diff --git a/thys/Word_Lib/Bitwise.thy b/thys/Word_Lib/Bitwise.thy
--- a/thys/Word_Lib/Bitwise.thy
+++ b/thys/Word_Lib/Bitwise.thy
@@ -1,507 +1,507 @@
 (*
  * 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 \<open>Helper constants used in defining addition\<close>
 
 definition xor3 :: "bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool"
   where "xor3 a b c = (a = (b = c))"
 
 definition carry :: "bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool"
   where "carry a b c = ((a \<and> (b \<or> c)) \<or> (b \<and> c))"
 
 lemma carry_simps:
   "carry True a b = (a \<or> b)"
   "carry a True b = (a \<or> b)"
   "carry a b True = (a \<or> b)"
   "carry False a b = (a \<and> b)"
   "carry a False b = (a \<and> b)"
   "carry a b False = (a \<and> 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 \<noteq> b)"
   "xor3 a False b = (a \<noteq> b)"
   "xor3 a b False = (a \<noteq> b)"
   by (simp_all add: xor3_def)
 
 text \<open>Breaking up word equalities into equalities on their
   bit lists. Equalities are generated and manipulated in the
   reverse order to \<^const>\<open>to_bl\<close>.\<close>
 
 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]"
   apply (rule_tac s="rev (to_bl (word_succ (0 :: 'a word)))" in trans)
    apply simp
   apply (simp only: rtb_rbl_ariths(1)[OF refl])
   apply simp
   apply (case_tac "LENGTH('a)")
    apply simp
   apply (simp add: takefill_alt)
   done
 
 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 \<Longrightarrow>
     \<forall>car. (if car then rbl_succ else id) (rbl_add xs ys) =
       (foldr (\<lambda>(x, y) res car. xor3 x y car # res (carry x y car)) (zip xs ys) (\<lambda>_. [])) car"
   apply (erule list_induct2)
    apply simp
   apply (simp only: rbl_add_carry_Cons)
   apply simp
   done
 
 lemma to_bl_plus_carry:
   "to_bl (x + y) =
     rev (foldr (\<lambda>(x, y) res car. xor3 x y car # res (carry x y car))
       (rev (zip (to_bl x) (to_bl y))) (\<lambda>_. []) False)"
   using rbl_add_suc_carry_fold[where xs="rev (to_bl x)" and ys="rev (to_bl y)"]
   apply (simp add: word_add_rbl[OF refl refl])
   apply (drule_tac x=False in spec)
   apply (simp add: zip_rev)
   done
 
 definition "rbl_plus cin xs ys =
   foldr (\<lambda>(x, y) res car. xor3 x y car # res (carry x y car)) (zip xs ys) (\<lambda>_. []) 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 \<noteq> x) # rbl_succ2 (x \<and> 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="\<lambda>a. 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 :: \<open>'a::len word\<close>
   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)))"
   apply (simp add: slice_take word_rev_tf rev_take)
   apply (cases "n < LENGTH('b)", simp_all)
   done
 
 lemma rbl_word_ucast:
   "rev (to_bl (ucast x :: 'a::len word)) = takefill False (LENGTH('a)) (rev (to_bl x))"
   apply (simp add: to_bl_ucast takefill_alt)
   apply (simp add: rev_drop)
   apply (cases "LENGTH('a) < LENGTH('b)")
    apply simp_all
   done
 
 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)))"
   apply (cases "n < size w")
    apply (simp add: bl_sshiftr takefill_last_def word_size
       takefill_alt rev_take last_rev
       drop_nonempty_def)
   apply (subgoal_tac "(w >>> n) = of_bl (replicate (size w) (msb w))")
    apply (simp add: word_size takefill_last_def takefill_alt
       last_rev word_msb_alt word_rev_tf
       drop_nonempty_def take_Cons')
    apply (case_tac "LENGTH('a)", simp_all)
   apply (rule word_eqI)
   apply (simp add: bit_simps word_size test_bit_of_bl
       msb_nth)
   done
 
 lemma nth_word_of_int:
   "bit (word_of_int x :: 'a::len word) n = (n < LENGTH('a) \<and> bit x n)"
   apply (simp add: test_bit_bl word_size to_bl_of_bin)
   apply (subst conj_cong[OF refl], erule bin_nth_bl)
   apply auto
   done
 
 lemma nth_scast:
   "bit (scast (x :: 'a::len word) :: 'b::len word) n =
     (n < LENGTH('b) \<and>
     (if n < LENGTH('a) - 1 then bit x n
      else bit x (LENGTH('a) - 1)))"
   apply transfer
   apply (auto simp add: bit_signed_take_bit_iff min_def)
   done
 
 lemma rbl_word_scast:
   "rev (to_bl (scast x :: 'a::len word)) = takefill_last False (LENGTH('a)) (rev (to_bl x))"
   apply (rule nth_equalityI)
    apply (simp add: word_size takefill_last_def)
   apply (clarsimp simp: nth_scast takefill_last_def
       nth_takefill word_size rev_nth to_bl_nth)
   apply (cases "LENGTH('b)")
    apply simp
   apply (clarsimp simp: less_Suc_eq_le linorder_not_less
       last_rev word_msb_alt[symmetric]
       msb_nth)
   done
 
 definition rbl_mul :: "bool list \<Rightarrow> bool list \<Rightarrow> bool list"
   where "rbl_mul xs ys = foldr (\<lambda>x sm. rbl_plus False (map ((\<and>) x) ys) (False # sm)) xs []"
 
 lemma rbl_mul_simps:
   "rbl_mul (x # xs) ys = rbl_plus False (map ((\<and>) x) ys) (False # rbl_mul xs ys)"
   "rbl_mul [] ys = []"
   by (simp_all add: rbl_mul_def)
 
 lemma takefill_le2: "length xs \<le> n \<Longrightarrow> takefill x m (takefill x n xs) = takefill x m xs"
   by (simp add: takefill_alt replicate_add[symmetric])
 
 lemma take_rbl_plus: "\<forall>n b. take n (rbl_plus b xs ys) = rbl_plus b (take n xs) (take n ys)"
   apply (simp add: rbl_plus_def take_zip[symmetric])
   apply (rule_tac list="zip xs ys" in list.induct)
    apply simp
   apply (clarsimp simp: split_def)
   apply (case_tac n, simp_all)
   done
 
 lemma word_rbl_mul_induct:
   "length xs \<le> size y \<Longrightarrow>
     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 ((\<and>) 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: "\<And>xs ys n. n = length ys \<Longrightarrow> zip (take n xs) ys = zip xs ys"
     by (rule nth_equalityI) simp_all
 
   from Cons show ?case
     apply (simp add: trans [OF of_bl_append add.commute]
         rbl_mul_simps rbl_word_plus' distrib_right mult_bit shiftl rbl_shiftl)
     apply (simp add: takefill_alt word_size rev_map take_rbl_plus min_def)
     apply (simp add: rbl_plus_def)
     apply (simp add: zip_take_triv)
     apply (simp only: mult.commute [of _ 2] to_bl_double_eq)
     apply (simp flip: butlast_rev add: take_butlast)
     done
 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 \<open>Breaking up inequalities into bitlist properties.\<close>
 
 definition
   "rev_bl_order F xs ys =
      (length xs = length ys \<and>
        ((xs = ys \<and> F)
           \<or> (\<exists>n < length xs. drop (Suc n) xs = drop (Suc n) ys
                    \<and> \<not> xs ! n \<and> ys ! n)))"
 
 lemma rev_bl_order_simps:
   "rev_bl_order F [] [] = F"
   "rev_bl_order F (x # xs) (y # ys) = rev_bl_order ((y \<and> \<not> x) \<or> ((y \<or> \<not> x) \<and> F)) xs ys"
    apply (simp_all add: rev_bl_order_def)
   apply (rule conj_cong[OF refl])
   apply (cases "xs = ys")
    apply (simp add: nth_Cons')
    apply blast
   apply (simp add: nth_Cons')
   apply safe
    apply (rule_tac x="n - 1" in exI)
    apply simp
   apply (rule_tac x="Suc n" in exI)
   apply simp
   done
 
 lemma rev_bl_order_rev_simp:
   "length xs = length ys \<Longrightarrow>
    rev_bl_order F (xs @ [x]) (ys @ [y]) = ((y \<and> \<not> x) \<or> ((y \<or> \<not> x) \<and> 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 \<Longrightarrow>
     rev_bl_order True xs ys = (bl_to_bin (rev xs) \<le> bl_to_bin (rev ys)) \<and>
     rev_bl_order False xs ys = (bl_to_bin (rev xs) < bl_to_bin (rev ys))"
   apply (induct xs ys rule: list_induct2)
    apply (simp_all add: rev_bl_order_simps bl_to_bin_app_cat concat_bit_Suc)
   apply (auto simp add: bl_to_bin_def add1_zle_eq)
   done
 
 lemma word_le_rbl: "x \<le> y \<longleftrightarrow> 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 \<longleftrightarrow> 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 \<longleftrightarrow> rev_bl_order True (map_last Not (rev (to_bl x))) (map_last Not (rev (to_bl y)))"
   using word_msb_alt[where w=x] word_msb_alt[where w=y]
   apply (simp add: word_sle_msb_le word_le_rbl)
   apply (subgoal_tac "length (to_bl x) = length (to_bl y)")
    apply (cases "to_bl x", simp)
    apply (cases "to_bl y", simp)
    apply (clarsimp simp: map_last_def rev_bl_order_rev_simp)
    apply auto
   done
 
 lemma word_sless_rbl:
   "x <s y \<longleftrightarrow> rev_bl_order False (map_last Not (rev (to_bl x))) (map_last Not (rev (to_bl y)))"
   using word_msb_alt[where w=x] word_msb_alt[where w=y]
   apply (simp add: word_sless_msb_less word_less_rbl)
   apply (subgoal_tac "length (to_bl x) = length (to_bl y)")
    apply (cases "to_bl x", simp)
    apply (cases "to_bl y", simp)
    apply (clarsimp simp: map_last_def rev_bl_order_rev_simp)
    apply auto
   done
 
 text \<open>Lemmas for unpacking \<^term>\<open>rev (to_bl n)\<close> for numerals n and also
   for irreducible values and expressions.\<close>
 
 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 \<le> i \<Longrightarrow> [i ..< j] = []"
   "i = x \<Longrightarrow> x < j \<Longrightarrow> [x + 1 ..< j] = xs \<Longrightarrow> [i ..< j] = (x # xs)"
   by (simp_all add: upt_eq_Cons_conv)
 
 
 text \<open>Tactic definition\<close>
 
 lemma if_bool_simps:
   "If p True y = (p \<or> y) \<and> If p False y = (\<not> p \<and> y) \<and>
     If p y True = (p \<longrightarrow> y) \<and> If p y False = (p \<and> y)"
   by auto
 
 ML \<open>
 structure Word_Bitwise_Tac =
 struct
 
 val word_ss = simpset_of \<^theory_context>\<open>Word\<close>;
 
 fun mk_nat_clist ns =
   fold_rev (Thm.mk_binop \<^cterm>\<open>Cons :: nat \<Rightarrow> _\<close>)
     ns \<^cterm>\<open>[] :: nat list\<close>;
 
 fun upt_conv ctxt ct =
   case Thm.term_of ct of
     \<^Const_>\<open>upt for n m\<close> =>
       let
         val (i, j) = apply2 (snd o HOLogic.dest_number) (n, m);
         val ns = map (Numeral.mk_cnumber \<^ctyp>\<open>nat\<close>) (i upto (j - 1))
           |> mk_nat_clist;
         val prop =
           Thm.mk_binop \<^cterm>\<open>(=) :: nat list \<Rightarrow> _\<close> ct ns
           |> Thm.apply \<^cterm>\<open>Trueprop\<close>;
       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;
 
 val expand_upt_simproc =
   Simplifier.make_simproc \<^context> "expand_upt"
    {lhss = [\<^term>\<open>upt x y\<close>], proc = K upt_conv};
 
 fun word_len_simproc_fn ctxt ct =
   (case Thm.term_of ct of
     \<^Const_>\<open>len_of _ for t\<close> =>
      (let
         val T = fastype_of t |> dest_Type |> snd |> the_single
         val n = Numeral.mk_cnumber \<^ctyp>\<open>nat\<close> (Word_Lib.dest_binT T);
         val prop =
           Thm.mk_binop \<^cterm>\<open>(=) :: nat \<Rightarrow> _\<close> ct n
           |> Thm.apply \<^cterm>\<open>Trueprop\<close>;
       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);
 
 val word_len_simproc =
   Simplifier.make_simproc \<^context> "word_len"
    {lhss = [\<^term>\<open>len_of x\<close>], proc = K word_len_simproc_fn};
 
 (* convert 5 or nat 5 to Suc 4 when n_sucs = 1, Suc (Suc 4) when n_sucs = 2,
    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>\<open>nat\<close> $ 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 \<^typ>\<open>nat\<close> 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 \<^context> "nat_get_Suc"
    {lhss = map (fn t => t $ \<^term>\<open>n :: nat\<close>) ts,
     proc = K (nat_get_Suc_simproc_fn n_sucs)};
 
 val no_split_ss =
   simpset_of (put_simpset HOL_ss \<^context>
     |> Splitter.del_split @{thm if_split});
 
 val expand_word_eq_sss =
   (simpset_of (put_simpset HOL_basic_ss \<^context> addsimps
        @{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 \<^context> addsimps
     @{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 \<^context> addsimps
     @{thms to_bl_numeral to_bl_neg_numeral to_bl_0 rbl_word_1},
    put_simpset no_split_ss \<^context> addsimps
     @{thms rev_rev_ident rev_replicate rev_map to_bl_upt word_size}
           addsimprocs [word_len_simproc],
    put_simpset no_split_ss \<^context> addsimps
     @{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}
           addsimprocs [expand_upt_simproc,
                        nat_get_Suc_simproc 4
                          [\<^term>\<open>replicate\<close>, \<^term>\<open>takefill x\<close>,
                           \<^term>\<open>drop\<close>, \<^term>\<open>bin_to_bl\<close>,
                           \<^term>\<open>takefill_last x\<close>,
                           \<^term>\<open>drop_nonempty x\<close>]],
     put_simpset no_split_ss \<^context> 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
 \<close>
 
 method_setup word_bitwise =
   \<open>Scan.succeed (fn ctxt => Method.SIMPLE_METHOD (Word_Bitwise_Tac.tac ctxt 1))\<close>
-  "decomposer for word equalities and inequalities into bit propositions"
+  "decomposer for word equalities and inequalities into bit propositions on concrete word lengths"
 
 end
diff --git a/thys/Word_Lib/Bitwise_Signed.thy b/thys/Word_Lib/Bitwise_Signed.thy
--- a/thys/Word_Lib/Bitwise_Signed.thy
+++ b/thys/Word_Lib/Bitwise_Signed.thy
@@ -1,30 +1,30 @@
 (*
  * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
  * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "Bitwise tactic for Signed Words"
 
 theory Bitwise_Signed
 imports
   "HOL-Library.Word"
   Bitwise
   Signed_Words
 begin
 
 ML \<open>fun bw_tac_signed ctxt = let
   val (ss, sss) = Word_Bitwise_Tac.expand_word_eq_sss
   val sss = nth_map 2 (fn ss => put_simpset ss ctxt addsimps @{thms len_signed} |> simpset_of) 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;
 \<close>
 
 method_setup word_bitwise_signed =
   \<open>Scan.succeed (fn ctxt => Method.SIMPLE_METHOD (bw_tac_signed ctxt 1))\<close>
-  "decomposer for word equalities and inequalities into bit propositions"
+  "decomposer for word equalities and inequalities into bit propositions on concrete word lengths"
 
 end
diff --git a/thys/Word_Lib/Boolean_Inequalities.thy b/thys/Word_Lib/Boolean_Inequalities.thy
new file mode 100644
--- /dev/null
+++ b/thys/Word_Lib/Boolean_Inequalities.thy
@@ -0,0 +1,139 @@
+(*
+ * Copyright 2023, Proofcraft Pty Ltd
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
+
+theory Boolean_Inequalities
+  imports Word_EqI
+begin
+
+section \<open>All inequalities between binary Boolean operations on @{typ "'a word"}\<close>
+
+text \<open>
+  Enumerates all binary functions resulting from Boolean operations on @{typ "'a word"}
+  and derives all inequalities of the form @{term "f x y \<le> g x y"} between them.
+  We leave out the trivial @{term "0 \<le> g x y"}, @{term "f x y \<le> -1"}, and
+  @{term "f x y \<le> f x y"}, because these are already readily available to the simplifier and
+  other methods.
+
+  This leaves 36 inequalities. Some of these are subsumed by each other, but we generate
+  the full list to avoid too much manual processing.
+
+  All inequalities produced here are in simp normal form.\<close>
+
+context
+  includes bit_operations_syntax
+begin
+
+(* The following are all 16 binary Boolean operations (on bool):
+   all_bool_funs \<equiv> [
+     \<lambda>x y. False,
+     \<lambda>x y. x \<and> y,
+     \<lambda>x y. x \<and> \<not>y,
+     \<lambda>x y. x,
+     \<lambda>x y. \<not>x \<and> y,
+     \<lambda>x y. y,
+     \<lambda>x y. x \<noteq> y,
+     \<lambda>x y. x \<or> y,
+     \<lambda>x y. \<not>(x \<or> y),
+     \<lambda>x y. x = y,
+     \<lambda>x y. \<not>y,
+     \<lambda>x y. x \<or> \<not>y,
+     \<lambda>x y. \<not>x,
+     \<lambda>x y. \<not>x \<or> y,
+     \<lambda>x y. \<not>(x \<and> y),
+     \<lambda>x y. True
+   ]
+
+  We can define the same for word operations:
+*)
+definition all_bool_word_funs :: "('a::len word \<Rightarrow> 'a word \<Rightarrow> 'a word) list" where
+  "all_bool_word_funs \<equiv> [
+     \<lambda>x y. 0,
+     \<lambda>x y. x AND y,
+     \<lambda>x y. x AND NOT y,
+     \<lambda>x y. x,
+     \<lambda>x y. NOT x AND y,
+     \<lambda>x y. y,
+     \<lambda>x y. x XOR y,
+     \<lambda>x y. x OR y,
+     \<lambda>x y. NOT (x OR y),
+     \<lambda>x y. NOT (x XOR y),
+     \<lambda>x y. NOT y,
+     \<lambda>x y. x OR NOT y,
+     \<lambda>x y. NOT x,
+     \<lambda>x y. NOT x OR y,
+     \<lambda>x y. NOT (x AND y),
+     \<lambda>x y. -1
+   ]"
+
+
+text \<open>
+  The inequalities on @{typ "'a word"} follow directly from implications on propositional
+  Boolean logic, which @{method simp} can solve automatically. This means, we can simply
+  enumerate all combinations, reduce from @{typ "'a word"} to @{typ bool}, and attempt to
+  solve by @{method simp} to get the complete list.\<close>
+local_setup \<open>
+let
+  (* derived from Numeral.mk_num, but returns a term, not syntax. *)
+  fun mk_num n =
+    if n > 0 then
+      (case Integer.quot_rem n 2 of
+        (0, 1) => \<^const>\<open>Num.One\<close>
+      | (n, 0) => \<^const>\<open>Num.Bit0\<close> $ mk_num n
+      | (n, 1) => \<^const>\<open>Num.Bit1\<close> $ mk_num n)
+    else raise Match
+
+  (* derived from Numeral.mk_number, but returns a term, not syntax. *)
+  fun mk_number n =
+    if n = 0 then \<^term>\<open>0::nat\<close>
+    else if n = 1 then \<^term>\<open>1::nat\<close>
+    else \<^term>\<open>numeral::num \<Rightarrow> nat\<close> $ mk_num n;
+
+  (* generic form of the goal statement *)
+  val goal = @{term "\<lambda>n m. (all_bool_word_funs!n) x y \<le> (all_bool_word_funs!m) x y"}
+  (* instance of the goal statement for a pair (i,j) of Boolean functions *)
+  fun stmt (i,j) = HOLogic.Trueprop $ (goal $ mk_number i $ mk_number j)
+
+  (* attempt to prove an inequality between functions i and j *)
+  fun le_thm ctxt (i,j) = Goal.prove ctxt ["x", "y"] [] (stmt (i,j)) (fn _ =>
+    (asm_full_simp_tac (ctxt addsimps [@{thm all_bool_word_funs_def}])
+     THEN_ALL_NEW resolve_tac ctxt @{thms word_leI}
+     THEN_ALL_NEW asm_full_simp_tac (ctxt addsimps @{thms word_eqI_simps bit_simps})) 1)
+
+  (* generate all combinations for (i,j), collect successful inequality theorems,
+     unfold all_bool_word_funs, and put into simp normal form. We leave out 0 (bottom)
+     and 15 (top), as well as reflexive thms to remove trivial lemmas from the list.*)
+  fun thms ctxt =
+    map_product (fn x => fn y => (x,y)) (1 upto 14) (1 upto 14)
+    |> filter (fn (x,y) => x <> y)
+    |> map_filter (try (le_thm ctxt))
+    |> map (Simplifier.simplify ctxt o Local_Defs.unfold ctxt @{thms all_bool_word_funs_def})
+in
+  fn ctxt =>
+    Local_Theory.notes [((Binding.name "word_bool_le_funs", []), [(thms ctxt, [])])] ctxt |> #2
+end
+\<close>
+
+(* Sanity checks: *)
+lemma
+  "x AND y \<le> x" for x :: "'a::len word"
+  by (rule word_bool_le_funs(1))
+
+lemma
+  "NOT x \<le> NOT x OR NOT y" for x :: "'a::len word"
+  by (rule word_bool_le_funs(36))
+
+lemma
+  "x XOR y \<le> NOT x OR NOT y" for x :: "'a::len word"
+  by (rule word_bool_le_funs)
+
+(* Example use when the goal is not in simp normal form: *)
+lemma word_xor_le_nand:
+  "x XOR y \<le> NOT (x AND y)" for x :: "'a::len word"
+  by (simp add: word_bool_le_funs)
+
+end
+
+end
\ No newline at end of file
diff --git a/thys/Word_Lib/Many_More.thy b/thys/Word_Lib/Many_More.thy
--- a/thys/Word_Lib/Many_More.thy
+++ b/thys/Word_Lib/Many_More.thy
@@ -1,693 +1,693 @@
 (*
  * Copyright Data61, CSIRO (ABN 41 687 119 230)
  *
  * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 theory Many_More
   imports
     Main
     "HOL-Library.Word"
     More_Word
     Even_More_List
 begin
 
 lemma nat_less_mult_monoish: "\<lbrakk> a < b; c < (d :: nat) \<rbrakk> \<Longrightarrow> (a + 1) * (c + 1) <= b * d"
   apply (drule Suc_leI)+
   apply (drule(1) mult_le_mono)
   apply simp
   done
 
