refactor(library/data/list): move theorems to separate modules per lean2
This commit is contained in:
Joe Hendrix
Leonardo de Moura
parent
3de9e722e1
commit
8e2cf491e5
3 changed files with 80 additions and 69 deletions
79
library/data/list/basic.lean
Normal file
79
library/data/list/basic.lean
Normal file
|
|
@ -0,0 +1,79 @@
|
|||
import init.data.list.basic
|
||||
import data.nat.order
|
||||
|
||||
universe variables u v w
|
||||
|
||||
namespace list
|
||||
|
||||
open nat
|
||||
|
||||
variables {α : Type u} {β : Type v} {φ : Type w}
|
||||
|
||||
/- length theorems -/
|
||||
|
||||
theorem length_append : ∀ (x y : list α), length (x ++ y) = length x + length y
|
||||
| [] l := eq.symm (nat.zero_add (length l))
|
||||
| (a::s) l :=
|
||||
calc succ (length (s ++ l))
|
||||
= succ (length s + length l) : congr_arg nat.succ (length_append s l)
|
||||
... = succ (length s) + length l : eq.symm (nat.succ_add (length s) (length l))
|
||||
|
||||
theorem length_concat (a : α) : ∀ (l : list α), length (concat l a) = succ (length l)
|
||||
| nil := rfl
|
||||
| (cons b l) := congr_arg succ (length_concat l)
|
||||
|
||||
theorem length_dropn
|
||||
: ∀ (i : ℕ) (l : list α), length (dropn i l) = length l - i
|
||||
| 0 l := rfl
|
||||
| (succ i) [] := eq.symm (nat.zero_sub_eq_zero (succ i))
|
||||
| (succ i) (x::l) := calc
|
||||
length (dropn (succ i) (x::l))
|
||||
= length l - i : length_dropn i l
|
||||
... = succ (length l) - succ i : nat.sub_eq_succ_sub_succ (length l) i
|
||||
|
||||
theorem length_map (f : α → β) : ∀ (a : list α), length (map f a) = length a
|
||||
| [] := rfl
|
||||
| (a :: l) := congr_arg succ (length_map l)
|
||||
|
||||
theorem length_repeat (a : α) : ∀ (n : ℕ), length (repeat a n) = n
|
||||
| 0 := eq.refl 0
|
||||
| (succ i) := congr_arg succ (length_repeat i)
|
||||
|
||||
/- firstn -/
|
||||
|
||||
def firstn : ℕ → list α → list α
|
||||
| 0 l := []
|
||||
| (succ n) [] := []
|
||||
| (succ n) (a::l) := a :: firstn n l
|
||||
|
||||
theorem length_firstn
|
||||
: ∀ (i : ℕ) (l : list α), length (firstn i l) = min i (length l)
|
||||
| 0 l := eq.symm (nat.zero_min (length l))
|
||||
| (succ n) [] := eq.symm (nat.min_zero (succ n))
|
||||
| (succ n) (a::l) :=
|
||||
calc succ (length (firstn n l)) = succ (min n (length l)) : congr_arg succ (length_firstn n l)
|
||||
... = min (succ n) (succ (length l))
|
||||
: eq.symm (nat.min_succ_succ n (length l))
|
||||
|
||||
/- decidable -/
|
||||
|
||||
definition has_decidable_eq [h : decidable_eq α]
|
||||
: ∀ (x y : list α), decidable (x = y)
|
||||
| nil nil := is_true rfl
|
||||
| nil (cons b s) := is_false (λ q, list.no_confusion q)
|
||||
| (cons a r) nil := is_false (λ q, list.no_confusion q)
|
||||
| (cons a r) (cons b s) :=
|
||||
match h a b with
|
||||
| (is_true h₁) :=
|
||||
match has_decidable_eq r s with
|
||||
| (is_true h₂) :=
|
||||
is_true (calc a :: r = b :: r : congr_arg (λc, c :: r) h₁
|
||||
... = b :: s : congr_arg (λt, b :: t) h₂)
|
||||
| (is_false h₂) :=
|
||||
is_false (λ q, list.no_confusion q (λ heq teq, h₂ teq))
|
||||
end
|
||||
| (is_false h₁) :=
|
||||
is_false (λ q, list.no_confusion q (λ heq teq, h₁ heq))
|
||||
end
|
||||
|
||||
end list
|
||||
|
|
@ -1,12 +1,4 @@
|
|||
/-
|
||||
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Author: Jeremy Avigad
|
||||
|
||||
This is a minimal port of functions from the lean2 list library.
