Theory "numeral_bit"

Parents bit

Signature

Constant	Type
BIT_MODF	:num -> (num -> bool -> bool) -> num -> num -> num -> num -> num
BIT_REV	:num -> num -> num -> num
FDUB	:(num -> num) -> num -> num
SFUNPOW	:(num -> num) -> num -> num -> num
iBITWISE	:num -> (bool -> bool -> bool) -> num -> num -> num
iDIV2	:num -> num
iLOG2	:num -> num
iMOD_2EXP	:num -> num -> num
iSUC	:num -> num

Definitions

BIT_MODF_def

⊢ (∀f x b e y. BIT_MODF 0 f x b e y = y) ∧
  ∀n f x b e y.
    BIT_MODF (SUC n) f x b e y =
    BIT_MODF n f (x DIV 2) (b + 1) (2 * e) (if f b (ODD x) then e + y else y)

BIT_REV_def

⊢ (∀x y. BIT_REV 0 x y = y) ∧
  ∀n x y. BIT_REV (SUC n) x y = BIT_REV n (x DIV 2) (2 * y + SBIT (ODD x) 0)

FDUB_def

⊢ (∀f. FDUB f 0 = 0) ∧ ∀f n. FDUB f (SUC n) = f (f (SUC n))

SFUNPOW_def

⊢ (∀f x. SFUNPOW f 0 x = x) ∧
  ∀f n x. SFUNPOW f (SUC n) x = if x = 0 then 0 else SFUNPOW f n (f x)

iBITWISE_def

⊢ numeral_bit$iBITWISE = BITWISE

iDIV2

⊢ numeral_bit$iDIV2 = DIV2

iLOG2_def

⊢ ∀n. numeral_bit$iLOG2 n = LOG2 (n + 1)

iMOD_2EXP

⊢ numeral_bit$iMOD_2EXP = MOD_2EXP

iSUC

⊢ numeral_bit$iSUC = SUC

Theorems

BIT_MODIFY_EVAL

⊢ ∀m f n. BIT_MODIFY m f n = BIT_MODF m f n 0 1 0

BIT_REVERSE_EVAL

⊢ ∀m n. BIT_REVERSE m n = BIT_REV m n 0

DIV_2EXP

⊢ ∀n x. DIV_2EXP n x = FUNPOW DIV2 n x

FDUB_FDUB

⊢ (FDUB (FDUB f) ZERO = ZERO) ∧
  (∀x. FDUB (FDUB f) (numeral_bit$iSUC x) =
       FDUB f (FDUB f (numeral_bit$iSUC x))) ∧
  (∀x. FDUB (FDUB f) (BIT1 x) = FDUB f (FDUB f (BIT1 x))) ∧
  ∀x. FDUB (FDUB f) (BIT2 x) = FDUB f (FDUB f (BIT2 x))

FDUB_iDIV2

⊢ ∀x. FDUB numeral_bit$iDIV2 x = numeral_bit$iDIV2 (numeral_bit$iDIV2 x)

FDUB_iDUB

⊢ ∀x. FDUB numeral$iDUB x = numeral$iDUB (numeral$iDUB x)

LOG_compute

⊢ ∀m n.
    LOG m n =
    if m < 2 ∨ (n = 0) then FAIL LOG $var$(base < 2 or n = 0) m n
    else if n < m then 0
    else SUC (LOG m (n DIV m))

LOWEST_SET_BIT

⊢ ∀n. n ≠ 0 ⇒
      (LOWEST_SET_BIT n = if ODD n then 0 else 1 + LOWEST_SET_BIT (DIV2 n))

LOWEST_SET_BIT_compute

⊢ (∀n. LOWEST_SET_BIT (NUMERAL (BIT2 n)) =
       SUC (LOWEST_SET_BIT (NUMERAL (SUC n)))) ∧
  ∀n. LOWEST_SET_BIT (NUMERAL (BIT1 n)) = 0