 context
   includes bit_operations_syntax
 begin
 
 lemma if_and_helper:
   "(If x v v') AND v'' = If x (v AND v'') (v' AND v'')"
   by (rule if_distrib)
 
 end
 
 lemma eq_eqI:
   "a = b \<Longrightarrow> (a = x) = (b = x)"
   by simp
 
 lemma map2_Cons_2_3:
   "(map2 f xs (y # ys) = (z # zs)) = (\<exists>x xs'. xs = x # xs' \<and> f x y = z \<and> map2 f xs' ys = zs)"
   by (case_tac xs, simp_all)
 
 lemma map2_xor_replicate_False:
   "map2 (\<lambda>x y. x \<longleftrightarrow> \<not> y) xs (replicate n False) = take n xs"
   apply (induct xs arbitrary: n, simp)
   apply (case_tac n; simp)
   done
 
 lemma plus_Collect_helper:
   "(+) x ` {xa. P (xa :: 'a :: len word)} = {xa. P (xa - x)}"
   by (fastforce simp add: image_def)
 
 lemma plus_Collect_helper2:
   "(+) (- x) ` {xa. P (xa :: 'a :: len word)} = {xa. P (x + xa)}"
   using plus_Collect_helper [of "- x" P] by (simp add: ac_simps)
 
 lemma range_subset_eq2:
   "{a :: 'a :: len word .. b} \<noteq> {} \<Longrightarrow> ({a .. b} \<subseteq> {c .. d}) = (c \<le> a \<and> b \<le> d)"
   by simp
 
 lemma nat_mod_power_lem:
   fixes a :: nat
   shows "1 < a \<Longrightarrow> a ^ n mod a ^ m = (if m \<le> n then 0 else a ^ n)"
   apply (clarsimp)
   apply (clarsimp simp add: le_iff_add power_add)
   done
 
 lemma i_hate_words_helper:
   "i \<le> (j - k :: nat) \<Longrightarrow> i \<le> j"
   by simp
 
 lemma i_hate_words:
   "unat (a :: 'a word) \<le> unat (b :: 'a :: len word) - Suc 0
     \<Longrightarrow> a \<noteq> -1"
   apply (frule i_hate_words_helper)
   apply (subst(asm) word_le_nat_alt[symmetric])
   apply (clarsimp simp only: word_minus_one_le)
   apply (simp only: linorder_not_less[symmetric])
   apply (erule notE)
   apply (rule diff_Suc_less)
   apply (subst neq0_conv[symmetric])
   apply (subst unat_eq_0)
   apply (rule notI, drule arg_cong[where f="(+) 1"])
   apply simp
   done
 
 lemma If_eq_obvious:
   "x \<noteq> z \<Longrightarrow> ((if P then x else y) = z) = (\<not> P \<and> y = z)"
   by simp
 
 lemma Some_to_the:
   "v = Some x \<Longrightarrow> x = the v"
   by simp
 
 lemma dom_if_Some:
   "dom (\<lambda>x. if P x then Some (f x) else g x) = {x. P x} \<union> dom g"
   by fastforce
 
 lemma dom_insert_absorb:
   "x \<in> dom f \<Longrightarrow> insert x (dom f) = dom f"
   by (fact insert_absorb)
 
 lemma emptyE2:
   "\<lbrakk> S = {}; x \<in> S \<rbrakk> \<Longrightarrow> P"
   by simp
 
 lemma ptr_add_image_multI:
   "\<lbrakk> \<And>x y. (x * val = y * val') = (x * val'' = y); x * val'' \<in> S \<rbrakk> \<Longrightarrow>
      ptr_add ptr (x * val) \<in> (\<lambda>p. ptr_add ptr (p * val')) ` S"
   by (auto simp add: image_iff) metis
 
 lemmas map_prod_split_imageI'
   = map_prod_imageI[where f="case_prod f" and g="case_prod g"
                     and a="(a, b)" and b="(c, d)" for a b c d f g]
 lemmas map_prod_split_imageI = map_prod_split_imageI'[simplified]
 
 lemma dom_if:
   "dom (\<lambda>a. if a \<in> addrs then Some (f a) else g a)  = addrs \<union> dom g"
   by (auto simp: dom_def split: if_split)
 
 lemmas arg_cong_Not = arg_cong [where f=Not]
 
 lemma drop_append_miracle:
   "n = length xs \<Longrightarrow> drop n (xs @ ys) = ys"
   by simp
 
 lemma foldr_does_nothing_to_xf:
   "\<lbrakk> \<And>x s. x \<in> set xs \<Longrightarrow> xf (f x s) = xf s \<rbrakk> \<Longrightarrow> xf (foldr f xs s) = xf s"
   by (induct xs, simp_all)
 
 lemma mod_mod_power_int:
   fixes k :: int
   shows "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ (min m n)"
   by (fact mod_exp_eq)
 
 lemma le_step_down_nat:"\<lbrakk>(i::nat) \<le> n; i = n \<longrightarrow> P; i \<le> n - 1 \<longrightarrow> P\<rbrakk> \<Longrightarrow> P"
   by arith
 
 lemma le_step_down_int:"\<lbrakk>(i::int) \<le> n; i = n \<longrightarrow> P; i \<le> n - 1 \<longrightarrow> P\<rbrakk> \<Longrightarrow> P"
   by arith
 
 lemma replicate_numeral [simp]: "replicate (numeral k) x = x # replicate (pred_numeral k) x"
   by (simp add: numeral_eq_Suc)
 
 lemma list_exhaust_size_gt0:
   assumes "\<And>a list. y = a # list \<Longrightarrow> P"
   shows "0 < length y \<Longrightarrow> P"
   apply (cases y)
    apply simp
   apply (rule assms)
   apply fastforce
   done
 
 lemma list_exhaust_size_eq0:
   assumes "y = [] \<Longrightarrow> P"
   shows "length y = 0 \<Longrightarrow> P"
   apply (cases y)
    apply (rule assms)
    apply simp
   apply simp
   done
 
 lemma size_Cons_lem_eq: "y = xa # list \<Longrightarrow> size y = Suc k \<Longrightarrow> size list = k"
   by auto
 
 lemma takeWhile_take_has_property:
   "n \<le> length (takeWhile P xs) \<Longrightarrow> \<forall>x \<in> set (take n xs). P x"
   by (induct xs arbitrary: n; simp split: if_split_asm) (case_tac n, simp_all)
 
 lemma takeWhile_take_has_property_nth:
   "\<lbrakk> n < length (takeWhile P xs) \<rbrakk> \<Longrightarrow> P (xs ! n)"
   by (induct xs arbitrary: n; simp split: if_split_asm) (case_tac n, simp_all)
 
-lemma takeWhile_replicate:
-  "takeWhile f (replicate len x) = (if f x then replicate len x else [])"
-  by (induct_tac len) auto
-
 lemma takeWhile_replicate_empty:
   "\<not> f x \<Longrightarrow> takeWhile f (replicate len x) = []"
-  by (simp add: takeWhile_replicate)
+  by simp
 
 lemma takeWhile_replicate_id:
   "f x \<Longrightarrow> takeWhile f (replicate len x) = replicate len x"
-  by (simp add: takeWhile_replicate)
+  by simp
+
+lemma takeWhile_all:
+  "length (takeWhile P xs) = length xs \<Longrightarrow> \<forall>x \<in> set xs. P x"
+  by (induct xs) (auto split: if_split_asm)
 
 lemma nth_rev: "n < length xs \<Longrightarrow> rev xs ! n = xs ! (length xs - 1 - n)"
   using rev_nth by simp
 
 lemma nth_rev_alt: "n < length ys \<Longrightarrow> ys ! n = rev ys ! (length ys - Suc n)"
   by (simp add: nth_rev)
 
 lemma hd_butlast: "length xs > 1 \<Longrightarrow> hd (butlast xs) = hd xs"
   by (cases xs) auto
 
 lemma split_upt_on_n:
   "n < m \<Longrightarrow> [0 ..< m] = [0 ..< n] @ [n] @ [Suc n ..< m]"
   by (metis append_Cons append_Nil less_Suc_eq_le less_imp_le_nat upt_add_eq_append'
             upt_rec zero_less_Suc)
 
 lemma drop_eq_mono:
   assumes le: "m \<le> n"
   assumes drop: "drop m xs = drop m ys"
   shows "drop n xs = drop n ys"
 proof -
   have ex: "\<exists>p. n = p + m" by (rule exI[of _ "n - m"]) (simp add: le)
   then obtain p where p: "n = p + m" by blast
   show ?thesis unfolding p drop_drop[symmetric] drop by simp
 qed
 
 lemma drop_Suc_nth:
   "n < length xs \<Longrightarrow> drop n xs = xs!n # drop (Suc n) xs"
   by (simp add: Cons_nth_drop_Suc)
 
 lemma and_len: "xs = ys \<Longrightarrow> xs = ys \<and> length xs = length ys"
   by auto
 
 lemma tl_if: "tl (if p then xs else ys) = (if p then tl xs else tl ys)"
   by auto
 
 lemma hd_if: "hd (if p then xs else ys) = (if p then hd xs else hd ys)"
   by auto
 
 lemma if_single: "(if xc then [xab] else [an]) = [if xc then xab else an]"
   by auto
 
 \<comment> \<open>note -- \<open>if_Cons\<close> can cause blowup in the size, if \<open>p\<close> is complex, so make a simproc\<close>
 lemma if_Cons: "(if p then x # xs else y # ys) = If p x y # If p xs ys"
   by auto
 
 lemma list_of_false:
   "True \<notin> set xs \<Longrightarrow> xs = replicate (length xs) False"
   by (induct xs, simp_all)
 
 lemma list_all2_induct [consumes 1, case_names Nil Cons]:
   assumes lall: "list_all2 Q xs ys"
   and     nilr: "P [] []"
   and    consr: "\<And>x xs y ys. \<lbrakk>list_all2 Q xs ys; Q x y; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)"
   shows  "P xs ys"
   using lall
 proof (induct rule: list_induct2 [OF list_all2_lengthD [OF lall]])
   case 1 then show ?case by auto fact+
 next
   case (2 x xs y ys)
 
   show ?case
   proof (rule consr)
     from "2.prems" show "list_all2 Q xs ys" and "Q x y" by simp_all
     then show "P xs ys" by (intro "2.hyps")
   qed
 qed
 
 lemma replicate_minus:
   "k < n \<Longrightarrow> replicate n False = replicate (n - k) False @ replicate k False"
   by (subst replicate_add [symmetric]) simp
 
 lemma cart_singleton_empty:
   "(S \<times> {e} = {}) = (S = {})"
   by blast
 
 lemma MinI:
   assumes fa: "finite A"
   and     ne: "A \<noteq> {}"
   and     xv: "m \<in> A"
   and    min: "\<forall>y \<in> A. m \<le> y"
   shows "Min A = m" using fa ne xv min
 proof (induct A arbitrary: m rule: finite_ne_induct)
   case singleton then show ?case by simp
 next
   case (insert y F)
 
   from insert.prems have yx: "m \<le> y" and fx: "\<forall>y \<in> F. m \<le> y" by auto
   have "m \<in> insert y F" by fact
   then show ?case
   proof
     assume mv: "m = y"
 
     have mlt: "m \<le> Min F"
       by (rule iffD2 [OF Min_ge_iff [OF insert.hyps(1) insert.hyps(2)] fx])
 
     show ?case
       apply (subst Min_insert [OF insert.hyps(1) insert.hyps(2)])
       apply (subst mv [symmetric])
       apply (auto simp: min_def mlt)
       done
   next
     assume "m \<in> F"
     then have mf: "Min F = m"
       by (rule insert.hyps(4) [OF _ fx])
 
     show ?case
       apply (subst Min_insert [OF insert.hyps(1) insert.hyps(2)])
       apply (subst mf)
       apply (rule iffD2 [OF _ yx])
       apply (auto simp: min_def)
       done
   qed
 qed
 
 lemma power_numeral: "a ^ numeral k = a * a ^ (pred_numeral k)"
   by (simp add: numeral_eq_Suc)
 
 lemma funpow_numeral [simp]: "f ^^ numeral k = f \<circ> f ^^ (pred_numeral k)"
   by (simp add: numeral_eq_Suc)
 
 lemma funpow_minus_simp: "0 < n \<Longrightarrow> f ^^ n = f \<circ> f ^^ (n - 1)"
   by (auto dest: gr0_implies_Suc)
 
 lemma rco_alt: "(f \<circ> g) ^^ n \<circ> f = f \<circ> (g \<circ> f) ^^ n"
   apply (rule ext)
   apply (induct n)
    apply (simp_all add: o_def)
   done
 
 lemma union_sub:
   "\<lbrakk>B \<subseteq> A; C \<subseteq> B\<rbrakk> \<Longrightarrow> (A - B) \<union> (B - C) = (A - C)"
   by fastforce
 
 lemma insert_sub:
   "x \<in> xs \<Longrightarrow> (insert x (xs - ys)) = (xs - (ys - {x}))"
   by blast
 
 lemma ran_upd:
   "\<lbrakk> inj_on f (dom f); f y = Some z \<rbrakk> \<Longrightarrow> ran (\<lambda>x. if x = y then None else f x) = ran f - {z}"
   unfolding ran_def
   apply (rule set_eqI)
   apply simp
   by (metis domI inj_on_eq_iff option.sel)
 
 lemma if_apply_def2:
   "(if P then F else G) = (\<lambda>x. (P \<longrightarrow> F x) \<and> (\<not> P \<longrightarrow> G x))"
   by simp
 
 lemma case_bool_If:
   "case_bool P Q b = (if b then P else Q)"
   by simp
 
 lemma if_f:
   "(if a then f b else f c) = f (if a then b else c)"
   by simp
 
 lemma size_if: "size (if p then xs else ys) = (if p then size xs else size ys)"
   by (fact if_distrib)
 
 lemma if_Not_x: "(if p then \<not> x else x) = (p = (\<not> x))"
   by auto
 
 lemma if_x_Not: "(if p then x else \<not> x) = (p = x)"
   by auto
 
 lemma if_same_and: "(If p x y \<and> If p u v) = (if p then x \<and> u else y \<and> v)"
   by auto
 
 lemma if_same_eq: "(If p x y  = (If p u v)) = (if p then x = u else y = v)"
   by auto
 
 lemma if_same_eq_not: "(If p x y = (\<not> If p u v)) = (if p then x = (\<not> u) else y = (\<not> v))"
   by auto
 
 lemma the_elemI: "y = {x} \<Longrightarrow> the_elem y = x"
   by simp
 
 lemma nonemptyE: "S \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> R) \<Longrightarrow> R"
   by auto
 
 lemmas xtr1 = xtrans(1)
 lemmas xtr2 = xtrans(2)
 lemmas xtr3 = xtrans(3)
 lemmas xtr4 = xtrans(4)
 lemmas xtr5 = xtrans(5)
 lemmas xtr6 = xtrans(6)
 lemmas xtr7 = xtrans(7)
 lemmas xtr8 = xtrans(8)
 
 lemmas if_fun_split = if_apply_def2
 
 lemma not_empty_eq:
   "(S \<noteq> {}) = (\<exists>x. x \<in> S)"
   by auto
 
 lemma range_subset_lower:
   fixes c :: "'a ::linorder"
   shows "\<lbrakk> {a..b} \<subseteq> {c..d}; x \<in> {a..b} \<rbrakk> \<Longrightarrow> c \<le> a"
   apply (frule (1) subsetD)
   apply (rule classical)
   apply clarsimp
   done
 
 lemma range_subset_upper:
   fixes c :: "'a ::linorder"
   shows "\<lbrakk> {a..b} \<subseteq> {c..d}; x \<in> {a..b} \<rbrakk> \<Longrightarrow> b \<le> d"
   apply (frule (1) subsetD)
   apply (rule classical)
   apply clarsimp
   done
 
 lemma range_subset_eq:
   fixes a::"'a::linorder"
   assumes non_empty: "a \<le> b"
   shows "({a..b} \<subseteq> {c..d}) = (c \<le> a \<and> b \<le> d)"
   apply (insert non_empty)
   apply (rule iffI)
    apply (frule range_subset_lower [where x=a], simp)
    apply (drule range_subset_upper [where x=a], simp)
    apply simp
   apply auto
   done
 
 lemma range_eq:
   fixes a::"'a::linorder"
   assumes non_empty: "a \<le> b"
   shows "({a..b} = {c..d}) = (a = c \<and> b = d)"
   by (metis atLeastatMost_subset_iff eq_iff non_empty)
 
 lemma range_strict_subset_eq:
   fixes a::"'a::linorder"
   assumes non_empty: "a \<le> b"
   shows "({a..b} \<subset> {c..d}) = (c \<le> a \<and> b \<le> d \<and> (a = c \<longrightarrow> b \<noteq> d))"
   apply (insert non_empty)
   apply (subst psubset_eq)
   apply (subst range_subset_eq, assumption+)
   apply (subst range_eq, assumption+)
   apply simp
   done
 
 lemma range_subsetI:
   fixes x :: "'a :: order"
   assumes xX: "X \<le> x"
   and     yY: "y \<le> Y"
   shows   "{x .. y} \<subseteq> {X .. Y}"
   using xX yY by auto
 
 lemma set_False [simp]:
   "(set bs \<subseteq> {False}) = (True \<notin> set bs)" by auto
 
 lemma int_not_emptyD:
   "A \<inter> B \<noteq> {} \<Longrightarrow> \<exists>x. x \<in> A \<and> x \<in> B"
   by (erule contrapos_np, clarsimp simp: disjoint_iff_not_equal)
 
 definition
   sum_map :: "('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> 'a + 'c \<Rightarrow> 'b + 'd" where
  "sum_map f g x \<equiv> case x of Inl v \<Rightarrow> Inl (f v) | Inr v' \<Rightarrow> Inr (g v')"
 
 lemma sum_map_simps[simp]:
   "sum_map f g (Inl v) = Inl (f v)"
   "sum_map f g (Inr w) = Inr (g w)"
   by (simp add: sum_map_def)+
 
 lemma if_Some_None_eq_None:
   "((if P then Some v else None) = None) = (\<not> P)"
   by simp
 
 lemma CollectPairFalse [iff]:
   "{(a,b). False} = {}"
   by (simp add: split_def)
 
 lemma if_conj_dist:
   "((if b then w else x) \<and> (if b then y else z) \<and> X) =
   ((if b then w \<and> y else x \<and> z) \<and> X)"
   by simp
 
 lemma if_P_True1:
   "Q \<Longrightarrow> (if P then True else Q)"
   by simp
 
 lemma if_P_True2:
   "Q \<Longrightarrow> (if P then Q else True)"
   by simp
 
 lemmas nat_simps = diff_add_inverse2 diff_add_inverse
 
 lemmas nat_iffs = le_add1 le_add2
 
 lemma nat_min_simps:
   "(a::nat) \<le> b \<Longrightarrow> min b a = a"
   "a \<le> b \<Longrightarrow> min a b = a"
   by simp_all
 
 lemmas zadd_diff_inverse =
   trans [OF diff_add_cancel [symmetric] add.commute]
 
 lemmas add_diff_cancel2 =
   add.commute [THEN diff_eq_eq [THEN iffD2]]
 
 lemmas mcl = mult_cancel_left [THEN iffD1, THEN make_pos_rule]
 
 lemma pl_pl_rels: "a + b = c + d \<Longrightarrow> a \<ge> c \<and> b \<le> d \<or> a \<le> c \<and> b \<ge> d"
   for a b c d :: nat
   by arith
 
 lemmas pl_pl_rels' = add.commute [THEN [2] trans, THEN pl_pl_rels]
 
 lemma iszero_minus:
   \<open>iszero (- z) \<longleftrightarrow> iszero z\<close>
   by (simp add: iszero_def)
 
 lemma diff_le_eq': "a - b \<le> c \<longleftrightarrow> a \<le> b + c"
   for a b c :: int
   by arith
 
 lemma zless2: "0 < (2 :: int)"
   by (fact zero_less_numeral)
 
 lemma zless2p: "0 < (2 ^ n :: int)"
   by arith
 
 lemma zle2p: "0 \<le> (2 ^ n :: int)"
   by arith
 
 lemma ex_eq_or: "(\<exists>m. n = Suc m \<and> (m = k \<or> P m)) \<longleftrightarrow> n = Suc k \<or> (\<exists>m. n = Suc m \<and> P m)"
   by auto
 
 lemma power_minus_simp: "0 < n \<Longrightarrow> a ^ n = a * a ^ (n - 1)"
   by (auto dest: gr0_implies_Suc)
 
 lemma n2s_ths:
   \<open>2 + n = Suc (Suc n)\<close>
   \<open>n + 2 = Suc (Suc n)\<close>
   by (fact add_2_eq_Suc add_2_eq_Suc')+
 
 lemma s2n_ths:
   \<open>Suc (Suc n) = 2 + n\<close>
   \<open>Suc (Suc n) = n + 2\<close>
   by simp_all
 
 lemma gt_or_eq_0: "0 < y \<or> 0 = y"
   for y :: nat
   by arith
 
 lemma sum_imp_diff: "j = k + i \<Longrightarrow> j - i = k"
   for k :: nat
   by simp
 
 lemma le_diff_eq': "a \<le> c - b \<longleftrightarrow> b + a \<le> c"
   for a b c :: int
   by arith
 
 lemma less_diff_eq': "a < c - b \<longleftrightarrow> b + a < c"
   for a b c :: int
   by arith
 
 lemma diff_less_eq': "a - b < c \<longleftrightarrow> a < b + c"
   for a b c :: int
   by arith
 
 lemma axxbyy: "a + m + m = b + n + n \<Longrightarrow> a = 0 \<or> a = 1 \<Longrightarrow> b = 0 \<or> b = 1 \<Longrightarrow> a = b \<and> m = n"
   for a b m n :: int
   by arith
 
 lemma minus_eq: "m - k = m \<longleftrightarrow> k = 0 \<or> m = 0"
   for k m :: nat
   by arith
 
 lemma pl_pl_mm: "a + b = c + d \<Longrightarrow> a - c = d - b"
   for a b c d :: nat
   by arith
 
 lemmas pl_pl_mm' = add.commute [THEN [2] trans, THEN pl_pl_mm]
 
 lemma less_le_mult': "w * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> (w + 1) * c \<le> b * c"
   for b c w :: int
   apply (rule mult_right_mono)
    apply (rule zless_imp_add1_zle)
    apply (erule (1) mult_right_less_imp_less)
   apply assumption
   done
 
 lemma less_le_mult: "w * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> w * c + c \<le> b * c"
   for b c w :: int
   using less_le_mult' [of w c b] by (simp add: algebra_simps)
 
