Theory "bitstring"

Signature

Definitions

extend_def

|- (∀v0 l. extend v0 0 l = l) ∧ ∀c n l. extend c (SUC n) l = extend c n (c::l)

boolify_def

|- (∀a. boolify a [] = a) ∧ ∀a n l. boolify a (n::l) = boolify ((n ≠ 0)::a) l

bitify_tupled_primitive_def

|- bitify_tupled =
   WFREC (@R. WF R ∧ (∀l a. R (0::a,l) (a,F::l)) ∧ ∀l a. R (1::a,l) (a,T::l))
     (λbitify_tupled a'.
        case a' of
          (a,[]) => I a
        | (a,T::l) => I (bitify_tupled (1::a,l))
        | (a,F::l) => I (bitify_tupled (0::a,l)))

bitify_curried_def

|- ∀x x1. bitify x x1 = bitify_tupled (x,x1)

n2v_def

|- ∀n.
     n2v n =
     if n = 0 then [F]
     else
       (let w = LOG2 n
        in
          PAD_LEFT F (w + 1) (boolify [] (num_to_bin_list (BITS w 0 n))))

v2n_def

|- ∀l. v2n l = num_from_bin_list (bitify [] l)

s2v_def

|- s2v = MAP (λc. (c = #"1") ∨ (c = #"T"))

v2s_def

|- v2s = MAP (λb. if b then #"1" else #"0")

zero_extend_def

|- ∀n v. zero_extend n v = PAD_LEFT F n v

sign_extend_def

|- ∀n v. sign_extend n v = PAD_LEFT (HD v) n v

fixwidth_def

|- ∀n v.
     fixwidth n v =
     (let l = LENGTH v in if l < n then zero_extend n v else DROP (l − n) v)

shiftl_def

|- ∀v m. shiftl v m = PAD_RIGHT F (LENGTH v + m) v

shiftr_def

|- ∀v m. shiftr v m = TAKE (LENGTH v − m) v

field_def

|- ∀h l v. field h l v = fixwidth (SUC h − l) (shiftr v l)

rotate_def

|- ∀v m.
     rotate v m =
     (let l = LENGTH v in
      let x = m MOD l
      in
        if x = 0 then v else field (x − 1) 0 v ++ field (l − 1) x v)

testbit_def

|- ∀b v. testbit b v ⇔ (field b b v = [T])

w2v_def

|- ∀w. w2v w = GENLIST (λi. w ' (dimindex (:α) − 1 − i)) (dimindex (:α))

v2w_def

|- ∀v. v2w v = FCP i. testbit i v

rev_count_list_def

|- ∀n. rev_count_list n = GENLIST (λi. n − 1 − i) n

modify_def

|- ∀f v. modify f v = MAP (UNCURRY f) (ZIP (rev_count_list (LENGTH v),v))

field_insert_def

|- ∀h l s.
     field_insert h l s =
     modify (λi. COND (l ≤ i ∧ i ≤ h) (testbit (i − l) s))

add_def

|- ∀a b.
     add a b =
     (let m = MAX (LENGTH a) (LENGTH b)
      in
        zero_extend m (n2v (v2n a + v2n b)))

bitwise_def

|- ∀f v1 v2.
     bitwise f v1 v2 =
     (let m = MAX (LENGTH v1) (LENGTH v2)
      in
        MAP (UNCURRY f) (ZIP (fixwidth m v1,fixwidth m v2)))

bnot_def

|- bnot = MAP $~

bor_def

|- bor = bitwise $\/

band_def

|- band = bitwise $/\

bxor_def

|- bxor = bitwise (λx y. x ⇎ y)

bnor_def

|- bnor = bitwise (λx y. ¬(x ∨ y))

bxnor_def

|- bxnor = bitwise $<=>

bnand_def

|- bnand = bitwise (λx y. ¬(x ∧ y))

replicate_def

|- ∀v n. replicate v n = FLAT (GENLIST (K v) n)

