Theory "logroot"

Signature

Definitions

ROOT

|- ∀r n. 0 < r ⇒ ROOT r n ** r ≤ n ∧ n < SUC (ROOT r n) ** r

LOG

|- ∀a n. 1 < a ∧ 0 < n ⇒ a ** LOG a n ≤ n ∧ n < a ** SUC (LOG a n)

SQRTd_def

|- ∀n. SQRTd n = (ROOT 2 n,n − ROOT 2 n * ROOT 2 n)

iSQRT0_def

|- ∀n.
     iSQRT0 n =
     (let p = SQRTd n in
      let d = SND p − FST p
      in
        if d = 0 then (2 * FST p,4 * SND p) else (SUC (2 * FST p),4 * d − 1))

iSQRT1_def

|- ∀n.
     iSQRT1 n =
     (let p = SQRTd n in
      let d = SUC (SND p) − FST p
      in
        if d = 0 then (2 * FST p,SUC (4 * SND p))
        else (SUC (2 * FST p),4 * (d − 1)))

iSQRT2_def

|- ∀n.
     iSQRT2 n =
     (let p = SQRTd n in
      let d = 2 * FST p in
      let c = SUC (2 * SND p) in
      let e = c − d
      in
        if e = 0 then (d,2 * c) else (SUC d,2 * e − 1))

iSQRT3_def

|- ∀n.
     iSQRT3 n =
     (let p = SQRTd n in
      let d = 2 * FST p in
      let c = SUC (2 * SND p) in
      let e = SUC c − d
      in
        if e = 0 then (d,SUC (2 * c)) else (SUC d,2 * (e − 1)))

Theorems

LT_EXP_ISO

|- ∀e a b. 1 < e ⇒ (a < b ⇔ e ** a < e ** b)

LE_EXP_ISO

|- ∀e a b. 1 < e ⇒ (a ≤ b ⇔ e ** a ≤ e ** b)

EXP_LT_ISO

|- ∀a b r. 0 < r ⇒ (a < b ⇔ a ** r < b ** r)

EXP_LE_ISO

|- ∀a b r. 0 < r ⇒ (a ≤ b ⇔ a ** r ≤ b ** r)

ROOT_exists

|- ∀r n. 0 < r ⇒ ∃rt. rt ** r ≤ n ∧ n < SUC rt ** r

ROOT_UNIQUE

|- ∀r n p. p ** r ≤ n ∧ n < SUC p ** r ⇒ (ROOT r n = p)

LOG_exists

|- ∃f. ∀a n. 1 < a ∧ 0 < n ⇒ a ** f a n ≤ n ∧ n < a ** SUC (f a n)

LOG_UNIQUE

|- ∀a n p. a ** p ≤ n ∧ n < a ** SUC p ⇒ (LOG a n = p)

LOG_ADD1

|- ∀n a b.
     0 < n ∧ 1 < a ∧ 0 < b ⇒
     (LOG a (a ** SUC n * b) = SUC (LOG a (a ** n * b)))

LOG_BASE

|- ∀a. 1 < a ⇒ (LOG a a = 1)

LOG_EXP

|- ∀n a b. 1 < a ∧ 0 < b ⇒ (LOG a (a ** n * b) = n + LOG a b)

LOG_1

|- ∀a. 1 < a ⇒ (LOG a 1 = 0)

LOG_DIV

|- ∀a x. 1 < a ∧ a ≤ x ⇒ (LOG a x = 1 + LOG a (x DIV a))

LOG_ADD

|- ∀a b c. 1 < a ∧ b < a ** c ⇒ (LOG a (b + a ** c) = c)

LOG_LE_MONO

|- ∀a x y. 1 < a ∧ 0 < x ⇒ x ≤ y ⇒ LOG a x ≤ LOG a y

LOG_RWT

|- ∀m n.
     1 < m ∧ 0 < n ⇒ (LOG m n = if n < m then 0 else SUC (LOG m (n DIV m)))

LOG_EQ_0

|- ∀a b. 1 < a ∧ 0 < b ⇒ ((LOG a b = 0) ⇔ b < a)

LOG_MULT

|- ∀b x. 1 < b ∧ 0 < x ⇒ (LOG b (b * x) = SUC (LOG b x))

LOG_add_digit

|- ∀b x y. 1 < b ∧ 0 < y ∧ x < b ⇒ (LOG b (b * y + x) = SUC (LOG b y))

ROOT_DIV

|- ∀r x y. 0 < r ∧ 0 < y ⇒ (ROOT r x DIV y = ROOT r (x DIV y ** r))

ROOT_LE_MONO

|- ∀r x y. 0 < r ⇒ x ≤ y ⇒ ROOT r x ≤ ROOT r y

EXP_MUL

|- ∀a b c. (a ** b) ** c = a ** (b * c)

LOG_ROOT

|- ∀a x r. 1 < a ∧ 0 < x ∧ 0 < r ⇒ (LOG a (ROOT r x) = LOG a x DIV r)

LOG_MOD

|- ∀n. 0 < n ⇒ (n = 2 ** LOG 2 n + n MOD 2 ** LOG 2 n)

ROOT_COMPUTE

|- ∀r n.
     0 < r ⇒
     (ROOT r 0 = 0) ∧
     (ROOT r n =
      (let x = 2 * ROOT r (n DIV 2 ** r)
       in
         if n < SUC x ** r then x else SUC x))

numeral_sqrt

|- (SQRTd ZERO = (0,0)) ∧ (SQRTd (BIT1 ZERO) = (1,0)) ∧
   (SQRTd (BIT2 ZERO) = (1,1)) ∧ (SQRTd (BIT1 (BIT1 n)) = iSQRT3 n) ∧
   (SQRTd (BIT2 (BIT1 n)) = iSQRT0 (SUC n)) ∧
   (SQRTd (BIT1 (BIT2 n)) = iSQRT1 (SUC n)) ∧
   (SQRTd (BIT2 (BIT2 n)) = iSQRT2 (SUC n)) ∧
   (SQRTd (SUC (BIT1 (BIT1 n))) = iSQRT0 (SUC n)) ∧
   (SQRTd (SUC (BIT2 (BIT1 n))) = iSQRT1 (SUC n)) ∧
   (SQRTd (SUC (BIT1 (BIT2 n))) = iSQRT2 (SUC n)) ∧
   (SQRTd (SUC (BIT2 (BIT2 n))) = iSQRT3 (SUC n))

numeral_root2

|- ROOT 2 (NUMERAL n) = FST (SQRTd n)