 lemmas less_le_mult_minus = iffD2 [OF le_diff_eq less_le_mult,
   simplified left_diff_distrib]
 
 lemma gen_minus: "0 < n \<Longrightarrow> f n = f (Suc (n - 1))"
   by auto
 
 lemma mpl_lem: "j \<le> i \<Longrightarrow> k < j \<Longrightarrow> i - j + k < i"
   for i j k :: nat
   by arith
 
 lemmas dme = div_mult_mod_eq
 lemmas dtle = div_times_less_eq_dividend
 lemmas th2 = order_trans [OF order_refl [THEN [2] mult_le_mono] div_times_less_eq_dividend]
 
 lemmas sdl = div_nat_eqI
 
 lemma given_quot: "f > 0 \<Longrightarrow> (f * l + (f - 1)) div f = l"
   for f l :: nat
   by (rule div_nat_eqI) (simp_all)
 
 lemma given_quot_alt: "f > 0 \<Longrightarrow> (l * f + f - Suc 0) div f = l"
   for f l :: nat
   apply (frule given_quot)
   apply (rule trans)
    prefer 2
    apply (erule asm_rl)
   apply (rule_tac f="\<lambda>n. n div f" in arg_cong)
   apply (simp add : ac_simps)
   done
 
 lemma x_power_minus_1:
   fixes x :: "'a :: {ab_group_add, power, numeral, one}"
   shows "x + (2::'a) ^ n - (1::'a) = x + (2 ^ n - 1)" by simp
 
 lemma nat_diff_add:
   fixes i :: nat
   shows "\<lbrakk> i + j = k \<rbrakk> \<Longrightarrow> i = k - j"
   by arith
 
 lemma pow_2_gt: "n \<ge> 2 \<Longrightarrow> (2::int) < 2 ^ n"
   by (induct n) auto
 
 lemma sum_to_zero:
   "(a :: 'a :: ring) + b = 0 \<Longrightarrow> a = (- b)"
   by (drule arg_cong[where f="\<lambda> x. x - a"], simp)
 
 lemma arith_is_1:
   "\<lbrakk> x \<le> Suc 0; x > 0 \<rbrakk> \<Longrightarrow> x = 1"
   by arith
 
 lemma suc_le_pow_2:
   "1 < (n::nat) \<Longrightarrow> Suc n < 2 ^ n"
   by (induct n; clarsimp)
      (case_tac "n = 1"; clarsimp)
 
 lemma nat_le_Suc_less_imp:
   "x < y \<Longrightarrow> x \<le> y - Suc 0"
   by arith
 
 lemma power_sub_int:
   "\<lbrakk> m \<le> n; 0 < b \<rbrakk> \<Longrightarrow> b ^ n div b ^ m = (b ^ (n - m) :: int)"
   apply (subgoal_tac "\<exists>n'. n = m + n'")
    apply (clarsimp simp: power_add)
   apply (rule exI[where x="n - m"])
   apply simp
   done
 
 lemma nat_Suc_less_le_imp:
   "(k::nat) < Suc n \<Longrightarrow> k \<le> n"
   by auto
 
 lemma nat_add_less_by_max:
   "\<lbrakk> (x::nat) \<le> xmax ; y < k - xmax \<rbrakk> \<Longrightarrow> x + y < k"
   by simp
 
 lemma nat_le_Suc_less:
   "0 < y \<Longrightarrow> (x \<le> y - Suc 0) = (x < y)"
   by arith
 
 lemma nat_power_minus_less:
   "a < 2 ^ (x - n) \<Longrightarrow> (a :: nat) < 2 ^ x"
   by (erule order_less_le_trans) simp
 
 lemma less_le_mult_nat':
   "w * c < b * c ==> 0 \<le> c ==> Suc w * c \<le> b * (c::nat)"
   apply (rule mult_right_mono)
    apply (rule Suc_leI)
    apply (erule (1) mult_right_less_imp_less)
   apply assumption
   done
 
 lemma less_le_mult_nat:
   \<open>0 < c \<and> w < b \<Longrightarrow> c + w * c \<le> b * c\<close> for b c w :: nat
   using less_le_mult_nat' [of w c b] by simp
 
 lemma p_assoc_help:
   fixes p :: "'a::{ring,power,numeral,one}"
   shows "p + 2^sz - 1 = p + (2^sz - 1)"
   by simp
 
 lemma pow_mono_leq_imp_lt:
   "x \<le> y \<Longrightarrow> x < 2 ^ y"
   by (simp add: le_less_trans)
 
 lemma small_powers_of_2:
   "x \<ge> 3 \<Longrightarrow> x < 2 ^ (x - 1)"
   by (induct x; simp add: suc_le_pow_2)
 
 lemma nat_less_power_trans2:
   fixes n :: nat
   shows "\<lbrakk>n < 2 ^ (m - k); k \<le> m\<rbrakk> \<Longrightarrow> n * 2 ^ k  < 2 ^ m"
   by (subst mult.commute, erule (1) nat_less_power_trans)
 
 lemma nat_move_sub_le: "(a::nat) + b \<le> c \<Longrightarrow> a \<le> c - b"
   by arith
 
 lemma plus_minus_one_rewrite:
   "v + (- 1 :: ('a :: {ring, one, uminus})) \<equiv> v - 1"
   by (simp add: field_simps)
 
 lemma Suc_0_lt_2p_len_of: "Suc 0 < 2 ^ LENGTH('a :: len)"
   by (metis One_nat_def len_gt_0 lessI numeral_2_eq_2 one_less_power)
 
 lemma bin_rest_code: "i div 2 = drop_bit 1 i" for i :: int
   by (simp add: drop_bit_eq_div)
 
 end
diff --git a/thys/Word_Lib/More_Bit_Ring.thy b/thys/Word_Lib/More_Bit_Ring.thy
new file mode 100644
--- /dev/null
+++ b/thys/Word_Lib/More_Bit_Ring.thy
@@ -0,0 +1,136 @@
+(*
+ * Copyright 2023, Proofcraft Pty Ltd
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
+
+theory More_Bit_Ring
+  imports Main
+begin
+
+(* Additional equalities on semiring_bit_operations and ring_bit_operations, in particular
+   relationships between Boolean and arithmetic operations. *)
+
+context semiring_bit_operations
+begin
+
+context
+  includes bit_operations_syntax
+begin
+
+lemma disjunctive_add2:
+  "(x AND y) = 0 \<Longrightarrow> x + y = x OR y"
+  by (metis disjunctive_add bit_0_eq bit_and_iff bot_apply bot_bool_def)
+
+end
+end
+
+context ring_bit_operations
+begin
+
+context
+  includes bit_operations_syntax
+begin
+
+lemma not_xor_is_eqv:
+  "NOT (x XOR y) = (x AND y) OR (NOT x AND NOT y)"
+  by (simp add: bit.xor_def bit.disj_conj_distrib or.commute)
+
+lemma not_xor_eq_xor_not:
+  "(NOT x) XOR y = x XOR (NOT y)"
+  by simp
+
+lemma not_minus:
+  "NOT (x - y) = y - x - 1"
+  by (simp add: not_eq_complement)
+
+lemma minus_not_eq_plus_1:
+  "- NOT x = x + 1"
+  by (simp add: minus_eq_not_plus_1)
+
+lemma not_minus_eq_minus_1:
+  "NOT (- x) = x - 1"
+  by (simp add: not_eq_complement)
+
+lemma and_plus_not_and:
+  "(x AND y) + (x AND NOT y) = x"
+  by (metis and.left_commute and.right_neutral bit.conj_cancel_right bit.conj_disj_distrib
+            bit.conj_zero_right bit.disj_cancel_right disjunctive_add2)
+
+lemma or_eq_and_not_plus:
+  "x OR y = (x AND NOT y) + y"
+  by (simp add: and.assoc bit.disj_conj_distrib2 disjunctive_add2)
+
+lemma and_eq_not_or_minus:
+  "x AND y = (NOT x OR y) - NOT x"
+  by (metis and.idem and_eq_not_not_or eq_diff_eq or.commute or.idem or_eq_and_not_plus)
+
+lemma and_not_eq_or_minus:
+  "x AND NOT y = (x OR y) - y"
+  by (simp add: or_eq_and_not_plus)
+
+lemma and_not_eq_minus_and:
+  "x AND NOT y = x - (x AND y)"
+  by (simp add: add.commute eq_diff_eq and_plus_not_and)
+
+lemma or_minus_eq_minus_and:
+  "(x OR y) - y = x - (x AND y)"
+  by (metis and_not_eq_minus_and and_not_eq_or_minus)
+
+lemma plus_eq_and_or:
+  "x + y = (x OR y) + (x AND y)"
+  using add_commute local.add.semigroup_axioms or_eq_and_not_plus semigroup.assoc
+  by (fastforce simp: and_plus_not_and)
+
+lemma xor_eq_or_minus_and:
+  "x XOR y = (x OR y) - (x AND y)"
+  by (metis (no_types) bit.de_Morgan_conj bit.xor_def2 bit_and_iff bit_or_iff disjunctive_diff)
+
+lemma not_xor_eq_and_plus_not_or:
+  "NOT (x XOR y) = (x AND y) + NOT (x OR y)"
+  by (metis (no_types, lifting) not_diff_distrib add.commute bit.de_Morgan_conj bit.xor_def2
+                                bit_and_iff bit_or_iff disjunctive_diff)
+
+lemma not_xor_eq_and_minus_or:
+  "NOT (x XOR y) = (x AND y) - (x OR y) - 1"
+  by (metis not_diff_distrib add.commute minus_diff_eq not_minus_eq_minus_1 not_xor_eq_and_plus_not_or)
+
+lemma plus_eq_xor_plus_carry:
+  "x + y = (x XOR y) + 2 * (x AND y)"
+  by (metis plus_eq_and_or add.commute add.left_commute diff_add_cancel mult_2 xor_eq_or_minus_and)
+
+lemma plus_eq_or_minus_xor:
+  "x + y = 2 * (x OR y) - (x XOR y)"
+  by (metis add_diff_cancel_left' diff_diff_eq2 local.mult_2 plus_eq_and_or xor_eq_or_minus_and)
+
+lemma plus_eq_minus_neg:
+  "x + y = x - NOT y - 1"
+  using add_commute local.not_diff_distrib not_minus
+  by auto
+
+lemma minus_eq_plus_neg:
+  "x - y = x + NOT y + 1"
+  by (simp add: add.semigroup_axioms diff_conv_add_uminus minus_eq_not_plus_1 semigroup.assoc)
+
+lemma minus_eq_and_not_minus_not_and:
+  "x - y = (x AND NOT y) - (NOT x AND y)"
+  by (metis bit.de_Morgan_conj bit.double_compl not_diff_distrib plus_eq_and_or)
+
+lemma minus_eq_xor_minus_not_and:
+  "x - y = (x XOR y) - 2 * (NOT x AND y)"
+  by (metis (no_types) bit.compl_eq_compl_iff bit.xor_compl_left not_diff_distrib
+                       plus_eq_xor_plus_carry)
+
+lemma minus_eq_and_not_minus_xor:
+  "x - y = 2 * (x AND NOT y) - (x XOR y)"
+  by (metis and.commute minus_diff_eq minus_eq_xor_minus_not_and xor.commute)
+
+lemma and_one_neq_simps[simp]:
+  "x AND 1 \<noteq> 0 \<longleftrightarrow> x AND 1 = 1"
+  "x AND 1 \<noteq> 1 \<longleftrightarrow> x AND 1 = 0"
+  by (clarsimp simp: and_one_eq)+
+
+end
+end
+
+end
\ No newline at end of file
diff --git a/thys/Word_Lib/More_Word.thy b/thys/Word_Lib/More_Word.thy
--- a/thys/Word_Lib/More_Word.thy
+++ b/thys/Word_Lib/More_Word.thy
@@ -1,2546 +1,2520 @@
 (*
  * Copyright Data61, CSIRO (ABN 41 687 119 230)
  *
  * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section \<open>Lemmas on words\<close>
 
 theory More_Word
   imports
     "HOL-Library.Word"
     More_Arithmetic
     More_Divides
+    More_Bit_Ring
 begin
 
 context
   includes bit_operations_syntax
 begin
 
 \<comment> \<open>problem posed by TPHOLs referee:
       criterion for overflow of addition of signed integers\<close>
 
 lemma sofl_test:
   \<open>sint x + sint y = sint (x + y) \<longleftrightarrow>
     drop_bit (size x - 1) ((x + y XOR x) AND (x + y XOR y)) = 0\<close>
   for x y :: \<open>'a::len word\<close>
 proof -
   obtain n where n: \<open>LENGTH('a) = Suc n\<close>
     by (cases \<open>LENGTH('a)\<close>) simp_all
   have *: \<open>sint x + sint y + 2 ^ Suc n > signed_take_bit n (sint x + sint y) \<Longrightarrow> sint x + sint y \<ge> - (2 ^ n)\<close>
     \<open>signed_take_bit n (sint x + sint y) > sint x + sint y - 2 ^ Suc n \<Longrightarrow> 2 ^ n > sint x + sint y\<close>
     using signed_take_bit_int_greater_eq [of \<open>sint x + sint y\<close> n] signed_take_bit_int_less_eq [of n \<open>sint x + sint y\<close>]
     by (auto intro: ccontr)
   have \<open>sint x + sint y = sint (x + y) \<longleftrightarrow>
     (sint (x + y) < 0 \<longleftrightarrow> sint x < 0) \<or>
     (sint (x + y) < 0 \<longleftrightarrow> sint y < 0)\<close>
     using sint_less [of x] sint_greater_eq [of x] sint_less [of y] sint_greater_eq [of y]
     signed_take_bit_int_eq_self [of \<open>LENGTH('a) - 1\<close> \<open>sint x + sint y\<close>]
     apply (auto simp add: not_less)
        apply (unfold sint_word_ariths)
        apply (subst signed_take_bit_int_eq_self)
          prefer 4
          apply (subst signed_take_bit_int_eq_self)
            prefer 7
            apply (subst signed_take_bit_int_eq_self)
              prefer 10
              apply (subst signed_take_bit_int_eq_self)
                apply (auto simp add: signed_take_bit_int_eq_self signed_take_bit_eq_take_bit_minus take_bit_Suc_from_most n not_less intro!: *)
     done
   then show ?thesis
     apply (simp only: One_nat_def word_size drop_bit_eq_zero_iff_not_bit_last bit_and_iff bit_xor_iff)
     apply (simp add: bit_last_iff)
     done
 qed
 
 lemma unat_power_lower [simp]:
   "unat ((2::'a::len word) ^ n) = 2 ^ n" if "n < LENGTH('a::len)"
   using that by transfer simp
 
 lemma unat_p2: "n < LENGTH('a :: len) \<Longrightarrow> unat (2 ^ n :: 'a word) = 2 ^ n"
   by (fact unat_power_lower)
 
 lemma word_div_lt_eq_0:
   "x < y \<Longrightarrow> x div y = 0" for x :: "'a :: len word"
-  by transfer simp
+  by (fact div_word_less)
 
 lemma word_div_eq_1_iff: "n div m = 1 \<longleftrightarrow> n \<ge> m \<and> unat n < 2 * unat (m :: 'a :: len word)"
   apply (simp only: word_arith_nat_defs word_le_nat_alt word_of_nat_eq_iff flip: nat_div_eq_Suc_0_iff)
   apply (simp flip: unat_div unsigned_take_bit_eq)
   done
 
 lemma AND_twice [simp]:
   "(w AND m) AND m = w AND m"
   by (fact and.right_idem)
 
 lemma word_combine_masks:
   "w AND m = z \<Longrightarrow> w AND m' = z' \<Longrightarrow> w AND (m OR m') = (z OR z')"
   for w m m' z z' :: \<open>'a::len word\<close>
   by (simp add: bit.conj_disj_distrib)
 
 lemma p2_gt_0:
   "(0 < (2 ^ n :: 'a :: len word)) = (n < LENGTH('a))"
   by (simp add : word_gt_0 not_le)
 
 lemma uint_2p_alt:
   \<open>n < LENGTH('a::len) \<Longrightarrow> uint ((2::'a::len word) ^ n) = 2 ^ n\<close>
   using p2_gt_0 [of n, where ?'a = 'a] by (simp add: uint_2p)
 
 lemma p2_eq_0:
   \<open>(2::'a::len word) ^ n = 0 \<longleftrightarrow> LENGTH('a::len) \<le> n\<close>
   by (fact exp_eq_zero_iff)
 
 lemma p2len:
   \<open>(2 :: 'a word) ^ LENGTH('a::len) = 0\<close>
-  by simp
+  by (fact word_pow_0)
 
 lemma neg_mask_is_div:
   "w AND NOT (mask n) = (w div 2^n) * 2^n"
   for w :: \<open>'a::len word\<close>
   by (rule bit_word_eqI)
     (auto simp add: bit_simps simp flip: push_bit_eq_mult drop_bit_eq_div)
 
 lemma neg_mask_is_div':
   "n < size w \<Longrightarrow> w AND NOT (mask n) = ((w div (2 ^ n)) * (2 ^ n))"
   for w :: \<open>'a::len word\<close>
   by (rule neg_mask_is_div)
 
 lemma and_mask_arith:
   "w AND mask n = (w * 2^(size w - n)) div 2^(size w - n)"
   for w :: \<open>'a::len word\<close>
   by (rule bit_word_eqI)
     (auto simp add: bit_simps word_size simp flip: push_bit_eq_mult drop_bit_eq_div)
 
 lemma and_mask_arith':
   "0 < n \<Longrightarrow> w AND mask n = (w * 2^(size w - n)) div 2^(size w - n)"
   for w :: \<open>'a::len word\<close>
   by (rule and_mask_arith)
 
 lemma mask_2pm1: "mask n = 2 ^ n - (1 :: 'a::len word)"
   by (fact mask_eq_decr_exp)
 
 lemma add_mask_fold:
   "x + 2 ^ n - 1 = x + mask n"
   for x :: \<open>'a::len word\<close>
   by (simp add: mask_eq_decr_exp)
 
 lemma word_and_mask_le_2pm1: "w AND mask n \<le> 2 ^ n - 1"
   for w :: \<open>'a::len word\<close>
   by (simp add: mask_2pm1[symmetric] word_and_le1)
 
 lemma is_aligned_AND_less_0:
   "u AND mask n = 0 \<Longrightarrow> v < 2^n \<Longrightarrow> u AND v = 0"
   for u v :: \<open>'a::len word\<close>
   apply (drule less_mask_eq)
   apply (simp flip: take_bit_eq_mask)
   apply (simp add: bit_eq_iff)
   apply (auto simp add: bit_simps)
   done
 
 lemma and_mask_eq_iff_le_mask:
   \<open>w AND mask n = w \<longleftrightarrow> w \<le> mask n\<close>
   for w :: \<open>'a::len word\<close>
   apply (simp flip: take_bit_eq_mask)
   apply (cases \<open>n \<ge> LENGTH('a)\<close>; transfer)
    apply (simp_all add: not_le min_def)
    apply (simp_all add: mask_eq_exp_minus_1)
   apply auto
    apply (metis take_bit_int_less_exp)
   apply (metis min_def nat_less_le take_bit_int_eq_self_iff take_bit_take_bit)
   done
 
 lemma less_eq_mask_iff_take_bit_eq_self:
   \<open>w \<le> mask n \<longleftrightarrow> take_bit n w = w\<close>
   for w :: \<open>'a::len word\<close>
   by (simp add: and_mask_eq_iff_le_mask take_bit_eq_mask)
 
 lemma NOT_eq:
   "NOT (x :: 'a :: len word) = - x - 1"
-  apply (cut_tac x = "x" in word_add_not)
-  apply (drule add.commute [THEN trans])
-  apply (drule eq_diff_eq [THEN iffD2])
-  by simp
+  by (fact not_eq_complement)
 
 lemma NOT_mask: "NOT (mask n :: 'a::len word) = - (2 ^ n)"
   by (simp add : NOT_eq mask_2pm1)
 
 lemma le_m1_iff_lt: "(x > (0 :: 'a :: len word)) = ((y \<le> x - 1) = (y < x))"
   by uint_arith
 
 lemma gt0_iff_gem1:
   \<open>0 < x \<longleftrightarrow> x - 1 < x\<close>
   for x :: \<open>'a::len word\<close>
   by (metis add.right_neutral diff_add_cancel less_irrefl measure_unat unat_arith_simps(2) word_neq_0_conv word_sub_less_iff)
 
 lemma power_2_ge_iff:
   \<open>2 ^ n - (1 :: 'a::len word) < 2 ^ n \<longleftrightarrow> n < LENGTH('a)\<close>
   using gt0_iff_gem1 p2_gt_0 by blast
 
 lemma le_mask_iff_lt_2n:
   "n < len_of TYPE ('a) = (((w :: 'a :: len word) \<le> mask n) = (w < 2 ^ n))"
   unfolding mask_2pm1 by (rule trans [OF p2_gt_0 [THEN sym] le_m1_iff_lt])
 
 lemma mask_lt_2pn:
   \<open>n < LENGTH('a) \<Longrightarrow> mask n < (2 :: 'a::len word) ^ n\<close>
   by (simp add: mask_eq_exp_minus_1 power_2_ge_iff)
 
 lemma word_unat_power:
   "(2 :: 'a :: len word) ^ n = of_nat (2 ^ n)"
   by simp
 
 lemma of_nat_mono_maybe:
   assumes xlt: "x < 2 ^ len_of TYPE ('a)"
   shows   "y < x \<Longrightarrow> of_nat y < (of_nat x :: 'a :: len word)"
   apply (subst word_less_nat_alt)
   apply (subst unat_of_nat)+
   apply (subst mod_less)
    apply (erule order_less_trans [OF _ xlt])
   apply (subst mod_less [OF xlt])
   apply assumption
   done
 
 lemma word_and_max_word:
   fixes a::"'a::len word"
   shows "x = - 1 \<Longrightarrow> a AND x = a"
   by simp
 
 lemma word_and_full_mask_simp:
   \<open>x AND mask LENGTH('a) = x\<close> for x :: \<open>'a::len word\<close>
   by (simp add: bit_eq_iff bit_simps)
 
 lemma of_int_uint:
   "of_int (uint x) = x"
   by (fact word_of_int_uint)
 
 corollary word_plus_and_or_coroll:
   "x AND y = 0 \<Longrightarrow> x + y = x OR y"
   for x y :: \<open>'a::len word\<close>
-  using word_plus_and_or[where x=x and y=y]
-  by simp
+  by (fact disjunctive_add2)
 
 corollary word_plus_and_or_coroll2:
   "(x AND w) + (x AND NOT w) = x"
   for x w :: \<open>'a::len word\<close>
-  apply (subst disjunctive_add)
-   apply (simp add: bit_simps)
-  apply (simp flip: bit.conj_disj_distrib)
-  done
+  by (fact and_plus_not_and)
 
