33 lines
1.1 KiB
Text
33 lines
1.1 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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Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad
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The order relation on the natural numbers.
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This is a minimal port of functions from the lean2 library.
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-/
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import algebra.order
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namespace nat
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/- min -/
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theorem zero_min (a : ℕ) : min 0 a = 0 := min_eq_left (zero_le a)
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theorem min_zero (a : ℕ) : min a 0 = 0 := min_eq_right (zero_le a)
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-- Distribute succ over min
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theorem min_succ_succ (x y : ℕ) : min (succ x) (succ y) = succ (min x y) :=
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let f : x ≤ y → min (succ x) (succ y) = succ (min x y) := λp,
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calc min (succ x) (succ y)
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= succ x : if_pos (succ_le_succ p)
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... = succ (min x y) : congr_arg succ (eq.symm (if_pos p)) in
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let g : ¬ (x ≤ y) → min (succ x) (succ y) = succ (min x y) := λp,
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calc min (succ x) (succ y)
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= succ y : if_neg (λeq, p (pred_le_pred eq))
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... = succ (min x y) : congr_arg succ (eq.symm (if_neg p)) in
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decidable.by_cases f g
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end nat
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