Package list-last-thm: list-last-thm

Information

name	list-last-thm
version	1.12
description	list-last-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-last-thm-1.12.tgz
Theory file list-last-thm.thy (included in the package tarball)

Theorems

⊦ ∀l. ¬(l = Data.List.[]) ⇒ Set.∈ (Data.List.last l) (Data.List.toSet l)

⊦ ∀p q.
Data.List.last (Data.List.@ p q) =
if q = Data.List.[] then Data.List.last p else Data.List.last q

⊦ (∀h. Data.List.last (Data.List.:: h Data.List.[]) = h) ∧
  ∀h k t.
    Data.List.last (Data.List.:: h (Data.List.:: k t)) =
    Data.List.last (Data.List.:: k t)

Input Type Operators

→
bool
Data
- List
  - Data.List.list
Set
- Set.set

Input Constants

=
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∨
  - ¬
  - cond
  - F
  - T
- List
  - Data.List.::
  - Data.List.@
  - Data.List.[]
  - Data.List.last
  - Data.List.toSet
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

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)

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

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r

⊦ ∀h t.
Data.List.last (Data.List.:: h t) =
if t = Data.List.[] then h else Data.List.last t

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

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

⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)

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

⊦ ∀x y s. Set.∈ x (Set.insert y s) ⇔ x = y ∨ Set.∈ x s

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

⊦ ∀P c x y. P (if c then x else y) ⇔ (c ⇒ P x) ∧ (¬c ⇒ P y)

⊦ (∀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)

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

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