Package list-replicate: The list replicate function

Information

name	list-replicate
version	1.45
description	The list replicate function
author	Joe Hurd <joe@gilith.com>
license	MIT
requires	bool list-length list-nth list-set natural set
show	Data.Bool Data.List Number.Natural Set

Files

Package tarball list-replicate-1.45.tgz
Theory file list-replicate.thy (included in the package tarball)

Defined Constant

Data
- List
  - replicate

Theorems

⊦ ∀x. replicate x 0 = []

⊦ ∀x n. length (replicate x n) = n

⊦ ∀x n. replicate x (suc n) = x :: replicate x n

⊦ ∀x n i. i < n ⇒ nth (replicate x n) i = x

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

⊦ ∀x n y. member y (replicate x n) ⇔ y = x ∧ ¬(n = 0)

Input Type Operators

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

Input Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∃!
  - ∨
  - ¬
  - cond
  - ⊥
  - ⊤
- List
  - ::
  - []
  - length
  - member
  - nth
  - toSet
Number
- Natural
  - <
  - suc
  - zero
Set
- ∅
- insert
- ∈

Assumptions

⊦ ⊤

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ length [] = 0

⊦ toSet [] = ∅

⊦ ∀t. t ⇒ t

⊦ ⊥ ⇔ ∀p. p

⊦ ∀x. ¬(x ∈ ∅)

⊦ ∀m. ¬(m < 0)

⊦ (¬) = λp. p ⇒ ⊥

⊦ (∃) = λp. p ((select) p)

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λp. p = λx. ⊤

⊦ ∀t. ⊥ ∧ t ⇔ ⊥

⊦ ∀t. ⊤ ∧ t ⇔ t

⊦ ∀t. t ∧ ⊥ ⇔ ⊥

⊦ ∀t. t ∧ ⊤ ⇔ t

⊦ ∀t. ⊥ ⇒ t ⇔ ⊤

⊦ ∀t. ⊤ ⇒ t ⇔ t

⊦ ∀t. t ⇒ ⊤ ⇔ ⊤

⊦ ∀t. ⊥ ∨ t ⇔ t

⊦ ∀t. ⊤ ∨ t ⇔ ⊤

⊦ ∀n. ¬(suc n = 0)

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀t₁ t₂. (if ⊥ then t₁ else t₂) = t₂

⊦ ∀t₁ t₂. (if ⊤ then t₁ else t₂) = t₁

⊦ ∀h t. nth (h :: t) 0 = h

⊦ ∀h t. length (h :: t) = suc (length t)

⊦ ∀m. m = 0 ∨ ∃n. m = suc n

⊦ ∀l x. member x l ⇔ x ∈ toSet l

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀x y. x ∈ insert y ∅ ⇔ x = y

⊦ ∀h t. toSet (h :: t) = insert h (toSet t)

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

⊦ ∀m n. suc m < suc n ⇔ m < n

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

⊦ ∀p. (∀x. ∃y. p x y) ⇔ ∃y. ∀x. p x (y x)

⊦ ∀p. p 0 ∧ (∀n. p n ⇒ p (suc n)) ⇒ ∀n. p n

⊦ (∃!) = λp. (∃) p ∧ ∀x y. p x ∧ p y ⇒ x = y

⊦ ∀e f. ∃!fn. fn 0 = e ∧ ∀n. fn (suc n) = f (fn n) n

⊦ ∀h t n. n < length t ⇒ nth (h :: t) (suc n) = nth t n

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