141 lines
4.7 KiB
Text
141 lines
4.7 KiB
Text
/-
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Copyright (c) 2014 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: Jeremy Avigad
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This is a minimal port of functions from the lean2 list library.
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-/
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import init.list
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import algebra.order
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import data.nat
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universe variables u v w
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namespace list
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open nat
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variables {α : Type u} {β : Type v} {φ : Type w}
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/- length theorems -/
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theorem length_append : ∀ (x y : list α), length (x ++ y) = length x + length y
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| [] l := eq.symm (nat.zero_add (length l))
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| (a::s) l :=
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calc nat.succ (length (s ++ l))
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= nat.succ (length s + length l) : congr_arg nat.succ (length_append s l)
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... = nat.succ (length s) + length l : eq.symm (nat.succ_add (length s) (length l))
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theorem length_repeat (a : α) : ∀ (n : ℕ), length (repeat a n) = n
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| 0 := eq.refl 0
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| (succ i) := congr_arg succ (length_repeat i)
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theorem length_map (f : α → β) : ∀ (a : list α), length (map f a) = length a
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| [] := rfl
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| (a :: l) := congr_arg succ (length_map l)
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theorem length_dropn
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: ∀ (i : ℕ) (l : list α), length (dropn i l) = length l - i
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| 0 l := rfl
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| (succ i) [] := eq.symm (nat.zero_sub_eq_zero (succ i))
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| (succ i) (x::l) := calc
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length (dropn (succ i) (x::l))
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= length l - i : length_dropn i l
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... = succ (length l) - succ i : nat.sub_eq_succ_sub_succ (length l) i
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/- firstn -/
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def firstn : ℕ → list α → list α
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| 0 l := []
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| (succ n) [] := []
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| (succ n) (a::l) := a :: firstn n l
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theorem length_firstn
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: ∀ (i : ℕ) (l : list α), length (firstn i l) = min i (length l)
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| 0 l := eq.symm (nat.min_zero_left (length l))
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| (succ n) [] := eq.symm (nat.min_zero_right (succ n))
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| (succ n) (a::l) :=
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calc succ (length (firstn n l)) = succ (min n (length l)) : congr_arg succ (length_firstn n l)
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... = min (succ n) (succ (length l))
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: eq.symm (nat.min_succ_succ n (length l))
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/- map₂ -/
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definition map₂ {α : Type u} {β : Type v} {φ : Type w} (f : α → β → φ) : list α → list β → list φ
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| [] l := []
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| l [] := []
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| (a::s) (b::t) := f a b :: map₂ s t
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theorem map₂_nil_1 {α : Type u} {β : Type v} {φ : Type w} (f : α → β → φ)
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: Π y, map₂ f nil y = nil
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| [] := eq.refl nil
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| (b::t) := eq.refl nil
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theorem map₂_nil_2 {α β φ : Type} (f : α → β → φ)
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: Π (x : list α), map₂ f x nil = nil
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| [] := eq.refl nil
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| (b::t) := eq.refl nil
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theorem length_map₂ {α β φ : Type} (f : α → β → φ)
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: Π x y, length (map₂ f x y) = min (length x) (length y)
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| [] y :=
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calc length (map₂ f nil y) = 0 : congr_arg length (map₂_nil_1 f y)
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... = min 0 (length y) : eq.symm (nat.min_zero_left (length y))
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| x [] :=
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calc length (map₂ f x nil) = 0 : congr_arg length (map₂_nil_2 f x)
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... = min (length x) 0 : eq.symm (nat.min_zero_right (length x))
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| (a::x) (b::y) :=
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calc succ (length (map₂ f x y))
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= succ (min (length x) (length y))
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: congr_arg succ (length_map₂ x y)
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... = min (succ (length x)) (succ (length y))
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: eq.symm (min_succ_succ (length x) (length y))
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section map_accumr
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variable {σ : Type}
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-- This runs a function over a list returning the intermediate results and a
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-- a final result.
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definition map_accumr (f : α → σ → σ × β) : list α → σ → (σ × list β)
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| [] c := (c, [])
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| (y::yr) c :=
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let r := map_accumr yr c in
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let z := f y (prod.fst r) in
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(prod.fst z, prod.snd z :: prod.snd r)
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theorem length_map_accumr
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: ∀ (f : α → σ → σ × β) (x : list α) (s : σ),
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length (prod.snd (map_accumr f x s)) = length x
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| f (a::x) s := congr_arg succ (length_map_accumr f x s)
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| f [] s := rfl
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end map_accumr
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section map_accumr₂
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-- This runs a function over two lists returning the intermediate results and a
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-- a final result.
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definition map_accumr₂ {α β σ φ : Type} (f : α → β → σ → σ × φ)
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: list α → list β → σ → σ × list φ
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| [] _ c := (c,[])
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| _ [] c := (c,[])
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| (x::xr) (y::yr) c :=
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let r := map_accumr₂ xr yr c in
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let q := f x y (prod.fst r) in
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(prod.fst q, prod.snd q :: (prod.snd r))
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theorem length_map_accumr₂ {α β σ φ : Type}
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: ∀ (f : α → β → σ → σ × φ) x y c,
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length (prod.snd (map_accumr₂ f x y c)) = min (length x) (length y)
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| f (a::x) (b::y) c := calc
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succ (length (prod.snd (map_accumr₂ f x y c)))
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= succ (min (length x) (length y))
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: congr_arg succ (length_map_accumr₂ f x y c)
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... = min (succ (length x)) (succ (length y))
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: eq.symm (min_succ_succ (length x) (length y))
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| f (a::x) [] c := rfl
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| f [] (b::y) c := rfl
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| f [] [] c := rfl
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end map_accumr₂
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end list
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