Package natural-min-max-thm: natural-min-max-thm

Information

Files

• Package tarball natural-min-max-thm-1.2.tgz
• Theory file natural-min-max-thm.thy (included in the package tarball)

Theorems

⊦ ∀n. Number.Natural.max Number.Numeral.zero n = n

⊦ ∀n. Number.Natural.max n Number.Numeral.zero = n

⊦ ∀n. Number.Natural.max n n = n

⊦ ∀n. Number.Natural.min Number.Numeral.zero n = Number.Numeral.zero

⊦ ∀n. Number.Natural.min n Number.Numeral.zero = Number.Numeral.zero

⊦ ∀n. Number.Natural.min n n = n

⊦ ∀m n. Number.Natural.≤ m (Number.Natural.max m n)

⊦ ∀m n. Number.Natural.≤ n (Number.Natural.max m n)

⊦ ∀m n. Number.Natural.≤ (Number.Natural.min m n) m

⊦ ∀m n. Number.Natural.≤ (Number.Natural.min m n) n

⊦ ∀m n. Number.Natural.max m n = Number.Natural.max n m

⊦ ∀m n. Number.Natural.min m n = Number.Natural.min n m

Input Type Operators

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

Input Constants

• =
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∨
    • ~
    • cond
    • F
    • T
• Number
  • Natural
    • Number.Natural.≤
    • Number.Natural.max
    • Number.Natural.min
  • Numeral
    • Number.Numeral.zero

Assumptions

⊦ T

⊦ ∀n. Number.Natural.≤ Number.Numeral.zero n

⊦ ∀n. Number.Natural.≤ n n

⊦ F ⇔ ∀p. p

⊦ (~) = λp. p ⇒ F

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λP. P = λx. T

⊦ ∀x. x = x ⇔ T

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀m n. Number.Natural.≤ m n ∨ Number.Natural.≤ n m

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ ∀m n. Number.Natural.max m n = if Number.Natural.≤ m n then n else m

⊦ ∀m n. Number.Natural.min m n = if Number.Natural.≤ m n then m else n

⊦ ∀m n. Number.Natural.≤ m n ∧ Number.Natural.≤ n m ⇔ m = n

⊦ ∀t1 t2. (if T then t1 else t2) = t1 ∧ (if F then t1 else t2) = t2

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

⊦ ∀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 ∨ F ⇔ t) ∧ (t ∨ t ⇔ t)

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