Bitwise.thy

(*
 * Copyright Thomas Sewell, NICTA and Sascha Boehme, TU Muenchen
 *
 * SPDX-License-Identifier: BSD-2-Clause
 *)

theory Bitwise
  imports
    "HOL-Library.Word"
    More_Arithmetic
    Reversed_Bit_Lists
    Bit_Shifts_Infix_Syntax

begin

text ‹Helper constants used in defining addition›

definition xor3 :: "bool ==> bool ==> bool ==> bool"
  where "xor3 a b c = (a = (b = c))"

definition carry :: "bool ==> bool ==> bool ==> bool"
  where "carry a b c = ((a ∧ (b ∨ c)) ∨ (b ∧ c))"

lemma carry_simps:
  "carry True a b = (a ∨ b)"
  "carry a True b = (a ∨ b)"
  "carry a b True = (a ∨ b)"
  "carry False a b = (a ∧ b)"
  "carry a False b = (a ∧ b)"
  "carry a b False = (a ∧ b)"
  by (auto simp add: carry_def)

lemma xor3_simps:
  "xor3 True a b = (a = b)"
  "xor3 a True b = (a = b)"
  "xor3 a b True = (a = b)"
  "xor3 False a b = (a ≠ b)"
  "xor3 a False b = (a ≠ b)"
  "xor3 a b False = (a ≠ b)"
  by (simp_all add: xor3_def)

text ‹Breaking up word equalities into equalities on their
 bit lists. Equalities are generated and manipulated in the
 reverse order to const‹to_bl›.›

lemma bl_word_sub: "to_bl (x - y) = to_bl (x + (- y))"
  by simp

lemma rbl_word_1: "rev (to_bl (1 :: 'a::len word)) = takefill False (LENGTH('a)) [True]"
  by (metis rev_rev_ident rev_singleton_conv word_1_bl word_rev_tf)

lemma rbl_word_if: "rev (to_bl (if P then x else y)) = map2 (If P) (rev (to_bl x)) (rev (to_bl y))"
  by (simp add: split_def)

lemma rbl_add_carry_Cons:
  "(if car then rbl_succ else id) (rbl_add (x # xs) (y # ys)) =
    xor3 x y car # (if carry x y car then rbl_succ else id) (rbl_add xs ys)"
  by (simp add: carry_def xor3_def)

lemma rbl_add_suc_carry_fold:
  "length xs = length ys ==>
    ∀car. (if car then rbl_succ else id) (rbl_add xs ys) =
      (foldr (λ(x, y) res car. xor3 x y car # res (carry x y car)) (zip xs ys) (λ_. [])) car"
proof (induction rule: list_induct2)
  case Nil
  then show ?case by simp
next
  case (Cons x xs y ys)
  then show ?case
    using rbl_add_carry_Cons by auto
qed

lemma to_bl_plus_carry:
  "to_bl (x + y) =
    rev (foldr (λ(x, y) res car. xor3 x y car # res (carry x y car))
      (rev (zip (to_bl x) (to_bl y))) (λ_. []) False)"
  using rbl_add_suc_carry_fold[where xs="rev (to_bl x)" and ys="rev (to_bl y)"]
  by (smt (verit) id_apply length_rev word_add_rbl word_rotate.lbl_lbl zip_rev)

definition "rbl_plus cin xs ys =
  foldr (λ(x, y) res car. xor3 x y car # res (carry x y car)) (zip xs ys) (λ_. []) cin"

lemma rbl_plus_simps:
  "rbl_plus cin (x # xs) (y # ys) = xor3 x y cin # rbl_plus (carry x y cin) xs ys"
  "rbl_plus cin [] ys = []"
  "rbl_plus cin xs [] = []"
  by (simp_all add: rbl_plus_def)

lemma rbl_word_plus: "rev (to_bl (x + y)) = rbl_plus False (rev (to_bl x)) (rev (to_bl y))"
  by (simp add: rbl_plus_def to_bl_plus_carry zip_rev)

definition "rbl_succ2 b xs = (if b then rbl_succ xs else xs)"