 lemma unat_mask_eq:
   \<open>unat (mask n :: 'a::len word) = mask (min LENGTH('a) n)\<close>
   by transfer (simp add: nat_mask_eq)
 
 lemma word_plus_mono_left:
   fixes x :: "'a :: len word"
   shows "\<lbrakk>y \<le> z; x \<le> x + z\<rbrakk> \<Longrightarrow> y + x \<le> z + x"
   by unat_arith
 
 lemma less_Suc_unat_less_bound:
   "n < Suc (unat (x :: 'a :: len word)) \<Longrightarrow> n < 2 ^ LENGTH('a)"
   by (auto elim!: order_less_le_trans intro: Suc_leI)
 
 lemma up_ucast_inj:
   "\<lbrakk> ucast x = (ucast y::'b::len word); LENGTH('a) \<le> len_of TYPE ('b) \<rbrakk> \<Longrightarrow> x = (y::'a::len word)"
   by transfer (simp add: min_def split: if_splits)
 
 lemmas ucast_up_inj = up_ucast_inj
 
 lemma up_ucast_inj_eq:
   "LENGTH('a) \<le> len_of TYPE ('b) \<Longrightarrow> (ucast x = (ucast y::'b::len word)) = (x = (y::'a::len word))"
   by (fastforce dest: up_ucast_inj)
 
 lemma no_plus_overflow_neg:
   "(x :: 'a :: len word) < -y \<Longrightarrow> x \<le> x + y"
   by (metis diff_minus_eq_add less_imp_le sub_wrap_lt)
 
 lemma ucast_ucast_eq:
   "\<lbrakk> ucast x = (ucast (ucast y::'a word)::'c::len word); LENGTH('a) \<le> LENGTH('b);
      LENGTH('b) \<le> LENGTH('c) \<rbrakk> \<Longrightarrow>
    x = ucast y" for x :: "'a::len word" and y :: "'b::len word"
   apply transfer
   apply (cases \<open>LENGTH('c) = LENGTH('a)\<close>)
    apply (auto simp add: min_def)
   done
 
 lemma ucast_0_I:
   "x = 0 \<Longrightarrow> ucast x = 0"
   by simp
 
 lemma word_add_offset_less:
   fixes x :: "'a :: len word"
   assumes yv: "y < 2 ^ n"
   and     xv: "x < 2 ^ m"
   and     mnv: "sz < LENGTH('a :: len)"
   and    xv': "x < 2 ^ (LENGTH('a :: len) - n)"
   and     mn: "sz = m + n"
   shows   "x * 2 ^ n + y < 2 ^ sz"
 proof (subst mn)
   from mnv mn have nv: "n < LENGTH('a)" and mv: "m < LENGTH('a)"  by auto
 
   have uy: "unat y < 2 ^ n"
     by (rule order_less_le_trans [OF unat_mono [OF yv] order_eq_refl],
         rule unat_power_lower[OF nv])
 
   have ux: "unat x < 2 ^ m"
     by (rule order_less_le_trans [OF unat_mono [OF xv] order_eq_refl],
         rule unat_power_lower[OF mv])
 
   then show "x * 2 ^ n + y < 2 ^ (m + n)" using ux uy nv mnv xv'
     apply (subst word_less_nat_alt)
     apply (subst unat_word_ariths)+
     apply (subst mod_less)
      apply simp
      apply (subst mult.commute)
      apply (rule nat_less_power_trans [OF _ order_less_imp_le [OF nv]])
      apply (rule order_less_le_trans [OF unat_mono [OF xv']])
      apply (cases "n = 0"; simp)
     apply (subst unat_power_lower[OF nv])
     apply (subst mod_less)
      apply (erule order_less_le_trans [OF nat_add_offset_less], assumption)
       apply (rule mn)
      apply simp
     apply (simp add: mn mnv)
     apply (erule nat_add_offset_less; simp)
     done
 qed
 
 lemma word_less_power_trans:
   fixes n :: "'a :: len word"
   assumes nv: "n < 2 ^ (m - k)"
   and     kv: "k \<le> m"
   and     mv: "m < len_of TYPE ('a)"
   shows "2 ^ k * n < 2 ^ m"
   using nv kv mv
   apply -
   apply (subst word_less_nat_alt)
   apply (subst unat_word_ariths)
   apply (subst mod_less)
    apply simp
    apply (rule nat_less_power_trans)
     apply (erule order_less_trans [OF unat_mono])
     apply simp
    apply simp
   apply simp
   apply (rule nat_less_power_trans)
    apply (subst unat_power_lower[where 'a = 'a, symmetric])
     apply simp
    apply (erule unat_mono)
   apply simp
   done
 
 lemma  word_less_power_trans2:
   fixes n :: "'a::len word"
   shows "\<lbrakk>n < 2 ^ (m - k); k \<le> m; m < LENGTH('a)\<rbrakk> \<Longrightarrow> n * 2 ^ k < 2 ^ m"
   by (subst field_simps, rule word_less_power_trans)
 
 lemma Suc_unat_diff_1:
   fixes x :: "'a :: len word"
   assumes lt: "1 \<le> x"
   shows "Suc (unat (x - 1)) = unat x"
 proof -
   have "0 < unat x"
     by (rule order_less_le_trans [where y = 1], simp, subst unat_1 [symmetric],
         rule iffD1 [OF word_le_nat_alt lt])
 
   then show ?thesis
     by ((subst unat_sub [OF lt])+, simp only:  unat_1)
 qed
 
 lemma word_eq_unatI:
   \<open>v = w\<close> if \<open>unat v = unat w\<close>
   using that by transfer (simp add: nat_eq_iff)
 
 lemma word_div_sub:
   fixes x :: "'a :: len word"
   assumes yx: "y \<le> x"
   and     y0: "0 < y"
   shows "(x - y) div y = x div y - 1"
   apply (rule word_eq_unatI)
   apply (subst unat_div)
   apply (subst unat_sub [OF yx])
   apply (subst unat_sub)
    apply (subst word_le_nat_alt)
    apply (subst unat_div)
    apply (subst le_div_geq)
      apply (rule order_le_less_trans [OF _ unat_mono [OF y0]])
      apply simp
     apply (subst word_le_nat_alt [symmetric], rule yx)
    apply simp
   apply (subst unat_div)
   apply (subst le_div_geq [OF _ iffD1 [OF word_le_nat_alt yx]])
    apply (rule order_le_less_trans [OF _ unat_mono [OF y0]])
    apply simp
   apply simp
   done
 
 lemma word_mult_less_mono1:
   fixes i :: "'a :: len word"
   assumes ij: "i < j"
   and    knz: "0 < k"
   and    ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
   shows  "i * k < j * k"
 proof -
   from ij ujk knz have jk: "unat i * unat k < 2 ^ len_of TYPE ('a)"
     by (auto intro: order_less_subst2 simp: word_less_nat_alt elim: mult_less_mono1)
 
   then show ?thesis using ujk knz ij
     by (auto simp: word_less_nat_alt iffD1 [OF unat_mult_lem])
 qed
 
 lemma word_mult_less_dest:
   fixes i :: "'a :: len word"
   assumes ij: "i * k < j * k"
   and    uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
   and    ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
   shows  "i < j"
   using uik ujk ij
   by (auto simp: word_less_nat_alt iffD1 [OF unat_mult_lem] elim: mult_less_mono1)
 
 lemma word_mult_less_cancel:
   fixes k :: "'a :: len word"
   assumes knz: "0 < k"
   and    uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
   and    ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
   shows "(i * k < j * k) = (i < j)"
   by (rule iffI [OF word_mult_less_dest [OF _ uik ujk] word_mult_less_mono1 [OF _ knz ujk]])
 
 lemma Suc_div_unat_helper:
   assumes szv: "sz < LENGTH('a :: len)"
   and   usszv: "us \<le> sz"
   shows "2 ^ (sz - us) = Suc (unat (((2::'a :: len word) ^ sz - 1) div 2 ^ us))"
 proof -
   note usv = order_le_less_trans [OF usszv szv]
 
   from usszv obtain q where qv: "sz = us + q" by (auto simp: le_iff_add)
 
   have "Suc (unat (((2:: 'a word) ^ sz - 1) div 2 ^ us)) =
     (2 ^ us + unat ((2:: 'a word) ^ sz - 1)) div 2 ^ us"
     apply (subst unat_div unat_power_lower[OF usv])+
     apply (subst div_add_self1, simp+)
     done
 
   also have "\<dots> = ((2 ^ us - 1) + 2 ^ sz) div 2 ^ us" using szv
     by (simp add: unat_minus_one)
 
   also have "\<dots> = 2 ^ q + ((2 ^ us - 1) div 2 ^ us)"
     apply (subst qv)
     apply (subst power_add)
     apply (subst div_mult_self2; simp)
     done
 
   also have "\<dots> = 2 ^ (sz - us)" using qv by simp
 
   finally show ?thesis ..
 qed
 
 lemma enum_word_nth_eq:
   \<open>(Enum.enum :: 'a::len word list) ! n = word_of_nat n\<close>
     if \<open>n < 2 ^ LENGTH('a)\<close>
     for n
   using that by (simp add: enum_word_def)
 
 lemma length_enum_word_eq:
   \<open>length (Enum.enum :: 'a::len word list) = 2 ^ LENGTH('a)\<close>
   by (simp add: enum_word_def)
 
 lemma unat_lt2p [iff]:
   \<open>unat x < 2 ^ LENGTH('a)\<close> for x :: \<open>'a::len word\<close>
   by transfer simp
 
 lemma of_nat_unat [simp]:
   "of_nat \<circ> unat = id"
   by (rule ext, simp)
 
 lemma Suc_unat_minus_one [simp]:
   "x \<noteq> 0 \<Longrightarrow> Suc (unat (x - 1)) = unat x"
   by (metis Suc_diff_1 unat_gt_0 unat_minus_one)
 
 lemma word_add_le_dest:
   fixes i :: "'a :: len word"
   assumes le: "i + k \<le> j + k"
   and    uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
   and    ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
   shows  "i \<le> j"
   using uik ujk le
   by (auto simp: word_le_nat_alt iffD1 [OF unat_add_lem] elim: add_le_mono1)
 
 lemma word_add_le_mono1:
   fixes i :: "'a :: len word"
   assumes ij: "i \<le> j"
   and    ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
   shows  "i + k \<le> j + k"
 proof -
   from ij ujk have jk: "unat i + unat k < 2 ^ len_of TYPE ('a)"
     by (auto elim: order_le_less_subst2 simp: word_le_nat_alt elim: add_le_mono1)
 
   then show ?thesis using ujk ij
     by (auto simp: word_le_nat_alt iffD1 [OF unat_add_lem])
 qed
 
 lemma word_add_le_mono2:
   fixes i :: "'a :: len word"
   shows "\<lbrakk>i \<le> j; unat j + unat k < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow> k + i \<le> k + j"
   by (subst field_simps, subst field_simps, erule (1) word_add_le_mono1)
 
 lemma word_add_le_iff:
   fixes i :: "'a :: len word"
   assumes uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
   and     ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
   shows  "(i + k \<le> j + k) = (i \<le> j)"
 proof
   assume "i \<le> j"
   show "i + k \<le> j + k" by (rule word_add_le_mono1) fact+
 next
   assume "i + k \<le> j + k"
   show "i \<le> j" by (rule word_add_le_dest) fact+
 qed
 
 lemma word_add_less_mono1:
   fixes i :: "'a :: len word"
   assumes ij: "i < j"
   and    ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
   shows  "i + k < j + k"
 proof -
   from ij ujk have jk: "unat i + unat k < 2 ^ len_of TYPE ('a)"
     by (auto elim: order_le_less_subst2 simp: word_less_nat_alt elim: add_less_mono1)
 
   then show ?thesis using ujk ij
     by (auto simp: word_less_nat_alt iffD1 [OF unat_add_lem])
 qed
 
 lemma word_add_less_dest:
   fixes i :: "'a :: len word"
   assumes le: "i + k < j + k"
   and    uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
   and    ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
   shows  "i < j"
   using uik ujk le
   by (auto simp: word_less_nat_alt iffD1 [OF unat_add_lem] elim: add_less_mono1)
 
 lemma word_add_less_iff:
   fixes i :: "'a :: len word"
   assumes uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
   and     ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
   shows  "(i + k < j + k) = (i < j)"
 proof
   assume "i < j"
   show "i + k < j + k" by (rule word_add_less_mono1) fact+
 next
   assume "i + k < j + k"
   show "i < j" by (rule word_add_less_dest) fact+
 qed
 
 lemma word_mult_less_iff:
   fixes i :: "'a :: len word"
   assumes knz: "0 < k"
   and     uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
   and     ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
   shows  "(i * k < j * k) = (i < j)"
   using assms by (rule word_mult_less_cancel)
 
 lemma word_le_imp_diff_le:
   fixes n :: "'a::len word"
   shows "\<lbrakk>k \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> n - k \<le> m"
   by (auto simp: unat_sub word_le_nat_alt)
 
 lemma word_less_imp_diff_less:
   fixes n :: "'a::len word"
   shows "\<lbrakk>k \<le> n; n < m\<rbrakk> \<Longrightarrow> n - k < m"
   by (clarsimp simp: unat_sub word_less_nat_alt
              intro!: less_imp_diff_less)
 
 lemma word_mult_le_mono1:
   fixes i :: "'a :: len word"
   assumes ij: "i \<le> j"
   and    knz: "0 < k"
   and    ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
   shows  "i * k \<le> j * k"
 proof -
   from ij ujk knz have jk: "unat i * unat k < 2 ^ len_of TYPE ('a)"
     by (auto elim: order_le_less_subst2 simp: word_le_nat_alt elim: mult_le_mono1)
 
   then show ?thesis using ujk knz ij
     by (auto simp: word_le_nat_alt iffD1 [OF unat_mult_lem])
 qed
 
 lemma word_mult_le_iff:
   fixes i :: "'a :: len word"
   assumes knz: "0 < k"
   and     uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
   and     ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
   shows  "(i * k \<le> j * k) = (i \<le> j)"
 proof
   assume "i \<le> j"
   show "i * k \<le> j * k" by (rule word_mult_le_mono1) fact+
 next
   assume p: "i * k \<le> j * k"
 
   have "0 < unat k" using knz by (simp add: word_less_nat_alt)
   then show "i \<le> j" using p
     by (clarsimp simp: word_le_nat_alt iffD1 [OF unat_mult_lem uik]
       iffD1 [OF unat_mult_lem ujk])
 qed
 
 lemma word_diff_less:
   fixes n :: "'a :: len word"
   shows "\<lbrakk>0 < n; 0 < m; n \<le> m\<rbrakk> \<Longrightarrow> m - n < m"
   apply (subst word_less_nat_alt)
   apply (subst unat_sub)
    apply assumption
   apply (rule diff_less)
    apply (simp_all add: word_less_nat_alt)
   done
 
 lemma word_add_increasing:
   fixes x :: "'a :: len word"
   shows "\<lbrakk> p + w \<le> x; p \<le> p + w \<rbrakk> \<Longrightarrow> p \<le> x"
   by unat_arith
 
 lemma word_random:
   fixes x :: "'a :: len word"
   shows "\<lbrakk> p \<le> p + x'; x \<le> x' \<rbrakk> \<Longrightarrow> p \<le> p + x"
   by unat_arith
 
 lemma word_sub_mono:
   "\<lbrakk> a \<le> c; d \<le> b; a - b \<le> a; c - d \<le> c \<rbrakk>
     \<Longrightarrow> (a - b) \<le> (c - d :: 'a :: len word)"
   by unat_arith
 
 lemma power_not_zero:
   "n < LENGTH('a::len) \<Longrightarrow> (2 :: 'a word) ^ n \<noteq> 0"
   by (metis p2_gt_0 word_neq_0_conv)
 
 lemma word_gt_a_gt_0:
   "a < n \<Longrightarrow> (0 :: 'a::len word) < n"
   apply (case_tac "n = 0")
    apply clarsimp
   apply (clarsimp simp: word_neq_0_conv)
   done
 
 lemma word_power_less_1 [simp]:
   "sz < LENGTH('a::len) \<Longrightarrow> (2::'a word) ^ sz - 1 < 2 ^ sz"
   apply (simp add: word_less_nat_alt)
   apply (subst unat_minus_one)
   apply simp_all
   done
 
 lemma word_sub_1_le:
-  "x \<noteq> 0 \<Longrightarrow> x - 1 \<le> (x :: ('a :: len) word)"
+  "x \<noteq> 0 \<Longrightarrow> x - 1 \<le> (x :: 'a :: len word)"
   apply (subst no_ulen_sub)
   apply simp
   apply (cases "uint x = 0")
    apply (simp add: uint_0_iff)
   apply (insert uint_ge_0[where x=x])
   apply arith
   done
 
 lemma push_bit_word_eq_nonzero:
   \<open>push_bit n w \<noteq> 0\<close> if \<open>w < 2 ^ m\<close> \<open>m + n < LENGTH('a)\<close> \<open>w \<noteq> 0\<close>
     for w :: \<open>'a::len word\<close>
   using that
   apply (simp only: word_neq_0_conv word_less_nat_alt
                     mod_0 unat_word_ariths
                     unat_power_lower word_le_nat_alt)
   apply (metis add_diff_cancel_right' gr0I gr_implies_not0 less_or_eq_imp_le min_def push_bit_eq_0_iff take_bit_nat_eq_self_iff take_bit_push_bit take_bit_take_bit unsigned_push_bit_eq)
   done
 
 lemma unat_less_power:
   fixes k :: "'a::len word"
   assumes szv: "sz < LENGTH('a)"
   and     kv:  "k < 2 ^ sz"
   shows   "unat k < 2 ^ sz"
   using szv unat_mono [OF kv] by simp
 
 lemma unat_mult_power_lem:
   assumes kv: "k < 2 ^ (LENGTH('a::len) - sz)"
   shows "unat (2 ^ sz * of_nat k :: (('a::len) word)) = 2 ^ sz * k"
 proof (cases \<open>sz < LENGTH('a)\<close>)
   case True
   with assms show ?thesis
     by (simp add: unat_word_ariths take_bit_eq_mod mod_simps unsigned_of_nat)
       (simp add: take_bit_nat_eq_self_iff nat_less_power_trans flip: take_bit_eq_mod)
 next
   case False
   with assms show ?thesis
     by simp
 qed
 
 lemma word_plus_mcs_4:
   "\<lbrakk>v + x \<le> w + x; x \<le> v + x\<rbrakk> \<Longrightarrow> v \<le> (w::'a::len word)"
   by uint_arith
 
 lemma word_plus_mcs_3:
   "\<lbrakk>v \<le> w; x \<le> w + x\<rbrakk> \<Longrightarrow> v + x \<le> w + (x::'a::len word)"
   by unat_arith
 
 lemma word_le_minus_one_leq:
   "x < y \<Longrightarrow> x \<le> y - 1" for x :: "'a :: len word"
   by transfer (metis le_less_trans less_irrefl take_bit_decr_eq take_bit_nonnegative zle_diff1_eq)
 
 lemma word_less_sub_le[simp]:
   fixes x :: "'a :: len word"
   assumes nv: "n < LENGTH('a)"
   shows "(x \<le> 2 ^ n - 1) = (x < 2 ^ n)"
   using le_less_trans word_le_minus_one_leq nv power_2_ge_iff by blast
 
 lemma unat_of_nat_len:
   "x < 2 ^ LENGTH('a) \<Longrightarrow> unat (of_nat x :: 'a::len word) = x"
   by (simp add: unsigned_of_nat take_bit_nat_eq_self_iff)
 
 lemma unat_of_nat_eq:
   "x < 2 ^ LENGTH('a) \<Longrightarrow> unat (of_nat x ::'a::len word) = x"
   by (rule unat_of_nat_len)
 
 lemma unat_eq_of_nat:
   "n < 2 ^ LENGTH('a) \<Longrightarrow> (unat (x :: 'a::len word) = n) = (x = of_nat n)"
   by transfer
     (auto simp add: take_bit_of_nat nat_eq_iff take_bit_nat_eq_self_iff intro: sym)
 
 lemma alignUp_div_helper:
   fixes a :: "'a::len word"
   assumes kv: "k < 2 ^ (LENGTH('a) - n)"
   and     xk: "x = 2 ^ n * of_nat k"
   and    le: "a \<le> x"
   and    sz: "n < LENGTH('a)"
   and   anz: "a mod 2 ^ n \<noteq> 0"
   shows "a div 2 ^ n < of_nat k"
 proof -
   have kn: "unat (of_nat k :: 'a word) * unat ((2::'a word) ^ n) < 2 ^ LENGTH('a)"
     using xk kv sz
     apply (subst unat_of_nat_eq)
      apply (erule order_less_le_trans)
      apply simp
     apply (subst unat_power_lower, simp)
     apply (subst mult.commute)
     apply (rule nat_less_power_trans)
      apply simp
     apply simp
     done
 
   have "unat a div 2 ^ n * 2 ^ n \<noteq> unat a"
   proof -
     have "unat a = unat a div 2 ^ n * 2 ^ n + unat a mod 2 ^ n"
       by (simp add: div_mult_mod_eq)
     also have "\<dots> \<noteq> unat a div 2 ^ n * 2 ^ n" using sz anz
       by (simp add: unat_arith_simps)
     finally show ?thesis ..
   qed
 
   then have "a div 2 ^ n * 2 ^ n < a" using sz anz
     apply (subst word_less_nat_alt)
     apply (subst unat_word_ariths)
     apply (subst unat_div)
     apply simp
     apply (rule order_le_less_trans [OF mod_less_eq_dividend])
     apply (erule order_le_neq_trans [OF div_mult_le])
     done
 
   also from xk le have "\<dots> \<le> of_nat k * 2 ^ n" by (simp add: field_simps)
   finally show ?thesis using sz kv
     apply -
     apply (erule word_mult_less_dest [OF _ _ kn])
     apply (simp add: unat_div)
     apply (rule order_le_less_trans [OF div_mult_le])
     apply (rule unat_lt2p)
     done
 qed
 
 lemma mask_out_sub_mask:
   "(x AND NOT (mask n)) = x - (x AND (mask n))"
   for x :: \<open>'a::len word\<close>
-  by (simp add: field_simps word_plus_and_or_coroll2)
+  by (fact and_not_eq_minus_and)
 
 lemma subtract_mask:
   "p - (p AND mask n) = (p AND NOT (mask n))"
   "p - (p AND NOT (mask n)) = (p AND mask n)"
   for p :: \<open>'a::len word\<close>
-  by (simp add: field_simps word_plus_and_or_coroll2)+
+  by (auto simp: and_not_eq_minus_and)
 
