31 lines
739 B
Plaintext
31 lines
739 B
Plaintext
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import Mathlib.Logic.Basic
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namespace List
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/--
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The length of any empty list is definitionally zero.
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-/
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theorem nil_length_eq_zero : @length α [] = 0 := rfl
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/--
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If the length of a list is greater than zero, it cannot be `List.nil`.
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-/
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theorem length_gt_zero_imp_not_nil : xs.length > 0 → xs ≠ [] := by
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intro h
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by_contra nh
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rw [nh] at h
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have : 0 > 0 := calc 0
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_ = length [] := by rw [←nil_length_eq_zero]
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_ > 0 := h
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simp at this
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/--
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Given a list `xs` of length `k`, produces a list of length `k - 1` where the
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`i`th member of the resulting list is `f xs[i] xs[i + 1]`.
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-/
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def pairwise (xs : List α) (f : α → α → β) : List β :=
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match xs.tail? with
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| none => []
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| some ys => zipWith f xs ys
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end List
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