55 lines
1.6 KiB
Text
55 lines
1.6 KiB
Text
/-
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Copyright (c) 2025 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: Kim Morrison
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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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import Init.Data.Array.Lemmas
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import Init.Data.List.OfFn
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/-!
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# Theorems about `Array.ofFn`
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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 Array
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@[simp] theorem ofFn_zero {f : Fin 0 → α} : ofFn f = #[] := by
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simp [ofFn, ofFn.go]
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theorem ofFn_succ {f : Fin (n+1) → α} :
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ofFn f = (ofFn (fun (i : Fin n) => f i.castSucc)).push (f ⟨n, by omega⟩) := by
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ext i h₁ h₂
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· simp
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· simp only [getElem_ofFn, getElem_push, size_ofFn, Fin.castSucc_mk, left_eq_dite_iff,
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Nat.not_lt]
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simp only [size_ofFn] at h₁
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intro h₃
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simp only [show i = n by omega]
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@[simp] theorem _root_.List.toArray_ofFn {f : Fin n → α} : (List.ofFn f).toArray = Array.ofFn f := by
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ext <;> simp
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@[simp] theorem toList_ofFn {f : Fin n → α} : (Array.ofFn f).toList = List.ofFn f := by
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apply List.ext_getElem <;> simp
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@[simp]
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theorem ofFn_eq_empty_iff {f : Fin n → α} : ofFn f = #[] ↔ n = 0 := by
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rw [← Array.toList_inj]
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simp
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@[simp 500]
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theorem mem_ofFn {n} {f : Fin n → α} {a : α} : a ∈ ofFn f ↔ ∃ i, f i = a := by
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constructor
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· intro w
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obtain ⟨i, h, rfl⟩ := getElem_of_mem w
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exact ⟨⟨i, by simpa using h⟩, by simp⟩
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· rintro ⟨i, rfl⟩
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apply mem_of_getElem (i := i) <;> simp
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end Array
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