Package natural-order-min-max-thm: Properties of natural number min and max functions

Information

Files

• Package tarball natural-order-min-max-thm-1.30.tgz
• Theory source file natural-order-min-max-thm.thy (included in the package tarball)

Theorems

⊦ (minimal n. ⊤) = 0

⊦ ∀n. max 0 n = n

⊦ ∀n. max n 0 = n

⊦ ∀n. max n n = n

⊦ ∀n. min 0 n = 0

⊦ ∀n. min n 0 = 0

⊦ ∀n. min n n = n

⊦ ∀m n. m ≤ max m n

⊦ ∀m n. n ≤ max m n

⊦ ∀m n. min m n ≤ m

⊦ ∀m n. min m n ≤ n

⊦ ∀m n. max m n = max n m

⊦ ∀m n. min m n = min n m

⊦ ∀m n. max (suc m) (suc n) = suc (max m n)

⊦ ∀m n. min (suc m) (suc n) = suc (min m n)

⊦ ∀m n p. m ≤ min n p ⇔ m ≤ n ∧ m ≤ p

⊦ ∀m n p. max n p ≤ m ⇔ n ≤ m ∧ p ≤ m

⊦ ∀p n. p n ∧ (∀m. m < n ⇒ ¬p m) ⇒ (minimal) p = n

External Type Operators

• →
• bool
• Number
  • Natural
    • natural

External Constants

• =
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ¬
    • cond
    • ⊥
    • ⊤
• Number
  • Natural
    • <
    • ≤
    • max
    • min
    • minimal
    • suc
    • zero

Assumptions

⊦ ⊤

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ ∀n. 0 ≤ n

⊦ ∀n. n ≤ n

⊦ ⊥ ⇔ ∀p. p

⊦ ∀m. ¬(m < 0)

⊦ (¬) = λp. 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

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t₁ t₂. (if ⊥ then t₁ else t₂) = t₂

⊦ ∀t₁ t₂. (if ⊤ then t₁ else t₂) = t₁

⊦ ∀m n. m < n ⇒ m ≤ n

⊦ ∀m n. m ≤ n ∨ n ≤ m

⊦ ∀m n. ¬(m < n) ⇔ n ≤ m

⊦ ∀m n. ¬(m ≤ n) ⇔ n < m

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀m n. suc m ≤ suc n ⇔ m ≤ n

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

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

⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n

⊦ ∀m n p. m ≤ n ∧ n ≤ p ⇒ m ≤ p

⊦ ∀p. (∃n. p n) ⇔ p ((minimal) p) ∧ ∀m. m < (minimal) p ⇒ ¬p m

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

OpenTheory package document generated by opentheory 1.2 (release 20140322)