Theory "gcd"

Signature

Definitions

is_gcd_def

⊢ ∀a b c.
    is_gcd a b c ⇔
    divides c a ∧ divides c b ∧ ∀d. divides d a ∧ divides d b ⇒ divides d c

lcm_def

⊢ ∀m n. lcm m n = if (m = 0) ∨ (n = 0) then 0 else m * n DIV gcd m n

Theorems

BINARY_GCD

⊢ ∀m n.
    (EVEN m ∧ EVEN n ⇒ (gcd m n = 2 * gcd (m DIV 2) (n DIV 2))) ∧
    (EVEN m ∧ ODD n ⇒ (gcd m n = gcd (m DIV 2) n))

FACTOR_OUT_GCD

⊢ ∀n m.
    n ≠ 0 ∧ m ≠ 0 ⇒
    ∃p q. (n = p * gcd n m) ∧ (m = q * gcd n m) ∧ (gcd p q = 1)

GCD_0L

⊢ ∀a. gcd 0 a = a

GCD_0R

⊢ ∀a. gcd a 0 = a

GCD_1

⊢ (gcd 1 x = 1) ∧ (gcd x 1 = 1)

GCD_ADD_L

⊢ ∀a b. gcd (a + b) a = gcd a b

GCD_ADD_L_THM

⊢ (∀a b. gcd (a + b) a = gcd a b) ∧ ∀a b. gcd (b + a) a = gcd a b

GCD_ADD_R

⊢ ∀a b. gcd a (a + b) = gcd a b

GCD_ADD_R_THM

⊢ (∀a b. gcd a (a + b) = gcd a b) ∧ ∀a b. gcd a (b + a) = gcd a b

GCD_CANCEL_MULT

⊢ ∀m n k. (gcd m k = 1) ⇒ (gcd m (k * n) = gcd m n)

GCD_COMMON_FACTOR

⊢ ∀m n k. gcd (k * m) (k * n) = k * gcd m n

GCD_EFFICIENTLY

⊢ ∀a b. gcd a b = if a = 0 then b else gcd (b MOD a) a

GCD_EQ_0

⊢ ∀n m. (gcd n m = 0) ⇔ (n = 0) ∧ (m = 0)

GCD_IS_GCD

⊢ ∀a b. is_gcd a b (gcd a b)

GCD_IS_GREATEST_COMMON_DIVISOR

⊢ ∀a b.
    divides (gcd a b) a ∧ divides (gcd a b) b ∧
    ∀d. divides d a ∧ divides d b ⇒ divides d (gcd a b)

GCD_REF

⊢ ∀a. gcd a a = a

GCD_SUCfree_ind

⊢ ∀P. (∀y. P 0 y) ∧ (∀x y. P x y ⇒ P y x) ∧ (∀x. P x x) ∧
      (∀x y. 0 < x ∧ 0 < y ∧ P x y ⇒ P x (x + y)) ⇒
      ∀m n. P m n

GCD_SYM

⊢ ∀a b. gcd a b = gcd b a

IS_GCD_0L

⊢ ∀a. is_gcd 0 a a

IS_GCD_0R

⊢ ∀a. is_gcd a 0 a

IS_GCD_MINUS_L

⊢ ∀a b c. b ≤ a ∧ is_gcd (a − b) b c ⇒ is_gcd a b c

IS_GCD_MINUS_R

⊢ ∀a b c. a ≤ b ∧ is_gcd a (b − a) c ⇒ is_gcd a b c

IS_GCD_REF

⊢ ∀a. is_gcd a a a

IS_GCD_SYM

⊢ ∀a b c. is_gcd a b c ⇔ is_gcd b a c

IS_GCD_UNIQUE

⊢ ∀a b c d. is_gcd a b c ∧ is_gcd a b d ⇒ (c = d)

LCM_0

⊢ (lcm 0 x = 0) ∧ (lcm x 0 = 0)

LCM_1

⊢ (lcm 1 x = x) ∧ (lcm x 1 = x)

