62 lines
2 KiB
Text
62 lines
2 KiB
Text
/-
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Copyright (c) 2014 Parikshit Khanna. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn
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Basic properties of lists.
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-/
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import init.data.list.basic
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universe variables u v w
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namespace list
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open nat
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variables {α : Type u} {β : Type v} {φ : Type w}
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/- length theorems -/
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@[simp]
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theorem length_append : ∀ (x y : list α), length (x ++ y) = length x + length y
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| [] l := eq.symm (nat.zero_add (length l))
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| (a::s) l :=
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calc succ (length (s ++ l))
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= succ (length s + length l) : congr_arg nat.succ (length_append s l)
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... = succ (length s) + length l : eq.symm (nat.succ_add (length s) (length l))
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theorem length_concat (a : α) : ∀ (l : list α), length (concat l a) = succ (length l)
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| nil := rfl
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| (cons b l) := congr_arg succ (length_concat l)
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@[simp]
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theorem length_taken
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: ∀ (i : ℕ) (l : list α), length (taken i l) = min i (length l)
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| 0 l := eq.symm (nat.zero_min (length l))
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| (succ n) [] := eq.symm (nat.min_zero (succ n))
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| (succ n) (a::l) :=
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calc succ (length (taken n l)) = succ (min n (length l)) : congr_arg succ (length_taken n l)
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... = min (succ n) (succ (length l))
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: eq.symm (nat.min_succ_succ n (length l))
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@[simp]
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theorem length_dropn
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: ∀ (i : ℕ) (l : list α), length (dropn i l) = length l - i
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| 0 l := rfl
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| (succ i) [] := eq.symm (nat.zero_sub_eq_zero (succ i))
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| (succ i) (x::l) := calc
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length (dropn (succ i) (x::l))
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= length l - i : length_dropn i l
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... = succ (length l) - succ i : nat.sub_eq_succ_sub_succ (length l) i
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@[simp]
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theorem length_map (f : α → β) : ∀ (a : list α), length (map f a) = length a
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| [] := rfl
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| (a :: l) := congr_arg succ (length_map l)
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@[simp]
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theorem length_repeat (a : α) : ∀ (n : ℕ), length (repeat a n) = n
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| 0 := eq.refl 0
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| (succ i) := congr_arg succ (length_repeat i)
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end list
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