194 lines
7 KiB
Text
194 lines
7 KiB
Text
-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Floris van Doorn, Leonardo de Moura
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import data.nat.basic data.empty data.prod
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open nat eq.ops prod
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inductive vector (T : Type) : ℕ → Type :=
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nil {} : vector T 0,
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cons : T → ∀{n}, vector T n → vector T (succ n)
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namespace vector
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notation a :: b := cons a b
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notation `[` l:(foldr `,` (h t, cons h t) nil) `]` := l
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variables {A B C : Type}
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variables {n m : nat}
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protected definition is_inhabited [instance] (H : inhabited A) (n : nat) : inhabited (vector A n) :=
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nat.rec_on n
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(inhabited.mk nil)
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(λ (n : nat) (iH : inhabited (vector A n)),
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inhabited.destruct H
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(λa, inhabited.destruct iH
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(λv, inhabited.mk (a :: v))))
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theorem z_cases_on {C : vector A 0 → Type} (v : vector A 0) (Hnil : C nil) : C v :=
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begin
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cases v,
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apply Hnil
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end
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theorem vector0_eq_nil (v : vector A 0) : v = nil :=
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z_cases_on v rfl
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protected definition destruct (v : vector A (succ n)) {P : Π {n : nat}, vector A (succ n) → Type}
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(H : Π {n : nat} (h : A) (t : vector A n), P (h :: t)) : P v :=
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begin
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cases v with (h', n', t'),
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apply (H h' t')
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end
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definition nz_cases_on := @destruct
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definition head (v : vector A (succ n)) : A :=
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destruct v (λ n h t, h)
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definition tail (v : vector A (succ n)) : vector A n :=
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destruct v (λ n h t, t)
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theorem head_cons (h : A) (t : vector A n) : head (h :: t) = h :=
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rfl
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theorem tail_cons (h : A) (t : vector A n) : tail (h :: t) = t :=
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rfl
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theorem eta (v : vector A (succ n)) : head v :: tail v = v :=
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-- TODO(Leo): replace 'head_cons h t ▸ tail_cons h t ▸ rfl' with rfl
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-- after issue #318 is fixed
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destruct v (λ n h t, head_cons h t ▸ tail_cons h t ▸ rfl)
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definition last : vector A (succ n) → A :=
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nat.rec_on n
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(λ (v : vector A (succ zero)), head v)
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(λ n₁ r v, r (tail v))
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theorem last_singleton (a : A) : last (a :: nil) = a :=
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rfl
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theorem last_cons (a : A) (v : vector A (succ n)) : last (a :: v) = last v :=
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rfl
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definition const (n : nat) (a : A) : vector A n :=
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nat.rec_on n
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nil
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(λ n₁ r, a :: r)
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theorem head_const (n : nat) (a : A) : head (const (succ n) a) = a :=
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rfl
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theorem last_const (n : nat) (a : A) : last (const (succ n) a) = a :=
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nat.induction_on n
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rfl
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(λ n₁ ih, ih)
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definition map (f : A → B) (v : vector A n) : vector B n :=
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nat.rec_on n
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(λ v, nil)
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(λ n₁ r v, f (head v) :: r (tail v))
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v
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theorem map_vnil (f : A → B) : map f nil = nil :=
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rfl
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theorem map_vcons (f : A → B) (h : A) (t : vector A n) : map f (h :: t) = f h :: map f t :=
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rfl
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definition map2 (f : A → B → C) (v₁ : vector A n) (v₂ : vector B n) : vector C n :=
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nat.rec_on n
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(λ v₁ v₂, nil)
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(λ n₁ r v₁ v₂, f (head v₁) (head v₂) :: r (tail v₁) (tail v₂))
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v₁ v₂
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theorem map2_vnil (f : A → B → C) : map2 f nil nil = nil :=
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rfl
