Structure bitstringTheory
signature bitstringTheory =
sig
type thm = Thm.thm
(* Definitions *)
val add_def : thm
val band_def : thm
val bitwise_def : thm
val bnand_def : thm
val bnor_def : thm
val bnot_def : thm
val boolify_def : thm
val bor_def : thm
val bxnor_def : thm
val bxor_def : thm
val extend_def : thm
val field_def : thm
val field_insert_def : thm
val fixwidth_def : thm
val modify_def : thm
val n2v_def : thm
val replicate_def : thm
val rev_count_list_def : thm
val rotate_def : thm
val s2v_def : thm
val shiftl_def : thm
val shiftr_def : thm
val sign_extend_def : thm
val testbit_def : thm
val v2n_def : thm
val v2s_def : thm
val v2w_def : thm
val w2v_def : thm
val zero_extend_def : thm
(* Theorems *)
val bit_v2w : thm
val bitify_def : thm
val bitify_ind : thm
val bitify_reverse_map : thm
val bitstring_nchotomy : thm
val boolify_reverse_map : thm
val el_field : thm
val el_fixwidth : thm
val el_rev_count_list : thm
val el_shiftr : thm
val el_sign_extend : thm
val el_w2v : thm
val el_zero_extend : thm
val every_bit_bitify : thm
val extend : thm
val extend_compute : thm
val extend_cons : thm
val extract_v2w : thm
val field_concat_left : thm
val field_concat_right : thm
val field_fixwidth : thm
val field_id_imp : thm
val fixwidth : thm
val fixwidth_REPLICATE : thm
val fixwidth_eq : thm
val fixwidth_fixwidth : thm
val fixwidth_id : thm
val fixwidth_id_imp : thm
val length_bitify : thm
val length_bitify_null : thm
val length_field : thm
val length_fixwidth : thm
val length_pad_left : thm
val length_pad_right : thm
val length_rev_count_list : thm
val length_rotate : thm
val length_shiftr : thm
val length_sign_extend : thm
val length_w2v : thm
val length_zero_extend : thm
val n2w_v2n : thm
val ops_to_n2w : thm
val ops_to_v2w : thm
val pad_left_extend : thm
val ranged_bitstring_nchotomy : thm
val reduce_and_v2w : thm
val reduce_or_v2w : thm
val shiftl_replicate_F : thm
val shiftr_0 : thm
val sw2sw_v2w : thm
val testbit : thm
val testbit_el : thm
val testbit_geq_len : thm
val testbit_w2v : thm
val v2n : thm
val v2n_APPEND : thm
val v2n_lt : thm
val v2n_n2v : thm
val v2w_11 : thm
val v2w_fixwidth : thm
val v2w_n2v : thm
val v2w_w2v : thm
val w2n_v2w : thm
val w2v_v2w : thm
val w2w_v2w : thm
val word_1comp_v2w : thm
val word_and_v2w : thm
val word_asr_v2w : thm
val word_bit_last_shiftr : thm
val word_bits_v2w : thm
val word_concat_v2w : thm
val word_concat_v2w_rwt : thm
val word_extract_v2w : thm
val word_index_v2w : thm
val word_join_v2w : thm
val word_join_v2w_rwt : thm
val word_lsb_v2w : thm
val word_lsl_v2w : thm
val word_lsr_v2w : thm
val word_modify_v2w : thm
val word_msb_v2w : thm
val word_nand_v2w : thm
val word_nor_v2w : thm
val word_or_v2w : thm
val word_reduce_v2w : thm
val word_reverse_v2w : thm
val word_ror_alt : thm
val word_ror_v2w : thm
val word_slice_v2w : thm
val word_xnor_v2w : thm
val word_xor_v2w : thm
val bitstring_grammars : type_grammar.grammar * term_grammar.grammar
(*
[words] Parent theory of "bitstring"
[add_def] Definition
⊢ ∀a b.