lemma rbl_succ2_simps:
  "rbl_succ2 b [] = []"
  "rbl_succ2 b (x # xs) = (b ≠ x) # rbl_succ2 (x ∧ b) xs"
  by (simp_all add: rbl_succ2_def)

lemma twos_complement: "- x = word_succ (not x)"
  using arg_cong[OF word_add_not[where x=x], where f="λa. a - x + 1"]
  by (simp add: word_succ_p1 word_sp_01[unfolded word_succ_p1] del: word_add_not)

lemma rbl_word_neg: "rev (to_bl (- x)) = rbl_succ2 True (map Not (rev (to_bl x)))"
  for x :: ‹'a::len word›
  by (simp add: twos_complement word_succ_rbl[OF refl] bl_word_not rev_map rbl_succ2_def)

lemma rbl_word_cat:
  "rev (to_bl (word_cat x y :: 'a::len word)) =
    takefill False (LENGTH('a)) (rev (to_bl y) @ rev (to_bl x))"
  by (simp add: word_cat_bl word_rev_tf)

lemma rbl_word_slice:
  "rev (to_bl (slice n w :: 'a::len word)) =
    takefill False (LENGTH('a)) (drop n (rev (to_bl w)))"
  by (simp add: drop_rev slice_take word_rev_tf)

lemma rbl_word_ucast:
  "rev (to_bl (ucast x :: 'a::len word)) = takefill False (LENGTH('a)) (rev (to_bl x))"
  by (simp add: takefill_alt ucast_bl word_rev_tf)

lemma rbl_shiftl:
  "rev (to_bl (w << n)) = takefill False (size w) (replicate n False @ rev (to_bl w))"
  by (simp add: bl_shiftl takefill_alt word_size rev_drop)

lemma rbl_shiftr:
  "rev (to_bl (w >> n)) = takefill False (size w) (drop n (rev (to_bl w)))"
  by (simp add: shiftr_slice rbl_word_slice word_size)

definition "drop_nonempty v n xs = (if n < length xs then drop n xs else [last (v # xs)])"

lemma drop_nonempty_simps:
  "drop_nonempty v (Suc n) (x # xs) = drop_nonempty x n xs"
  "drop_nonempty v 0 (x # xs) = (x # xs)"
  "drop_nonempty v n [] = [v]"
  by (simp_all add: drop_nonempty_def)

definition "takefill_last x n xs = takefill (last (x # xs)) n xs"

lemma takefill_last_simps:
  "takefill_last z (Suc n) (x # xs) = x # takefill_last x n xs"
  "takefill_last z 0 xs = []"
  "takefill_last z n [] = replicate n z"
  by (simp_all add: takefill_last_def) (simp_all add: takefill_alt)

lemma rbl_sshiftr:
  "rev (to_bl (w >>> n)) = takefill_last False (size w) (drop_nonempty False n (rev (to_bl w)))"
proof (cases "n < size w")
  case True
  then show ?thesis
    by (simp add: bl_sshiftr takefill_last_def word_size takefill_alt
           rev_take last_rev drop_nonempty_def)
next
  case False
  then have 🍋: "(w >>> n) = of_bl (replicate (size w) (msb w))"
    by (intro word_eqI) (simp add: bit_simps word_size msb_nth)
  with False show ?thesis
   apply (simp add: word_size takefill_last_def takefill_alt
      last_rev word_msb_alt word_rev_tf drop_nonempty_def take_Cons')
    by (metis Suc_pred len_gt_0 replicate_Suc)
qed

lemma nth_word_of_int:
  "bit (word_of_int x :: 'a::len word) n = (n < LENGTH('a) ∧ bit x n)"
  by (simp add: bit_word_of_int_iff)

lemma nth_scast:
  "bit (scast (x :: 'a::len word) :: 'b::len word) n =
    (n < LENGTH('b) ∧
    (if n < LENGTH('a) - 1 then bit x n
     else bit x (LENGTH('a) - 1)))"
  by (simp add: bit_signed_iff)

