88 lines
2.5 KiB
Text
88 lines
2.5 KiB
Text
/-
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Copyright (c) 2017 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Leonardo de Moura
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-/
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universes u
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/--
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A difference list is a function that, given a list, returns the original
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contents of the difference list prepended to the given list.
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This structure supports `O(1)` `append` and `concat` operations on lists, making it
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useful for append-heavy uses such as logging and pretty printing.
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-/
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structure dlist (α : Type u) :=
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(apply : list α → list α)
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(inv : ∀ l, apply l = apply [] ++ l)
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namespace dlist
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open function
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variables {α : Type u}
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local notation `♯`:max := by abstract {intros, rsimp}
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/-- Convert a list to a dlist -/
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def of_list (l : list α) : dlist α :=
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⟨append l, ♯⟩
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/-- Convert a dlist to a list -/
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def to_list : dlist α → list α
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| ⟨xs, _⟩ := xs []
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/-- Create a dlist containing no elements -/
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def empty : dlist α :=
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⟨id, ♯⟩
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local notation a `::_`:max := list.cons a
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/-- Create dlist with a single element -/
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def singleton (x : α) : dlist α :=
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⟨x::_, ♯⟩
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local attribute [simp] function.comp
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/-- `O(1)` Prepend a single element to a dlist -/
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def cons (x : α) : dlist α → dlist α
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| ⟨xs, h⟩ := ⟨x::_ ∘ xs, ♯⟩
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/-- `O(1)` Append a single element to a dlist -/
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def concat (x : α) : dlist α → dlist α
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| ⟨xs, h⟩ := ⟨xs ∘ x::_, ♯⟩
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/-- `O(1)` Append dlists -/
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protected def append : dlist α → dlist α → dlist α
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| ⟨xs, h₁⟩ ⟨ys, h₂⟩ := ⟨xs ∘ ys, ♯⟩
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instance : has_append (dlist α) :=
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⟨dlist.append⟩
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local attribute [simp] of_list to_list empty singleton cons concat dlist.append
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lemma to_list_of_list (l : list α) : to_list (of_list l) = l :=
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by cases l; simp
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lemma of_list_to_list (l : dlist α) : of_list (to_list l) = l :=
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begin
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cases l with xs,
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assert h : append (xs []) = xs,
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{intros, apply funext, intros x, simp [inv x]},
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simp [h]
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end
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lemma to_list_empty : to_list (@empty α) = [] :=
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by simp
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lemma to_list_singleton (x : α) : to_list (singleton x) = [x] :=
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by simp
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lemma to_list_append (l₁ l₂ : dlist α) : to_list (l₁ ++ l₂) = to_list l₁ ++ to_list l₂ :=
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show to_list (dlist.append l₁ l₂) = to_list l₁ ++ to_list l₂, from
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by cases l₁; cases l₂; simp; rsimp
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lemma to_list_cons (x : α) (l : dlist α) : to_list (cons x l) = x :: to_list l :=
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by cases l; rsimp
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lemma to_list_concat (x : α) (l : dlist α) : to_list (concat x l) = to_list l ++ [x] :=
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by cases l; simp; rsimp
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end dlist
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