128 lines
3.5 KiB
Text
128 lines
3.5 KiB
Text
/-
|
|
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|
|
|
Module: data.stream
|
|
Author: Leonardo de Moura
|
|
-/
|
|
import data.nat
|
|
open nat
|
|
|
|
definition stream (A : Type) := nat → A
|
|
|
|
namespace stream
|
|
variables {A B C : Type}
|
|
|
|
definition cons (a : A) (s : stream A) : stream A :=
|
|
λ i,
|
|
match i with
|
|
| 0 := a
|
|
| succ n := s n
|
|
end
|
|
|
|
definition head (s : stream A) : A :=
|
|
s 0
|
|
|
|
definition tail (s : stream A) : stream A :=
|
|
λ i, s (i+1)
|
|
|
|
definition nth_tail (n : nat) (s : stream A) : stream A :=
|
|
λ i, s (i+n)
|
|
|
|
definition nth (n : nat) (s : stream A) : A :=
|
|
s n
|
|
|
|
protected theorem eta (s : stream A) : cons (head s) (tail s) = s :=
|
|
funext (λ i, begin cases i, repeat reflexivity end)
|
|
|
|
theorem tail_nth_tail (n : nat) (s : stream A) : tail (nth_tail n s) = nth_tail n (tail s) :=
|
|
funext (λ i, begin esimp [tail, nth_tail], congruence, rewrite add.right_comm end)
|
|
|
|
theorem nth_nth_tail (n m : nat) (s : stream A) : nth n (nth_tail m s) = nth (n+m) s :=
|
|
rfl
|
|
|
|
theorem nth_tail_nth_tail (n m : nat) (s : stream A) : nth_tail n (nth_tail m s) = nth_tail (n+m) s :=
|
|
funext (λ i, begin esimp [nth_tail], rewrite add.assoc end)
|
|
|
|
theorem nth_succ (n : nat) (s : stream A) : nth (succ n) s = nth n (tail s) :=
|
|
rfl
|
|
|
|
protected theorem ext {s₁ s₂ : stream A} : (∀ n, nth n s₁ = nth n s₂) → s₁ = s₂ :=
|
|
assume h, funext h
|
|
|
|
protected definition all (p : A → Prop) (s : stream A) := ∀ n, p (nth n s)
|
|
|
|
protected definition any (p : A → Prop) (s : stream A) := ∃ n, p (nth n s)
|
|
|
|
theorem all_def (p : A → Prop) (s : stream A) : stream.all p s = ∀ n, p (nth n s) :=
|
|
rfl
|
|
|
|
theorem any_def (p : A → Prop) (s : stream A) : stream.any p s = ∃ n, p (nth n s) :=
|
|
rfl
|
|
|
|
section map
|
|
variable (f : A → B)
|
|
|
|
definition map (s : stream A) : stream B :=
|
|
λ n, f (nth n s)
|
|
|
|
theorem nth_tail_map (n : nat) (s : stream A) : nth_tail n (map f s) = map f (nth_tail n s) :=
|
|
stream.ext (λ i, rfl)
|
|
|
|
theorem nth_map (n : nat) (s : stream A) : nth n (map f s) = f (nth n s) :=
|
|
rfl
|
|
end map
|
|
|
|
section zip
|
|
variable (f : A → B → C)
|
|
|
|
definition zip (s₁ : stream A) (s₂ : stream B) : stream C :=
|
|
λ n, f (nth n s₁) (nth n s₂)
|
|
|
|
theorem nth_tail_zip (n : nat) (s₁ : stream A) (s₂ : stream B) : nth_tail n (zip f s₁ s₂) = zip f (nth_tail n s₁) (nth_tail n s₂) :=
|
|
stream.ext (λ i, rfl)
|
|
|
|
theorem nth_zip (n : nat) (s₁ : stream A) (s₂ : stream B) : nth n (zip f s₁ s₂) = f (nth n s₁) (nth n s₂) :=
|
|
rfl
|
|
end zip
|
|
|
|
definition repeat (a : A) : stream A :=
|
|
λ n, a
|
|
|
|
theorem nth_repeat (n : nat) (a : A) : nth n (repeat a) = a :=
|
|
rfl
|
|
|
|
theorem nth_tail_repeat (n : nat) (a : A) : nth_tail n (repeat a) = repeat a :=
|
|
stream.ext (λ i, rfl)
|
|
|
|
definition iterate (f : A → A) (a : A) : stream A :=
|
|
λ n, nat.rec_on n a (λ n r, f r)
|
|
|
|
theorem head_iterate (f : A → A) (a : A) : head (iterate f a) = a :=
|
|
rfl
|
|
|
|
theorem tail_iterate (f : A → A) (a : A) : tail (iterate f a) = iterate f (f a) :=
|
|
begin
|
|
apply funext, intro n,
|
|
induction n with n' IH,
|
|
{reflexivity},
|
|
{esimp [tail, iterate] at *,
|
|
rewrite add_one at *,
|
|
esimp at *, rewrite IH}
|
|
end
|
|
|
|
theorem nth_zero_iterate (f : A → A) (a : A) : nth 0 (iterate f a) = a :=
|
|
rfl
|
|
|
|
theorem nth_succ_iterate (n : nat) (f : A → A) (a : A) : nth (succ n) (iterate f a) = nth n (iterate f (f a)) :=
|
|
by rewrite [nth_succ, tail_iterate]
|
|
|
|
theorem map_iterate (f : A → A) (a : A) : iterate f (f a) = map f (iterate f a) :=
|
|
begin
|
|
apply funext, intro n,
|
|
induction n with n' IH,
|
|
{reflexivity},
|
|
{esimp [map, iterate, nth] at *,
|
|
rewrite IH}
|
|
end
|
|
end stream
|