lemma rbl_word_scast:
  "rev (to_bl (scast x :: 'a::len word)) = takefill_last False (LENGTH('a)) (rev (to_bl x))"
proof (rule nth_equalityI)
  show "length (rev (to_bl (scast x::'a word))) = length (takefill_last False (len_of (TYPE('a)::'a itself)) (rev (to_bl x)))"
    by (simp add: word_size takefill_last_def)
next
  fix i
  assume "i < length (rev (to_bl (scast x::'a word)))"
  then show "rev (to_bl (scast x::'a word)) ! i = takefill_last False (LENGTH('a)) (rev (to_bl x)) ! i"
  apply (cases "LENGTH('b)")
     apply (auto simp: nth_scast takefill_last_def nth_takefill word_size rev_nth
        to_bl_nth less_Suc_eq_le last_rev msb_nth simp flip: word_msb_alt)
  done
qed

definition rbl_mul :: "bool list ==> bool list ==> bool list"
  where "rbl_mul xs ys = foldr (λx sm. rbl_plus False (map ((∧) x) ys) (False # sm)) xs []"

lemma rbl_mul_simps:
  "rbl_mul (x # xs) ys = rbl_plus False (map ((∧) x) ys) (False # rbl_mul xs ys)"
  "rbl_mul [] ys = []"
  by (simp_all add: rbl_mul_def)

lemma takefill_le2: "length xs ≤ n ==> takefill x m (takefill x n xs) = takefill x m xs"
  by (simp add: takefill_alt replicate_add[symmetric])

lemma take_rbl_plus: "∀n b. take n (rbl_plus b xs ys) = rbl_plus b (take n xs) (take n ys)"
  unfolding rbl_plus_def take_zip[symmetric]
  by (rule list.induct) (auto simp: take_Cons' split_def)

lemma word_rbl_mul_induct:
  "length xs ≤ size y ==>
    rbl_mul xs (rev (to_bl y)) = take (length xs) (rev (to_bl (of_bl (rev xs) * y)))"
  for y :: "'a::len word"
proof (induct xs)
  case Nil
  show ?case by (simp add: rbl_mul_simps)
next
  case (Cons z zs)

  have rbl_word_plus': "to_bl (x + y) = rev (rbl_plus False (rev (to_bl x)) (rev (to_bl y)))"
    for x y :: "'a word"
    by (simp add: rbl_word_plus[symmetric])

  have mult_bit: "to_bl (of_bl [z] * y) = map ((∧) z) (to_bl y)"
    by (cases z) (simp cong: map_cong, simp add: map_replicate_const cong: map_cong)

  have shiftl: "of_bl xs * 2 * y = (of_bl xs * y) << 1" for xs
    by (simp add: push_bit_eq_mult shiftl_def)

  have zip_take_triv: "∧xs ys n. n = length ys ==> zip (take n xs) ys = zip xs ys"
    by (rule nth_equalityI) simp_all

  from Cons
  have "rbl_plus False (map ((∧) z) (rev (to_bl y)))
         (False # take (length zs) (rev (to_bl (of_bl (rev zs) * y)))) =
        rbl_plus False
         (take (Suc (length zs)) (map ((∧) z) (rev (to_bl y))))
         (take (Suc (length zs)) (rev (to_bl (of_bl (rev zs) * y * 2))))"
    unfolding word_size
    by (simp add: rbl_plus_def zip_take_triv mult.commute [of _ 2] to_bl_double_eq take_butlast
        flip: butlast_rev)
  with Cons show ?case
    by (simp add: trans [OF of_bl_append add.commute]
        rbl_mul_simps rbl_word_plus' distrib_right mult_bit shiftl rev_map take_rbl_plus)
qed

lemma rbl_word_mul: "rev (to_bl (x * y)) = rbl_mul (rev (to_bl x)) (rev (to_bl y))"
  for x :: "'a::len word"
  using word_rbl_mul_induct[where xs="rev (to_bl x)" and y=y] by (simp add: word_size)

