Structure numeral_bitTheory
signature numeral_bitTheory =
sig
type thm = Thm.thm
(* Definitions *)
val BIT_MODF_def : thm
val BIT_REV_def : thm
val FDUB_def : thm
val SFUNPOW_def : thm
val iBITWISE_def : thm
val iDIV2 : thm
val iLOG2_def : thm
val iMOD_2EXP : thm
val iSUC : thm
(* Theorems *)
val BIT_MODIFY_EVAL : thm
val BIT_REVERSE_EVAL : thm
val DIV_2EXP : thm
val FDUB_FDUB : thm
val FDUB_iDIV2 : thm
val FDUB_iDUB : thm
val LOG_compute : thm
val LOWEST_SET_BIT : thm
val LOWEST_SET_BIT_compute : thm
val MOD_2EXP : thm
val MOD_2EXP_EQ : thm
val MOD_2EXP_MAX : thm
val NUMERAL_BITWISE : thm
val NUMERAL_BIT_MODF : thm
val NUMERAL_BIT_MODIFY : thm
val NUMERAL_BIT_REV : thm
val NUMERAL_BIT_REVERSE : thm
val NUMERAL_DIV_2EXP : thm
val NUMERAL_SFUNPOW_FDUB : thm
val NUMERAL_SFUNPOW_iDIV2 : thm
val NUMERAL_SFUNPOW_iDUB : thm
val NUMERAL_TIMES_2EXP : thm
val NUMERAL_iDIV2 : thm
val iBITWISE : thm
val iDUB_NUMERAL : thm
val numeral_ilog2 : thm
val numeral_imod_2exp : thm
val numeral_log2 : thm
val numeral_mod2 : thm
val numeral_bit_grammars : type_grammar.grammar * term_grammar.grammar
(*
[bit] Parent theory of "numeral_bit"
[BIT_MODF_def] Definition
|- (∀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] Definition
|- (∀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] Definition
|- (∀f. FDUB f 0 = 0) ∧ ∀f n. FDUB f (SUC n) = f (f (SUC n))
[SFUNPOW_def] Definition
|- (∀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] Definition
|- numeral_bit$iBITWISE = BITWISE
[iDIV2] Definition
|- numeral_bit$iDIV2 = DIV2
[iLOG2_def] Definition
|- ∀n. numeral_bit$iLOG2 n = LOG2 (n + 1)
[iMOD_2EXP] Definition
|- numeral_bit$iMOD_2EXP = MOD_2EXP
[iSUC] Definition
|- numeral_bit$iSUC = SUC
[BIT_MODIFY_EVAL] Theorem
|- ∀m f n. BIT_MODIFY m f n = BIT_MODF m f n 0 1 0
[BIT_REVERSE_EVAL] Theorem
|- ∀m n. BIT_REVERSE m n = BIT_REV m n 0
[DIV_2EXP] Theorem
|- ∀n x. DIV_2EXP n x = FUNPOW DIV2 n x
[FDUB_FDUB] Theorem
|- (FDUB (FDUB f) ZERO = ZERO) ∧
(∀x.
FDUB (FDUB f) (numeral_bit$iSUC x) =
FDUB f (FDUB f (numeral_bit$iSUC x))) ∧
(∀x. FDUB (FDUB f) (BIT1 x) = FDUB f (FDUB f (BIT1 x))) ∧
∀x. FDUB (FDUB f) (BIT2 x) = FDUB f (FDUB f (BIT2 x))
[FDUB_iDIV2] Theorem
|- ∀x.
FDUB numeral_bit$iDIV2 x =
numeral_bit$iDIV2 (numeral_bit$iDIV2 x)
[FDUB_iDUB] Theorem
|- ∀x. FDUB numeral$iDUB x = numeral$iDUB (numeral$iDUB x)
[LOG_compute] Theorem
|- ∀m n.
LOG m n =
if m < 2 ∨ (n = 0) then FAIL LOG base < 2 or n = 0 m n
else if n < m then 0
else SUC (LOG m (n DIV m))
[LOWEST_SET_BIT] Theorem
|- ∀n.
n ≠ 0 ⇒
(LOWEST_SET_BIT n =
if ODD n then 0 else 1 + LOWEST_SET_BIT (DIV2 n))
[LOWEST_SET_BIT_compute] Theorem
|- (∀n.
LOWEST_SET_BIT (NUMERAL (BIT2 n)) =
SUC (LOWEST_SET_BIT (NUMERAL (SUC n)))) ∧
∀n. LOWEST_SET_BIT (NUMERAL (BIT1 n)) = 0
[MOD_2EXP] Theorem
|- (∀x. MOD_2EXP x 0 = 0) ∧
∀x n. MOD_2EXP x (NUMERAL n) = NUMERAL (numeral_bit$iMOD_2EXP x n)
[MOD_2EXP_EQ] Theorem
|- (∀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] Theorem
|- (∀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] Theorem
|- (∀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] Theorem
|- (∀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] Theorem
|- (∀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] Theorem
|- (∀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] Theorem
|- (∀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] Theorem
|- (∀n. DIV_2EXP n 0 = 0) ∧
∀n x.
DIV_2EXP n (NUMERAL x) = NUMERAL (SFUNPOW numeral_bit$iDIV2 n x)
[NUMERAL_SFUNPOW_FDUB] Theorem
|- ∀f.
(∀x. SFUNPOW (FDUB f) 0 x = x) ∧
(∀y. SFUNPOW (FDUB f) y 0 = 0) ∧
(∀n x.
SFUNPOW (FDUB f) (NUMERAL (BIT1 n)) x =
SFUNPOW (FDUB (FDUB f)) (NUMERAL n) (FDUB f x)) ∧
∀n x.
SFUNPOW (FDUB f) (NUMERAL (BIT2 n)) x =
SFUNPOW (FDUB (FDUB f)) (NUMERAL n) (FDUB f (FDUB f x))
[NUMERAL_SFUNPOW_iDIV2] Theorem
|- (∀x. SFUNPOW numeral_bit$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] Theorem
|- (∀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$iDUB x))
[NUMERAL_TIMES_2EXP] Theorem
|- (∀n. TIMES_2EXP n 0 = 0) ∧
∀n x.
TIMES_2EXP n (NUMERAL x) = NUMERAL (SFUNPOW numeral$iDUB n x)
[NUMERAL_iDIV2] Theorem
|- (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$iSUC n) = NUMERAL (SUC n))
[iBITWISE] Theorem
|- (∀opr a b. numeral_bit$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] Theorem
|- numeral$iDUB (NUMERAL i) = NUMERAL (numeral$iDUB i)
[numeral_ilog2] Theorem
|- (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$iLOG2 n
[numeral_imod_2exp] Theorem
|- (∀n. numeral_bit$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$iMOD_2EXP (NUMERAL (BIT1 x)) (SUC n))
[numeral_log2] Theorem
|- (∀n.
LOG2 (NUMERAL (BIT1 n)) = numeral_bit$iLOG2 (numeral$iDUB n)) ∧
∀n. LOG2 (NUMERAL (BIT2 n)) = numeral_bit$iLOG2 (BIT1 n)
[numeral_mod2] Theorem
|- (0 MOD 2 = 0) ∧ (∀n. NUMERAL (BIT1 n) MOD 2 = 1) ∧
∀n. NUMERAL (BIT2 n) MOD 2 = 0
*)
end
HOL 4, Kananaskis-10