Package list-replicate-thm: list-replicate-thm

Information

name	list-replicate-thm
version	1.11
description	list-replicate-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-replicate-thm-1.11.tgz
Theory file list-replicate-thm.thy (included in the package tarball)

Theorems

⊦ ∀n x. Data.List.length (Data.List.replicate n x) = n

⊦ ∀n x i.
Number.Natural.< i n ⇒ Data.List.nth i (Data.List.replicate n x) = x

⊦ ∀n x.
Data.List.toSet (Data.List.replicate n x) =
if n = 0 then Set.∅ else Set.insert x Set.∅

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.length
  - Data.List.nth
  - Data.List.replicate
  - Data.List.toSet
Number
- Natural
  - Number.Natural.<
  - Number.Natural.suc
  - Number.Natural.zero
Set
- Set.∅
- Set.insert

Assumptions

⊦ T

⊦ F ⇔ ∀p. p

⊦ (¬) = λp. p ⇒ F

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λp. p = λx. T

⊦ ∀x. x = x ⇔ T

⊦ ∀n. ¬(Number.Natural.suc n = 0)

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀h t. Data.List.nth 0 (Data.List.:: h t) = h

⊦ ∀m. m = 0 ∨ ∃n. m = Number.Natural.suc n

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

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

⊦ ∀m n.
Number.Natural.< (Number.Natural.suc m) (Number.Natural.suc n) ⇔
Number.Natural.< m n

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

⊦ ∀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)

⊦ ∀P. P 0 ∧ (∀n. P n ⇒ P (Number.Natural.suc n)) ⇒ ∀n. P n

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

⊦ ∀h t n.
    Number.Natural.< n (Data.List.length t) ⇒
    Data.List.nth (Number.Natural.suc n) (Data.List.:: h t) =
    Data.List.nth n t

⊦ (∀x. Data.List.replicate 0 x = Data.List.[]) ∧
  ∀x n.
    Data.List.replicate (Number.Natural.suc n) x =
    Data.List.:: x (Data.List.replicate n x)

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

⊦ (∀m. Number.Natural.< m 0 ⇔ F) ∧
  ∀m n.
    Number.Natural.< m (Number.Natural.suc n) ⇔
    m = n ∨ Number.Natural.< m n

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