text ‹Breaking up inequalities into bitlist properties.›

definition
  "rev_bl_order F xs ys =
     (length xs = length ys ∧
       ((xs = ys ∧ F)
          ∨ (∃n < length xs. drop (Suc n) xs = drop (Suc n) ys
                   ∧ ¬ xs ! n ∧ ys ! n)))"

lemma rev_bl_order_simps:
  "rev_bl_order F [] [] = F"
  "rev_bl_order F (x # xs) (y # ys) = rev_bl_order ((y ∧ ¬ x) ∨ ((y ∨ ¬ x) ∧ F)) xs ys"
   apply (simp_all add: rev_bl_order_def)
  using less_Suc_eq_0_disj by fastforce

lemma rev_bl_order_rev_simp:
  "length xs = length ys ==>
   rev_bl_order F (xs @ [x]) (ys @ [y]) = ((y ∧ ¬ x) ∨ ((y ∨ ¬ x) ∧ rev_bl_order F xs ys))"
  by (induct arbitrary: F rule: list_induct2) (auto simp: rev_bl_order_simps)

lemma rev_bl_order_bl_to_bin:
  "length xs = length ys ==>
    rev_bl_order True xs ys = (bl_to_bin (rev xs) ≤ bl_to_bin (rev ys)) ∧
    rev_bl_order False xs ys = (bl_to_bin (rev xs) < bl_to_bin (rev ys))"
proof (induct xs ys rule: list_induct2)
  case Nil
  then show ?case
    by (auto simp: rev_bl_order_simps(1))
next
  case (Cons x xs y ys)
  then show ?case
    apply (simp add: rev_bl_order_simps bl_to_bin_app_cat)
    apply (auto simp add: bl_to_bin_def add1_zle_eq concat_bit_Suc)
    done
qed

lemma word_le_rbl: "x ≤ y ⟷ rev_bl_order True (rev (to_bl x)) (rev (to_bl y))"
  for x y :: "'a::len word"
  by (simp add: rev_bl_order_bl_to_bin word_le_def)

lemma word_less_rbl: "x < y ⟷ rev_bl_order False (rev (to_bl x)) (rev (to_bl y))"
  for x y :: "'a::len word"
  by (simp add: word_less_alt rev_bl_order_bl_to_bin)

definition "map_last f xs = (if xs = [] then [] else butlast xs @ [f (last xs)])"

lemma map_last_simps:
  "map_last f [] = []"
  "map_last f [x] = [f x]"
  "map_last f (x # y # zs) = x # map_last f (y # zs)"
  by (simp_all add: map_last_def)

lemma word_sle_rbl:
  "x <=s y ⟷ rev_bl_order True (map_last Not (rev (to_bl x))) (map_last Not (rev (to_bl y)))"
proof -
  have "length (to_bl x) = length (to_bl y)"
    by auto
  with word_msb_alt[where w=x] word_msb_alt[where w=y]
  show ?thesis
    unfolding word_sle_msb_le word_le_rbl
    by (cases "to_bl x"; cases "to_bl y"; auto simp: map_last_def rev_bl_order_rev_simp)
qed

lemma word_sless_rbl:
  "x <s y ⟷ rev_bl_order False (map_last Not (rev (to_bl x))) (map_last Not (rev (to_bl y)))"
  by (metis (no_types, lifting) rev_bl_order_def signed.less_le signed.not_less word_sle_rbl)

text ‹Lemmas for unpacking term‹rev (to_bl n)› for numerals n and also
 for irreducible values and expressions.›