Theorems

extend_def_compute

|- (∀v0 l. extend v0 0 l = l) ∧
   (∀c n l.
      extend c (NUMERAL (BIT1 n)) l =
      extend c (NUMERAL (BIT1 n) − 1) (c::l)) ∧
   ∀c n l. extend c (NUMERAL (BIT2 n)) l = extend c (NUMERAL (BIT1 n)) (c::l)

bitify_ind

|- ∀P.
     (∀a. P a []) ∧ (∀a l. P (0::a) l ⇒ P a (F::l)) ∧
     (∀a l. P (1::a) l ⇒ P a (T::l)) ⇒
     ∀v v1. P v v1

bitify_def

|- (∀a. bitify a [] = a) ∧ (∀l a. bitify a (F::l) = bitify (0::a) l) ∧
   ∀l a. bitify a (T::l) = bitify (1::a) l

extend_cons

|- ∀n c l. extend c (SUC n) l = c::extend c n l

pad_left_extend

|- ∀n l c. PAD_LEFT c n l = extend c (n − LENGTH l) l

extend

|- (∀n v. zero_extend n v = extend F (n − LENGTH v) v) ∧
   ∀n v. sign_extend n v = extend (HD v) (n − LENGTH v) v

fixwidth

|- ∀n v.
     fixwidth n v =
     (let l = LENGTH v
      in
        if l < n then extend F (n − l) v else DROP (l − n) v)

fixwidth_id

|- ∀w. fixwidth (LENGTH w) w = w

fixwidth_id_imp

|- ∀n w. (n = LENGTH w) ⇒ (fixwidth n w = w)

boolify_reverse_map

|- ∀v a. boolify a v = REVERSE (MAP (λn. n ≠ 0) v) ++ a

bitify_reverse_map

|- ∀v a. bitify a v = REVERSE (MAP (λb. if b then 1 else 0) v) ++ a

every_bit_bitify

|- ∀v. EVERY ($> 2) (bitify [] v)

length_pad_left

|- ∀x n a. LENGTH (PAD_LEFT x n a) = if LENGTH a < n then n else LENGTH a

length_pad_right

|- ∀x n a. LENGTH (PAD_RIGHT x n a) = if LENGTH a < n then n else LENGTH a

length_zero_extend

|- ∀n v. LENGTH v < n ⇒ (LENGTH (zero_extend n v) = n)

length_sign_extend

|- ∀n v. LENGTH v < n ⇒ (LENGTH (sign_extend n v) = n)

length_fixwidth

|- ∀n v. LENGTH (fixwidth n v) = n

length_field

|- ∀h l v. LENGTH (field h l v) = SUC h − l

length_bitify

|- ∀v l. LENGTH (bitify l v) = LENGTH l + LENGTH v

length_bitify_null

|- ∀v l. LENGTH (bitify [] v) = LENGTH v

length_shiftr

|- ∀v n. LENGTH (shiftr v n) = LENGTH v − n

length_rev_count_list

|- ∀n. LENGTH (rev_count_list n) = n

length_w2v

|- ∀w. LENGTH (w2v w) = dimindex (:α)

el_rev_count_list

|- ∀n i. i < n ⇒ (EL i (rev_count_list n) = n − 1 − i)

el_zero_extend

|- ∀n i v.
     EL i (zero_extend n v) ⇔ n − LENGTH v ≤ i ∧ EL (i − (n − LENGTH v)) v

el_sign_extend

|- ∀n i v.
     EL i (sign_extend n v) =
     if i < n − LENGTH v then EL 0 v else EL (i − (n − LENGTH v)) v

el_fixwidth

|- ∀i n w.
     i < n ⇒
     (EL i (fixwidth n w) ⇔
      if LENGTH w < n then n − LENGTH w ≤ i ∧ EL (i − (n − LENGTH w)) w
      else EL (i + (LENGTH w − n)) w)

el_field

|- ∀v h l i.
     i < SUC h − l ⇒
     (EL i (field h l v) ⇔ SUC h ≤ i + LENGTH v ∧ EL (i + LENGTH v − SUC h) v)

el_w2v

|- ∀w n. n < dimindex (:α) ⇒ (EL n (w2v w) ⇔ w ' (dimindex (:α) − 1 − n))

el_shiftr

|- ∀i v n d.
     n < d ∧ i < d − n ∧ 0 < d ⇒
     (EL i (shiftr (fixwidth d v) n) ⇔
      d ≤ i + LENGTH v ∧ EL (i + LENGTH v − d) v)

