Package natural-order-def: natural-order-def

Information

Files

• Package tarball natural-order-def-1.4.tgz
• Theory file natural-order-def.thy (included in the package tarball)

Defined Constants

• Number
  • Natural
    • Number.Natural.<
    • Number.Natural.≤
    • Number.Natural.>
    • Number.Natural.≥

Theorems

⊦ ∀n m. Number.Natural.> m n ⇔ Number.Natural.< n m

⊦ ∀n m. Number.Natural.≥ m n ⇔ Number.Natural.≤ n m

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

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

Input Type Operators

• →
• bool
• Number
  • Natural
    • Number.Natural.natural

Input Constants

• =
• select
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • F
    • T
• Number
  • Natural
    • Number.Natural.suc
    • Number.Natural.zero

Assumptions

⊦ T

⊦ (∃) = λP. P ((select) P)

⊦ (∀) = λp. p = λx. T

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

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