 lemma take_bit_word_eq_self_iff:
   \<open>take_bit n w = w \<longleftrightarrow> n \<ge> LENGTH('a) \<or> w < 2 ^ n\<close>
   for w :: \<open>'a::len word\<close>
   using take_bit_int_eq_self_iff [of n \<open>take_bit LENGTH('a) (uint w)\<close>]
   by (transfer fixing: n) auto
 
 lemma word_power_increasing:
-  assumes x: "2 ^ x < (2 ^ y::'a::len word)" "x < LENGTH('a::len)" "y < LENGTH('a::len)"
+  assumes x: "2 ^ x < (2 ^ y::'a::len word)" "x < LENGTH('a)" "y < LENGTH('a)"
   shows "x < y" using x
   using assms by transfer simp
 
 lemma mask_twice:
   "(x AND mask n) AND mask m = x AND mask (min m n)"
   for x :: \<open>'a::len word\<close>
   by (simp flip: take_bit_eq_mask)
 
 lemma plus_one_helper[elim!]:
   "x < n + (1 :: 'a :: len word) \<Longrightarrow> x \<le> n"
   apply (simp add: word_less_nat_alt word_le_nat_alt field_simps)
   apply (case_tac "1 + n = 0")
    apply simp_all
   apply (subst(asm) unatSuc, assumption)
   apply arith
   done
 
 lemma plus_one_helper2:
   "\<lbrakk> x \<le> n; n + 1 \<noteq> 0 \<rbrakk> \<Longrightarrow> x < n + (1 :: 'a :: len word)"
   by (simp add: word_less_nat_alt word_le_nat_alt field_simps
                 unatSuc)
 
 lemma less_x_plus_1:
-  fixes x :: "'a :: len word" shows
-  "x \<noteq> - 1 \<Longrightarrow> (y < (x + 1)) = (y < x \<or> y = x)"
-  apply (rule iffI)
-   apply (rule disjCI)
-   apply (drule plus_one_helper)
-   apply simp
-  apply (subgoal_tac "x < x + 1")
-   apply (erule disjE, simp_all)
-  apply (rule plus_one_helper2 [OF order_refl])
-  apply (rule notI, drule max_word_wrap)
-  apply simp
-  done
+  "x \<noteq> - 1 \<Longrightarrow> (y < x + 1) = (y < x \<or> y = x)" for x :: "'a::len word"
+  by (meson max_word_wrap plus_one_helper plus_one_helper2 word_le_less_eq)
 
 lemma word_Suc_leq:
   fixes k::"'a::len word" shows "k \<noteq> - 1 \<Longrightarrow> x < k + 1 \<longleftrightarrow> x \<le> k"
   using less_x_plus_1 word_le_less_eq by auto
 
 lemma word_Suc_le:
    fixes k::"'a::len word" shows "x \<noteq> - 1 \<Longrightarrow> x + 1 \<le> k \<longleftrightarrow> x < k"
   by (meson not_less word_Suc_leq)
 
 lemma word_lessThan_Suc_atMost:
   \<open>{..< k + 1} = {..k}\<close> if \<open>k \<noteq> - 1\<close> for k :: \<open>'a::len word\<close>
   using that by (simp add: lessThan_def atMost_def word_Suc_leq)
 
 lemma word_atLeastLessThan_Suc_atLeastAtMost:
   \<open>{l ..< u + 1} = {l..u}\<close> if \<open>u \<noteq> - 1\<close> for l :: \<open>'a::len word\<close>
   using that by (simp add: atLeastAtMost_def atLeastLessThan_def word_lessThan_Suc_atMost)
 
 lemma word_atLeastAtMost_Suc_greaterThanAtMost:
   \<open>{m<..u} = {m + 1..u}\<close> if \<open>m \<noteq> - 1\<close> for m :: \<open>'a::len word\<close>
   using that by (simp add: greaterThanAtMost_def greaterThan_def atLeastAtMost_def atLeast_def word_Suc_le)
 
 lemma word_atLeastLessThan_Suc_atLeastAtMost_union:
   fixes l::"'a::len word"
   assumes "m \<noteq> - 1" and "l \<le> m" and "m \<le> u"
   shows "{l..m} \<union> {m+1..u} = {l..u}"
-  proof -
+proof -
   from ivl_disj_un_two(8)[OF assms(2) assms(3)] have "{l..u} = {l..m} \<union> {m<..u}" by blast
   with assms show ?thesis by(simp add: word_atLeastAtMost_Suc_greaterThanAtMost)
-  qed
+qed
 
 lemma max_word_less_eq_iff [simp]:
   \<open>- 1 \<le> w \<longleftrightarrow> w = - 1\<close> for w :: \<open>'a::len word\<close>
   by (fact word_order.extremum_unique)
 
 lemma word_or_zero:
   "(a OR b = 0) = (a = 0 \<and> b = 0)"
   for a b :: \<open>'a::len word\<close>
   by (fact or_eq_0_iff)
 
 lemma word_2p_mult_inc:
   assumes x: "2 * 2 ^ n < (2::'a::len word) * 2 ^ m"
   assumes suc_n: "Suc n < LENGTH('a::len)"
   shows "2^n < (2::'a::len word)^m"
   by (smt suc_n le_less_trans lessI nat_less_le nat_mult_less_cancel_disj p2_gt_0
           power_Suc power_Suc unat_power_lower word_less_nat_alt x)
 
 lemma power_overflow:
   "n \<ge> LENGTH('a) \<Longrightarrow> 2 ^ n = (0 :: 'a::len word)"
   by simp
 
 lemmas extra_sle_sless_unfolds [simp] =
     word_sle_eq[where a=0 and b=1]
     word_sle_eq[where a=0 and b="numeral n"]
     word_sle_eq[where a=1 and b=0]
     word_sle_eq[where a=1 and b="numeral n"]
     word_sle_eq[where a="numeral n" and b=0]
     word_sle_eq[where a="numeral n" and b=1]
     word_sless_alt[where a=0 and b=1]
     word_sless_alt[where a=0 and b="numeral n"]
     word_sless_alt[where a=1 and b=0]
     word_sless_alt[where a=1 and b="numeral n"]
     word_sless_alt[where a="numeral n" and b=0]
     word_sless_alt[where a="numeral n" and b=1]
   for n
 
 lemma word_sint_1:
   "sint (1::'a::len word) = (if LENGTH('a) = 1 then -1 else 1)"
   by (fact signed_1)
 
 lemma ucast_of_nat:
   "is_down (ucast :: 'a :: len word \<Rightarrow> 'b :: len word)
     \<Longrightarrow> ucast (of_nat n :: 'a word) = (of_nat n :: 'b word)"
   by transfer simp
 
 lemma scast_1':
   "(scast (1::'a::len word) :: 'b::len word) =
    (word_of_int (signed_take_bit (LENGTH('a::len) - Suc 0) (1::int)))"
   by transfer simp
 
 lemma scast_1:
   "(scast (1::'a::len word) :: 'b::len word) = (if LENGTH('a) = 1 then -1 else 1)"
   by (fact signed_1)
 
 lemma unat_minus_one_word:
   "unat (-1 :: 'a :: len word) = 2 ^ LENGTH('a) - 1"
   by (simp add: mask_eq_exp_minus_1 unsigned_minus_1_eq_mask)
 
 lemmas word_diff_ls'' = word_diff_ls [where xa=x and x=x for x]
 lemmas word_diff_ls' = word_diff_ls'' [simplified]
 
 lemmas word_l_diffs' = word_l_diffs [where xa=x and x=x for x]
 lemmas word_l_diffs = word_l_diffs' [simplified]
 
 lemma two_power_increasing:
   "\<lbrakk> n \<le> m; m < LENGTH('a) \<rbrakk> \<Longrightarrow> (2 :: 'a :: len word) ^ n \<le> 2 ^ m"
   by (simp add: word_le_nat_alt)
 
 lemma word_leq_le_minus_one:
   "\<lbrakk> x \<le> y; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x - 1 < (y :: 'a :: len word)"
   apply (simp add: word_less_nat_alt word_le_nat_alt)
   apply (subst unat_minus_one)
    apply assumption
   apply (cases "unat x")
    apply (simp add: unat_eq_zero)
   apply arith
   done
 
 lemma neg_mask_combine:
   "NOT(mask a) AND NOT(mask b) = NOT(mask (max a b) :: 'a::len word)"
   by (rule bit_word_eqI) (auto simp add: bit_simps)
 
 lemma neg_mask_twice:
   "x AND NOT(mask n) AND NOT(mask m) = x AND NOT(mask (max n m))"
   for x :: \<open>'a::len word\<close>
   by (rule bit_word_eqI) (auto simp add: bit_simps)
 
 lemma multiple_mask_trivia:
   "n \<ge> m \<Longrightarrow> (x AND NOT(mask n)) + (x AND mask n AND NOT(mask m)) = x AND NOT(mask m)"
   for x :: \<open>'a::len word\<close>
   apply (rule trans[rotated], rule_tac w="mask n" in word_plus_and_or_coroll2)
   apply (simp add: word_bw_assocs word_bw_comms word_bw_lcs neg_mask_twice
                    max_absorb2)
   done
 
 lemma word_of_nat_less:
   "\<lbrakk> n < unat x \<rbrakk> \<Longrightarrow> of_nat n < x"
   apply (simp add: word_less_nat_alt)
   apply (erule order_le_less_trans[rotated])
   apply (simp add: unsigned_of_nat take_bit_eq_mod)
   done
 
 lemma unat_mask:
   "unat (mask n :: 'a :: len word) = 2 ^ (min n (LENGTH('a))) - 1"
   apply (subst min.commute)
   apply (simp add: mask_eq_decr_exp not_less min_def  split: if_split_asm)
   apply (intro conjI impI)
    apply (simp add: unat_sub_if_size)
    apply (simp add: power_overflow word_size)
   apply (simp add: unat_sub_if_size)
   done
 
 lemma mask_over_length:
   "LENGTH('a) \<le> n \<Longrightarrow> mask n = (-1::'a::len word)"
   by (simp add: mask_eq_decr_exp)
 
 lemma Suc_2p_unat_mask:
   "n < LENGTH('a) \<Longrightarrow> Suc (2 ^ n * k + unat (mask n :: 'a::len word)) = 2 ^ n * (k+1)"
   by (simp add: unat_mask)
 
 lemma sint_of_nat_ge_zero:
   "x < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> sint (of_nat x :: 'a :: len word) \<ge> 0"
   by (simp add: bit_iff_odd signed_of_nat)
 
 lemma int_eq_sint:
   "x < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> sint (of_nat x :: 'a :: len word) = int x"
   apply transfer
   apply (rule signed_take_bit_int_eq_self)
    apply simp_all
   apply (metis negative_zle numeral_power_eq_of_nat_cancel_iff)
   done
 
 lemma sint_of_nat_le:
   "\<lbrakk> b < 2 ^ (LENGTH('a) - 1); a \<le> b \<rbrakk>
    \<Longrightarrow> sint (of_nat a :: 'a :: len word) \<le> sint (of_nat b :: 'a :: len word)"
   apply (cases \<open>LENGTH('a)\<close>)
   apply simp_all
   apply transfer
   apply (subst signed_take_bit_eq_if_positive)
    apply (simp add: bit_simps)
   apply (metis bit_take_bit_iff nat_less_le order_less_le_trans take_bit_nat_eq_self_iff)
   apply (subst signed_take_bit_eq_if_positive)
     apply (simp add: bit_simps)
   apply (metis bit_take_bit_iff nat_less_le take_bit_nat_eq_self_iff)
     apply (simp flip: of_nat_take_bit add: take_bit_nat_eq_self)
   done
 
 lemma word_le_not_less:
   "((b::'a::len word) \<le> a) = (\<not>(a < b))"
   by fastforce
 
 lemma less_is_non_zero_p1:
   fixes a :: "'a :: len word"
   shows "a < k \<Longrightarrow> a + 1 \<noteq> 0"
   apply (erule contrapos_pn)
   apply (drule max_word_wrap)
   apply (simp add: not_less)
   done
 
 lemma unat_add_lem':
   "(unat x + unat y < 2 ^ LENGTH('a)) \<Longrightarrow>
     (unat (x + y :: 'a :: len word) = unat x + unat y)"
   by (subst unat_add_lem[symmetric], assumption)
 
 lemma word_less_two_pow_divI:
   "\<lbrakk> (x :: 'a::len word) < 2 ^ (n - m); m \<le> n; n < LENGTH('a) \<rbrakk> \<Longrightarrow> x < 2 ^ n div 2 ^ m"
   apply (simp add: word_less_nat_alt)
   apply (subst unat_word_ariths)
   apply (subst mod_less)
    apply (rule order_le_less_trans [OF div_le_dividend])
    apply (rule unat_lt2p)
   apply (simp add: power_sub)
   done
 
 lemma word_less_two_pow_divD:
   "\<lbrakk> (x :: 'a::len word) < 2 ^ n div 2 ^ m \<rbrakk>
      \<Longrightarrow> n \<ge> m \<and> (x < 2 ^ (n - m))"
   apply (cases "n < LENGTH('a)")
    apply (cases "m < LENGTH('a)")
     apply (simp add: word_less_nat_alt)
     apply (subst(asm) unat_word_ariths)
     apply (subst(asm) mod_less)
      apply (rule order_le_less_trans [OF div_le_dividend])
      apply (rule unat_lt2p)
     apply (clarsimp dest!: less_two_pow_divD)
    apply (simp add: power_overflow)
    apply (simp add: word_div_def)
   apply (simp add: power_overflow word_div_def)
   done
 
 lemma of_nat_less_two_pow_div_set:
   "\<lbrakk> n < LENGTH('a) \<rbrakk> \<Longrightarrow>
    {x. x < (2 ^ n div 2 ^ m :: 'a::len word)}
       = of_nat ` {k. k < 2 ^ n div 2 ^ m}"
   apply (simp add: image_def)
   apply (safe dest!: word_less_two_pow_divD less_two_pow_divD
              intro!: word_less_two_pow_divI)
    apply (rule_tac x="unat x" in exI)
    apply (simp add: power_sub[symmetric])
    apply (subst unat_power_lower[symmetric, where 'a='a])
     apply simp
    apply (erule unat_mono)
   apply (subst word_unat_power)
   apply (rule of_nat_mono_maybe)
    apply (rule power_strict_increasing)
     apply simp
    apply simp
   apply assumption
   done
 
 lemma ucast_less:
   "LENGTH('b) < LENGTH('a) \<Longrightarrow>
    (ucast (x :: 'b :: len word) :: ('a :: len word)) < 2 ^ LENGTH('b)"
   by transfer simp
 
 lemma ucast_range_less:
   "LENGTH('a :: len) < LENGTH('b :: len) \<Longrightarrow>
    range (ucast :: 'a word \<Rightarrow> 'b word) = {x. x < 2 ^ len_of TYPE ('a)}"
   apply safe
    apply (erule ucast_less)
   apply (simp add: image_def)
   apply (rule_tac x="ucast x" in exI)
   apply (rule bit_word_eqI)
   apply (auto simp add: bit_simps)
   apply (metis bit_take_bit_iff take_bit_word_eq_self_iff)
   done
 
 lemma word_power_less_diff:
   "\<lbrakk>2 ^ n * q < (2::'a::len word) ^ m; q < 2 ^ (LENGTH('a) - n)\<rbrakk> \<Longrightarrow> q < 2 ^ (m - n)"
   apply (case_tac "m \<ge> LENGTH('a)")
    apply (simp add: power_overflow)
   apply (case_tac "n \<ge> LENGTH('a)")
    apply (simp add: power_overflow)
   apply (cases "n = 0")
    apply simp
   apply (subst word_less_nat_alt)
   apply (subst unat_power_lower)
    apply simp
   apply (rule nat_power_less_diff)
   apply (simp add: word_less_nat_alt)
   apply (subst (asm) iffD1 [OF unat_mult_lem])
    apply (simp add:nat_less_power_trans)
   apply simp
   done
 
 lemma word_less_sub_1:
   "x < (y :: 'a :: len word) \<Longrightarrow> x \<le> y - 1"
   by (fact word_le_minus_one_leq)
 
 lemma word_sub_mono2:
-  "\<lbrakk> a + b \<le> c + d; c \<le> a; b \<le> a + b; d \<le> c + d \<rbrakk>
-    \<Longrightarrow> b \<le> (d :: 'a :: len word)"
-  apply (drule(1) word_sub_mono)
-    apply simp
-   apply simp
-  apply simp
-  done
+  "\<lbrakk> a + b \<le> c + d; c \<le> a; b \<le> a + b; d \<le> c + d \<rbrakk> \<Longrightarrow> b \<le> (d :: 'a :: len word)"
+  by (drule(1) word_sub_mono; simp)
 
 lemma word_not_le:
   "(\<not> x \<le> (y :: 'a :: len word)) = (y < x)"
-  by fastforce
+  by (fact not_le)
 
 lemma word_subset_less:
   "\<lbrakk> {x .. x + r - 1} \<subseteq> {y .. y + s - 1};
      x \<le> x + r - 1; y \<le> y + (s :: 'a :: len word) - 1;
      s \<noteq> 0 \<rbrakk>
      \<Longrightarrow> r \<le> s"
   apply (frule subsetD[where c=x])
    apply simp
   apply (drule subsetD[where c="x + r - 1"])
    apply simp
   apply (clarsimp simp: add_diff_eq[symmetric])
   apply (drule(1) word_sub_mono2)
     apply (simp_all add: olen_add_eqv[symmetric])
   apply (erule word_le_minus_cancel)
   apply (rule ccontr)
   apply (simp add: word_not_le)
   done
 
 lemma uint_power_lower:
   "n < LENGTH('a) \<Longrightarrow> uint (2 ^ n :: 'a :: len word) = (2 ^ n :: int)"
   by (rule uint_2p_alt)
 
 lemma power_le_mono:
   "\<lbrakk>2 ^ n \<le> (2::'a::len word) ^ m; n < LENGTH('a); m < LENGTH('a)\<rbrakk>
    \<Longrightarrow> n \<le> m"
   apply (clarsimp simp add: le_less)
   apply safe
   apply (simp add: word_less_nat_alt)
   apply (simp only: uint_arith_simps(3))
   apply (drule uint_power_lower)+
   apply simp
   done
 
 lemma two_power_eq:
   "\<lbrakk>n < LENGTH('a); m < LENGTH('a)\<rbrakk>
    \<Longrightarrow> ((2::'a::len word) ^ n = 2 ^ m) = (n = m)"
   apply safe
   apply (rule order_antisym)
    apply (simp add: power_le_mono[where 'a='a])+
   done
 
 lemma unat_less_helper:
   "x < of_nat n \<Longrightarrow> unat x < n"
   apply (simp add: word_less_nat_alt)
   apply (erule order_less_le_trans)
   apply (simp add: take_bit_eq_mod unsigned_of_nat)
   done
 
 lemma nat_uint_less_helper:
   "nat (uint y) = z \<Longrightarrow> x < y \<Longrightarrow> nat (uint x) < z"
   apply (erule subst)
   apply (subst unat_eq_nat_uint [symmetric])
   apply (subst unat_eq_nat_uint [symmetric])
   by (simp add: unat_mono)
 
 lemma of_nat_0:
   "\<lbrakk>of_nat n = (0::'a::len word); n < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow> n = 0"
   by (auto simp add: word_of_nat_eq_0_iff)
 
 lemma of_nat_inj:
   "\<lbrakk>x < 2 ^ LENGTH('a); y < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow>
    (of_nat x = (of_nat y :: 'a :: len word)) = (x = y)"
   by (metis unat_of_nat_len)
 
 lemma div_to_mult_word_lt:
   "\<lbrakk> (x :: 'a :: len word) \<le> y div z \<rbrakk> \<Longrightarrow> x * z \<le> y"
   apply (cases "z = 0")
    apply simp
   apply (simp add: word_neq_0_conv)
   apply (rule order_trans)
    apply (erule(1) word_mult_le_mono1)
    apply (simp add: unat_div)
    apply (rule order_le_less_trans [OF div_mult_le])
    apply simp
   apply (rule word_div_mult_le)
   done
 
 lemma ucast_ucast_mask:
   "(ucast :: 'a :: len word \<Rightarrow> 'b :: len word) (ucast x) = x AND mask (len_of TYPE ('a))"
   apply (simp flip: take_bit_eq_mask)
   apply transfer
   apply (simp add: ac_simps)
   done
 
 lemma ucast_ucast_len:
   "\<lbrakk> x < 2 ^ LENGTH('b) \<rbrakk> \<Longrightarrow> ucast (ucast x::'b::len word) = (x::'a::len word)"
   apply (subst ucast_ucast_mask)
   apply (erule less_mask_eq)
   done
 
 lemma ucast_ucast_id:
   "LENGTH('a) < LENGTH('b) \<Longrightarrow> ucast (ucast (x::'a::len word)::'b::len word) = x"
   by (auto intro: ucast_up_ucast_id simp: is_up_def source_size_def target_size_def word_size)
 
 lemma unat_ucast:
   "unat (ucast x :: ('a :: len) word) = unat x mod 2 ^ (LENGTH('a))"
 proof -
   have \<open>2 ^ LENGTH('a) = nat (2 ^ LENGTH('a))\<close>
     by simp
   moreover have \<open>unat (ucast x :: 'a word) = unat x mod nat (2 ^ LENGTH('a))\<close>
     by transfer (simp flip: nat_mod_distrib take_bit_eq_mod)
   ultimately show ?thesis
     by (simp only:)
 qed
 
 lemma ucast_less_ucast:
   "LENGTH('a) \<le> LENGTH('b) \<Longrightarrow>
    (ucast x < ((ucast (y :: 'a::len word)) :: 'b::len word)) = (x < y)"
   apply (simp add: word_less_nat_alt unat_ucast)
   apply (subst mod_less)
    apply(rule less_le_trans[OF unat_lt2p], simp)
   apply (subst mod_less)
    apply(rule less_le_trans[OF unat_lt2p], simp)
   apply simp
   done
 