MOD_2EXP

⊢ (∀x. MOD_2EXP x 0 = 0) ∧
  ∀x n. MOD_2EXP x (NUMERAL n) = NUMERAL (numeral_bit$iMOD_2EXP x n)

MOD_2EXP_EQ

⊢ (∀a b. MOD_2EXP_EQ 0 a b ⇔ T) ∧
  (∀n a b.
     MOD_2EXP_EQ (SUC n) a b ⇔
     (ODD a ⇔ ODD b) ∧ MOD_2EXP_EQ n (DIV2 a) (DIV2 b)) ∧
  ∀n a. MOD_2EXP_EQ n a a ⇔ T

MOD_2EXP_MAX

⊢ (∀a. MOD_2EXP_MAX 0 a ⇔ T) ∧
  ∀n a. MOD_2EXP_MAX (SUC n) a ⇔ ODD a ∧ MOD_2EXP_MAX n (DIV2 a)

NUMERAL_BITWISE

⊢ (∀x f a. BITWISE x f 0 0 = NUMERAL (numeral_bit$iBITWISE x f 0 0)) ∧
  (∀x f a.
     BITWISE x f (NUMERAL a) 0 =
     NUMERAL (numeral_bit$iBITWISE x f (NUMERAL a) 0)) ∧
  (∀x f b.
     BITWISE x f 0 (NUMERAL b) =
     NUMERAL (numeral_bit$iBITWISE x f 0 (NUMERAL b))) ∧
  ∀x f a b.
    BITWISE x f (NUMERAL a) (NUMERAL b) =
    NUMERAL (numeral_bit$iBITWISE x f (NUMERAL a) (NUMERAL b))

NUMERAL_BIT_MODF

⊢ (∀f x b e y. BIT_MODF 0 f x b e y = y) ∧
  (∀n f b e y.
     BIT_MODF (NUMERAL (BIT1 n)) f 0 b (NUMERAL e) y =
     BIT_MODF (NUMERAL (BIT1 n) − 1) f 0 (b + 1) (NUMERAL (numeral$iDUB e))
       (if f b F then NUMERAL e + y else y)) ∧
  (∀n f b e y.
     BIT_MODF (NUMERAL (BIT2 n)) f 0 b (NUMERAL e) y =
     BIT_MODF (NUMERAL (BIT1 n)) f 0 (b + 1) (NUMERAL (numeral$iDUB e))
       (if f b F then NUMERAL e + y else y)) ∧
  (∀n f x b e y.
     BIT_MODF (NUMERAL (BIT1 n)) f (NUMERAL x) b (NUMERAL e) y =
     BIT_MODF (NUMERAL (BIT1 n) − 1) f (DIV2 (NUMERAL x)) (b + 1)
       (NUMERAL (numeral$iDUB e)) (if f b (ODD x) then NUMERAL e + y else y)) ∧
  ∀n f x b e y.
    BIT_MODF (NUMERAL (BIT2 n)) f (NUMERAL x) b (NUMERAL e) y =
    BIT_MODF (NUMERAL (BIT1 n)) f (DIV2 (NUMERAL x)) (b + 1)
      (NUMERAL (numeral$iDUB e)) (if f b (ODD x) then NUMERAL e + y else y)

NUMERAL_BIT_MODIFY

⊢ (∀m f. BIT_MODIFY (NUMERAL m) f 0 = BIT_MODF (NUMERAL m) f 0 0 1 0) ∧
  ∀m f n.
    BIT_MODIFY (NUMERAL m) f (NUMERAL n) =
    BIT_MODF (NUMERAL m) f (NUMERAL n) 0 1 0

