EOL
1.3 KiB
Text
43 lines
No EOL
1.3 KiB
Text
-- 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
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namespace pair
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inductive pair (A : Type) (B : Type) : Type :=
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| mk_pair : A → B → pair A B
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section
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parameter {A : Type}
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parameter {B : Type}
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definition fst [inline] (p : pair A B) := pair_rec (λ x y, x) p
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definition snd [inline] (p : pair A B) := pair_rec (λ x y, y) p
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theorem pair_inhabited (H1 : inhabited A) (H2 : inhabited B) : inhabited (pair A B)
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:= inhabited_elim H1 (λ a, inhabited_elim H2 (λ b, inhabited_intro (mk_pair a b)))
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theorem fst_mk_pair (a : A) (b : B) : fst (mk_pair a b) = a
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:= refl a
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theorem snd_mk_pair (a : A) (b : B) : snd (mk_pair a b) = b
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:= refl b
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theorem pair_ext (p : pair A B) : mk_pair (fst p) (snd p) = p
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:= pair_rec (λ x y, refl (mk_pair x y)) p
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theorem pair_ext_eq {p1 p2 : pair A B} (H1 : fst p1 = fst p2) (H2 : snd p1 = snd p2) : p1 = p2
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:= calc p1 = mk_pair (fst p1) (snd p1) : symm (pair_ext p1)
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... = mk_pair (fst p2) (snd p1) : {H1}
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... = mk_pair (fst p2) (snd p2) : {H2}
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... = p2 : pair_ext p2
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end
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instance pair_inhabited
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precedence `×`:30
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infixr × := pair
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-- notation for n-ary tuples
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notation `(` h `,` t:(foldl `,` (e r, mk_pair r e) h) `)` := t
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end |