Theory "numeral"

Parents arithmetic

Signature

Constant	Type
exactlog	:num -> num
iBIT_cases	:num -> α -> (num -> α) -> (num -> α) -> α
iDUB	:num -> num
iSQR	:num -> num
iSUB	:bool -> num -> num -> num
iZ	:num -> num
iiSUC	:num -> num
internal_mult	:num -> num -> num
onecount	:num -> num -> num
texp_help	:num -> num -> num

Definitions

texp_help_def

⊢ (∀acc. numeral$texp_help 0 acc = BIT2 acc) ∧
  ∀n acc. numeral$texp_help (SUC n) acc = numeral$texp_help n (BIT1 acc)

onecount_def

⊢ (∀x. numeral$onecount ZERO x = x) ∧
  (∀n x. numeral$onecount (BIT1 n) x = numeral$onecount n (SUC x)) ∧
  ∀n x. numeral$onecount (BIT2 n) x = ZERO

iZ

⊢ ∀x. numeral$iZ x = x

iSUB_DEF

⊢ (∀b x. numeral$iSUB b ZERO x = ZERO) ∧
  (∀b n x.
       numeral$iSUB b (BIT1 n) x =
       if b then
         iBIT_cases x (BIT1 n) (λm. numeral$iDUB (numeral$iSUB T n m))
           (λm. BIT1 (numeral$iSUB F n m))
       else
         iBIT_cases x (numeral$iDUB n) (λm. BIT1 (numeral$iSUB F n m))
           (λm. numeral$iDUB (numeral$iSUB F n m))) ∧
  ∀b n x.
      numeral$iSUB b (BIT2 n) x =
      if b then
        iBIT_cases x (BIT2 n) (λm. BIT1 (numeral$iSUB T n m))
          (λm. numeral$iDUB (numeral$iSUB T n m))
      else
        iBIT_cases x (BIT1 n) (λm. numeral$iDUB (numeral$iSUB T n m))
          (λm. BIT1 (numeral$iSUB F n m))

iSQR

⊢ ∀x. numeral$iSQR x = x * x

internal_mult_def

⊢ internal_mult = $*

iiSUC

⊢ ∀n. numeral$iiSUC n = SUC (SUC n)

iDUB

⊢ ∀x. numeral$iDUB x = x + x

iBIT_cases

⊢ (∀zf bf1 bf2. iBIT_cases ZERO zf bf1 bf2 = zf) ∧
  (∀n zf bf1 bf2. iBIT_cases (BIT1 n) zf bf1 bf2 = bf1 n) ∧
  ∀n zf bf1 bf2. iBIT_cases (BIT2 n) zf bf1 bf2 = bf2 n

exactlog_def

⊢ (numeral$exactlog ZERO = ZERO) ∧ (∀n. numeral$exactlog (BIT1 n) = ZERO) ∧
  ∀n.
      numeral$exactlog (BIT2 n) =
      (let x = numeral$onecount n ZERO in if x = ZERO then ZERO else BIT1 x)

Theorems

TWO_EXP_THM

⊢ (2 ** 0 = 1) ∧
  (2 ** NUMERAL (BIT1 n) = NUMERAL (numeral$texp_help (PRE (BIT1 n)) ZERO)) ∧
  (2 ** NUMERAL (BIT2 n) = NUMERAL (numeral$texp_help (BIT1 n) ZERO))

texp_help_thm

⊢ ∀n a. numeral$texp_help n a = (a + 1) * 2 ** (n + 1)

texp_help0

⊢ numeral$texp_help n 0 = 2 ** (n + 1)

onecount_characterisation

⊢ ∀n a.
      0 < numeral$onecount n a ∧ 0 < n ⇒
      (n = 2 ** (numeral$onecount n a − a) − 1)

numeral_texp_help

⊢ (numeral$texp_help ZERO acc = BIT2 acc) ∧
  (numeral$texp_help (BIT1 n) acc =
   numeral$texp_help (PRE (BIT1 n)) (BIT1 acc)) ∧
  (numeral$texp_help (BIT2 n) acc = numeral$texp_help (BIT1 n) (BIT1 acc))

numeral_suc

⊢ (SUC ZERO = BIT1 ZERO) ∧ (∀n. SUC (BIT1 n) = BIT2 n) ∧
  ∀n. SUC (BIT2 n) = BIT1 (SUC n)