 \<comment> \<open>This weaker version was previously called @{text ucast_less_ucast}. We retain it to
     support existing proofs.\<close>
 lemmas ucast_less_ucast_weak = ucast_less_ucast[OF order.strict_implies_order]
 
 lemma unat_Suc2:
   fixes n :: "'a :: len word"
   shows
   "n \<noteq> -1 \<Longrightarrow> unat (n + 1) = Suc (unat n)"
   apply (subst add.commute, rule unatSuc)
   apply (subst eq_diff_eq[symmetric], simp add: minus_equation_iff)
   done
 
 lemma word_div_1:
   "(n :: 'a :: len word) div 1 = n"
   by (fact bits_div_by_1)
 
 lemma word_minus_one_le:
   "-1 \<le> (x :: 'a :: len word) = (x = -1)"
   by (fact word_order.extremum_unique)
 
 lemma up_scast_inj:
-      "\<lbrakk> scast x = (scast y :: 'b :: len word); size x \<le> LENGTH('b) \<rbrakk>
-         \<Longrightarrow> x = y"
+  "\<lbrakk> scast x = (scast y :: 'b :: len word); size x \<le> LENGTH('b) \<rbrakk> \<Longrightarrow> x = y"
   apply transfer
-  apply (cases \<open>LENGTH('a)\<close>)
-  apply simp_all
+  apply (cases \<open>LENGTH('a)\<close>; simp)
   apply (metis order_refl take_bit_signed_take_bit take_bit_tightened)
   done
 
 lemma up_scast_inj_eq:
   "LENGTH('a) \<le> len_of TYPE ('b) \<Longrightarrow>
   (scast x = (scast y::'b::len word)) = (x = (y::'a::len word))"
   by (fastforce dest: up_scast_inj simp: word_size)
 
 lemma word_le_add:
   fixes x :: "'a :: len word"
   shows "x \<le> y \<Longrightarrow> \<exists>n. y = x + of_nat n"
   by (rule exI [where x = "unat (y - x)"]) simp
 
 lemma word_plus_mcs_4':
-  fixes x :: "'a :: len word"
-  shows "\<lbrakk>x + v \<le> x + w; x \<le> x + v\<rbrakk> \<Longrightarrow> v \<le> w"
-  apply (rule word_plus_mcs_4)
-   apply (simp add: add.commute)
-  apply (simp add: add.commute)
-  done
+  "\<lbrakk>x + v \<le> x + w; x \<le> x + v\<rbrakk> \<Longrightarrow> v \<le> w" for x :: "'a::len word"
+  by (rule word_plus_mcs_4; simp add: add.commute)
 
 lemma unat_eq_1:
   \<open>unat x = Suc 0 \<longleftrightarrow> x = 1\<close>
   by (auto intro!: unsigned_word_eqI [where ?'a = nat])
 
 lemma word_unat_Rep_inject1:
   \<open>unat x = unat 1 \<longleftrightarrow> x = 1\<close>
   by (simp add: unat_eq_1)
 
 lemma and_not_mask_twice:
   "(w AND NOT (mask n)) AND NOT (mask m) = w AND NOT (mask (max m n))"
   for w :: \<open>'a::len word\<close>
   by (rule bit_word_eqI) (auto simp add: bit_simps)
 
 lemma word_less_cases:
   "x < y \<Longrightarrow> x = y - 1 \<or> x < y - (1 ::'a::len word)"
   apply (drule word_less_sub_1)
   apply (drule order_le_imp_less_or_eq)
   apply auto
   done
 
 lemma mask_and_mask:
   "mask a AND mask b = (mask (min a b) :: 'a::len word)"
   by (simp flip: take_bit_eq_mask ac_simps)
 
 lemma mask_eq_0_eq_x:
   "(x AND w = 0) = (x AND NOT w = x)"
   for x w :: \<open>'a::len word\<close>
   using word_plus_and_or_coroll2[where x=x and w=w]
   by auto
 
 lemma mask_eq_x_eq_0:
   "(x AND w = x) = (x AND NOT w = 0)"
   for x w :: \<open>'a::len word\<close>
   using word_plus_and_or_coroll2[where x=x and w=w]
   by auto
 
 lemma compl_of_1: "NOT 1 = (-2 :: 'a :: len word)"
   by (fact not_one_eq)
 
 lemma split_word_eq_on_mask:
   "(x = y) = (x AND m = y AND m \<and> x AND NOT m = y AND NOT m)"
   for x y m :: \<open>'a::len word\<close>
   apply transfer
   apply (simp add: bit_eq_iff)
   apply (auto simp add: bit_simps ac_simps)
   done
 
 lemma word_FF_is_mask:
   "0xFF = (mask 8 :: 'a::len word)"
   by (simp add: mask_eq_decr_exp)
 
 lemma word_1FF_is_mask:
   "0x1FF = (mask 9 :: 'a::len word)"
   by (simp add: mask_eq_decr_exp)
 
 lemma ucast_of_nat_small:
   "x < 2 ^ LENGTH('a) \<Longrightarrow> ucast (of_nat x :: 'a :: len word) = (of_nat x :: 'b :: len word)"
   apply transfer
   apply (auto simp add: take_bit_of_nat min_def not_le)
   apply (metis linorder_not_less min_def take_bit_nat_eq_self take_bit_take_bit)
   done
 
 lemma word_le_make_less:
   fixes x :: "'a :: len word"
   shows "y \<noteq> -1 \<Longrightarrow> (x \<le> y) = (x < (y + 1))"
   apply safe
   apply (erule plus_one_helper2)
   apply (simp add: eq_diff_eq[symmetric])
   done
 
 lemmas finite_word = finite [where 'a="'a::len word"]
 
 lemma word_to_1_set:
   "{0 ..< (1 :: 'a :: len word)} = {0}"
   by fastforce
 
 lemma word_leq_minus_one_le:
   fixes x :: "'a::len word"
   shows "\<lbrakk>y \<noteq> 0; x \<le> y - 1 \<rbrakk> \<Longrightarrow> x < y"
   using le_m1_iff_lt word_neq_0_conv by blast
 
 lemma word_count_from_top:
   "n \<noteq> 0 \<Longrightarrow> {0 ..< n :: 'a :: len word} = {0 ..< n - 1} \<union> {n - 1}"
   apply (rule set_eqI, rule iffI)
    apply simp
    apply (drule word_le_minus_one_leq)
    apply (rule disjCI)
    apply simp
   apply simp
   apply (erule word_leq_minus_one_le)
   apply fastforce
   done
 
 lemma word_minus_one_le_leq:
   "\<lbrakk> x - 1 < y \<rbrakk> \<Longrightarrow> x \<le> (y :: 'a :: len word)"
   apply (cases "x = 0")
    apply simp
   apply (simp add: word_less_nat_alt word_le_nat_alt)
   apply (subst(asm) unat_minus_one)
    apply (simp add: word_less_nat_alt)
   apply (cases "unat x")
    apply (simp add: unat_eq_zero)
   apply arith
   done
 
 lemma word_must_wrap:
   "\<lbrakk> x \<le> n - 1; n \<le> x \<rbrakk> \<Longrightarrow> n = (0 :: 'a :: len word)"
   using dual_order.trans sub_wrap word_less_1 by blast
 
 lemma range_subset_card:
   "\<lbrakk> {a :: 'a :: len word .. b} \<subseteq> {c .. d}; b \<ge> a \<rbrakk> \<Longrightarrow> d \<ge> c \<and> d - c \<ge> b - a"
   using word_sub_le word_sub_mono by fastforce
 
 lemma less_1_simp:
   "n - 1 < m = (n \<le> (m :: 'a :: len word) \<and> n \<noteq> 0)"
   by unat_arith
 
 lemma word_power_mod_div:
   fixes x :: "'a::len word"
   shows "\<lbrakk> n < LENGTH('a); m < LENGTH('a)\<rbrakk>
   \<Longrightarrow> x mod 2 ^ n div 2 ^ m = x div 2 ^ m mod 2 ^ (n - m)"
   apply (simp add: word_arith_nat_div unat_mod power_mod_div)
   apply (subst unat_arith_simps(3))
   apply (subst unat_mod)
   apply (subst unat_of_nat)+
   apply (simp add: mod_mod_power min.commute)
   done
 
 lemma word_range_minus_1':
   fixes a :: "'a :: len word"
   shows "a \<noteq> 0 \<Longrightarrow> {a - 1<..b} = {a..b}"
   by (simp add: greaterThanAtMost_def atLeastAtMost_def greaterThan_def atLeast_def less_1_simp)
 
 lemma word_range_minus_1:
   fixes a :: "'a :: len word"
   shows "b \<noteq> 0 \<Longrightarrow> {a..b - 1} = {a..<b}"
   apply (simp add: atLeastLessThan_def atLeastAtMost_def atMost_def lessThan_def)
   apply (rule arg_cong [where f = "\<lambda>x. {a..} \<inter> x"])
   apply rule
    apply clarsimp
    apply (erule contrapos_pp)
    apply (simp add: linorder_not_less linorder_not_le word_must_wrap)
   apply (clarsimp)
   apply (drule word_le_minus_one_leq)
   apply (auto simp: word_less_sub_1)
   done
 
 lemma ucast_nat_def:
   "of_nat (unat x) = (ucast :: 'a :: len word \<Rightarrow> 'b :: len word) x"
   by transfer simp
 
 lemma overflow_plus_one_self:
   "(1 + p \<le> p) = (p = (-1 :: 'a :: len word))"
   apply rule
   apply (rule ccontr)
    apply (drule plus_one_helper2)
    apply (rule notI)
    apply (drule arg_cong[where f="\<lambda>x. x - 1"])
    apply simp
    apply (simp add: field_simps)
   apply simp
   done
 
 lemma plus_1_less:
   "(x + 1 \<le> (x :: 'a :: len word)) = (x = -1)"
   apply (rule iffI)
    apply (rule ccontr)
    apply (cut_tac plus_one_helper2[where x=x, OF order_refl])
     apply simp
    apply clarsimp
    apply (drule arg_cong[where f="\<lambda>x. x - 1"])
    apply simp
   apply simp
   done
 
 lemma pos_mult_pos_ge:
   "[|x > (0::int); n>=0 |] ==> n * x >= n*1"
   apply (simp only: mult_left_mono)
   done
 
 lemma word_plus_strict_mono_right:
   fixes x :: "'a :: len word"
   shows "\<lbrakk>y < z; x \<le> x + z\<rbrakk> \<Longrightarrow> x + y < x + z"
   by unat_arith
 
 lemma word_div_mult:
   "0 < c \<Longrightarrow> a < b * c \<Longrightarrow> a div c < b" for a b c :: "'a::len word"
   by (rule classical)
      (use div_to_mult_word_lt [of b a c] in
       \<open>auto simp add: word_less_nat_alt word_le_nat_alt unat_div\<close>)
 
 lemma word_less_power_trans_ofnat:
   "\<lbrakk>n < 2 ^ (m - k); k \<le> m; m < LENGTH('a)\<rbrakk>
    \<Longrightarrow> of_nat n * 2 ^ k < (2::'a::len word) ^ m"
   apply (subst mult.commute)
   apply (rule word_less_power_trans)
     apply (simp_all add: word_less_nat_alt unsigned_of_nat)
   using take_bit_nat_less_eq_self
   apply (rule le_less_trans)
   apply assumption
   done
 
 lemma word_1_le_power:
   "n < LENGTH('a) \<Longrightarrow> (1 :: 'a :: len word) \<le> 2 ^ n"
   by (rule inc_le[where i=0, simplified], erule iffD2[OF p2_gt_0])
 
 lemma unat_1_0:
   "1 \<le> (x::'a::len word) = (0 < unat x)"
   by (auto simp add: word_le_nat_alt)
 
 lemma x_less_2_0_1':
   fixes x :: "'a::len word"
   shows "\<lbrakk>LENGTH('a) \<noteq> 1; x < 2\<rbrakk> \<Longrightarrow> x = 0 \<or> x = 1"
-  apply (cases \<open>2 \<le> LENGTH('a)\<close>)
-  apply simp_all
+  apply (cases \<open>2 \<le> LENGTH('a)\<close>; simp)
   apply transfer
-  apply auto
-  apply (metis add.commute add.right_neutral even_two_times_div_two mod_div_trivial mod_pos_pos_trivial mult.commute mult_zero_left not_less not_take_bit_negative odd_two_times_div_two_succ)
+  apply clarsimp
+  apply (metis add.commute add.right_neutral even_two_times_div_two mod_div_trivial
+               mod_pos_pos_trivial mult.commute mult_zero_left not_less not_take_bit_negative
+               odd_two_times_div_two_succ)
   done
 
 lemmas word_add_le_iff2 = word_add_le_iff [folded no_olen_add_nat]
 
 lemma of_nat_power:
   shows "\<lbrakk> p < 2 ^ x; x < len_of TYPE ('a) \<rbrakk> \<Longrightarrow> of_nat p < (2 :: 'a :: len word) ^ x"
   apply (rule order_less_le_trans)
    apply (rule of_nat_mono_maybe)
     apply (erule power_strict_increasing)
     apply simp
    apply assumption
   apply (simp add: word_unat_power del: of_nat_power)
   done
 
 lemma of_nat_n_less_equal_power_2:
   "n < LENGTH('a::len) \<Longrightarrow> ((of_nat n)::'a word) < 2 ^ n"
   apply (induct n)
    apply clarsimp
   apply clarsimp
   apply (metis of_nat_power n_less_equal_power_2 of_nat_Suc power_Suc)
   done
 
 lemma eq_mask_less:
   fixes w :: "'a::len word"
   assumes eqm: "w = w AND mask n"
   and      sz: "n < len_of TYPE ('a)"
   shows "w < (2::'a word) ^ n"
   by (subst eqm, rule and_mask_less' [OF sz])
 
 lemma of_nat_mono_maybe':
   fixes Y :: "nat"
   assumes xlt: "x < 2 ^ len_of TYPE ('a)"
   assumes ylt: "y < 2 ^ len_of TYPE ('a)"
   shows   "(y < x) = (of_nat y < (of_nat x :: 'a :: len word))"
   apply (subst word_less_nat_alt)
   apply (subst unat_of_nat)+
   apply (subst mod_less)
    apply (rule ylt)
   apply (subst mod_less)
    apply (rule xlt)
   apply simp
   done
 
 lemma of_nat_mono_maybe_le:
   "\<lbrakk>x < 2 ^ LENGTH('a); y < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow>
   (y \<le> x) = ((of_nat y :: 'a :: len word) \<le> of_nat x)"
   apply (clarsimp simp: le_less)
   apply (rule disj_cong)
    apply (rule of_nat_mono_maybe', assumption+)
   apply auto
   using of_nat_inj apply blast
   done
 
 lemma mask_AND_NOT_mask:
   "(w AND NOT (mask n)) AND mask n = 0"
   for w :: \<open>'a::len word\<close>
   by (rule bit_word_eqI) (simp add: bit_simps)
 
 lemma AND_NOT_mask_plus_AND_mask_eq:
   "(w AND NOT (mask n)) + (w AND mask n) = w"
   for w :: \<open>'a::len word\<close>
   apply (subst disjunctive_add)
   apply (auto simp add: bit_simps)
   apply (rule bit_word_eqI)
   apply (auto simp add: bit_simps)
   done
 
 lemma mask_eqI:
   fixes x :: "'a :: len word"
   assumes m1: "x AND mask n = y AND mask n"
   and     m2: "x AND NOT (mask n) = y AND NOT (mask n)"
   shows "x = y"
 proof -
   have *: \<open>x = x AND mask n OR x AND NOT (mask n)\<close> for x :: \<open>'a word\<close>
     by (rule bit_word_eqI) (auto simp add: bit_simps)
   from assms * [of x] * [of y] show ?thesis
     by simp
 qed
 
 lemma neq_0_no_wrap:
   fixes x :: "'a :: len word"
   shows "\<lbrakk> x \<le> x + y; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x + y \<noteq> 0"
   by clarsimp
 
 lemma unatSuc2:
   fixes n :: "'a :: len word"
   shows "n + 1 \<noteq> 0 \<Longrightarrow> unat (n + 1) = Suc (unat n)"
   by (simp add: add.commute unatSuc)
 
 lemma word_of_nat_le:
   "n \<le> unat x \<Longrightarrow> of_nat n \<le> x"
   apply (simp add: word_le_nat_alt unat_of_nat)
   apply (erule order_trans[rotated])
   apply (simp add: take_bit_eq_mod)
   done
 
 lemma word_unat_less_le:
   "a \<le> of_nat b \<Longrightarrow> unat a \<le> b"
   by (metis eq_iff le_cases le_unat_uoi word_of_nat_le)
 
 lemma mask_Suc_0 : "mask (Suc 0) = (1 :: 'a::len word)"
   by (simp add: mask_eq_decr_exp)
 
 lemma bool_mask':
   fixes x :: "'a :: len word"
   shows "2 < LENGTH('a) \<Longrightarrow> (0 < x AND 1) = (x AND 1 = 1)"
   by (simp add: and_one_eq mod_2_eq_odd)
 
 lemma ucast_ucast_add:
   fixes x :: "'a :: len word"
   fixes y :: "'b :: len word"
   shows
   "LENGTH('b) \<ge> LENGTH('a) \<Longrightarrow>
     ucast (ucast x + y) = x + ucast y"
   apply transfer
   apply simp
   apply (subst (2) take_bit_add [symmetric])
   apply (subst take_bit_add [symmetric])
   apply simp
   done
 
 lemma lt1_neq0:
   fixes x :: "'a :: len word"
   shows "(1 \<le> x) = (x \<noteq> 0)" by unat_arith
 
 lemma word_plus_one_nonzero:
   fixes x :: "'a :: len word"
   shows "\<lbrakk>x \<le> x + y; y \<noteq> 0\<rbrakk> \<Longrightarrow> x + 1 \<noteq> 0"
   apply (subst lt1_neq0 [symmetric])
   apply (subst olen_add_eqv [symmetric])
   apply (erule word_random)
   apply (simp add: lt1_neq0)
   done
 
 lemma word_sub_plus_one_nonzero:
   fixes n :: "'a :: len word"
   shows "\<lbrakk>n' \<le> n; n' \<noteq> 0\<rbrakk> \<Longrightarrow> (n - n') + 1 \<noteq> 0"
   apply (subst lt1_neq0 [symmetric])
   apply (subst olen_add_eqv [symmetric])
   apply (rule word_random [where x' = n'])
    apply simp
    apply (erule word_sub_le)
   apply (simp add: lt1_neq0)
   done
 
 lemma word_le_minus_mono_right:
   fixes x :: "'a :: len word"
   shows "\<lbrakk> z \<le> y; y \<le> x; z \<le> x \<rbrakk> \<Longrightarrow> x - y \<le> x - z"
   apply (rule word_sub_mono)
      apply simp
     apply assumption
    apply (erule word_sub_le)
   apply (erule word_sub_le)
   done
 
 lemma word_0_sle_from_less:
   \<open>0 \<le>s x\<close> if \<open>x < 2 ^ (LENGTH('a) - 1)\<close> for x :: \<open>'a::len word\<close>
   using that
   apply transfer
   apply (cases \<open>LENGTH('a)\<close>)
    apply simp_all
   apply (metis bit_take_bit_iff min_def nat_less_le not_less_eq take_bit_int_eq_self_iff take_bit_take_bit)
   done
 
 lemma ucast_sub_ucast:
   fixes x :: "'a::len word"
   assumes "y \<le> x"
   assumes T: "LENGTH('a) \<le> LENGTH('b)"
   shows "ucast (x - y) = (ucast x - ucast y :: 'b::len word)"
 proof -
   from T
   have P: "unat x < 2 ^ LENGTH('b)" "unat y < 2 ^ LENGTH('b)"
     by (fastforce intro!: less_le_trans[OF unat_lt2p])+
   then show ?thesis
     by (simp add: unat_arith_simps unat_ucast assms[simplified unat_arith_simps])
 qed
 
 lemma word_1_0:
   "\<lbrakk>a + (1::('a::len) word) \<le> b; a < of_nat x\<rbrakk> \<Longrightarrow> a < b"
   apply transfer
   apply (subst (asm) take_bit_incr_eq)
    apply (auto simp add: diff_less_eq)
   using take_bit_int_less_exp le_less_trans by blast
 
 lemma unat_of_nat_less:"\<lbrakk> a < b; unat b = c \<rbrakk> \<Longrightarrow> a < of_nat c"
   by fastforce
 
 lemma word_le_plus_1: "\<lbrakk> (y::('a::len) word) < y + n; a < n \<rbrakk> \<Longrightarrow> y + a \<le> y + a + 1"
   by unat_arith
 
 lemma word_le_plus:"\<lbrakk>(a::('a::len) word) < a + b; c < b\<rbrakk> \<Longrightarrow> a \<le> a + c"
   by (metis order_less_imp_le word_random)
 
 lemma sint_minus1 [simp]: "(sint x = -1) = (x = -1)"
   apply (cases \<open>LENGTH('a)\<close>)
    apply simp_all
   apply transfer
   apply (simp only: flip: signed_take_bit_eq_iff_take_bit_eq)
   apply simp
   done
 
 lemma sint_0 [simp]: "(sint x = 0) = (x = 0)"
   by (fact signed_eq_0_iff)
 
