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 ;
         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 ;
         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 ;
         d = 2 * FST p ;
         c = SUC (2 * SND p) ;
         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 ;
         d = 2 * FST p ;
         c = SUC (2 * SND p) ;
         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)