numeral_sub

⊢ ∀n m. NUMERAL (n − m) = if m < n then NUMERAL (numeral$iSUB T n m) else 0

numeral_pre

⊢ (PRE ZERO = ZERO) ∧ (PRE (BIT1 ZERO) = ZERO) ∧
  (∀n. PRE (BIT1 (BIT1 n)) = BIT2 (PRE (BIT1 n))) ∧
  (∀n. PRE (BIT1 (BIT2 n)) = BIT2 (BIT1 n)) ∧ ∀n. PRE (BIT2 n) = BIT1 n

numeral_mult

⊢ ∀n m.
      (ZERO * n = ZERO) ∧ (n * ZERO = ZERO) ∧
      (BIT1 n * m = numeral$iZ (numeral$iDUB (n * m) + m)) ∧
      (BIT2 n * m = numeral$iDUB (numeral$iZ (n * m + m)))

numeral_MIN

⊢ (MIN 0 x = 0) ∧ (MIN x 0 = 0) ∧
  (MIN (NUMERAL x) (NUMERAL y) = NUMERAL (if x < y then x else y))

numeral_MAX

⊢ (MAX 0 x = x) ∧ (MAX x 0 = x) ∧
  (MAX (NUMERAL x) (NUMERAL y) = NUMERAL (if x < y then y else x))

numeral_lte

⊢ ∀n m.
      (ZERO ≤ n ⇔ T) ∧ (BIT1 n ≤ ZERO ⇔ F) ∧ (BIT2 n ≤ ZERO ⇔ F) ∧
      (BIT1 n ≤ BIT1 m ⇔ n ≤ m) ∧ (BIT1 n ≤ BIT2 m ⇔ n ≤ m) ∧
      (BIT2 n ≤ BIT1 m ⇔ ¬(m ≤ n)) ∧ (BIT2 n ≤ BIT2 m ⇔ n ≤ m)

numeral_lt

⊢ ∀n m.
      (ZERO < BIT1 n ⇔ T) ∧ (ZERO < BIT2 n ⇔ T) ∧ (n < ZERO ⇔ F) ∧
      (BIT1 n < BIT1 m ⇔ n < m) ∧ (BIT2 n < BIT2 m ⇔ n < m) ∧
      (BIT1 n < BIT2 m ⇔ ¬(m < n)) ∧ (BIT2 n < BIT1 m ⇔ n < m)

numeral_iisuc

⊢ (numeral$iiSUC ZERO = BIT2 ZERO) ∧ (numeral$iiSUC (BIT1 n) = BIT1 (SUC n)) ∧
  (numeral$iiSUC (BIT2 n) = BIT2 (SUC n))

numeral_funpow

⊢ (FUNPOW f 0 x = x) ∧
  (FUNPOW f (NUMERAL (BIT1 n)) x = FUNPOW f (PRE (NUMERAL (BIT1 n))) (f x)) ∧
  (FUNPOW f (NUMERAL (BIT2 n)) x = FUNPOW f (NUMERAL (BIT1 n)) (f x))

numeral_fact

⊢ (FACT 0 = 1) ∧
  (∀n.
       FACT (NUMERAL (BIT1 n)) =
       NUMERAL (BIT1 n) * FACT (PRE (NUMERAL (BIT1 n)))) ∧
  ∀n. FACT (NUMERAL (BIT2 n)) = NUMERAL (BIT2 n) * FACT (NUMERAL (BIT1 n))

numeral_exp

⊢ (∀n. n ** ZERO = BIT1 ZERO) ∧
  (∀n m. n ** BIT1 m = n * numeral$iSQR (n ** m)) ∧
  ∀n m. n ** BIT2 m = numeral$iSQR n * numeral$iSQR (n ** m)

numeral_evenodd

⊢ ∀n.
      EVEN ZERO ∧ EVEN (BIT2 n) ∧ ¬EVEN (BIT1 n) ∧ ¬ODD ZERO ∧ ¬ODD (BIT2 n) ∧
      ODD (BIT1 n)

numeral_eq

⊢ ∀n m.
      ((ZERO = BIT1 n) ⇔ F) ∧ ((BIT1 n = ZERO) ⇔ F) ∧ ((ZERO = BIT2 n) ⇔ F) ∧
      ((BIT2 n = ZERO) ⇔ F) ∧ ((BIT1 n = BIT2 m) ⇔ F) ∧
      ((BIT2 n = BIT1 m) ⇔ F) ∧ ((BIT1 n = BIT1 m) ⇔ (n = m)) ∧
      ((BIT2 n = BIT2 m) ⇔ (n = m))

numeral_div2

⊢ (DIV2 0 = 0) ∧ (∀n. DIV2 (NUMERAL (BIT1 n)) = NUMERAL n) ∧
  ∀n. DIV2 (NUMERAL (BIT2 n)) = NUMERAL (SUC n)