add a b =
(let
m = MAX (LENGTH a) (LENGTH b)
in
zero_extend m (n2v (v2n a + v2n b)))
[band_def] Definition
⊢ band = bitwise $/\
[bitwise_def] Definition
⊢ ∀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)))
[bnand_def] Definition
⊢ bnand = bitwise (λx y. ¬(x ∧ y))
[bnor_def] Definition
⊢ bnor = bitwise (λx y. ¬(x ∨ y))
[bnot_def] Definition
⊢ bnot = MAP $¬
[boolify_def] Definition
⊢ (∀a. boolify a [] = a) ∧
∀a n l. boolify a (n::l) = boolify ((n ≠ 0)::a) l
[bor_def] Definition
⊢ bor = bitwise $\/
[bxnor_def] Definition
⊢ bxnor = bitwise $<=>
[bxor_def] Definition
⊢ bxor = bitwise (λx y. x ⇎ y)
[extend_def] Definition
⊢ (∀v0 l. extend v0 0 l = l) ∧
∀c n l. extend c (SUC n) l = extend c n (c::l)
[field_def] Definition
⊢ ∀h l v. field h l v = fixwidth (SUC h − l) (shiftr v l)
[field_insert_def] Definition
⊢ ∀h l s.
field_insert h l s =
modify (λi. COND (l ≤ i ∧ i ≤ h) (testbit (i − l) s))
[fixwidth_def] Definition
⊢ ∀n v.
fixwidth n v =
(let
l = LENGTH v
in
if l < n then zero_extend n v else DROP (l − n) v)
[modify_def] Definition
⊢ ∀f v.
modify f v = MAP (UNCURRY f) (ZIP (rev_count_list (LENGTH v),v))
[n2v_def] Definition
⊢ ∀n. n2v n = boolify [] (n2l 2 n)
[replicate_def] Definition
⊢ ∀v n. replicate v n = FLAT (GENLIST (K v) n)
[rev_count_list_def] Definition
⊢ ∀n. rev_count_list n = GENLIST (λi. n − 1 − i) n
[rotate_def] Definition
⊢ ∀v m.
rotate v m =
(let
l = LENGTH v;
x = m MOD l
in
if l = 0 ∨ x = 0 then v
else field (x − 1) 0 v ⧺ field (l − 1) x v)
[s2v_def] Definition
⊢ s2v = MAP (λc. c = #"1" ∨ c = #"T")
[shiftl_def] Definition
⊢ ∀v m. shiftl v m = PAD_RIGHT F (LENGTH v + m) v
[shiftr_def] Definition
⊢ ∀v m. shiftr v m = TAKE (LENGTH v − m) v
[sign_extend_def] Definition
⊢ ∀n v. sign_extend n v = PAD_LEFT (HD v) n v
[testbit_def] Definition
⊢ ∀b v. testbit b v ⇔ field b b v = [T]
[v2n_def] Definition
⊢ ∀l. v2n l = l2n 2 (bitify [] l)
[v2s_def] Definition
⊢ v2s = MAP (λb. if b then #"1" else #"0")
[v2w_def] Definition
⊢ ∀v. v2w v = FCP i. testbit i v
[w2v_def] Definition
⊢ ∀w. w2v w =
GENLIST (λi. w ' (dimindex (:α) − 1 − i)) (dimindex (:α))
[zero_extend_def] Definition
⊢ ∀n v. zero_extend n v = PAD_LEFT F n v
[bit_v2w] Theorem
⊢ ∀n v. word_bit n (v2w v) ⇔ n < dimindex (:α) ∧ testbit n v
[bitify_def] Theorem
⊢ (∀a. bitify a [] = a) ∧ (∀l a. bitify a (F::l) = bitify (0::a) l) ∧
∀l a. bitify a (T::l) = bitify (1::a) l
[bitify_ind] Theorem
⊢ ∀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_reverse_map] Theorem
⊢ ∀v a. bitify a v = REVERSE (MAP (λb. if b then 1 else 0) v) ⧺ a
[bitstring_nchotomy] Theorem
⊢ ∀w. ∃v. w = v2w v
[boolify_reverse_map] Theorem
⊢ ∀v a. boolify a v = REVERSE (MAP (λn. n ≠ 0) v) ⧺ a
[el_field] Theorem
⊢ ∀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_fixwidth] Theorem
⊢ ∀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_rev_count_list] Theorem
⊢ ∀n i. i < n ⇒ EL i (rev_count_list n) = n − 1 − i
[el_shiftr] Theorem
⊢ ∀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)
[el_sign_extend] Theorem
⊢ ∀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_w2v] Theorem
⊢ ∀w n.