shiftr_0

|- ∀v. shiftr v 0 = v

field_fixwidth

|- ∀h v. field h 0 v = fixwidth (SUC h) v

testbit

|- ∀b v. testbit b v ⇔ (let n = LENGTH v in b < n ∧ EL (n − 1 − b) v)

testbit_geq_len

|- ∀v i. LENGTH v ≤ i ⇒ ¬testbit i v

testbit_el

|- ∀v i. i < LENGTH v ⇒ (testbit i v ⇔ EL (LENGTH v − 1 − i) v)

bit_v2w

|- ∀n v. word_bit n (v2w v) ⇔ n < dimindex (:α) ∧ testbit n v

word_index_v2w

|- ∀v i.
     v2w v ' i ⇔
     if i < dimindex (:α) then testbit i v
     else FAIL $' index too large (v2w v) i

testbit_w2v

|- ∀n w. testbit n (w2v w) ⇔ word_bit n w

w2v_v2w

|- ∀v. w2v (v2w v) = fixwidth (dimindex (:α)) v

v2w_w2v

|- ∀w. v2w (w2v w) = w

v2w_fixwidth

|- ∀v. v2w (fixwidth (dimindex (:α)) v) = v2w v

fixwidth_fixwidth

|- ∀n v. fixwidth n (fixwidth n v) = fixwidth n v

bitstring_nchotomy

|- ∀w. ∃v. w = v2w v

ranged_bitstring_nchotomy

|- ∀w. ∃v. (w = v2w v) ∧ Abbrev (LENGTH v = dimindex (:α))

fixwidth_eq

|- ∀n v w.
     (fixwidth n v = fixwidth n w) ⇔ ∀i. i < n ⇒ (testbit i v ⇔ testbit i w)

v2w_11

|- ∀v w.
     (v2w v = v2w w) ⇔
     (fixwidth (dimindex (:α)) v = fixwidth (dimindex (:α)) w)

field_concat_right

|- ∀h a b. (LENGTH b = SUC h) ⇒ (field h 0 (a ++ b) = b)

field_concat_left

|- ∀h l a b.
     l ≤ h ∧ LENGTH b ≤ l ⇒
     (field h l (a ++ b) = field (h − LENGTH b) (l − LENGTH b) a)

field_id_imp

|- ∀n v. (SUC n = LENGTH v) ⇒ (field n 0 v = v)

shiftl_replicate_F

|- ∀v n. shiftl v n = v ++ replicate [F] n

word_lsb_v2w

|- ∀v. word_lsb (v2w v) ⇔ v ≠ [] ∧ LAST v

word_msb_v2w

|- ∀v. word_msb (v2w v) ⇔ testbit (dimindex (:α) − 1) v

w2w_v2w

|- ∀v.
     w2w (v2w v) =
     v2w
       (fixwidth
          (if dimindex (:β) < dimindex (:α) then dimindex (:β)
           else dimindex (:α)) v)

sw2sw_v2w

|- ∀v.
     sw2sw (v2w v) =
     if dimindex (:α) < dimindex (:β) then
       v2w (sign_extend (dimindex (:β)) (fixwidth (dimindex (:α)) v))
     else v2w (fixwidth (dimindex (:β)) v)

n2w_v2n

|- ∀v. n2w (v2n v) = v2w v

v2n_n2v

|- ∀n. v2n (n2v n) = n

v2w_n2v

|- ∀n. v2w (n2v n) = n2w n

w2n_v2w

|- ∀v. w2n (v2w v) = MOD_2EXP (dimindex (:α)) (v2n v)

v2n_lt

|- ∀v. v2n v < 2 ** LENGTH v

word_and_v2w

|- ∀v w. v2w v && v2w w = v2w (band v w)

word_or_v2w

|- ∀v w. v2w v ‖ v2w w = v2w (bor v w)

word_xor_v2w

|- ∀v w. v2w v ⊕ v2w w = v2w (bxor v w)

word_nand_v2w

|- ∀v w.
     v2w v ~&& v2w w =
     v2w (bnand (fixwidth (dimindex (:α)) v) (fixwidth (dimindex (:α)) w))

word_nor_v2w

|- ∀v w.
     v2w v ~|| v2w w =
     v2w (bnor (fixwidth (dimindex (:α)) v) (fixwidth (dimindex (:α)) w))

word_xnor_v2w

|- ∀v w.
     v2w v ~?? v2w w =
     v2w (bxnor (fixwidth (dimindex (:α)) v) (fixwidth (dimindex (:α)) w))