numeral_distrib

⊢ (∀n. 0 + n = n) ∧ (∀n. n + 0 = n) ∧
  (∀n m. NUMERAL n + NUMERAL m = NUMERAL (numeral$iZ (n + m))) ∧
  (∀n. 0 * n = 0) ∧ (∀n. n * 0 = 0) ∧
  (∀n m. NUMERAL n * NUMERAL m = NUMERAL (n * m)) ∧ (∀n. 0 − n = 0) ∧
  (∀n. n − 0 = n) ∧ (∀n m. NUMERAL n − NUMERAL m = NUMERAL (n − m)) ∧
  (∀n. 0 ** NUMERAL (BIT1 n) = 0) ∧ (∀n. 0 ** NUMERAL (BIT2 n) = 0) ∧
  (∀n. n ** 0 = 1) ∧ (∀n m. NUMERAL n ** NUMERAL m = NUMERAL (n ** m)) ∧
  (SUC 0 = 1) ∧ (∀n. SUC (NUMERAL n) = NUMERAL (SUC n)) ∧ (PRE 0 = 0) ∧
  (∀n. PRE (NUMERAL n) = NUMERAL (PRE n)) ∧
  (∀n. (NUMERAL n = 0) ⇔ (n = ZERO)) ∧ (∀n. (0 = NUMERAL n) ⇔ (n = ZERO)) ∧
  (∀n m. (NUMERAL n = NUMERAL m) ⇔ (n = m)) ∧ (∀n. n < 0 ⇔ F) ∧
  (∀n. 0 < NUMERAL n ⇔ ZERO < n) ∧ (∀n m. NUMERAL n < NUMERAL m ⇔ n < m) ∧
  (∀n. 0 > n ⇔ F) ∧ (∀n. NUMERAL n > 0 ⇔ ZERO < n) ∧
  (∀n m. NUMERAL n > NUMERAL m ⇔ m < n) ∧ (∀n. 0 ≤ n ⇔ T) ∧
  (∀n. NUMERAL n ≤ 0 ⇔ n ≤ ZERO) ∧ (∀n m. NUMERAL n ≤ NUMERAL m ⇔ n ≤ m) ∧
  (∀n. n ≥ 0 ⇔ T) ∧ (∀n. 0 ≥ n ⇔ (n = 0)) ∧
  (∀n m. NUMERAL n ≥ NUMERAL m ⇔ m ≤ n) ∧ (∀n. ODD (NUMERAL n) ⇔ ODD n) ∧
  (∀n. EVEN (NUMERAL n) ⇔ EVEN n) ∧ ¬ODD 0 ∧ EVEN 0

numeral_add

⊢ ∀n m.
      (numeral$iZ (ZERO + n) = n) ∧ (numeral$iZ (n + ZERO) = n) ∧
      (numeral$iZ (BIT1 n + BIT1 m) = BIT2 (numeral$iZ (n + m))) ∧
      (numeral$iZ (BIT1 n + BIT2 m) = BIT1 (SUC (n + m))) ∧
      (numeral$iZ (BIT2 n + BIT1 m) = BIT1 (SUC (n + m))) ∧
      (numeral$iZ (BIT2 n + BIT2 m) = BIT2 (SUC (n + m))) ∧
      (SUC (ZERO + n) = SUC n) ∧ (SUC (n + ZERO) = SUC n) ∧
      (SUC (BIT1 n + BIT1 m) = BIT1 (SUC (n + m))) ∧
      (SUC (BIT1 n + BIT2 m) = BIT2 (SUC (n + m))) ∧
      (SUC (BIT2 n + BIT1 m) = BIT2 (SUC (n + m))) ∧
      (SUC (BIT2 n + BIT2 m) = BIT1 (numeral$iiSUC (n + m))) ∧
      (numeral$iiSUC (ZERO + n) = numeral$iiSUC n) ∧
      (numeral$iiSUC (n + ZERO) = numeral$iiSUC n) ∧
      (numeral$iiSUC (BIT1 n + BIT1 m) = BIT2 (SUC (n + m))) ∧
      (numeral$iiSUC (BIT1 n + BIT2 m) = BIT1 (numeral$iiSUC (n + m))) ∧
      (numeral$iiSUC (BIT2 n + BIT1 m) = BIT1 (numeral$iiSUC (n + m))) ∧
      (numeral$iiSUC (BIT2 n + BIT2 m) = BIT2 (numeral$iiSUC (n + m)))

