Package natural-order-min-max-thm: Properties of natural number min and max functions
Information
| name | natural-order-min-max-thm |
| version | 1.17 |
| description | Properties of natural number min and max functions |
| author | Joe Hurd <joe@gilith.com> |
| license | HOLLight |
| provenance | HOL Light theory extracted on 2012-02-07 |
| requires | bool natural-order-thm natural-order-min-max-def |
| show | Data.Bool Number.Natural |
Files
- Package tarball natural-order-min-max-thm-1.17.tgz
- Theory file natural-order-min-max-thm.thy (included in the package tarball)
Theorems
⊦ ∀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
Input Type Operators
- →
- bool
- Number
- Natural
- natural
- Natural
Input Constants
- =
- Data
- Bool
- ∀
- ∧
- ⇒
- ∨
- ¬
- cond
- ⊥
- ⊤
- Bool
- Number
- Natural
- ≤
- max
- min
- zero
- Natural
Assumptions
⊦ ⊤
⊦ ∀n. 0 ≤ n
⊦ ∀n. n ≤ n
⊦ ⊥ ⇔ ∀p. p
⊦ (¬) = λp. p ⇒ ⊥
⊦ ∀t. (∀x. t) ⇔ t
⊦ (∀) = λp. p = λx. ⊤
⊦ ∀t. ⊤ ∧ t ⇔ t
⊦ ∀t. ⊥ ⇒ t ⇔ ⊤
⊦ ∀t. ⊥ ∨ t ⇔ t
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t1 t2. (if ⊤ then t1 else t2) = t1
⊦ ∀m n. m ≤ n ∨ n ≤ m
⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤
⊦ ∀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
⊦ ∀p c x y. p (if c then x else y) ⇔ (c ⇒ p x) ∧ (¬c ⇒ p y)