word_1comp_v2w

|- ∀v. ¬v2w v = v2w (bnot (fixwidth (dimindex (:α)) v))

word_lsl_v2w

|- ∀n v. v2w v ≪ n = v2w (shiftl v n)

word_lsr_v2w

|- ∀n v. v2w v ⋙ n = v2w (shiftr (fixwidth (dimindex (:α)) v) n)

word_modify_v2w

|- ∀f v. word_modify f (v2w v) = v2w (modify f (fixwidth (dimindex (:α)) v))

word_bits_v2w

|- ∀h l v. (h -- l) (v2w v) = v2w (field h l (fixwidth (dimindex (:α)) v))

word_extract_v2w

|- ∀h l v. (h >< l) (v2w v) = w2w ((h -- l) (v2w v))

word_slice_v2w

|- ∀h l v.
     (h '' l) (v2w v) =
     v2w (shiftl (field h l (fixwidth (dimindex (:α)) v)) l)

word_asr_v2w

|- ∀n v.
     v2w v ≫ n =
     (let l = fixwidth (dimindex (:α)) v
      in
        v2w
          (sign_extend (dimindex (:α))
             (if dimindex (:α) ≤ n then [HD l] else shiftr l n)))

word_ror_v2w

|- ∀n v. v2w v ⇄ n = v2w (rotate (fixwidth (dimindex (:α)) v) n)

word_reverse_v2w

|- ∀v. word_reverse (v2w v) = v2w (REVERSE (fixwidth (dimindex (:α)) v))

word_join_v2w

|- ∀v1 v2.
     FINITE 𝕌(:α) ∧ FINITE 𝕌(:β) ⇒
     (word_join (v2w v1) (v2w v2) = v2w (v1 ++ fixwidth (dimindex (:β)) v2))

word_concat_v2w

|- ∀v1 v2.
     FINITE 𝕌(:α) ∧ FINITE 𝕌(:β) ⇒
     (v2w v1 @@ v2w v2 =
      v2w
        (fixwidth (MIN (dimindex (:γ)) (dimindex (:α) + dimindex (:β)))
           (v1 ++ fixwidth (dimindex (:β)) v2)))

word_join_v2w_rwt

|- ∀v1 v2.
     word_join (v2w v1) (v2w v2) =
     if FINITE 𝕌(:α) ∧ FINITE 𝕌(:β) then
       v2w (v1 ++ fixwidth (dimindex (:β)) v2)
     else FAIL word_join bad domain (v2w v1) (v2w v2)

word_concat_v2w_rwt

|- ∀v1 v2.
     v2w v1 @@ v2w v2 =
     if FINITE 𝕌(:α) ∧ FINITE 𝕌(:β) then
       v2w
         (fixwidth (MIN (dimindex (:γ)) (dimindex (:α) + dimindex (:β)))
            (v1 ++ fixwidth (dimindex (:β)) v2))
     else FAIL $@@ bad domain (v2w v1) (v2w v2)

word_reduce_v2w

|- ∀f v.
     word_reduce f (v2w v) =
     (let l = fixwidth (dimindex (:α)) v in v2w [FOLDL f (HD l) (TL l)])

reduce_and_v2w

|- ∀v. reduce_and (v2w v) = word_reduce $/\ (v2w v)

reduce_or_v2w

|- ∀v. reduce_or (v2w v) = word_reduce $\/ (v2w v)

extract_v2w

|- ∀h l v.
     LENGTH v ≤ dimindex (:α) ∧ (dimindex (:β) = SUC h − l) ∧
     dimindex (:β) < dimindex (:α) ⇒
     ((h >< l) (v2w v) = v2w (field h l v))

word_bit_last_shiftr

|- ∀i v.
     i < dimindex (:α) ⇒
     (word_bit i (v2w v) ⇔ (let l = shiftr v i in ¬NULL l ∧ LAST l))