|
||||
-/
|
||||
import init.data.list.basic
|
||||
|
||||
import data.nat.order
|
||||
|
||||
universe variables u v w
|
||||
|
|
@ -17,67 +9,6 @@ open nat
|
|||
|
||||
variables {α : Type u} {β : Type v} {φ : Type w}
|
||||
|
||||
/- length theorems -/
|
||||
|
||||
theorem length_append : ∀ (x y : list α), length (x ++ y) = length x + length y
|
||||
| [] l := eq.symm (nat.zero_add (length l))
|
||||
| (a::s) l :=
|
||||
calc nat.succ (length (s ++ l))
|
||||
= nat.succ (length s + length l) : congr_arg nat.succ (length_append s l)
|
||||
... = nat.succ (length s) + length l : eq.symm (nat.succ_add (length s) (length l))
|
||||
|
||||
theorem length_repeat (a : α) : ∀ (n : ℕ), length (repeat a n) = n
|
||||
| 0 := eq.refl 0
|
||||
| (succ i) := congr_arg succ (length_repeat i)
|
||||
|
||||
theorem length_map (f : α → β) : ∀ (a : list α), length (map f a) = length a
|
||||
| [] := rfl
|
||||
| (a :: l) := congr_arg succ (length_map l)
|
||||
|
||||
theorem length_dropn
|
||||
: ∀ (i : ℕ) (l : list α), length (dropn i l) = length l - i
|
||||
| 0 l := rfl
|
||||
| (succ i) [] := eq.symm (nat.zero_sub_eq_zero (succ i))
|
||||
| (succ i) (x::l) := calc
|
||||
length (dropn (succ i) (x::l))
|
||||
= length l - i : length_dropn i l
|
||||
... = succ (length l) - succ i : nat.sub_eq_succ_sub_succ (length l) i
|
||||
|
||||
definition has_decidable_eq [h : decidable_eq α]
|
||||
: ∀ (x y : list α), decidable (x = y)
|
||||
| nil nil := is_true rfl
|
||||
| nil (cons b s) := is_false (λ q, list.no_confusion q)
|
||||
| (cons a r) nil := is_false (λ q, list.no_confusion q)
|
||||
| (cons a r) (cons b s) :=
|
||||
match h a b with
|
||||
| (is_true h₁) :=
|
||||
match has_decidable_eq r s with
|
||||
| (is_true h₂) :=
|
||||
is_true (calc a :: r = b :: r : congr_arg (λc, c :: r) h₁
|
||||
... = b :: s : congr_arg (λt, b :: t) h₂)
|
||||
| (is_false h₂) :=
|
||||
is_false (λ q, list.no_confusion q (λ heq teq, h₂ teq))
|
||||
end
|
||||
| (is_false h₁) :=
|
||||
is_false (λ q, list.no_confusion q (λ heq teq, h₁ heq))
|
||||
end
|
||||
|
||||
/- firstn -/
|
||||
|
||||
def firstn : ℕ → list α → list α
|
||||
| 0 l := []
|
||||
| (succ n) [] := []
|
||||
| (succ n) (a::l) := a :: firstn n l
|
||||
|
||||
theorem length_firstn
|
||||
: ∀ (i : ℕ) (l : list α), length (firstn i l) = min i (length l)
|
||||
| 0 l := eq.symm (nat.zero_min (length l))
|
||||
| (succ n) [] := eq.symm (nat.min_zero (succ n))
|
||||
| (succ n) (a::l) :=
|
||||
calc succ (length (firstn n l)) = succ (min n (length l)) : congr_arg succ (length_firstn n l)
|
||||
... = min (succ n) (succ (length l))
|
||||
: eq.symm (nat.min_succ_succ n (length l))
|
||||
|
||||
/- map₂ -/
|
||||
|
||||
definition map₂ {α : Type u} {β : Type v} {φ : Type w} (f : α → β → φ) : list α → list β → list φ
|
||||
1
library/data/list/default.lean
Normal file
1
library/data/list/default.lean
Normal file
|
|
@ -0,0 +1 @@
|
|||
import .basic .comb
|
||||
Loading…
Add table
Reference in a new issue