n < dimindex (:α) ⇒ (EL n (w2v w) ⇔ w ' (dimindex (:α) − 1 − n))
[el_zero_extend] Theorem
⊢ ∀n i v.
EL i (zero_extend n v) ⇔
n − LENGTH v ≤ i ∧ EL (i − (n − LENGTH v)) v
[every_bit_bitify] Theorem
⊢ ∀v. EVERY ($> 2) (bitify [] v)
[extend] Theorem
⊢ (∀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
[extend_compute] Theorem
⊢ (∀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)
[extend_cons] Theorem
⊢ ∀n c l. extend c (SUC n) l = c::extend c n l
[extract_v2w] Theorem
⊢ ∀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)
[field_concat_left] Theorem
⊢ ∀h l a b.
l ≤ h ∧ LENGTH b ≤ l ⇒
field h l (a ⧺ b) = field (h − LENGTH b) (l − LENGTH b) a
[field_concat_right] Theorem
⊢ ∀h a b. LENGTH b = SUC h ⇒ field h 0 (a ⧺ b) = b
[field_fixwidth] Theorem
⊢ ∀h v. field h 0 v = fixwidth (SUC h) v
[field_id_imp] Theorem
⊢ ∀n v. SUC n = LENGTH v ⇒ field n 0 v = v
[fixwidth] Theorem
⊢ ∀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_REPLICATE] Theorem
⊢ ∀len l.
fixwidth len l =
if LENGTH l ≤ len then REPLICATE (len − LENGTH l) F ⧺ l
else DROP (LENGTH l − len) l
[fixwidth_eq] Theorem
⊢ ∀n v w.
fixwidth n v = fixwidth n w ⇔
∀i. i < n ⇒ (testbit i v ⇔ testbit i w)
[fixwidth_fixwidth] Theorem
⊢ ∀n v. fixwidth n (fixwidth n v) = fixwidth n v
[fixwidth_id] Theorem
⊢ ∀w. fixwidth (LENGTH w) w = w
[fixwidth_id_imp] Theorem
⊢ ∀n w. n = LENGTH w ⇒ fixwidth n w = w
[length_bitify] Theorem
⊢ ∀v l. LENGTH (bitify l v) = LENGTH l + LENGTH v
[length_bitify_null] Theorem
⊢ ∀v l. LENGTH (bitify [] v) = LENGTH v
[length_field] Theorem
⊢ ∀h l v. LENGTH (field h l v) = SUC h − l
[length_fixwidth] Theorem
⊢ ∀n v. LENGTH (fixwidth n v) = n
[length_pad_left] Theorem
⊢ ∀x n a.
LENGTH (PAD_LEFT x n a) = if LENGTH a < n then n else LENGTH a
[length_pad_right] Theorem
⊢ ∀x n a.
