- 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))
- MOD_2EXP
-
|- (∀x. MOD_2EXP x 0 = 0) ∧
∀x n. MOD_2EXP x (NUMERAL n) = NUMERAL (numeral_bit$iMOD_2EXP x n)
- DIV_2EXP
-
|- ∀n x. DIV_2EXP n x = FUNPOW DIV2 n x
- numeral_mod2
-
|- (0 MOD 2 = 0) ∧ (∀n. NUMERAL (BIT1 n) MOD 2 = 1) ∧
∀n. NUMERAL (BIT2 n) MOD 2 = 0
- iDUB_NUMERAL
-
|- numeral$iDUB (NUMERAL i) = NUMERAL (numeral$iDUB i)
- BIT_REVERSE_EVAL
-
|- ∀m n. BIT_REVERSE m n = BIT_REV m n 0
- BIT_MODIFY_EVAL
-
|- ∀m f n. BIT_MODIFY m f n = BIT_MODF m f n 0 1 0
- iBITWISE
-
|- (∀opr a b. numeral_bit$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)
- 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_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_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_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))
- NUMERAL_DIV_2EXP
-
|- (∀n. DIV_2EXP n 0 = 0) ∧
∀n x. DIV_2EXP n (NUMERAL x) = NUMERAL (SFUNPOW numeral_bit$iDIV2 n x)
- NUMERAL_SFUNPOW_iDIV2
-
|- (∀x. SFUNPOW numeral_bit$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$iDUB x))
- NUMERAL_SFUNPOW_FDUB
-
|- ∀f.
(∀x. SFUNPOW (FDUB f) 0 x = x) ∧ (∀y. SFUNPOW (FDUB f) y 0 = 0) ∧
(∀n x.
SFUNPOW (FDUB f) (NUMERAL (BIT1 n)) x =
SFUNPOW (FDUB (FDUB f)) (NUMERAL n) (FDUB f x)) ∧
∀n x.
SFUNPOW (FDUB f) (NUMERAL (BIT2 n)) x =
SFUNPOW (FDUB (FDUB f)) (NUMERAL n) (FDUB f (FDUB f x))
- FDUB_iDIV2
-
|- ∀x. FDUB numeral_bit$iDIV2 x = numeral_bit$iDIV2 (numeral_bit$iDIV2 x)
- FDUB_iDUB
-
|- ∀x. FDUB numeral$iDUB x = numeral$iDUB (numeral$iDUB x)
- FDUB_FDUB
-
|- (FDUB (FDUB f) ZERO = ZERO) ∧
(∀x.
FDUB (FDUB f) (numeral_bit$iSUC x) =
FDUB f (FDUB f (numeral_bit$iSUC x))) ∧
(∀x. FDUB (FDUB f) (BIT1 x) = FDUB f (FDUB f (BIT1 x))) ∧
∀x. FDUB (FDUB f) (BIT2 x) = FDUB f (FDUB f (BIT2 x))
- LOG_compute
-
|- ∀m n.
LOG m n =
if m < 2 ∨ (n = 0) then FAIL LOG base < 2 or n = 0 m n
else if n < m then 0
else SUC (LOG m (n DIV m))
- numeral_ilog2
-
|- (numeral_bit$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_log2
-
|- (∀n. LOG2 (NUMERAL (BIT1 n)) = numeral_bit$iLOG2 (numeral$iDUB n)) ∧
∀n. LOG2 (NUMERAL (BIT2 n)) = numeral_bit$iLOG2 (BIT1 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)
- 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