NUMERAL_BIT_REV

⊢ (∀x y. BIT_REV 0 x y = y) ∧
  (∀n y.
     BIT_REV (NUMERAL (BIT1 n)) 0 y =
     BIT_REV (NUMERAL (BIT1 n) − 1) 0 (numeral$iDUB y)) ∧
  (∀n y.
     BIT_REV (NUMERAL (BIT2 n)) 0 y =
     BIT_REV (NUMERAL (BIT1 n)) 0 (numeral$iDUB y)) ∧
  (∀n x y.
     BIT_REV (NUMERAL (BIT1 n)) (NUMERAL x) y =
     BIT_REV (NUMERAL (BIT1 n) − 1) (DIV2 (NUMERAL x))
       (if ODD x then BIT1 y else numeral$iDUB y)) ∧
  ∀n x y.
    BIT_REV (NUMERAL (BIT2 n)) (NUMERAL x) y =
    BIT_REV (NUMERAL (BIT1 n)) (DIV2 (NUMERAL x))
      (if ODD x then BIT1 y else numeral$iDUB y)

NUMERAL_BIT_REVERSE

⊢ (∀m. BIT_REVERSE (NUMERAL m) 0 = NUMERAL (BIT_REV (NUMERAL m) 0 ZERO)) ∧
  ∀n m.
    BIT_REVERSE (NUMERAL m) (NUMERAL n) =
    NUMERAL (BIT_REV (NUMERAL m) (NUMERAL n) ZERO)

NUMERAL_DIV_2EXP

⊢ (∀n. DIV_2EXP n 0 = 0) ∧
  ∀n x. DIV_2EXP n (NUMERAL x) = NUMERAL (SFUNPOW numeral_bit$iDIV2 n x)

NUMERAL_SFUNPOW_FDUB

⊢ ∀f. (∀x. SFUNPOW (FDUB f) 0 x = x) ∧ (∀y. SFUNPOW (FDUB f) y 0 = 0) ∧
      (∀n x.
         SFUNPOW (FDUB f) (NUMERAL (BIT1 n)) x =
         SFUNPOW (FDUB (FDUB f)) (NUMERAL n) (FDUB f x)) ∧
      ∀n x.
        SFUNPOW (FDUB f) (NUMERAL (BIT2 n)) x =
        SFUNPOW (FDUB (FDUB f)) (NUMERAL n) (FDUB f (FDUB f x))

NUMERAL_SFUNPOW_iDIV2

⊢ (∀x. SFUNPOW numeral_bit$iDIV2 0 x = x) ∧
  (∀y. SFUNPOW numeral_bit$iDIV2 y 0 = 0) ∧
  (∀n x.
     SFUNPOW numeral_bit$iDIV2 (NUMERAL (BIT1 n)) x =
     SFUNPOW (FDUB numeral_bit$iDIV2) (NUMERAL n) (numeral_bit$iDIV2 x)) ∧
  ∀n x.
    SFUNPOW numeral_bit$iDIV2 (NUMERAL (BIT2 n)) x =
    SFUNPOW (FDUB numeral_bit$iDIV2) (NUMERAL n)
      (numeral_bit$iDIV2 (numeral_bit$iDIV2 x))

NUMERAL_SFUNPOW_iDUB

⊢ (∀x. SFUNPOW numeral$iDUB 0 x = x) ∧ (∀y. SFUNPOW numeral$iDUB y 0 = 0) ∧
  (∀n x.
     SFUNPOW numeral$iDUB (NUMERAL (BIT1 n)) x =
     SFUNPOW (FDUB numeral$iDUB) (NUMERAL n) (numeral$iDUB x)) ∧
  ∀n x.
    SFUNPOW numeral$iDUB (NUMERAL (BIT2 n)) x =
    SFUNPOW (FDUB numeral$iDUB) (NUMERAL n) (numeral$iDUB (numeral$iDUB x))

NUMERAL_TIMES_2EXP

⊢ (∀n. TIMES_2EXP n 0 = 0) ∧
  ∀n x. TIMES_2EXP n (NUMERAL x) = NUMERAL (SFUNPOW numeral$iDUB n x)

