Package natural-add-thm: Properties of natural number addition
Information
| name | natural-add-thm |
| version | 1.47 |
| description | Properties of natural number addition |
| author | Joe Leslie-Hurd <joe@gilith.com> |
| license | HOLLight |
| provenance | HOL Light theory extracted on 2012-11-10 |
| requires | bool natural-add-def natural-def natural-numeral natural-order |
| show | Data.Bool Number.Natural |
Files
- Package tarball natural-add-thm-1.47.tgz
- Theory file natural-add-thm.thy (included in the package tarball)
Theorems
⊦ ∀m. m + 0 = m
⊦ ∀m n. m ≤ m + n
⊦ ∀m n. n ≤ m + n
⊦ ∀m. suc m = m + 1
⊦ ∀m n. m + n = n + m
⊦ ∀m n. m + suc n = suc (m + n)
⊦ ∀m n. m < m + n ⇔ 0 < n
⊦ ∀m n. n < m + n ⇔ 0 < m
⊦ ∀m n. m + n = m ⇔ n = 0
⊦ ∀m n. m + n = n ⇔ m = 0
⊦ ∀m n. m ≤ n ⇔ ∃d. n = m + d
⊦ ∀m n. m < n ⇔ ∃d. n = m + suc d
⊦ ∀m n p. m + (n + p) = m + n + p
⊦ ∀m n p. m + n = m + p ⇔ n = p
⊦ ∀m n p. m + p = n + p ⇔ m = n
⊦ ∀m n p. m + n < m + p ⇔ n < p
⊦ ∀m n p. n + m < p + m ⇔ n < p
⊦ ∀m n p. m + n ≤ m + p ⇔ n ≤ p
⊦ ∀m n p. m + p ≤ n + p ⇔ m ≤ n
⊦ ∀m n. m + n = 0 ⇔ m = 0 ∧ n = 0
⊦ ∀m n p q. m < p ∧ n < q ⇒ m + n < p + q
⊦ ∀m n p q. m < p ∧ n ≤ q ⇒ m + n < p + q
⊦ ∀m n p q. m ≤ p ∧ n < q ⇒ m + n < p + q
⊦ ∀m n p q. m ≤ p ∧ n ≤ q ⇒ m + n ≤ p + q
⊦ ∀p q. (∃b. ∀i. p i ≤ q i + b) ⇔ ∃b n. ∀i. n ≤ i ⇒ p i ≤ q i + b
Input Type Operators
- →
- bool
- Number
- Natural
- natural
- Natural
Input Constants
- =
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∨
- ¬
- ⊥
- ⊤
- Bool
- Number
- Natural
- +
- <
- ≤
- bit0
- bit1
- suc
- zero
- Natural
Assumptions
⊦ ⊤
⊦ ¬⊥ ⇔ ⊤
⊦ ¬⊤ ⇔ ⊥
⊦ bit0 0 = 0
⊦ ∀n. 0 ≤ n
⊦ ∀n. n ≤ n
⊦ ⊥ ⇔ ∀p. p
⊦ ∀t. t ∨ ¬t
⊦ ∀m. ¬(m < 0)
⊦ (¬) = λp. p ⇒ ⊥
⊦ ∀a. ∃x. x = a
⊦ ∀t. (∀x. t) ⇔ t
⊦ ∀t. (∃x. t) ⇔ t
⊦ ∀t. (λx. t x) = t
⊦ (∀) = λp. p = λx. ⊤
⊦ ∀t. ¬¬t ⇔ t
⊦ ∀t. (⊤ ⇔ t) ⇔ t
⊦ ∀t. ⊥ ∧ t ⇔ ⊥
⊦ ∀t. ⊤ ∧ t ⇔ t
⊦ ∀t. t ∧ ⊤ ⇔ t
⊦ ∀t. t ∧ t ⇔ t
⊦ ∀t. ⊥ ⇒ t ⇔ ⊤
⊦ ∀t. ⊤ ⇒ t ⇔ t
⊦ ∀t. t ⇒ ⊤ ⇔ ⊤
⊦ ∀t. ⊥ ∨ t ⇔ t
⊦ ∀t. ⊤ ∨ t ⇔ ⊤
⊦ ∀t. t ∨ ⊥ ⇔ t
⊦ ∀t. t ∨ ⊤ ⇔ ⊤
⊦ ∀n. ¬(suc n = 0)
⊦ ∀n. 0 + n = n
⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t
⊦ ∀t. t ⇒ ⊥ ⇔ ¬t
⊦ ∀n. bit1 n = suc (bit0 n)
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)
⊦ ∀m. m ≤ 0 ⇔ m = 0
⊦ ∀n. bit0 (suc n) = suc (suc (bit0 n))
⊦ ∀t1 t2. t1 ∧ t2 ⇔ t2 ∧ t1
⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1
⊦ ∀m n. m < n ⇒ m ≤ n
⊦ ∀m n. ¬(m ≤ n) ⇔ n < m
⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤
⊦ ∀p. ¬(∀x. p x) ⇔ ∃x. ¬p x
⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q
⊦ ∀m n. suc m + n = suc (m + n)
⊦ ∀m n. suc m = suc n ⇔ m = n
⊦ ∀m n. suc m < suc n ⇔ m < n
⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r
⊦ ∀p q. (∃x. p ∧ q x) ⇔ p ∧ ∃x. q x
⊦ ∀p q. p ∨ (∃x. q x) ⇔ ∃x. p ∨ q x
⊦ ∀m n. m < suc n ⇔ m = n ∨ m < n
⊦ ∀p q. (∃x. p x) ⇒ q ⇔ ∀x. p x ⇒ q
⊦ ∀t1 t2 t3. (t1 ∨ t2) ∨ t3 ⇔ t1 ∨ t2 ∨ t3
⊦ ∀m n p. m ≤ n ∧ n ≤ p ⇒ m ≤ p
⊦ ∀m n. m ≤ suc n ⇔ m = suc n ∨ m ≤ n
⊦ ∀p. p 0 ∧ (∀n. p n ⇒ p (suc n)) ⇒ ∀n. p n
⊦ ∀p q. (∀x. p x ∧ q x) ⇔ (∀x. p x) ∧ ∀x. q x
⊦ ∀p q. (∃x. p x) ∨ (∃x. q x) ⇔ ∃x. p x ∨ q x