Package natural-min-max-thm: natural-min-max-thm
Information
| name | natural-min-max-thm |
| version | 1.5 |
| description | natural-min-max-thm |
| author | Joe Hurd <joe@gilith.com> |
| license | HOLLight |
| provenance | HOL Light theory extracted on 2011-09-21 |
| show | Data.Bool |
Files
- Package tarball natural-min-max-thm-1.5.tgz
- Theory file natural-min-max-thm.thy (included in the package tarball)
Theorems
⊦ ∀n. Number.Natural.max 0 n = n
⊦ ∀n. Number.Natural.max n 0 = n
⊦ ∀n. Number.Natural.max n n = n
⊦ ∀n. Number.Natural.min 0 n = 0
⊦ ∀n. Number.Natural.min n 0 = 0
⊦ ∀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
- Natural
Input Constants
- =
- Data
- Bool
- ∀
- ∧
- ⇒
- ∨
- ¬
- cond
- F
- T
- Bool
- Number
- Natural
- Number.Natural.≤
- Number.Natural.max
- Number.Natural.min
- Number.Natural.zero
- Natural
Assumptions
⊦ T
⊦ ∀n. Number.Natural.≤ 0 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)