 (* It is not always that case that "sint 1 = 1", because of 1-bit word sizes.
  * This lemma produces the different cases. *)
 lemma sint_1_cases:
   P if \<open>\<lbrakk> len_of TYPE ('a::len) = 1; (a::'a word) = 0; sint a = 0 \<rbrakk> \<Longrightarrow> P\<close>
      \<open>\<lbrakk> len_of TYPE ('a) = 1; a = 1; sint (1 :: 'a word) = -1 \<rbrakk> \<Longrightarrow> P\<close>
      \<open>\<lbrakk> len_of TYPE ('a) > 1; sint (1 :: 'a word) = 1 \<rbrakk> \<Longrightarrow> P\<close>
 proof (cases \<open>LENGTH('a) = 1\<close>)
   case True
   then have \<open>a = 0 \<or> a = 1\<close>
     by transfer auto
   with True that show ?thesis
     by auto
 next
   case False
   with that show ?thesis
     by (simp add: less_le Suc_le_eq)
 qed
 
 lemma sint_int_min:
   "sint (- (2 ^ (LENGTH('a) - Suc 0)) :: ('a::len) word) = - (2 ^ (LENGTH('a) - Suc 0))"
   apply (cases \<open>LENGTH('a)\<close>)
    apply simp_all
   apply transfer
   apply (simp add: signed_take_bit_int_eq_self)
   done
 
 lemma sint_int_max_plus_1:
   "sint (2 ^ (LENGTH('a) - Suc 0) :: ('a::len) word) = - (2 ^ (LENGTH('a) - Suc 0))"
   apply (cases \<open>LENGTH('a)\<close>)
    apply simp_all
   apply (subst word_of_int_2p [symmetric])
   apply (subst int_word_sint)
   apply simp
   done
 
 lemma uint_range':
   \<open>0 \<le> uint x \<and> uint x < 2 ^ LENGTH('a)\<close>
   for x :: \<open>'a::len word\<close>
   by transfer simp
 
 lemma sint_of_int_eq:
   "\<lbrakk> - (2 ^ (LENGTH('a) - 1)) \<le> x; x < 2 ^ (LENGTH('a) - 1) \<rbrakk> \<Longrightarrow> sint (of_int x :: ('a::len) word) = x"
   by (simp add: signed_take_bit_int_eq_self signed_of_int)
 
 lemma of_int_sint:
   "of_int (sint a) = a"
   by simp
 
 lemma sint_ucast_eq_uint:
     "\<lbrakk> \<not> is_down (ucast :: ('a::len word \<Rightarrow> 'b::len word)) \<rbrakk>
             \<Longrightarrow> sint ((ucast :: ('a::len word \<Rightarrow> 'b::len word)) x) = uint x"
   apply transfer
   apply (simp add: signed_take_bit_take_bit)
   done
 
 lemma word_less_nowrapI':
   "(x :: 'a :: len word) \<le> z - k \<Longrightarrow> k \<le> z \<Longrightarrow> 0 < k \<Longrightarrow> x < x + k"
   by uint_arith
 
 lemma mask_plus_1:
   "mask n + 1 = (2 ^ n :: 'a::len word)"
   by (clarsimp simp: mask_eq_decr_exp)
 
 lemma unat_inj: "inj unat"
   by (metis eq_iff injI word_le_nat_alt)
 
 lemma unat_ucast_upcast:
   "is_up (ucast :: 'b word \<Rightarrow> 'a word)
       \<Longrightarrow> unat (ucast x :: ('a::len) word) = unat (x :: ('b::len) word)"
   unfolding ucast_eq unat_eq_nat_uint
   apply transfer
   apply simp
   done
 
 lemma ucast_mono:
   "\<lbrakk> (x :: 'b :: len word) < y; y < 2 ^ LENGTH('a) \<rbrakk>
    \<Longrightarrow> ucast x < ((ucast y) :: 'a :: len word)"
   apply (simp only: flip: ucast_nat_def)
   apply (rule of_nat_mono_maybe)
   apply (rule unat_less_helper)
   apply simp
   apply (simp add: word_less_nat_alt)
   done
 
 lemma ucast_mono_le:
   "\<lbrakk>x \<le> y; y < 2 ^ LENGTH('b)\<rbrakk> \<Longrightarrow> (ucast (x :: 'a :: len word) :: 'b :: len word) \<le> ucast y"
   apply (simp only: flip: ucast_nat_def)
   apply (subst of_nat_mono_maybe_le[symmetric])
     apply (rule unat_less_helper)
     apply simp
    apply (rule unat_less_helper)
    apply (erule le_less_trans)
   apply (simp_all add: word_le_nat_alt)
   done
 
 lemma ucast_mono_le':
   "\<lbrakk> unat y < 2 ^ LENGTH('b); LENGTH('b::len) < LENGTH('a::len); x \<le> y \<rbrakk>
    \<Longrightarrow> ucast x \<le> (ucast y :: 'b word)" for x y :: \<open>'a::len word\<close>
   by (auto simp: word_less_nat_alt intro: ucast_mono_le)
 
 lemma neg_mask_add_mask:
   "((x:: 'a :: len word) AND NOT (mask n)) + (2 ^ n - 1) = x OR mask n"
   unfolding mask_2pm1 [symmetric]
   apply (subst word_plus_and_or_coroll; rule bit_word_eqI)
    apply (auto simp add: bit_simps)
   done
 
 lemma le_step_down_word:"\<lbrakk>(i::('a::len) word) \<le> n; i = n \<longrightarrow> P; i \<le> n - 1 \<longrightarrow> P\<rbrakk> \<Longrightarrow> P"
   by unat_arith
 
 lemma le_step_down_word_2:
   fixes x :: "'a::len word"
   shows "\<lbrakk>x \<le>  y; x \<noteq> y\<rbrakk> \<Longrightarrow> x \<le> y - 1"
   by (subst (asm) word_le_less_eq,
       clarsimp,
       simp add: word_le_minus_one_leq)
 
 lemma NOT_mask_AND_mask[simp]: "(w AND mask n) AND NOT (mask n) = 0"
   by (rule bit_eqI) (simp add: bit_simps)
 
 lemma and_and_not[simp]:"(a AND b) AND NOT b = 0"
   for a b :: \<open>'a::len word\<close>
   apply (subst word_bw_assocs(1))
   apply clarsimp
   done
 
 lemma ex_mask_1[simp]: "(\<exists>x. mask x = (1 :: 'a::len word))"
   apply (rule_tac x=1 in exI)
   apply (simp add:mask_eq_decr_exp)
   done
 
 lemma not_switch:"NOT a = x \<Longrightarrow> a = NOT x"
   by auto
 
 lemma test_bit_eq_iff: "bit u = bit v \<longleftrightarrow> u = v"
   for u v :: "'a::len word"
   by (auto intro: bit_eqI simp add: fun_eq_iff)
 
 lemma test_bit_size: "bit w n \<Longrightarrow> n < size w"
   for w :: "'a::len word"
   by transfer simp
 
-lemma word_eq_iff: "x = y \<longleftrightarrow> (\<forall>n<LENGTH('a). bit x n = bit y n)" (is \<open>?P \<longleftrightarrow> ?Q\<close>)
+lemma word_eq_iff: "x = y \<longleftrightarrow> (\<forall>n<LENGTH('a). bit x n = bit y n)"
   for x y :: "'a::len word"
   by transfer (auto simp add: bit_eq_iff bit_take_bit_iff)
 
 lemma word_eqI: "(\<And>n. n < size u \<longrightarrow> bit u n = bit v n) \<Longrightarrow> u = v"
   for u :: "'a::len word"
   by (simp add: word_size word_eq_iff)
 
 lemma word_eqD: "u = v \<Longrightarrow> bit u x = bit v x"
   for u v :: "'a::len word"
   by simp
 
 lemma test_bit_bin': "bit w n \<longleftrightarrow> n < size w \<and> bit (uint w) n"
   by transfer (simp add: bit_take_bit_iff)
 
 lemmas test_bit_bin = test_bit_bin' [unfolded word_size]
 
 lemma word_test_bit_def:
   \<open>bit a = bit (uint a)\<close>
   by transfer (simp add: fun_eq_iff bit_take_bit_iff)
 
 lemmas test_bit_def' = word_test_bit_def [THEN fun_cong]
 
 lemma word_test_bit_transfer [transfer_rule]:
   "(rel_fun pcr_word (rel_fun (=) (=)))
     (\<lambda>x n. n < LENGTH('a) \<and> bit x n) (bit :: 'a::len word \<Rightarrow> _)"
   by transfer_prover
 
 lemma test_bit_wi:
   "bit (word_of_int x :: 'a::len word) n \<longleftrightarrow> n < LENGTH('a) \<and> bit x n"
   by transfer simp
 
 lemma word_ops_nth_size:
   "n < size x \<Longrightarrow>
     bit (x OR y) n = (bit x n | bit y n) \<and>
     bit (x AND y) n = (bit x n \<and> bit y n) \<and>
     bit (x XOR y) n = (bit x n \<noteq> bit y n) \<and>
     bit (NOT x) n = (\<not> bit x n)"
   for x :: "'a::len word"
   by transfer (simp add: bit_or_iff bit_and_iff bit_xor_iff bit_not_iff)
 
 lemma word_ao_nth:
   "bit (x OR y) n = (bit x n | bit y n) \<and>
     bit (x AND y) n = (bit x n \<and> bit y n)"
   for x :: "'a::len word"
   by transfer (auto simp add: bit_or_iff bit_and_iff)
 
 lemmas lsb0 = len_gt_0 [THEN word_ops_nth_size [unfolded word_size]]
 
 lemma nth_sint:
   fixes w :: "'a::len word"
   defines "l \<equiv> LENGTH('a)"
   shows "bit (sint w) n = (if n < l - 1 then bit w n else bit w (l - 1))"
   unfolding sint_uint l_def
   by (auto simp: bit_signed_take_bit_iff word_test_bit_def not_less min_def)
 
 lemma test_bit_2p: "bit (word_of_int (2 ^ n)::'a::len word) m \<longleftrightarrow> m = n \<and> m < LENGTH('a)"
   by transfer (auto simp add: bit_exp_iff)
 
 lemma nth_w2p: "bit ((2::'a::len word) ^ n) m \<longleftrightarrow> m = n \<and> m < LENGTH('a::len)"
   by transfer (auto simp add: bit_exp_iff)
 
 lemma bang_is_le: "bit x m \<Longrightarrow> 2 ^ m \<le> x"
   for x :: "'a::len word"
   apply (rule xtrans(3))
    apply (rule_tac [2] y = "x" in le_word_or2)
   apply (rule word_eqI)
   apply (auto simp add: word_ao_nth nth_w2p word_size)
   done
 
 lemmas msb0 = len_gt_0 [THEN diff_Suc_less, THEN word_ops_nth_size [unfolded word_size]]
 lemmas msb1 = msb0 [where i = 0]
 
 lemma test_bit_1 [iff]: "bit (1 :: 'a::len word) n \<longleftrightarrow> n = 0"
   by transfer (auto simp add: bit_1_iff)
 
 lemma nth_0: "\<not> bit (0 :: 'a::len word) n"
   by transfer simp
 
 lemma nth_minus1: "bit (-1 :: 'a::len word) n \<longleftrightarrow> n < LENGTH('a)"
   by transfer simp
 
-lemma nth_ucast:
+lemma nth_ucast_weak:
   "bit (ucast w::'a::len word) n = (bit w n \<and> n < LENGTH('a))"
   by transfer (simp add: bit_take_bit_iff ac_simps)
 
-lemma drop_bit_numeral_bit0_1 [simp]:
-  \<open>drop_bit (Suc 0) (numeral k) =
-    (word_of_int (drop_bit (Suc 0) (take_bit LENGTH('a) (numeral k))) :: 'a::len word)\<close>
-  by (metis Word_eq_word_of_int drop_bit_word.abs_eq of_int_numeral)
+lemma nth_ucast:
+  "bit (ucast (w::'a::len word)::'b::len word) n =
+   (bit w n \<and> n < min LENGTH('a) LENGTH('b))"
+  by (auto simp: not_le nth_ucast_weak dest: bit_imp_le_length)
 
 lemma nth_mask:
   \<open>bit (mask n :: 'a::len word) i \<longleftrightarrow> i < n \<and> i < size (mask n :: 'a word)\<close>
   by (auto simp add: word_size Word.bit_mask_iff)
 
 lemma nth_slice: "bit (slice n w :: 'a::len word) m = (bit w (m + n) \<and> m < LENGTH('a))"
   apply (auto simp add: bit_simps less_diff_conv dest: bit_imp_le_length)
   using bit_imp_le_length
   apply fastforce
   done
 
 lemma test_bit_cat [OF refl]:
   "wc = word_cat a b \<Longrightarrow> bit wc n = (n < size wc \<and>
     (if n < size b then bit b n else bit a (n - size b)))"
   apply (simp add: word_size not_less; transfer)
        apply (auto simp add: bit_concat_bit_iff bit_take_bit_iff)
   done
 
 \<comment> \<open>keep quantifiers for use in simplification\<close>
 lemma test_bit_split':
   "word_split c = (a, b) \<longrightarrow>
     (\<forall>n m.
       bit b n = (n < size b \<and> bit c n) \<and>
       bit a m = (m < size a \<and> bit c (m + size b)))"
   by (auto simp add: word_split_bin' bit_unsigned_iff word_size bit_drop_bit_eq ac_simps
            dest: bit_imp_le_length)
 
 lemma test_bit_split:
   "word_split c = (a, b) \<Longrightarrow>
     (\<forall>n::nat. bit b n \<longleftrightarrow> n < size b \<and> bit c n) \<and>
     (\<forall>m::nat. bit a m \<longleftrightarrow> m < size a \<and> bit c (m + size b))"
   by (simp add: test_bit_split')
 
 lemma test_bit_split_eq:
   "word_split c = (a, b) \<longleftrightarrow>
     ((\<forall>n::nat. bit b n = (n < size b \<and> bit c n)) \<and>
      (\<forall>m::nat. bit a m = (m < size a \<and> bit c (m + size b))))"
   apply (rule_tac iffI)
    apply (rule_tac conjI)
     apply (erule test_bit_split [THEN conjunct1])
    apply (erule test_bit_split [THEN conjunct2])
   apply (case_tac "word_split c")
   apply (frule test_bit_split)
   apply (erule trans)
   apply (fastforce intro!: word_eqI simp add: word_size)
   done
 
 lemma test_bit_rcat:
   "sw = size (hd wl) \<Longrightarrow> rc = word_rcat wl \<Longrightarrow> bit rc n =
     (n < size rc \<and> n div sw < size wl \<and> bit ((rev wl) ! (n div sw)) (n mod sw))"
   for wl :: "'a::len word list"
   by (simp add: word_size word_rcat_def rev_map bit_horner_sum_uint_exp_iff bit_simps not_le)
 
 lemmas test_bit_cong = arg_cong [where f = "bit", THEN fun_cong]
 
 lemma max_test_bit: "bit (- 1::'a::len word) n \<longleftrightarrow> n < LENGTH('a)"
   by (fact nth_minus1)
 
 lemma map_nth_0 [simp]: "map (bit (0::'a::len word)) xs = replicate (length xs) False"
   by (simp flip: map_replicate_const)
 
 lemma word_and_1:
   "n AND 1 = (if bit n 0 then 1 else 0)" for n :: "_ word"
   by (rule bit_word_eqI) (auto simp add: bit_and_iff bit_1_iff intro: gr0I)
 
 lemma test_bit_1':
   "bit (1 :: 'a :: len word) n \<longleftrightarrow> 0 < LENGTH('a) \<and> n = 0"
   by simp
 
 lemma nth_w2p_same:
   "bit (2^n :: 'a :: len word) n = (n < LENGTH('a))"
   by (simp add: nth_w2p)
 
 lemma word_leI:
   "(\<And>n.  \<lbrakk>n < size (u::'a::len word); bit u n \<rbrakk> \<Longrightarrow> bit (v::'a::len word) n) \<Longrightarrow> u <= v"
   apply (rule order_trans [of u \<open>u AND v\<close> v])
    apply (rule eq_refl)
    apply (rule bit_word_eqI)
    apply (auto simp add: bit_simps word_and_le1 word_size)
   done
 
 lemma bang_eq:
   fixes x :: "'a::len word"
   shows "(x = y) = (\<forall>n. bit x n = bit y n)"
   by (auto intro!: bit_eqI)
 
 lemma neg_mask_test_bit:
   "bit (NOT(mask n) :: 'a :: len word) m = (n \<le> m \<and> m < LENGTH('a))"
   by (auto simp add: bit_simps)
 
 lemma upper_bits_unset_is_l2p:
   \<open>(\<forall>n' \<ge> n. n' < LENGTH('a) \<longrightarrow> \<not> bit p n') \<longleftrightarrow> (p < 2 ^ n)\<close> (is \<open>?P \<longleftrightarrow> ?Q\<close>)
     if \<open>n < LENGTH('a)\<close>
     for p :: "'a :: len word"
 proof
   assume ?Q
   then show ?P
     by (meson bang_is_le le_less_trans not_le word_power_increasing)
 next
   assume ?P
   have \<open>take_bit n p = p\<close>
   proof (rule bit_word_eqI)
     fix q
     assume \<open>q < LENGTH('a)\<close>
     show \<open>bit (take_bit n p) q \<longleftrightarrow> bit p q\<close>
     proof (cases \<open>q < n\<close>)
       case True
       then show ?thesis
         by (auto simp add: bit_simps)
     next
       case False
       then have \<open>n \<le> q\<close>
         by simp
       with \<open>?P\<close> \<open>q < LENGTH('a)\<close> have \<open>\<not> bit p q\<close>
         by simp
       then show ?thesis
         by (simp add: bit_simps)
     qed
   qed
   with that show ?Q
     using take_bit_word_eq_self_iff [of n p] by auto
 qed
 
 lemma less_2p_is_upper_bits_unset:
   "p < 2 ^ n \<longleftrightarrow> n < LENGTH('a) \<and> (\<forall>n' \<ge> n. n' < LENGTH('a) \<longrightarrow> \<not> bit p n')" for p :: "'a :: len word"
   by (meson le_less_trans le_mask_iff_lt_2n upper_bits_unset_is_l2p word_zero_le)
 
 lemma test_bit_over:
   "n \<ge> size (x::'a::len word) \<Longrightarrow> (bit x n) = False"
   by transfer auto
 
 lemma le_mask_high_bits:
   "w \<le> mask n \<longleftrightarrow> (\<forall>i \<in> {n ..< size w}. \<not> bit w i)"
   for w :: \<open>'a::len word\<close>
   apply (auto simp add: bit_simps word_size less_eq_mask_iff_take_bit_eq_self)
    apply (metis bit_take_bit_iff leD)
   apply (metis atLeastLessThan_iff leI take_bit_word_eq_self_iff upper_bits_unset_is_l2p)
   done
 
 lemma test_bit_conj_lt:
   "(bit x m \<and> m < LENGTH('a)) = bit x m" for x :: "'a :: len word"
   using test_bit_bin by blast
 
 lemma neg_test_bit:
   "bit (NOT x) n = (\<not> bit x n \<and> n < LENGTH('a))" for x :: "'a::len word"
   by (cases "n < LENGTH('a)") (auto simp add: test_bit_over word_ops_nth_size word_size)
 
 lemma nth_bounded:
   "\<lbrakk>bit (x :: 'a :: len word) n; x < 2 ^ m; m \<le> len_of TYPE ('a)\<rbrakk> \<Longrightarrow> n < m"
   apply (rule ccontr)
   apply (auto simp add: not_less)
   apply (meson bit_imp_le_length bit_uint_iff less_2p_is_upper_bits_unset test_bit_bin)
   done
 
 lemma and_neq_0_is_nth:
   \<open>x AND y \<noteq> 0 \<longleftrightarrow> bit x n\<close> if \<open>y = 2 ^ n\<close> for x y :: \<open>'a::len word\<close>
   apply (simp add: bit_eq_iff bit_simps)
   using that apply (simp add: bit_simps not_le)
   apply transfer
   apply auto
   done
 
 lemma nth_is_and_neq_0:
   "bit (x::'a::len word) n = (x AND 2 ^ n \<noteq> 0)"
   by (subst and_neq_0_is_nth; rule refl)
 
 lemma max_word_not_less [simp]:
    "\<not> - 1 < x" for x :: \<open>'a::len word\<close>
   by (fact word_order.extremum_strict)
 
 lemma bit_twiddle_min:
   "(y::'a::len word) XOR (((x::'a::len word) XOR y) AND (if x < y then -1 else 0)) = min x y"
   by (rule bit_eqI) (auto simp add: bit_simps)
 
 lemma bit_twiddle_max:
   "(x::'a::len word) XOR (((x::'a::len word) XOR y) AND (if x < y then -1 else 0)) = max x y"
   by (rule bit_eqI) (auto simp add: bit_simps max_def)
 
 lemma swap_with_xor:
   "\<lbrakk>(x::'a::len word) = a XOR b; y = b XOR x; z = x XOR y\<rbrakk> \<Longrightarrow> z = b \<and> y = a"
   by (auto intro: bit_word_eqI simp add: bit_simps)
 
 lemma le_mask_imp_and_mask:
   "(x::'a::len word) \<le> mask n \<Longrightarrow> x AND mask n = x"
   by (metis and_mask_eq_iff_le_mask)
 
 lemma or_not_mask_nop:
   "((x::'a::len word) OR NOT (mask n)) AND mask n = x AND mask n"
   by (metis word_and_not word_ao_dist2 word_bw_comms(1) word_log_esimps(3))
 
 lemma mask_subsume:
   "\<lbrakk>n \<le> m\<rbrakk> \<Longrightarrow> ((x::'a::len word) OR y AND mask n) AND NOT (mask m) = x AND NOT (mask m)"
   by (rule bit_word_eqI) (auto simp add: bit_simps word_size)
 
 lemma and_mask_0_iff_le_mask:
   fixes w :: "'a::len word"
   shows "(w AND NOT(mask n) = 0) = (w \<le> mask n)"
   by (simp add: mask_eq_0_eq_x le_mask_imp_and_mask and_mask_eq_iff_le_mask)
 
 lemma mask_twice2:
   "n \<le> m \<Longrightarrow> ((x::'a::len word) AND mask m) AND mask n = x AND mask n"
   by (metis mask_twice min_def)
 
 lemma uint_2_id:
   "LENGTH('a) \<ge> 2 \<Longrightarrow> uint (2::('a::len) word) = 2"
   by simp
 
 lemma div_of_0_id[simp]:"(0::('a::len) word) div n = 0"
   by (simp add: word_div_def)
 
 lemma degenerate_word:"LENGTH('a) = 1 \<Longrightarrow> (x::('a::len) word) = 0 \<or> x = 1"
   by (metis One_nat_def less_irrefl_nat sint_1_cases)
 