lemma rev_bin_to_bl_simps:
  "rev (bin_to_bl 0 x) = []"
  "rev (bin_to_bl (Suc n) (numeral (num.Bit0 nm))) = False # rev (bin_to_bl n (numeral nm))"
  "rev (bin_to_bl (Suc n) (numeral (num.Bit1 nm))) = True # rev (bin_to_bl n (numeral nm))"
  "rev (bin_to_bl (Suc n) (numeral (num.One))) = True # replicate n False"
  "rev (bin_to_bl (Suc n) (- numeral (num.Bit0 nm))) = False # rev (bin_to_bl n (- numeral nm))"
  "rev (bin_to_bl (Suc n) (- numeral (num.Bit1 nm))) =
    True # rev (bin_to_bl n (- numeral (nm + num.One)))"
  "rev (bin_to_bl (Suc n) (- numeral (num.One))) = True # replicate n True"
  "rev (bin_to_bl (Suc n) (- numeral (num.Bit0 nm + num.One))) =
    True # rev (bin_to_bl n (- numeral (nm + num.One)))"
  "rev (bin_to_bl (Suc n) (- numeral (num.Bit1 nm + num.One))) =
    False # rev (bin_to_bl n (- numeral (nm + num.One)))"
  "rev (bin_to_bl (Suc n) (- numeral (num.One + num.One))) =
    False # rev (bin_to_bl n (- numeral num.One))"
  by (simp_all add: bin_to_bl_aux_append bin_to_bl_zero_aux bin_to_bl_minus1_aux replicate_append_same)

lemma to_bl_upt: "to_bl x = rev (map (bit x) [0 ..< size x])"
  by (simp add: to_bl_eq_rev word_size rev_map)

lemma rev_to_bl_upt: "rev (to_bl x) = map (bit x) [0 ..< size x]"
  by (simp add: to_bl_upt)

lemma upt_eq_list_intros:
  "j ≤ i ==> [i ..< j] = []"
  "i = x ==> x < j ==> [x + 1 ..< j] = xs ==> [i ..< j] = (x # xs)"
  by (simp_all add: upt_eq_Cons_conv)


text ‹Tactic definition›

lemma if_bool_simps:
  "If p True y = (p ∨ y) ∧ If p False y = (¬ p ∧ y) ∧
    If p y True = (p ⟶ y) ∧ If p y False = (p ∧ y)"
  by auto

ML ‹
  Word_Bitwise_Tac =
 

  word_ss = simpset_of 🍋‹Word›;

  mk_nat_clist ns =
 fold_rev (Thm.mk_binop 🍋‹Cons :: nat ==> _›)
 ns 🍋‹[] :: nat list›;

  upt_conv ctxt ct =
 case Thm.term_of ct of
 🍋‹upt for n m› =>
 let
 val (i, j) = apply2 (snd o HOLogic.dest_number) (n, m);
 val ns = map (Numeral.mk_cnumber 🍋‹nat›) (i upto (j - 1))
 |> mk_nat_clist;
 val prop =
 Thm.mk_binop 🍋‹(=) :: nat list ==> _› ct ns
 |> Thm.apply 🍋‹Trueprop›;
 in
 try (fn () =>
 Goal.prove_internal ctxt [] prop
 (K (REPEAT_DETERM (resolve_tac ctxt @{thms upt_eq_list_intros} 1
 ORELSE simp_tac (put_simpset word_ss ctxt) 1))) |> mk_meta_eq) ()
 end
 | _ => NONE;

  expand_upt_simproc = 🍋‹passive expand_upt ("upt x y") = ‹K upt_conv››;

  word_len_simproc_fn ctxt ct =
 (case Thm.term_of ct of
 🍋‹len_of _ for t› =>
 (let
 val T = fastype_of t |> dest_Type |> snd |> the_single
 val n = Numeral.mk_cnumber 🍋‹nat› (Word_Lib.dest_binT T);
 val prop =
 Thm.mk_binop 🍋‹(=) :: nat ==> _› ct n
 |> Thm.apply 🍋‹Trueprop›;
 in
 Goal.prove_internal ctxt [] prop (K (simp_tac (put_simpset word_ss ctxt) 1))
 |> mk_meta_eq |> SOME
 end handle TERM _ => NONE | TYPE _ => NONE)
 | _ => NONE);

  word_len_simproc =
 🍋‹passive word_len ("len_of x") = ‹K word_len_simproc_fn››;

 
   or just 5 (discarding nat) when n_sucs = 0 *)