ops_to_v2w

|- (∀v n. v2w v ‖ n2w n = v2w v ‖ v2w (n2v n)) ∧
   (∀v n. n2w n ‖ v2w v = v2w (n2v n) ‖ v2w v) ∧
   (∀v n. v2w v && n2w n = v2w v && v2w (n2v n)) ∧
   (∀v n. n2w n && v2w v = v2w (n2v n) && v2w v) ∧
   (∀v n. v2w v ⊕ n2w n = v2w v ⊕ v2w (n2v n)) ∧
   (∀v n. n2w n ⊕ v2w v = v2w (n2v n) ⊕ v2w v) ∧
   (∀v n. v2w v ~|| n2w n = v2w v ~|| v2w (n2v n)) ∧
   (∀v n. n2w n ~|| v2w v = v2w (n2v n) ~|| v2w v) ∧
   (∀v n. v2w v ~&& n2w n = v2w v ~&& v2w (n2v n)) ∧
   (∀v n. n2w n ~&& v2w v = v2w (n2v n) ~&& v2w v) ∧
   (∀v n. v2w v ~?? n2w n = v2w v ~?? v2w (n2v n)) ∧
   (∀v n. n2w n ~?? v2w v = v2w (n2v n) ~?? v2w v) ∧
   (∀v n. v2w v @@ n2w n = v2w v @@ v2w (n2v n)) ∧
   (∀v n. n2w n @@ v2w v = v2w (n2v n) @@ v2w v) ∧
   (∀v n. word_join (v2w v) (n2w n) = word_join (v2w v) (v2w (n2v n))) ∧
   ∀v n. word_join (n2w n) (v2w v) = word_join (v2w (n2v n)) (v2w v)

ops_to_n2w

|- (∀v. -v2w v = -n2w (v2n v)) ∧
   (∀v. word_log2 (v2w v) = word_log2 (n2w (v2n v))) ∧
   (∀v n. (v2w v = n2w n) ⇔ (n2w (v2n v) = n2w n)) ∧
   (∀v n. (n2w n = v2w v) ⇔ (n2w n = n2w (v2n v))) ∧
   (∀v w. v2w v + w = n2w (v2n v) + w) ∧ (∀v w. w + v2w v = w + n2w (v2n v)) ∧
   (∀v w. v2w v − w = n2w (v2n v) − w) ∧ (∀v w. w − v2w v = w − n2w (v2n v)) ∧
   (∀v w. v2w v * w = n2w (v2n v) * w) ∧ (∀v w. w * v2w v = w * n2w (v2n v)) ∧
   (∀v w. v2w v / w = n2w (v2n v) / w) ∧ (∀v w. w / v2w v = w / n2w (v2n v)) ∧
   (∀v w. v2w v // w = n2w (v2n v) // w) ∧
   (∀v w. w // v2w v = w // n2w (v2n v)) ∧
   (∀v w. word_mod (v2w v) w = word_mod (n2w (v2n v)) w) ∧
   (∀v w. word_mod w (v2w v) = word_mod w (n2w (v2n v))) ∧
   (∀v w. v2w v < w ⇔ n2w (v2n v) < w) ∧ (∀v w. w < v2w v ⇔ w < n2w (v2n v)) ∧
   (∀v w. v2w v > w ⇔ n2w (v2n v) > w) ∧ (∀v w. w > v2w v ⇔ w > n2w (v2n v)) ∧
   (∀v w. v2w v ≤ w ⇔ n2w (v2n v) ≤ w) ∧ (∀v w. w ≤ v2w v ⇔ w ≤ n2w (v2n v)) ∧
   (∀v w. v2w v ≥ w ⇔ n2w (v2n v) ≥ w) ∧ (∀v w. w ≥ v2w v ⇔ w ≥ n2w (v2n v)) ∧
   (∀v w. v2w v <₊ w ⇔ n2w (v2n v) <₊ w) ∧
   (∀v w. w <₊ v2w v ⇔ w <₊ n2w (v2n v)) ∧
   (∀v w. v2w v >₊ w ⇔ n2w (v2n v) >₊ w) ∧
   (∀v w. w >₊ v2w v ⇔ w >₊ n2w (v2n v)) ∧
   (∀v w. v2w v ≤₊ w ⇔ n2w (v2n v) ≤₊ w) ∧
   (∀v w. w ≤₊ v2w v ⇔ w ≤₊ n2w (v2n v)) ∧
   (∀v w. v2w v ≥₊ w ⇔ n2w (v2n v) ≥₊ w) ∧ ∀v w. w ≥₊ v2w v ⇔ w ≥₊ n2w (v2n v)