b2f70dad52
This PR introduces the basic theory of permutations of `Array`s and proves `Array.swap_perm`. The API falls well short of what is available for `List` at this point.
65 lines
2.1 KiB
Text
65 lines
2.1 KiB
Text
/-
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Copyright (c) 2024 Lean FRO. 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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prelude
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import Init.Data.List.Nat.Perm
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import Init.Data.Array.Lemmas
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namespace Array
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open List
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/--
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`Perm as bs` asserts that `as` and `bs` are permutations of each other.
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This is a wrapper around `List.Perm`, and for now has much less API.
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For more complicated verification, use `perm_iff_toList_perm` and the `List` API.
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-/
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def Perm (as bs : Array α) : Prop :=
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as.toList ~ bs.toList
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@[inherit_doc] scoped infixl:50 " ~ " => Perm
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theorem perm_iff_toList_perm {as bs : Array α} : as ~ bs ↔ as.toList ~ bs.toList := Iff.rfl
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@[simp] theorem perm_toArray (as bs : List α) : as.toArray ~ bs.toArray ↔ as ~ bs := by
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simp [perm_iff_toList_perm]
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@[simp, refl] protected theorem Perm.refl (l : Array α) : l ~ l := by
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cases l
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simp
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protected theorem Perm.rfl {l : List α} : l ~ l := .refl _
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theorem Perm.of_eq {l₁ l₂ : Array α} (h : l₁ = l₂) : l₁ ~ l₂ := h ▸ .rfl
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protected theorem Perm.symm {l₁ l₂ : Array α} (h : l₁ ~ l₂) : l₂ ~ l₁ := by
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cases l₁; cases l₂
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simp only [perm_toArray] at h
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simpa using h.symm
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protected theorem Perm.trans {l₁ l₂ l₃ : Array α} (h₁ : l₁ ~ l₂) (h₂ : l₂ ~ l₃) : l₁ ~ l₃ := by
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cases l₁; cases l₂; cases l₃
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simp only [perm_toArray] at h₁ h₂
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simpa using h₁.trans h₂
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instance : Trans (Perm (α := α)) (Perm (α := α)) (Perm (α := α)) where
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trans h₁ h₂ := Perm.trans h₁ h₂
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theorem perm_comm {l₁ l₂ : Array α} : l₁ ~ l₂ ↔ l₂ ~ l₁ := ⟨Perm.symm, Perm.symm⟩
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theorem Perm.push (x y : α) {l₁ l₂ : Array α} (p : l₁ ~ l₂) :
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(l₁.push x).push y ~ (l₂.push y).push x := by
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cases l₁; cases l₂
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simp only [perm_toArray] at p
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simp only [push_toArray, List.append_assoc, singleton_append, perm_toArray]
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exact p.append (Perm.swap' _ _ Perm.nil)
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theorem swap_perm {as : Array α} {i j : Nat} (h₁ : i < as.size) (h₂ : j < as.size) :
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as.swap i j ~ as := by
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simp only [swap, perm_iff_toList_perm, toList_set]
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apply set_set_perm
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end Array
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