iSUB_THM

⊢ ∀b n m.
      (numeral$iSUB b ZERO x = ZERO) ∧ (numeral$iSUB T n ZERO = n) ∧
      (numeral$iSUB F (BIT1 n) ZERO = numeral$iDUB n) ∧
      (numeral$iSUB T (BIT1 n) (BIT1 m) = numeral$iDUB (numeral$iSUB T n m)) ∧
      (numeral$iSUB F (BIT1 n) (BIT1 m) = BIT1 (numeral$iSUB F n m)) ∧
      (numeral$iSUB T (BIT1 n) (BIT2 m) = BIT1 (numeral$iSUB F n m)) ∧
      (numeral$iSUB F (BIT1 n) (BIT2 m) = numeral$iDUB (numeral$iSUB F n m)) ∧
      (numeral$iSUB F (BIT2 n) ZERO = BIT1 n) ∧
      (numeral$iSUB T (BIT2 n) (BIT1 m) = BIT1 (numeral$iSUB T n m)) ∧
      (numeral$iSUB F (BIT2 n) (BIT1 m) = numeral$iDUB (numeral$iSUB T n m)) ∧
      (numeral$iSUB T (BIT2 n) (BIT2 m) = numeral$iDUB (numeral$iSUB T n m)) ∧
      (numeral$iSUB F (BIT2 n) (BIT2 m) = BIT1 (numeral$iSUB F n m))

internal_mult_characterisation

⊢ ∀n m.
      (internal_mult ZERO n = ZERO) ∧ (internal_mult n ZERO = ZERO) ∧
      (internal_mult (BIT1 n) m =
       numeral$iZ (numeral$iDUB (internal_mult n m) + m)) ∧
      (internal_mult (BIT2 n) m =
       numeral$iDUB (numeral$iZ (internal_mult n m + m)))

iDUB_removal

⊢ ∀n.
      (numeral$iDUB (BIT1 n) = BIT2 (numeral$iDUB n)) ∧
      (numeral$iDUB (BIT2 n) = BIT2 (BIT1 n)) ∧ (numeral$iDUB ZERO = ZERO)

exactlog_characterisation

⊢ ∀n m. (numeral$exactlog n = BIT1 m) ⇒ (n = 2 ** (m + 1))

enumeral_mult

⊢ (ZERO * n = ZERO) ∧ (n * ZERO = ZERO) ∧
  (BIT1 x * BIT1 y = internal_mult (BIT1 x) (BIT1 y)) ∧
  (BIT1 x * BIT2 y =
   (let
      n = numeral$exactlog (BIT2 y)
    in
      if ODD n then numeral$texp_help (DIV2 n) (PRE (BIT1 x))
      else internal_mult (BIT1 x) (BIT2 y))) ∧
  (BIT2 x * BIT1 y =
   (let
      m = numeral$exactlog (BIT2 x)
    in
      if ODD m then numeral$texp_help (DIV2 m) (PRE (BIT1 y))
      else internal_mult (BIT2 x) (BIT1 y))) ∧
  (BIT2 x * BIT2 y =
   (let
      m = numeral$exactlog (BIT2 x) ;
      n = numeral$exactlog (BIT2 y)
    in
      if ODD m then numeral$texp_help (DIV2 m) (PRE (BIT2 y))
      else if ODD n then numeral$texp_help (DIV2 n) (PRE (BIT2 x))
      else internal_mult (BIT2 x) (BIT2 y)))

divmod_POS

⊢ ∀n.
      0 < n ⇒
      (DIVMOD (a,m,n) =
       if m < n then (a,m)
       else (let q = findq (1,m,n) in DIVMOD (a + q,m − n * q,n)))

DIVMOD_NUMERAL_CALC

⊢ (∀m n. m DIV BIT1 n = FST (DIVMOD (ZERO,m,BIT1 n))) ∧
  (∀m n. m DIV BIT2 n = FST (DIVMOD (ZERO,m,BIT2 n))) ∧
  (∀m n. m MOD BIT1 n = SND (DIVMOD (ZERO,m,BIT1 n))) ∧
  ∀m n. m MOD BIT2 n = SND (DIVMOD (ZERO,m,BIT2 n))

DIV2_BIT1

⊢ DIV2 (BIT1 x) = x

bit_initiality

⊢ ∀zf b1f b2f.
      ∃f.
          (f ZERO = zf) ∧ (∀n. f (BIT1 n) = b1f n (f n)) ∧
          ∀n. f (BIT2 n) = b2f n (f n)

bit_induction

⊢ ∀P. P ZERO ∧ (∀n. P n ⇒ P (BIT1 n)) ∧ (∀n. P n ⇒ P (BIT2 n)) ⇒ ∀n. P n