Package natural-divides-lcm: Natural number least common multiple

Information

Files

• Package tarball natural-divides-lcm-1.0.tgz
• Theory source file natural-divides-lcm.thy (included in the package tarball)

Defined Constant

• Number
  • Natural
    • lcm

Theorems

⊦ ∀a. lcm 0 a = 0

⊦ ∀a. lcm a 0 = 0

⊦ ∀a. lcm a a = a

⊦ ∀a. lcm a 1 = a

⊦ ∀a. lcm 1 a = a

⊦ ∀a b. divides a (lcm a b)

⊦ ∀a b. divides b (lcm a b)

⊦ ∀a b. lcm a b = lcm b a

⊦ ∀a b. lcm a b = a ⇔ divides b a

⊦ ∀a b. lcm b a = a ⇔ divides b a

⊦ ∀a b c. divides a b ⇒ divides a (lcm b c)

⊦ ∀a b c. divides a b ⇒ divides a (lcm c b)

⊦ ∀a b. lcm a b * gcd a b = a * b

⊦ ∀a b c. lcm (lcm a b) c = lcm a (lcm b c)

⊦ ∀a b. lcm a b = 0 ⇔ a = 0 ∨ b = 0

⊦ ∀a b c. divides (lcm a b) c ⇔ divides a c ∧ divides b c

⊦ ∀a b c. gcd a (lcm b c) = lcm (gcd a b) (gcd a c)

⊦ ∀a b c. lcm a (gcd b c) = gcd (lcm a b) (lcm a c)

⊦ ∀a b c. gcd (lcm b c) a = lcm (gcd b a) (gcd c a)

⊦ ∀a b c. lcm (gcd b c) a = gcd (lcm b a) (lcm c a)

⊦ ∀a b c. lcm (a * b) (a * c) = a * lcm b c

⊦ ∀a b c. lcm (b * a) (c * a) = lcm b c * a

⊦ ∀a b c. divides a c ∧ divides b c ⇒ divides (lcm a b) c

⊦ ∀a b. lcm a b = 1 ⇔ a = 1 ∧ b = 1

⊦ ∀a b l.
    divides a l ∧ divides b l ∧
    (∀c. divides a c ∧ divides b c ⇒ divides l c) ⇒ lcm a b = l

External Type Operators

• →
• bool
• Number
  • Natural
    • natural

External Constants

• =
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ¬
    • cond
    • ⊥
    • ⊤
• Number
  • Natural
    • *
    • +
    • <
    • ≤
    • bit0
    • bit1
    • div
    • divides
    • gcd
    • suc
    • zero

Assumptions

⊦ ⊤

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ bit0 0 = 0

⊦ ∀t. t ⇒ t

⊦ ∀a. divides a 0

⊦ ∀a. divides a a

⊦ ⊥ ⇔ ∀p. p

⊦ ∀t. t ∨ ¬t

⊦ ∀a. divides 1 a

⊦ (¬) = λp. p ⇒ ⊥

⊦ ∀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

⊦ ∀n. 0 * n = 0

⊦ ∀m. m * 0 = 0

⊦ ∀n. 0 + n = n

⊦ ∀a. gcd a 0 = a

⊦ ∀a. gcd a a = a

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ ∀n. bit1 n = suc (bit0 n)

⊦ ∀m. m * 1 = m

⊦ ∀a b. divides (gcd a b) a

⊦ ∀a b. divides (gcd a b) b

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀t₁ t₂. (if ⊥ then t₁ else t₂) = t₂

⊦ ∀t₁ t₂. (if ⊤ then t₁ else t₂) = t₁

⊦ ∀n. bit0 (suc n) = suc (suc (bit0 n))

⊦ ∀x y. x = y ⇒ y = x

⊦ ∀m n. m * n = n * m

⊦ ∀m n. m + n = n + m

⊦ ∀a b. gcd a b = gcd b a

⊦ ∀a. divides a 1 ⇔ a = 1

⊦ ∀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. m + suc n = suc (m + n)

⊦ ∀m n. suc m + n = suc (m + n)

⊦ ∀t₁ t₂. ¬(t₁ ∧ t₂) ⇔ ¬t₁ ∨ ¬t₂

⊦ ∀a b c. divides a b ⇒ divides a (b * c)

⊦ ∀a b c. divides b a ⇒ divides (gcd c b) a

⊦ ∀a b c. divides (a * b) c ⇒ divides a c

⊦ ∀a b. divides a b ⇔ ∃c. c * a = b

⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r

⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n

⊦ ∀a b. divides a b ∧ divides b a ⇒ a = b

⊦ ∀m n. m < n ⇔ ∃d. n = m + suc d

⊦ ∀p q r. p ⇒ q ⇒ r ⇔ p ∧ q ⇒ r

⊦ ∀m n p. m * (n * p) = m * n * p

⊦ ∀a b c. divides a b ∧ divides b c ⇒ divides a c

⊦ ∀m n. m * n = 0 ⇔ m = 0 ∨ n = 0

⊦ ∀a b. gcd a b = 0 ⇔ a = 0 ∧ b = 0

⊦ ∀p. p 0 ∧ (∀n. p n ⇒ p (suc n)) ⇒ ∀n. p n

⊦ ∀a b c. divides c (gcd a b) ⇔ divides c a ∧ divides c b

⊦ ∀a b c. divides (gcd a (b * c)) (gcd a b * gcd a c)

⊦ ∀a b c. gcd (a * b) (a * c) = a * gcd b c

⊦ ∀a b. ¬(a = 0) ⇒ (divides a b ⇔ (b div a) * a = b)

⊦ ∀m n p. m * n = m * p ⇔ m = 0 ∨ n = p

⊦ ∀m n p. m * p = n * p ⇔ m = n ∨ p = 0

⊦ ∀a b c. divides (a * b) (a * c) ⇔ a = 0 ∨ divides b c

⊦ ∀a b c. divides (b * a) (c * a) ⇔ a = 0 ∨ divides b c

⊦ ∀a b c d. divides a c ∧ divides b d ⇒ divides (a * b) (c * d)

⊦ ∀a b c. gcd b c = 1 ∧ divides b a ∧ divides c a ⇒ divides (b * c) a

⊦ ∀p.
    p 0 0 ∧ (∀a b. gcd a b = 1 ⇒ p a b) ∧
    (∀c a b. ¬(c = 0) ∧ p a b ⇒ p (c * a) (c * b)) ⇒ ∀a b. p a b

OpenTheory package document generated by opentheory 1.3 (release 20150502)