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