fun nat_get_Suc_simproc_fn n_sucs ctxt ct =
  let
    val (f, arg) = dest_comb (Thm.term_of ct);
    val n =
      (case arg of term‹nat› $ n => n | n => n)
      |> HOLogic.dest_number |> snd;
    val (i, j) = if n > n_sucs then (n_sucs, n - n_sucs) else (n, 0);
    val arg' = funpow i HOLogic.mk_Suc (HOLogic.mk_number 🍋‹nat› j);
    val _ = if arg = arg' then raise TERM ("", []) else ();
    fun propfn g =
      HOLogic.mk_eq (g arg, g arg')
      |> HOLogic.mk_Trueprop |> Thm.cterm_of ctxt;
    val eq1 =
      Goal.prove_internal ctxt [] (propfn I)
        (K (simp_tac (put_simpset word_ss ctxt) 1));
  in
    Goal.prove_internal ctxt [] (propfn (curry (op $) f))
      (K (simp_tac (put_simpset HOL_ss ctxt addsimps [eq1]) 1))
    |> mk_meta_eq |> SOME
  end handle TERM _ => NONE;

fun nat_get_Suc_simproc n_sucs ts =
  Simplifier.make_simproc 🍋
   {name = "nat_get_Suc",
    kind = Simproc,
    lhss = map (fn t => t $ term‹n :: nat›) ts,
    proc = K (nat_get_Suc_simproc_fn n_sucs),
    identifier = []};

val no_split_ss =
  simpset_of (put_simpset HOL_ss 🍋
    |> Splitter.del_split @{thm if_split});

val expand_word_eq_sss =
  (simpset_of (put_simpset HOL_basic_ss 🍋
    |> Simplifier.add_simps @{thms word_eq_rbl_eq word_le_rbl word_less_rbl word_sle_rbl word_sless_rbl}),
  map simpset_of [
    put_simpset no_split_ss 🍋
      |> Simplifier.add_simps
        @{thms rbl_word_plus rbl_word_and rbl_word_or rbl_word_not
                                rbl_word_neg bl_word_sub rbl_word_xor
                                rbl_word_cat rbl_word_slice rbl_word_scast
                                rbl_word_ucast rbl_shiftl rbl_shiftr rbl_sshiftr
                                rbl_word_if},
    put_simpset no_split_ss 🍋
      |> Simplifier.add_simps @{thms to_bl_numeral to_bl_neg_numeral to_bl_0 rbl_word_1},
    put_simpset no_split_ss 🍋
      |> Simplifier.add_simps @{thms rev_rev_ident rev_replicate rev_map to_bl_upt word_size}
      |> Simplifier.add_proc word_len_simproc,
    put_simpset no_split_ss 🍋
      |> Simplifier.add_simps
        @{thms list.simps split_conv replicate.simps list.map
                                zip_Cons_Cons zip_Nil drop_Suc_Cons drop_0 drop_Nil
                                foldr.simps list.map zip.simps(1) zip_Nil zip_Cons_Cons takefill_Suc_Cons
                                takefill_Suc_Nil takefill.Z rbl_succ2_simps
                                rbl_plus_simps rev_bin_to_bl_simps append.simps
                                takefill_last_simps drop_nonempty_simps
                                rev_bl_order_simps}
      |> fold Simplifier.add_proc
          [expand_upt_simproc,
           nat_get_Suc_simproc 4
             [term‹replicate›, term‹takefill x›,
              term‹drop›, term‹bin_to_bl›,
              term‹takefill_last x›,
              term‹drop_nonempty x›]],
    put_simpset no_split_ss 🍋 addsimps @{thms xor3_simps carry_simps if_bool_simps}
  ])

fun tac ctxt =
  let
    val (ss, sss) = expand_word_eq_sss;
  in
    foldr1 (op THEN_ALL_NEW)
      ((CHANGED o safe_full_simp_tac (put_simpset ss ctxt)) ::
        map (fn ss => safe_full_simp_tac (put_simpset ss ctxt)) sss)
  end;

end
›

method_setup word_bitwise =
  ‹Scan.succeed (fn ctxt => Method.SIMPLE_METHOD (Word_Bitwise_Tac.tac ctxt 1))›
  "decomposer for word equalities and inequalities into bit propositions on concrete word lengths"

end