Theory "primeFactor"

Signature

Definitions

PRIME_FACTORS_def

|- ∀n.
     0 < n ⇒
     FINITE_BAG (PRIME_FACTORS n) ∧ (∀m. m ⋲ PRIME_FACTORS n ⇒ prime m) ∧
     (n = BAG_GEN_PROD (PRIME_FACTORS n) 1)

Theorems

PRIME_FACTORS_EXIST

|- ∀n.
     0 < n ⇒ ∃b. FINITE_BAG b ∧ (∀m. m ⋲ b ⇒ prime m) ∧ (n = BAG_GEN_PROD b 1)

UNIQUE_PRIME_FACTORS

|- ∀n b1 b2.
     (FINITE_BAG b1 ∧ (∀m. m ⋲ b1 ⇒ prime m) ∧ (n = BAG_GEN_PROD b1 1)) ∧
     FINITE_BAG b2 ∧ (∀m. m ⋲ b2 ⇒ prime m) ∧ (n = BAG_GEN_PROD b2 1) ⇒
     (b1 = b2)

PRIME_FACTORIZATION

|- ∀n.
     0 < n ⇒
     ∀b.
       FINITE_BAG b ∧ (∀x. x ⋲ b ⇒ prime x) ∧ (BAG_GEN_PROD b 1 = n) ⇒
       (b = PRIME_FACTORS n)

PRIME_FACTORS_1

|- PRIME_FACTORS 1 = {||}

PRIME_FACTOR_DIVIDES

|- ∀x n. 0 < n ∧ x ⋲ PRIME_FACTORS n ⇒ divides x n

DIVISOR_IN_PRIME_FACTORS

|- ∀p n. 0 < n ∧ prime p ∧ divides p n ⇒ p ⋲ PRIME_FACTORS n

PRIME_FACTORS_MULT

|- ∀a b.
     0 < a ∧ 0 < b ⇒
     (PRIME_FACTORS (a * b) = PRIME_FACTORS a ⊎ PRIME_FACTORS b)

FACTORS_prime

|- ∀p. prime p ⇒ (PRIME_FACTORS p = {|p|})

PRIME_FACTORS_EXP

|- ∀p e. prime p ⇒ (PRIME_FACTORS (p ** e) p = e)