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

gcd_tupled_primitive_def

|- gcd_tupled =
   WFREC
     (@R.
        WF R ∧ (∀x y. ¬(y ≤ x) ⇒ R (SUC x,y − x) (SUC x,SUC y)) ∧
        ∀x y. y ≤ x ⇒ R (x − y,SUC y) (SUC x,SUC y))
     (λgcd_tupled a.
        case a of
          (0,y) => I y
        | (SUC x,0) => I (SUC x)
        | (SUC x,SUC y') =>
            I
              (if y' ≤ x then gcd_tupled (x − y',SUC y')
               else gcd_tupled (SUC x,y' − x)))

gcd_curried_def

|- ∀x x1. gcd x x1 = gcd_tupled (x,x1)

lcm_def

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

Theorems

IS_GCD_UNIQUE

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

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_0R

|- ∀a. is_gcd a 0 a

IS_GCD_0L

|- ∀a. is_gcd 0 a a

PRIME_IS_GCD

|- ∀p b. prime p ⇒ divides p b ∨ is_gcd p b 1

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

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

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_def_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_IS_GCD

|- ∀a b. is_gcd a b (gcd a b)

GCD_REF

|- ∀a. gcd a a = a

GCD_SYM

|- ∀a b. gcd a b = gcd b a

GCD_0R

|- ∀a. gcd a 0 = a

GCD_0L

|- ∀a. gcd 0 a = a

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_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_EQ_0

|- ∀n m. (gcd n m = 0) ⇔ (n = 0) ∧ (m = 0)

GCD_1

|- (gcd 1 x = 1) ∧ (gcd x 1 = 1)

PRIME_GCD

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

L_EUCLIDES

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

P_EUCLIDES

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

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_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

LINEAR_GCD

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

GCD_EFFICIENTLY

|- ∀a b. gcd a b = if a = 0 then b else gcd (b MOD a) 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_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_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

GCD_COMMON_FACTOR

|- ∀m n k. gcd (k * m) (k * n) = k * gcd m n

GCD_CANCEL_MULT

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

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))