Package list-reverse: Definitions and theorems about the list reverse function

Information

name	list-reverse
version	1.12
description	Definitions and theorems about the list reverse function
author	Joe Hurd <joe@gilith.com>
license	MIT
show	Data.Bool Data.List

Files

Package tarball list-reverse-1.12.tgz
Theory file list-reverse.thy (included in the package tarball)

Defined Constant

Data
- List
  - reverse

Theorems

⊦ ∀l. reverse (reverse l) = l

⊦ ∀l. length (reverse l) = length l

⊦ ∀l. toSet (reverse l) = toSet l

⊦ ∀l m. reverse (l @ m) = reverse m @ reverse l

⊦ reverse [] = [] ∧ ∀x l. reverse (x :: l) = reverse l @ x :: []

Input Type Operators

→
bool
Data
- List
  - list
Number
- Natural
  - Number.Natural.natural
Set
- Set.set

Input Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - T
- List
  - ::
  - @
  - []
  - length
  - toSet
Number
- Natural
  - Number.Natural.+
  - Number.Natural.suc
  - Number.Natural.zero
Set
- Set.∅
- Set.insert
- Set.∪

Assumptions

⊦ T

⊦ (∃) = λP. P ((select) P)

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λp. p = λx. T

⊦ ∀x. x = x ⇔ T

⊦ ∀l. l @ [] = l

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀s t. Set.∪ s t = Set.∪ t s

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ ∀x s. Set.∪ (Set.insert x Set.∅) s = Set.insert x s

⊦ ∀l m. length (l @ m) = Number.Natural.+ (length l) (length m)

⊦ ∀l1 l2. toSet (l1 @ l2) = Set.∪ (toSet l1) (toSet l2)

⊦ ∀l m n. l @ m @ n = (l @ m) @ n

⊦ length [] = 0 ∧ ∀h t. length (h :: t) = Number.Natural.suc (length t)

⊦ toSet [] = Set.∅ ∧ ∀h t. toSet (h :: t) = Set.insert h (toSet t)

⊦ ∀P. P [] ∧ (∀a0 a1. P a1 ⇒ P (a0 :: a1)) ⇒ ∀x. P x

⊦ ∀NIL' CONS'.
∃fn. fn [] = NIL' ∧ ∀a0 a1. fn (a0 :: a1) = CONS' a0 a1 (fn a1)

⊦ (∀l. [] @ l = l) ∧ ∀l h t. (h :: t) @ l = h :: t @ l

⊦ (∀n. Number.Natural.+ 0 n = n) ∧ (∀m. Number.Natural.+ m 0 = m) ∧
  (∀m n.
     Number.Natural.+ (Number.Natural.suc m) n =
     Number.Natural.suc (Number.Natural.+ m n)) ∧
  ∀m n.
    Number.Natural.+ m (Number.Natural.suc n) =
    Number.Natural.suc (Number.Natural.+ m n)