Package list-append-thm: list-append-thm

Information

name	list-append-thm
version	1.7
description	list-append-thm
author	Joe Hurd <joe@gilith.com>
license	HOLLight
provenance	HOL Light theory extracted on 2011-09-21
show	Data.Bool

Files

Package tarball list-append-thm-1.7.tgz
Theory file list-append-thm.thy (included in the package tarball)

Theorems

⊦ ∀l. Data.List.@ l Data.List.[] = l

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

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

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

⊦ ∀l m n. Data.List.@ l (Data.List.@ m n) = Data.List.@ (Data.List.@ l m) n

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

⊦ ∀l m.
Data.List.@ l m = Data.List.[] ⇔ l = Data.List.[] ∧ m = Data.List.[]

Input Type Operators

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

Input Constants

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

Assumptions

⊦ T

⊦ F ⇔ ∀p. p

⊦ (¬) = λp. p ⇒ F

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λp. p = λx. T

⊦ ∀x. x = x ⇔ T

⊦ ∀l. Data.List.null l ⇔ l = Data.List.[]

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀h t. ¬(Data.List.:: h t = Data.List.[])

⊦ ∀h t. Data.List.head (Data.List.:: h t) = h

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ ∀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

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

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

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

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

⊦ ∀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)