NUMERAL_iDIV2

⊢ (numeral_bit$iDIV2 ZERO = ZERO) ∧
  (numeral_bit$iDIV2 (numeral_bit$iSUC ZERO) = ZERO) ∧
  (numeral_bit$iDIV2 (BIT1 n) = n) ∧
  (numeral_bit$iDIV2 (numeral_bit$iSUC (BIT1 n)) = numeral_bit$iSUC n) ∧
  (numeral_bit$iDIV2 (BIT2 n) = numeral_bit$iSUC n) ∧
  (numeral_bit$iDIV2 (numeral_bit$iSUC (BIT2 n)) = numeral_bit$iSUC n) ∧
  (NUMERAL (numeral_bit$iSUC n) = NUMERAL (SUC n))

iBITWISE

⊢ (∀opr a b. numeral_bit$iBITWISE 0 opr a b = ZERO) ∧
  (∀x opr a b.
     numeral_bit$iBITWISE (NUMERAL (BIT1 x)) opr a b =
     (let
        w = numeral_bit$iBITWISE (NUMERAL (BIT1 x) − 1) opr (DIV2 a) (DIV2 b)
      in
        if opr (ODD a) (ODD b) then BIT1 w else numeral$iDUB w)) ∧
  ∀x opr a b.
    numeral_bit$iBITWISE (NUMERAL (BIT2 x)) opr a b =
    (let
       w = numeral_bit$iBITWISE (NUMERAL (BIT1 x)) opr (DIV2 a) (DIV2 b)
     in
       if opr (ODD a) (ODD b) then BIT1 w else numeral$iDUB w)

iDUB_NUMERAL

⊢ numeral$iDUB (NUMERAL i) = NUMERAL (numeral$iDUB i)

numeral_ilog2

⊢ (numeral_bit$iLOG2 ZERO = 0) ∧
  (∀n. numeral_bit$iLOG2 (BIT1 n) = 1 + numeral_bit$iLOG2 n) ∧
  ∀n. numeral_bit$iLOG2 (BIT2 n) = 1 + numeral_bit$iLOG2 n

numeral_imod_2exp

⊢ (∀n. numeral_bit$iMOD_2EXP 0 n = ZERO) ∧
  (∀x n. numeral_bit$iMOD_2EXP x ZERO = ZERO) ∧
  (∀x n.
     numeral_bit$iMOD_2EXP (NUMERAL (BIT1 x)) (BIT1 n) =
     BIT1 (numeral_bit$iMOD_2EXP (NUMERAL (BIT1 x) − 1) n)) ∧
  (∀x n.
     numeral_bit$iMOD_2EXP (NUMERAL (BIT2 x)) (BIT1 n) =
     BIT1 (numeral_bit$iMOD_2EXP (NUMERAL (BIT1 x)) n)) ∧
  (∀x n.
     numeral_bit$iMOD_2EXP (NUMERAL (BIT1 x)) (BIT2 n) =
     numeral$iDUB (numeral_bit$iMOD_2EXP (NUMERAL (BIT1 x) − 1) (SUC n))) ∧
  ∀x n.
    numeral_bit$iMOD_2EXP (NUMERAL (BIT2 x)) (BIT2 n) =
    numeral$iDUB (numeral_bit$iMOD_2EXP (NUMERAL (BIT1 x)) (SUC n))

numeral_log2

⊢ (∀n. LOG2 (NUMERAL (BIT1 n)) = numeral_bit$iLOG2 (numeral$iDUB n)) ∧
  ∀n. LOG2 (NUMERAL (BIT2 n)) = numeral_bit$iLOG2 (BIT1 n)

numeral_mod2

⊢ (0 MOD 2 = 0) ∧ (∀n. NUMERAL (BIT1 n) MOD 2 = 1) ∧
  ∀n. NUMERAL (BIT2 n) MOD 2 = 0