LCM_COMM

⊢ lcm a b = lcm b a

LCM_IS_LEAST_COMMON_MULTIPLE

⊢ divides m (lcm m n) ∧ divides n (lcm m n) ∧
  ∀p. divides m p ∧ divides n p ⇒ divides (lcm m n) p

LCM_LE

⊢ 0 < m ∧ 0 < n ⇒ m ≤ lcm m n ∧ m ≤ lcm n m

LCM_LEAST

⊢ 0 < m ∧ 0 < n ⇒ ∀p. 0 < p ∧ p < lcm m n ⇒ ¬divides m p ∨ ¬divides n p

LINEAR_GCD

⊢ ∀n m. n ≠ 0 ⇒ ∃p q. p * n = q * m + gcd m n

L_EUCLIDES

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

PRIME_GCD

⊢ ∀p b. prime p ⇒ divides p b ∨ (gcd p b = 1)

PRIME_IS_GCD

⊢ ∀p b. prime p ⇒ divides p b ∨ is_gcd p b 1

P_EUCLIDES

⊢ ∀p a b. prime p ∧ divides p (a * b) ⇒ divides p a ∨ divides p b

gcd_compute

⊢ (∀y. gcd 0 y = y) ∧ (∀x. gcd (NUMERAL (BIT1 x)) 0 = NUMERAL (BIT1 x)) ∧
  (∀x. gcd (NUMERAL (BIT2 x)) 0 = NUMERAL (BIT2 x)) ∧
  (∀y x.
     gcd (NUMERAL (BIT1 x)) (NUMERAL (BIT1 y)) =
     if NUMERAL (BIT1 y) − 1 ≤ NUMERAL (BIT1 x) − 1 then
       gcd (NUMERAL (BIT1 x) − 1 − (NUMERAL (BIT1 y) − 1)) (NUMERAL (BIT1 y))
     else
       gcd (NUMERAL (BIT1 x)) (NUMERAL (BIT1 y) − 1 − (NUMERAL (BIT1 x) − 1))) ∧
  (∀y x.
     gcd (NUMERAL (BIT2 x)) (NUMERAL (BIT1 y)) =
     if NUMERAL (BIT1 y) − 1 ≤ NUMERAL (BIT1 x) then
       gcd (NUMERAL (BIT1 x) − (NUMERAL (BIT1 y) − 1)) (NUMERAL (BIT1 y))
     else gcd (NUMERAL (BIT2 x)) (NUMERAL (BIT1 y) − 1 − NUMERAL (BIT1 x))) ∧
  (∀y x.
     gcd (NUMERAL (BIT1 x)) (NUMERAL (BIT2 y)) =
     if NUMERAL (BIT1 y) ≤ NUMERAL (BIT1 x) − 1 then
       gcd (NUMERAL (BIT1 x) − 1 − NUMERAL (BIT1 y)) (NUMERAL (BIT2 y))
     else gcd (NUMERAL (BIT1 x)) (NUMERAL (BIT1 y) − (NUMERAL (BIT1 x) − 1))) ∧
  ∀y x.
    gcd (NUMERAL (BIT2 x)) (NUMERAL (BIT2 y)) =
    if NUMERAL (BIT1 y) ≤ NUMERAL (BIT1 x) then
      gcd (NUMERAL (BIT1 x) − NUMERAL (BIT1 y)) (NUMERAL (BIT2 y))
    else gcd (NUMERAL (BIT2 x)) (NUMERAL (BIT1 y) − NUMERAL (BIT1 x))

gcd_def

⊢ (∀y. gcd 0 y = y) ∧ (∀x. gcd (SUC x) 0 = SUC x) ∧
  ∀y x.
    gcd (SUC x) (SUC y) =
    if y ≤ x then gcd (x − y) (SUC y) else gcd (SUC x) (y − x)

gcd_ind

⊢ ∀P. (∀y. P 0 y) ∧ (∀x. P (SUC x) 0) ∧
      (∀x y.
         (¬(y ≤ x) ⇒ P (SUC x) (y − x)) ∧ (y ≤ x ⇒ P (x − y) (SUC y)) ⇒
         P (SUC x) (SUC y)) ⇒
      ∀v v1. P v v1