LENGTH (PAD_RIGHT x n a) = if LENGTH a < n then n else LENGTH a
[length_rev_count_list] Theorem
⊢ ∀n. LENGTH (rev_count_list n) = n
[length_rotate] Theorem
⊢ ∀v n. LENGTH (rotate v n) = LENGTH v
[length_shiftr] Theorem
⊢ ∀v n. LENGTH (shiftr v n) = LENGTH v − n
[length_sign_extend] Theorem
⊢ ∀n v. LENGTH v ≤ n ⇒ LENGTH (sign_extend n v) = n
[length_w2v] Theorem
⊢ ∀w. LENGTH (w2v w) = dimindex (:α)
[length_zero_extend] Theorem
⊢ ∀n v. LENGTH v ≤ n ⇒ LENGTH (zero_extend n v) = n
[n2w_v2n] Theorem
⊢ ∀v. n2w (v2n v) = v2w v
[ops_to_n2w] Theorem
⊢ (∀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)
[ops_to_v2w] Theorem
⊢ (∀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)
[pad_left_extend] Theorem
⊢ ∀n l c. PAD_LEFT c n l = extend c (n − LENGTH l) l
[ranged_bitstring_nchotomy] Theorem
⊢ ∀w. ∃v. w = v2w v ∧ Abbrev (LENGTH v = dimindex (:α))
[reduce_and_v2w] Theorem
⊢ ∀v. reduce_and (v2w v) = word_reduce $/\ (v2w v)
[reduce_or_v2w] Theorem
⊢ ∀v. reduce_or (v2w v) = word_reduce $\/ (v2w v)
[shiftl_replicate_F] Theorem
⊢ ∀v n. shiftl v n = v ⧺ replicate [F] n
[shiftr_0] Theorem
⊢ ∀v. shiftr v 0 = v
[sw2sw_v2w] Theorem
⊢ ∀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)
[testbit] Theorem
⊢ ∀b v. testbit b v ⇔ (let n = LENGTH v in b < n ∧ EL (n − 1 − b) v)
[testbit_el] Theorem
⊢ ∀v i. i < LENGTH v ⇒ (testbit i v ⇔ EL (LENGTH v − 1 − i) v)
[testbit_geq_len] Theorem
⊢ ∀v i. LENGTH v ≤ i ⇒ ¬testbit i v
[testbit_w2v] Theorem
⊢ ∀n w. testbit n (w2v w) ⇔ word_bit n w
[v2n] Theorem
⊢ v2n [] = 0 ∧
v2n (b::bs) = if b then 2 ** LENGTH bs + v2n bs else v2n bs
[v2n_APPEND] Theorem
⊢ ∀a b. v2n (a ⧺ b) = v2n b + 2 ** LENGTH b * v2n a
[v2n_lt] Theorem
⊢ ∀v. v2n v < 2 ** LENGTH v
[v2n_n2v] Theorem
⊢ ∀n. v2n (n2v n) = n
[v2w_11] Theorem
⊢ ∀v w.
v2w v = v2w w ⇔
fixwidth (dimindex (:α)) v = fixwidth (dimindex (:α)) w
[v2w_fixwidth] Theorem
⊢ ∀v. v2w (fixwidth (dimindex (:α)) v) = v2w v
[v2w_n2v] Theorem
⊢ ∀n. v2w (n2v n) = n2w n
[v2w_w2v] Theorem
⊢ ∀w. v2w (w2v w) = w
[w2n_v2w] Theorem
⊢ ∀v. w2n (v2w v) = MOD_2EXP (dimindex (:α)) (v2n v)
[w2v_v2w] Theorem
⊢ ∀v. w2v (v2w v) = fixwidth (dimindex (:α)) v
[w2w_v2w] Theorem
⊢ ∀v. w2w (v2w v) =
v2w
(fixwidth
(if dimindex (:β) < dimindex (:α) then dimindex (:β)
else dimindex (:α)) v)
[word_1comp_v2w] Theorem
⊢ ∀v. ¬v2w v = v2w (bnot (fixwidth (dimindex (:α)) v))
[word_and_v2w] Theorem
⊢ ∀v w. v2w v && v2w w = v2w (band v w)
[word_asr_v2w] Theorem
⊢ ∀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_bit_last_shiftr] Theorem
⊢ ∀i v.
i < dimindex (:α) ⇒
(word_bit i (v2w v) ⇔ (let l = shiftr v i in ¬NULL l ∧ LAST l))
[word_bits_v2w] Theorem
⊢ ∀h l v.
