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

Information

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

Files

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

Defined Constant

Data
- List
  - @

Theorems

⊦ ∀l. l @ [] = l

⊦ ∀l m. null (l @ m) ⇔ null l ∧ null m

⊦ ∀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

⊦ ∀l m. head (l @ m) = if l = [] then head m else head l

⊦ ∀l m. l @ m = [] ⇔ l = [] ∧ m = []

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

Input Type Operators

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

Input Constants

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

Assumptions

⊦ T

⊦ F ⇔ ∀p. p

⊦ (¬) = λp. p ⇒ F

⊦ (∃) = λP. P ((select) P)

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λp. p = λx. T

⊦ ∀x. x = x ⇔ T

⊦ ∀l. null l ⇔ l = []

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀h t. ¬(h :: t = [])

⊦ ∀h t. head (h :: t) = h

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

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

⊦ (∀s. Set.∪ Set.∅ s = s) ∧ ∀s. Set.∪ s Set.∅ = s

⊦ ∀P. (∀x y. P x y) ⇔ ∀y x. P x y

⊦ ∀s t u. Set.∪ (Set.∪ s t) u = Set.∪ s (Set.∪ t u)

⊦ ∀t1 t2. (if T then t1 else t2) = t1 ∧ (if F then t1 else t2) = t2

⊦ 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)

⊦ ∀t. ((T ⇔ t) ⇔ t) ∧ ((t ⇔ T) ⇔ t) ∧ ((F ⇔ t) ⇔ ¬t) ∧ ((t ⇔ F) ⇔ ¬t)

⊦ ∀t. (T ∧ t ⇔ t) ∧ (t ∧ T ⇔ t) ∧ (F ∧ t ⇔ F) ∧ (t ∧ F ⇔ F) ∧ (t ∧ t ⇔ t)

⊦ (∀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)