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theorem map2_vcons (f : A → B → C) (h₁ : A) (h₂ : B) (t₁ : vector A n) (t₂ : vector B n) :
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map2 f (h₁ :: t₁) (h₂ :: t₂) = f h₁ h₂ :: map2 f t₁ t₂ :=
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rfl
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definition append (w : vector A n) (v : vector A m) : vector A (n ⊕ m) :=
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rec_on w
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v
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(λ (a₁ : A) (n₁ : nat) (v₁ : vector A n₁) (r₁ : vector A (n₁ ⊕ m)), a₁ :: r₁)
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theorem append_nil (v : vector A n) : append nil v = v :=
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rfl
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theorem append_cons (h : A) (t : vector A n) (v : vector A m) :
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append (h :: t) v = h :: (append t v) :=
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rfl
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definition unzip : vector (A × B) n → vector A n × vector B n :=
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nat.rec_on n
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(λ v, (nil, nil))
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(λ a r v,
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let t := r (tail v) in
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(pr₁ (head v) :: pr₁ t, pr₂ (head v) :: pr₂ t))
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definition zip : vector A n → vector B n → vector (A × B) n :=
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nat.rec_on n
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(λ v₁ v₂, nil)
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(λ a r v₁ v₂, (head v₁, head v₂) :: r (tail v₁) (tail v₂))
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theorem unzip_zip : ∀ (v₁ : vector A n) (v₂ : vector B n), unzip (zip v₁ v₂) = (v₁, v₂) :=
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nat.induction_on n
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(λ (v₁ : vector A zero) (v₂ : vector B zero),
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z_cases_on v₁ (z_cases_on v₂ rfl))
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(λ (n₁ : nat) (ih : ∀ (v₁ : vector A n₁) (v₂ : vector B n₁), unzip (zip v₁ v₂) = (v₁, v₂))
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(v₁ : vector A (succ n₁)) (v₂ : vector B (succ n₁)), calc
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unzip (zip v₁ v₂) = unzip ((head v₁, head v₂) :: zip (tail v₁) (tail v₂)) : rfl
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... = (head v₁ :: pr₁ (unzip (zip (tail v₁) (tail v₂))),
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head v₂ :: pr₂ (unzip (zip (tail v₁) (tail v₂)))) : rfl
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... = (head v₁ :: pr₁ (tail v₁, tail v₂),
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head v₂ :: pr₂ (tail v₁, tail v₂)) : ih
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... = (head v₁ :: tail v₁, head v₂ :: tail v₂) : rfl
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... = (v₁, head v₂ :: tail v₂) : vector.eta
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... = (v₁, v₂) : vector.eta)
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theorem zip_unzip : ∀ (v : vector (A × B) n), zip (pr₁ (unzip v)) (pr₂ (unzip v)) = v :=
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nat.induction_on n
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(λ (v : vector (A × B) zero),
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z_cases_on v rfl)
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(λ (n₁ : nat) (ih : ∀ v, zip (pr₁ (unzip v)) (pr₂ (unzip v)) = v) (v : vector (A × B) (succ n₁)), calc
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zip (pr₁ (unzip v)) (pr₂ (unzip v)) = zip (pr₁ (head v) :: pr₁ (unzip (tail v)))
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(pr₂ (head v) :: pr₂ (unzip (tail v))) : rfl
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... = (pr₁ (head v), pr₂ (head v)) :: zip (pr₁ (unzip (tail v))) (pr₂ (unzip (tail v))) : rfl
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... = (pr₁ (head v), pr₂ (head v)) :: tail v : ih
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... = head v :: tail v : prod.eta
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... = v : vector.eta)
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-- Length
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-- ------
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definition length (v : vector A n) :=
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n
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theorem length_nil : length (@nil A) = 0 :=
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rfl
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theorem length_cons (a : A) (v : vector A n) : length (a :: v) = succ (length v) :=
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rfl
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theorem length_append (v₁ : vector A n) (v₂ : vector A m) : length (append v₁ v₂) = length v₁ + length v₂ :=
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calc length (append v₁ v₂) = length v₁ ⊕ length v₂ : rfl
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... = length v₁ + length v₂ : add_eq_addl
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-- Concat
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-- ------
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definition concat (v : vector A n) (a : A) : vector A (succ n) :=
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vector.rec_on v
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(a :: nil)
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(λ h n t r, h :: r)
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theorem concat_nil (a : A) : concat nil a = a :: nil :=
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rfl
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theorem last_concat (v : vector A n) (a : A) : last (concat v a) = a :=
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vector.induction_on v
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rfl
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(λ h n t ih, calc
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last (concat (h :: t) a) = last (concat t a) : rfl
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... = a : ih)
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end vector
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