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

Information

Files

• Package tarball list-interval-1.0.tgz
• Theory file list-interval.thy (included in the package tarball)

Defined Constant

• Data
  • List
    • interval

Theorems

⊦ ∀m n. length (interval m n) = n

⊦ ∀m n i. i < n ⇒ nth i (interval m n) = m + i

⊦ (∀m. interval m 0 = []) ∧
  ∀m n. interval m (suc n) = m :: interval (suc m) n

Input Type Operators

• →
• bool
• Data
  • List
    • list
• Number
  • Natural
    • natural

Input Constants

• =
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ¬
    • select
    • F
    • T
  • List
    • ::
    • []
    • length
    • nth
• Number
  • Natural
    • +
    • <
    • suc
  • Numeral
    • zero

Assumptions

⊦ T

⊦ (∃) = λP. P ((select) P)

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λP. P = λx. T

⊦ ∀x. x = x ⇔ T

⊦ ∀m. m + 0 = m

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀m. m = 0 ∨ ∃n. m = suc n

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

⊦ ∀m n. m + suc n = suc (m + n)

⊦ ∀m n. suc m = suc n ⇔ m = n

⊦ ∀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

⊦ length [] = 0 ∧ ∀h t. length (h :: t) = suc (length t)

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

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

⊦ (∀n. 0 + n = n) ∧ ∀m n. suc m + n = suc (m + n)

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

⊦ (∀h t. nth 0 (h :: t) = h) ∧ ∀h t n. nth (suc n) (h :: t) = nth n 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 ⇒ t ⇔ T) ∧ (t ⇒ F ⇔ ¬t)