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

Information

Files

• Package tarball list-replicate-thm-1.0.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

Input Type Operators

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

Input Constants

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

Assumptions

⊦ T

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λP. P = λx. T

⊦ ∀x. x = x ⇔ T

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀m. m = Number.Numeral.zero ∨ ∃n. m = Number.Natural.suc n

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

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

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

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

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

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

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

⊦ (∀h t. Data.List.nth Number.Numeral.zero (Data.List.:: h t) = h) ∧
  ∀h t n.
    Data.List.nth (Number.Natural.suc n) (Data.List.:: h t) =
    Data.List.nth n t

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