39 lines
1.2 KiB
Text
39 lines
1.2 KiB
Text
/-
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Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Markus Himmel
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-/
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module
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prelude
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import all Init.Data.Array.Basic
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public import Init.Data.Array.Lemmas
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public section
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/-!
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These lemmas are used in the internals of HashMap.
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They should find a new home and/or be reformulated.
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-/
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set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
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set_option linter.indexVariables true -- Enforce naming conventions for index variables.
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namespace List
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theorem exists_of_set {i : Nat} {a' : α} {l : List α} (h : i < l.length) :
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∃ l₁ l₂, l = l₁ ++ l[i] :: l₂ ∧ l₁.length = i ∧ l.set i a' = l₁ ++ a' :: l₂ := by
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refine ⟨l.take i, l.drop (i + 1), ⟨by simp, ⟨length_take_of_le (Nat.le_of_lt h), ?_⟩⟩⟩
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simp [set_eq_take_append_cons_drop, h]
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end List
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namespace Array
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theorem exists_of_uset {xs : Array α} {i d} (h) :
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∃ l₁ l₂, xs.toList = l₁ ++ xs[i] :: l₂ ∧ List.length l₁ = i.toNat ∧
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(xs.uset i d h).toList = l₁ ++ d :: l₂ := by
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simpa only [ugetElem_eq_getElem, ← getElem_toList, uset, toList_set] using
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List.exists_of_set _
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end Array
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