Package natural-mult: Definitions and theorems about natural number multiplication
Information
| name | natural-mult |
| version | 1.11 |
| description | Definitions and theorems about natural number multiplication |
| author | Joe Hurd <joe@gilith.com> |
| license | MIT |
| show | Data.Bool Number.Natural |
Files
- Package tarball natural-mult-1.11.tgz
- Theory file natural-mult.thy (included in the package tarball)
Defined Constant
- Number
- Natural
- *
- Natural
Theorems
⊦ ∀n. n ≤ n * n
⊦ ∀m. m * 0 = 0
⊦ ∀m n. m * n = n * m
⊦ ∀n. 2 * n = n + n
⊦ ∀m n. m * suc n = m + m * n
⊦ ∀A B. (∀n. A * n ≤ B) ⇔ A = 0
⊦ ∀m n p. m * (n * p) = m * n * p
⊦ ∀m n. 0 < m * n ⇔ 0 < m ∧ 0 < n
⊦ ∀m n. m * n = 0 ⇔ m = 0 ∨ n = 0
⊦ ∀m n p. m * (n + p) = m * n + m * p
⊦ ∀m n p. (m + n) * p = m * p + n * p
⊦ ∀A B C. (∀n. A * n ≤ B * n + C) ⇔ A ≤ B
⊦ ∀m n. m * n = 1 ⇔ m = 1 ∧ n = 1
⊦ ∀m n p. m * n = m * p ⇔ m = 0 ∨ n = p
⊦ ∀m n p. m * p = n * p ⇔ m = n ∨ p = 0
⊦ ∀m n p. m * n ≤ m * p ⇔ m = 0 ∨ n ≤ p
⊦ ∀m n p. m * p ≤ n * p ⇔ m ≤ n ∨ p = 0
⊦ ∀m n p. m * n < m * p ⇔ ¬(m = 0) ∧ n < p
⊦ ∀m n p. m * p < n * p ⇔ m < n ∧ ¬(p = 0)
⊦ ∀m n p. ¬(m = 0) ∧ n < p ⇒ m * n < m * p
⊦ (∀n. 0 * n = 0) ∧ ∀m n. suc m * n = m * n + n
⊦ ∀m n p q. m < n ∧ p < q ⇒ m * p < n * q
⊦ ∀m n p q. m ≤ n ∧ p ≤ q ⇒ m * p ≤ n * q
⊦ ∀P. (∃B. ∀n. P n ≤ B) ⇔ ∃A B. ∀n. n * P n ≤ A * n + B
⊦ ∀m n p.
m * n = n * m ∧ m * n * p = m * (n * p) ∧ m * (n * p) = n * (m * p)
⊦ ∀P A B.
P 0 0 = 0 ∧ (∀m n. P m n ≤ A * (m + n) + B) ⇒
∃B. ∀m n. P m n ≤ B * (m + n)
⊦ (∀n. 0 * n = 0) ∧ (∀m. m * 0 = 0) ∧ (∀n. 1 * n = n) ∧ (∀m. m * 1 = m) ∧
(∀m n. suc m * n = m * n + n) ∧ ∀m n. m * suc n = m + m * n
Input Type Operators
- →
- bool
- Number
- Natural
- natural
- Natural
Input Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∨
- ¬
- F
- T
- Bool
- Number
- Natural
- +
- <
- ≤
- bit0
- bit1
- suc
- zero
- Natural
Assumptions
⊦ T
⊦ ∀n. 0 ≤ n
⊦ ∀n. n ≤ n
⊦ F ⇔ ∀p. p
⊦ ∀t. t ∨ ¬t
⊦ ∀n. ¬(n < n)
⊦ ∀n. 0 < suc n
⊦ (¬) = λp. p ⇒ F
⊦ (∃) = λP. P ((select) P)
⊦ ∀t. (∀x. t) ⇔ t
⊦ ∀t. (λx. t x) = t
⊦ (∀) = λp. p = λx. T
⊦ ∀x. x = x ⇔ T
⊦ ∀n. ¬(suc n = 0)
⊦ ∀n. bit0 n = n + n
⊦ ∀m n. m ≤ m + n
⊦ ∀m n. n ≤ m + n
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)
⊦ ∀n. bit1 n = suc (n + n)
⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)
⊦ ∀t1 t2. t1 ∧ t2 ⇔ t2 ∧ t1
⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1
⊦ ∀m n. m + n = n + m
⊦ ∀m n. m < n ⇒ m ≤ n
⊦ ∀m n. ¬(m ≤ n) ⇔ n < m
⊦ ∀m n. m < suc n ⇔ m ≤ n
⊦ (∧) = λp q. (λf. f p q) = λf. f T T
⊦ ∀P. ¬(∀x. P x) ⇔ ∃x. ¬P x
⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q
⊦ ∀m n. suc m = suc n ⇔ m = n
⊦ ∀m n. suc m < suc n ⇔ m < n
⊦ ∀m n. suc m ≤ suc n ⇔ m ≤ n
⊦ ∀m n. m ≤ n ⇔ ∃d. n = m + d
⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r
⊦ ∀m n. m < n ⇔ ∃d. n = m + suc d
⊦ ∀P Q. P ∨ (∃x. Q x) ⇔ ∃x. P ∨ Q x
⊦ ∀m n. m < n ⇔ m ≤ n ∧ ¬(m = n)
⊦ ∀m n p. m + (n + p) = m + n + p
⊦ ∀m n p. m + n = m + p ⇔ n = p
⊦ ∀m n p. m + n < m + p ⇔ n < p
⊦ ∀m n p. m + n ≤ m + p ⇔ n ≤ p
⊦ ∀m n p. m ≤ n ∧ n < p ⇒ m < p
⊦ ∀m n p. m ≤ n ∧ n ≤ p ⇒ m ≤ p
⊦ ∀m n. m + n = 0 ⇔ m = 0 ∧ n = 0
⊦ ∀P. P 0 ∧ (∀n. P n ⇒ P (suc n)) ⇒ ∀n. P n
⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)
⊦ ∀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
⊦ ∀e f. ∃fn. fn 0 = e ∧ ∀n. fn (suc n) = f (fn n) n
⊦ (∀m. m < 0 ⇔ F) ∧ ∀m n. m < suc n ⇔ m = n ∨ m < n
⊦ (∀m. m ≤ 0 ⇔ m = 0) ∧ ∀m n. m ≤ suc n ⇔ m = suc n ∨ m ≤ n
⊦ ∀t. ((T ⇔ t) ⇔ t) ∧ ((t ⇔ T) ⇔ t) ∧ ((F ⇔ t) ⇔ ¬t) ∧ ((t ⇔ F) ⇔ ¬t)
⊦ ∀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)
⊦ (∀n. 0 + n = n) ∧ (∀m. m + 0 = m) ∧ (∀m n. suc m + n = suc (m + n)) ∧
∀m n. m + suc n = suc (m + n)
⊦ ∀p q r.
(p ∨ q ⇔ q ∨ p) ∧ ((p ∨ q) ∨ r ⇔ p ∨ q ∨ r) ∧ (p ∨ q ∨ r ⇔ q ∨ p ∨ r) ∧
(p ∨ p ⇔ p) ∧ (p ∨ p ∨ q ⇔ p ∨ q)