(h -- l) (v2w v) = v2w (field h l (fixwidth (dimindex (:α)) v))
[word_concat_v2w] Theorem
⊢ ∀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_concat_v2w_rwt] Theorem
⊢ ∀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 $@@ $var$(bad domain) (v2w v1) (v2w v2)
[word_extract_v2w] Theorem
⊢ ∀h l v. (h >< l) (v2w v) = w2w ((h -- l) (v2w v))
[word_index_v2w] Theorem
⊢ ∀v i.
v2w v ' i ⇔
if i < dimindex (:α) then testbit i v
else FAIL $' $var$(index too large) (v2w v) i
[word_join_v2w] Theorem
⊢ ∀v1 v2.
FINITE 𝕌(:α) ∧ FINITE 𝕌(:β) ⇒
word_join (v2w v1) (v2w v2) =
v2w (v1 ⧺ fixwidth (dimindex (:β)) v2)
[word_join_v2w_rwt] Theorem
⊢ ∀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 $var$(bad domain) (v2w v1) (v2w v2)
[word_lsb_v2w] Theorem
⊢ ∀v. word_lsb (v2w v) ⇔ v ≠ [] ∧ LAST v
[word_lsl_v2w] Theorem
⊢ ∀n v. v2w v ≪ n = v2w (shiftl v n)
[word_lsr_v2w] Theorem
⊢ ∀n v. v2w v ⋙ n = v2w (shiftr (fixwidth (dimindex (:α)) v) n)
[word_modify_v2w] Theorem
⊢ ∀f v.
word_modify f (v2w v) =
v2w (modify f (fixwidth (dimindex (:α)) v))
[word_msb_v2w] Theorem
⊢ ∀v. word_msb (v2w v) ⇔ testbit (dimindex (:α) − 1) v
[word_nand_v2w] Theorem
⊢ ∀v w.
v2w v ~&& v2w w =
v2w
(bnand (fixwidth (dimindex (:α)) v)
(fixwidth (dimindex (:α)) w))
[word_nor_v2w] Theorem
⊢ ∀v w.
v2w v ~|| v2w w =
v2w
(bnor (fixwidth (dimindex (:α)) v) (fixwidth (dimindex (:α)) w))
[word_or_v2w] Theorem
⊢ ∀v w. v2w v ‖ v2w w = v2w (bor v w)
[word_reduce_v2w] Theorem
⊢ ∀f v.
word_reduce f (v2w v) =
(let
l = fixwidth (dimindex (:α)) v
in
v2w [FOLDL f (HD l) (TL l)])
[word_reverse_v2w] Theorem
⊢ ∀v. word_reverse (v2w v) =
v2w (REVERSE (fixwidth (dimindex (:α)) v))
[word_ror_alt] Theorem
⊢ ∀r a.
a ⇄ r =
(let d = dimindex (:α) in a ≪ (d − r MOD d) ‖ a ⋙ (r MOD d))
[word_ror_v2w] Theorem
⊢ ∀n v. v2w v ⇄ n = v2w (rotate (fixwidth (dimindex (:α)) v) n)
[word_slice_v2w] Theorem
⊢ ∀h l v.
(h '' l) (v2w v) =
v2w (shiftl (field h l (fixwidth (dimindex (:α)) v)) l)
[word_xnor_v2w] Theorem
⊢ ∀v w.
v2w v ~?? v2w w =
v2w
(bxnor (fixwidth (dimindex (:α)) v)
(fixwidth (dimindex (:α)) w))
[word_xor_v2w] Theorem
⊢ ∀v w. v2w v ⊕ v2w w = v2w (bxor v w)
*)
end
HOL 4, Trindemossen-1