-lemma div_by_0_word:"(x::('a::len) word) div 0 = 0"
-  by (metis div_0 div_by_0 unat_0 word_arith_nat_defs(6) word_div_1)
+lemma div_by_0_word: "(x::'a::len word) div 0 = 0"
+  by (fact bits_div_by_0)
 
 lemma div_less_dividend_word:"\<lbrakk>x \<noteq> 0; n \<noteq> 1\<rbrakk> \<Longrightarrow> (x::('a::len) word) div n < x"
   apply (cases \<open>n = 0\<close>)
    apply clarsimp
    apply (simp add:word_neq_0_conv)
   apply (subst word_arith_nat_div)
   apply (rule word_of_nat_less)
   apply (rule div_less_dividend)
   using unat_eq_zero word_unat_Rep_inject1 apply force
   apply (simp add:unat_gt_0)
   done
 
 lemma word_less_div:
   fixes x :: "('a::len) word"
     and y :: "('a::len) word"
   shows "x div y = 0 \<Longrightarrow> y = 0 \<or> x < y"
   apply (case_tac "y = 0", clarsimp+)
   by (metis One_nat_def Suc_le_mono le0 le_div_geq not_less unat_0 unat_div unat_gt_0 word_less_nat_alt zero_less_one)
 
 lemma not_degenerate_imp_2_neq_0:"LENGTH('a) > 1 \<Longrightarrow> (2::('a::len) word) \<noteq> 0"
   by (metis numerals(1) power_not_zero power_zero_numeral)
 
 lemma word_overflow:"(x::('a::len) word) + 1 > x \<or> x + 1 = 0"
   apply clarsimp
   by (metis diff_0 eq_diff_eq less_x_plus_1)
 
 lemma word_overflow_unat:"unat ((x::('a::len) word) + 1) = unat x + 1 \<or> x + 1 = 0"
   by (metis Suc_eq_plus1 add.commute unatSuc)
 
 lemma even_word_imp_odd_next:"even (unat (x::('a::len) word)) \<Longrightarrow> x + 1 = 0 \<or> odd (unat (x + 1))"
   apply (cut_tac x=x in word_overflow_unat)
   apply clarsimp
   done
 
 lemma odd_word_imp_even_next:"odd (unat (x::('a::len) word)) \<Longrightarrow> x + 1 = 0 \<or> even (unat (x + 1))"
   apply (cut_tac x=x in word_overflow_unat)
   apply clarsimp
   done
 
 lemma overflow_imp_lsb:"(x::('a::len) word) + 1 = 0 \<Longrightarrow> bit x 0"
   using even_plus_one_iff [of x] by (simp add: bit_0)
 
 lemma odd_iff_lsb:"odd (unat (x::('a::len) word)) = bit x 0"
   by transfer (simp add: even_nat_iff bit_0)
 
 lemma of_nat_neq_iff_word:
       "x mod 2 ^ LENGTH('a) \<noteq> y mod 2 ^ LENGTH('a) \<Longrightarrow>
          (((of_nat x)::('a::len) word) \<noteq> of_nat y) = (x \<noteq> y)"
   apply (rule iffI)
    apply (case_tac "x = y")
    apply (subst (asm) of_nat_eq_iff[symmetric])
    apply auto
   apply (case_tac "((of_nat x)::('a::len) word) = of_nat y")
   apply auto
    apply (metis unat_of_nat)
   done
 
 lemma lsb_this_or_next: "\<not> (bit ((x::('a::len) word) + 1) 0) \<Longrightarrow> bit x 0"
   by (simp add: bit_0)
 
 lemma mask_or_not_mask:
   "x AND mask n OR x AND NOT (mask n) = x"
   for x :: \<open>'a::len word\<close>
   apply (subst word_oa_dist, simp)
   apply (subst word_oa_dist2, simp)
   done
 
 lemma word_gr0_conv_Suc: "(m::'a::len word) > 0 \<Longrightarrow> \<exists>n. m = n + 1"
   by (metis add.commute add_minus_cancel)
 
 lemma revcast_down_us [OF refl]:
   "rc = revcast \<Longrightarrow> source_size rc = target_size rc + n \<Longrightarrow> rc w = ucast (signed_drop_bit n w)"
   for w :: "'a::len word"
   apply (simp add: source_size_def target_size_def)
   apply (rule bit_word_eqI)
   apply (simp add: bit_simps ac_simps)
   done
 
 lemma revcast_down_ss [OF refl]:
   "rc = revcast \<Longrightarrow> source_size rc = target_size rc + n \<Longrightarrow> rc w = scast (signed_drop_bit n w)"
   for w :: "'a::len word"
   apply (simp add: source_size_def target_size_def)
   apply (rule bit_word_eqI)
   apply (simp add: bit_simps ac_simps)
   done
 
 lemma revcast_down_uu [OF refl]:
   "rc = revcast \<Longrightarrow> source_size rc = target_size rc + n \<Longrightarrow> rc w = ucast (drop_bit n w)"
   for w :: "'a::len word"
   apply (simp add: source_size_def target_size_def)
   apply (rule bit_word_eqI)
   apply (simp add: bit_simps ac_simps)
   done
 
 lemma revcast_down_su [OF refl]:
   "rc = revcast \<Longrightarrow> source_size rc = target_size rc + n \<Longrightarrow> rc w = scast (drop_bit n w)"
   for w :: "'a::len word"
   apply (simp add: source_size_def target_size_def)
   apply (rule bit_word_eqI)
   apply (simp add: bit_simps ac_simps)
   done
 
 lemma cast_down_rev [OF refl]:
   "uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> uc w = revcast (push_bit n w)"
   for w :: "'a::len word"
   apply (simp add: source_size_def target_size_def)
   apply (rule bit_word_eqI)
   apply (simp add: bit_simps)
   done
 
 lemma revcast_up [OF refl]:
   "rc = revcast \<Longrightarrow> source_size rc + n = target_size rc \<Longrightarrow>
     rc w = push_bit n (ucast w :: 'a::len word)"
   apply (simp add: source_size_def target_size_def)
   apply (rule bit_word_eqI)
   apply (simp add: bit_simps)
   apply auto
   apply (metis add.commute add_diff_cancel_right)
   apply (metis diff_add_inverse2 diff_diff_add)
   done
 
 lemmas rc1 = revcast_up [THEN
   revcast_rev_ucast [symmetric, THEN trans, THEN word_rev_gal, symmetric]]
 lemmas rc2 = revcast_down_uu [THEN
   revcast_rev_ucast [symmetric, THEN trans, THEN word_rev_gal, symmetric]]
 
 lemma word_ops_nth:
   fixes x y :: \<open>'a::len word\<close>
   shows
   word_or_nth:  "bit (x OR y) n = (bit x n \<or> bit y n)" and
   word_and_nth: "bit (x AND y) n = (bit x n \<and> bit y n)" and
   word_xor_nth: "bit (x XOR y) n = (bit x n \<noteq> bit y n)"
   by (simp_all add: bit_simps)
 
 lemma word_power_nonzero:
   "\<lbrakk> (x :: 'a::len word) < 2 ^ (LENGTH('a) - n); n < LENGTH('a); x \<noteq> 0 \<rbrakk>
   \<Longrightarrow> x * 2 ^ n \<noteq> 0"
   by (metis Word.word_div_mult bits_div_0 len_gt_0 len_of_finite_2_def nat_mult_power_less_eq
     p2_gt_0 unat_mono unat_power_lower word_gt_a_gt_0)
 
 lemma less_1_helper:
   "n \<le> m \<Longrightarrow> (n - 1 :: int) < m"
   by arith
 
 lemma div_power_helper:
   "\<lbrakk> x \<le> y; y < LENGTH('a) \<rbrakk> \<Longrightarrow> (2 ^ y - 1) div (2 ^ x :: 'a::len word) = 2 ^ (y - x) - 1"
   apply (simp flip: mask_eq_exp_minus_1 drop_bit_eq_div)
   apply (rule bit_word_eqI)
   apply (auto simp add: bit_simps not_le)
   done
 
 lemma max_word_mask:
   "(- 1 :: 'a::len word) = mask LENGTH('a)"
   by (fact minus_1_eq_mask)
 
 lemmas mask_len_max = max_word_mask[symmetric]
 
 lemma mask_out_first_mask_some:
   "\<lbrakk> x AND NOT (mask n) = y; n \<le> m \<rbrakk> \<Longrightarrow> x AND NOT (mask m) = y AND NOT (mask m)"
   for x y :: \<open>'a::len word\<close>
   by (rule bit_word_eqI) (auto simp add: bit_simps word_size)
 
 lemma mask_lower_twice:
   "n \<le> m \<Longrightarrow> (x AND NOT (mask n)) AND NOT (mask m) = x AND NOT (mask m)"
   for x :: \<open>'a::len word\<close>
   by (rule bit_word_eqI) (auto simp add: bit_simps word_size)
 
 lemma mask_lower_twice2:
   "(a AND NOT (mask n)) AND NOT (mask m) = a AND NOT (mask (max n m))"
   for a :: \<open>'a::len word\<close>
   by (rule bit_word_eqI) (auto simp add: bit_simps)
 
 lemma ucast_and_neg_mask:
   "ucast (x AND NOT (mask n)) = ucast x AND NOT (mask n)"
   apply (rule bit_word_eqI)
   apply (auto simp add: bit_simps dest: bit_imp_le_length)
   done
 
 lemma ucast_and_mask:
   "ucast (x AND mask n) = ucast x AND mask n"
   apply (rule bit_word_eqI)
   apply (auto simp add: bit_simps dest: bit_imp_le_length)
   done
 
 lemma ucast_mask_drop:
   "LENGTH('a :: len) \<le> n \<Longrightarrow> (ucast (x AND mask n) :: 'a word) = ucast x"
   apply (rule bit_word_eqI)
   apply (auto simp add: bit_simps dest: bit_imp_le_length)
   done
 
 lemma mask_exceed:
   "n \<ge> LENGTH('a) \<Longrightarrow> (x::'a::len word) AND NOT (mask n) = 0"
   by (rule bit_word_eqI) (simp add: bit_simps)
 
 lemma word_add_no_overflow:"(x::'a::len word) < - 1 \<Longrightarrow> x < x + 1"
   using less_x_plus_1 order_less_le by blast
 
 lemma lt_plus_1_le_word:
   fixes x :: "'a::len word"
   assumes bound:"n < unat (maxBound::'a word)"
   shows "x < 1 + of_nat n = (x \<le> of_nat n)"
   by (metis add.commute bound max_word_max word_Suc_leq word_not_le word_of_nat_less)
 
 lemma unat_ucast_up_simp:
   fixes x :: "'a::len word"
   assumes "LENGTH('a) \<le> LENGTH('b)"
   shows "unat (ucast x :: 'b::len word) = unat x"
   apply (rule bit_eqI)
   using assms apply (auto simp add: bit_simps dest: bit_imp_le_length)
   done
 
 lemma unat_ucast_less_no_overflow:
   "\<lbrakk>n < 2 ^ LENGTH('a); unat f < n\<rbrakk> \<Longrightarrow> (f::('a::len) word) < of_nat n"
   by (erule (1)  order_le_less_trans[OF _ of_nat_mono_maybe,rotated]) simp
 
 lemma unat_ucast_less_no_overflow_simp:
   "n < 2 ^ LENGTH('a) \<Longrightarrow> (unat f < n) = ((f::('a::len) word) < of_nat n)"
   using unat_less_helper unat_ucast_less_no_overflow by blast
 
 lemma unat_ucast_no_overflow_le:
   assumes no_overflow: "unat b < (2 :: nat) ^ LENGTH('a)"
   and upward_cast: "LENGTH('a) < LENGTH('b)"
   shows "(ucast (f::'a::len word) < (b :: 'b :: len word)) = (unat f < unat b)"
 proof -
   have LR: "ucast f < b \<Longrightarrow> unat f < unat b"
     apply (rule unat_less_helper)
     apply (simp add:ucast_nat_def)
     apply (rule_tac 'b1 = 'b in  ucast_less_ucast[OF order.strict_implies_order, THEN iffD1])
      apply (rule upward_cast)
     apply (simp add: ucast_ucast_mask less_mask_eq word_less_nat_alt
                      unat_power_lower[OF upward_cast] no_overflow)
     done
   have RL: "unat f < unat b \<Longrightarrow> ucast f < b"
   proof-
     assume ineq: "unat f < unat b"
     have "ucast (f::'a::len word) < ((ucast (ucast b ::'a::len word)) :: 'b :: len word)"
       apply (simp add: ucast_less_ucast[OF order.strict_implies_order] upward_cast)
       apply (simp only: flip: ucast_nat_def)
       apply (rule unat_ucast_less_no_overflow[OF no_overflow ineq])
       done
     then show ?thesis
       apply (rule order_less_le_trans)
       apply (simp add:ucast_ucast_mask word_and_le2)
       done
   qed
   then show ?thesis by (simp add:RL LR iffI)
 qed
 
 lemmas ucast_up_mono = ucast_less_ucast[THEN iffD2]
 
 lemma minus_one_word:
   "(-1 :: 'a :: len word) = 2 ^ LENGTH('a) - 1"
   by simp
 
 lemma le_2p_upper_bits:
   "\<lbrakk> (p::'a::len word) \<le> 2^n - 1; n < LENGTH('a) \<rbrakk> \<Longrightarrow>
   \<forall>n'\<ge>n. n' < LENGTH('a) \<longrightarrow> \<not> bit p n'"
   by (subst upper_bits_unset_is_l2p; simp)
 
 lemma le2p_bits_unset:
   "p \<le> 2 ^ n - 1 \<Longrightarrow> \<forall>n'\<ge>n. n' < LENGTH('a) \<longrightarrow> \<not> bit (p::'a::len word) n'"
   using upper_bits_unset_is_l2p [where p=p]
   by (cases "n < LENGTH('a)") auto
 
 lemma complement_nth_w2p:
   shows "n' < LENGTH('a) \<Longrightarrow> bit (NOT (2 ^ n :: 'a::len word)) n' = (n' \<noteq> n)"
   by (fastforce simp: word_ops_nth_size word_size nth_w2p)
 
 lemma word_unat_and_lt:
   "unat x < n \<or> unat y < n \<Longrightarrow> unat (x AND y) < n"
   by (meson le_less_trans word_and_le1 word_and_le2 word_le_nat_alt)
 
 lemma word_unat_mask_lt:
   "m \<le> size w \<Longrightarrow> unat ((w::'a::len word) AND mask m) < 2 ^ m"
   by (rule word_unat_and_lt) (simp add: unat_mask word_size)
 
 lemma word_sless_sint_le:"x <s y \<Longrightarrow> sint x \<le> sint y - 1"
   by (metis word_sless_alt zle_diff1_eq)
 
 lemma upper_trivial:
   fixes x :: "'a::len word"
   shows "x \<noteq> 2 ^ LENGTH('a) - 1 \<Longrightarrow> x < 2 ^ LENGTH('a) - 1"
   by (simp add: less_le)
 
 lemma constraint_expand:
   fixes x :: "'a::len word"
   shows "x \<in> {y. lower \<le> y \<and> y \<le> upper} = (lower \<le> x \<and> x \<le> upper)"
   by (rule mem_Collect_eq)
 
 lemma card_map_elide:
   "card ((of_nat :: nat \<Rightarrow> 'a::len word) ` {0..<n}) = card {0..<n}"
   if "n \<le> CARD('a::len word)"
 proof -
   let ?of_nat = "of_nat :: nat \<Rightarrow> 'a word"
   have "inj_on ?of_nat {i. i < CARD('a word)}"
     by (rule inj_onI) (simp add: card_word of_nat_inj)
   moreover have "{0..<n} \<subseteq> {i. i < CARD('a word)}"
     using that by auto
   ultimately have "inj_on ?of_nat {0..<n}"
     by (rule inj_on_subset)
   then show ?thesis
     by (simp add: card_image)
 qed
 
 lemma card_map_elide2:
   "n \<le> CARD('a::len word) \<Longrightarrow> card ((of_nat::nat \<Rightarrow> 'a::len word) ` {0..<n}) = n"
   by (subst card_map_elide) clarsimp+
 
 lemma eq_ucast_ucast_eq:
   "LENGTH('b) \<le> LENGTH('a) \<Longrightarrow> x = ucast y \<Longrightarrow> ucast x = y"
   for x :: "'a::len word" and y :: "'b::len word"
   by transfer simp
 
 lemma le_ucast_ucast_le:
   "x \<le> ucast y \<Longrightarrow> ucast x \<le> y"
   for x :: "'a::len word" and y :: "'b::len word"
   by (smt le_unat_uoi linorder_not_less order_less_imp_le ucast_nat_def unat_arith_simps(1))
 
 lemma less_ucast_ucast_less:
   "LENGTH('b) \<le> LENGTH('a) \<Longrightarrow> x < ucast y \<Longrightarrow> ucast x < y"
   for x :: "'a::len word" and y :: "'b::len word"
   by (metis ucast_nat_def unat_mono unat_ucast_up_simp word_of_nat_less)
 
 lemma ucast_le_ucast:
   "LENGTH('a) \<le> LENGTH('b) \<Longrightarrow> (ucast x \<le> (ucast y::'b::len word)) = (x \<le> y)"
   for x :: "'a::len word"
   by (simp add: unat_arith_simps(1) unat_ucast_up_simp)
 
 lemmas ucast_up_mono_le = ucast_le_ucast[THEN iffD2]
 
 lemma ucast_or_distrib:
   fixes x :: "'a::len word"
   fixes y :: "'a::len word"
   shows "(ucast (x OR y) :: ('b::len) word) = ucast x OR ucast y"
   by (fact unsigned_or_eq)
 
 lemma word_exists_nth:
   "(w::'a::len word) \<noteq> 0 \<Longrightarrow> \<exists>i. bit w i"
   by (auto simp add: bit_eq_iff)
 
 lemma max_word_not_0 [simp]:
   "- 1 \<noteq> (0 :: 'a::len word)"
   by simp
 
 lemma unat_max_word_pos[simp]: "0 < unat (- 1 :: 'a::len word)"
   using unat_gt_0 [of \<open>- 1 :: 'a::len word\<close>] by simp
 
 
 (* Miscellaneous conditional injectivity rules. *)
 
 lemma mult_pow2_inj:
   assumes ws: "m + n \<le> LENGTH('a)"
   assumes le: "x \<le> mask m" "y \<le> mask m"
   assumes eq: "x * 2 ^ n = y * (2 ^ n::'a::len word)"
   shows "x = y"
 proof (rule bit_word_eqI)
   fix q
   assume \<open>q < LENGTH('a)\<close>
   from eq have \<open>push_bit n x = push_bit n y\<close>
     by (simp add: push_bit_eq_mult)
   moreover from le have \<open>take_bit m x = x\<close> \<open>take_bit m y = y\<close>
     by (simp_all add: less_eq_mask_iff_take_bit_eq_self)
   ultimately have \<open>push_bit n (take_bit m x) = push_bit n (take_bit m y)\<close>
     by simp_all
   with \<open>q < LENGTH('a)\<close> ws show \<open>bit x q \<longleftrightarrow> bit y q\<close>
     apply (simp add: push_bit_take_bit)
     unfolding bit_eq_iff
     apply (simp add: bit_simps not_le)
     apply (metis (full_types) \<open>take_bit m x = x\<close> \<open>take_bit m y = y\<close> add.commute add_diff_cancel_right' add_less_cancel_right bit_take_bit_iff le_add2 less_le_trans)
     done
 qed
 
 lemma word_of_nat_inj:
   assumes bounded: "x < 2 ^ LENGTH('a)" "y < 2 ^ LENGTH('a)"
   assumes of_nats: "of_nat x = (of_nat y :: 'a::len word)"
   shows "x = y"
   by (rule contrapos_pp[OF of_nats]; cases "x < y"; cases "y < x")
      (auto dest: bounded[THEN of_nat_mono_maybe])
 
 lemma word_of_int_bin_cat_eq_iff:
   "(word_of_int (concat_bit LENGTH('b) (uint b) (uint a))::'c::len word) =
   word_of_int (concat_bit LENGTH('b) (uint d) (uint c)) \<longleftrightarrow> b = d \<and> a = c"
   if "LENGTH('a) + LENGTH('b) \<le> LENGTH('c)"
   for a::"'a::len word" and b::"'b::len word"
 proof -
   from that show ?thesis
   using that concat_bit_eq_iff [of \<open>LENGTH('b)\<close> \<open>uint b\<close> \<open>uint a\<close> \<open>uint d\<close> \<open>uint c\<close>]
   apply (simp add: word_of_int_eq_iff take_bit_int_eq_self flip: word_eq_iff_unsigned)
   apply (simp add: concat_bit_def take_bit_int_eq_self bintr_uint take_bit_push_bit)
   done
 qed
 
 lemma word_cat_inj: "(word_cat a b::'c::len word) = word_cat c d \<longleftrightarrow> a = c \<and> b = d"
   if "LENGTH('a) + LENGTH('b) \<le> LENGTH('c)"
   for a::"'a::len word" and b::"'b::len word"
   using word_of_int_bin_cat_eq_iff [OF that, of b a d c]
   by (simp add: word_cat_eq' ac_simps)
 
 lemma p2_eq_1: "2 ^ n = (1::'a::len word) \<longleftrightarrow> n = 0"
 proof -
   have "2 ^ n = (1::'a word) \<Longrightarrow> n = 0"
     by (metis One_nat_def not_less one_less_numeral_iff p2_eq_0 p2_gt_0 power_0 power_0
         power_inject_exp semiring_norm(76) unat_power_lower zero_neq_one)
   then show ?thesis by auto
 qed
 
 end
 
 lemmas word_div_less = div_word_less
 
 (* FIXME: move to Word distribution? *)
 lemma bin_nth_minus_Bit0[simp]:
   "0 < n \<Longrightarrow> bit (numeral (num.Bit0 w) :: int) n = bit (numeral w :: int) (n - 1)"
   by (cases n; simp)
 
 lemma bin_nth_minus_Bit1[simp]:
   "0 < n \<Longrightarrow> bit (numeral (num.Bit1 w) :: int) n = bit (numeral w :: int) (n - 1)"
   by (cases n; simp)
 
 lemma word_mod_by_0: "k mod (0::'a::len word) = k"
   by (simp add: word_arith_nat_mod)
 
 end
diff --git a/thys/Word_Lib/More_Word_Operations.thy b/thys/Word_Lib/More_Word_Operations.thy
--- a/thys/Word_Lib/More_Word_Operations.thy
+++ b/thys/Word_Lib/More_Word_Operations.thy
@@ -1,1023 +1,1122 @@
 (*
  * Copyright Data61, CSIRO (ABN 41 687 119 230)
  *
  * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section \<open>Misc word operations\<close>
 
 theory More_Word_Operations
   imports
     "HOL-Library.Word"
     Aligned
     Reversed_Bit_Lists
     More_Misc
     Signed_Words
     Word_Lemmas
+    Many_More
     Word_EqI
 begin
 
 context
   includes bit_operations_syntax
 begin
 
 definition
   ptr_add :: "'a :: len word \<Rightarrow> nat \<Rightarrow> 'a word" where
   "ptr_add ptr n \<equiv> ptr + of_nat n"
 
 definition
   alignUp :: "'a::len word \<Rightarrow> nat \<Rightarrow> 'a word" where
  "alignUp x n \<equiv> x + 2 ^ n - 1 AND NOT (2 ^ n - 1)"
 
 lemma alignUp_unfold:
   \<open>alignUp w n = (w + mask n) AND NOT (mask n)\<close>
   by (simp add: alignUp_def mask_eq_exp_minus_1 add_mask_fold)
 
 (* standard notation for blocks of 2^n-1 words, usually aligned;
    abbreviation so it simplifies directly *)
 abbreviation mask_range :: "'a::len word \<Rightarrow> nat \<Rightarrow> 'a word set" where
   "mask_range p n \<equiv> {p .. p + mask n}"
 
 definition
   w2byte :: "'a :: len word \<Rightarrow> 8 word" where
   "w2byte \<equiv> ucast"
 
 (* Count leading zeros  *)
 definition
   word_clz :: "'a::len word \<Rightarrow> nat"
 where
   "word_clz w \<equiv> length (takeWhile Not (to_bl w))"
 
 (* Count trailing zeros  *)
 definition
   word_ctz :: "'a::len word \<Rightarrow> nat"
 where
   "word_ctz w \<equiv> length (takeWhile Not (rev (to_bl w)))"
 
 lemma word_ctz_unfold:
   \<open>word_ctz w = length (takeWhile (Not \<circ> bit w) [0..<LENGTH('a)])\<close> for w :: \<open>'a::len word\<close>
   by (simp add: word_ctz_def rev_to_bl_eq takeWhile_map)
 
 lemma word_ctz_unfold':
   \<open>word_ctz w = Min (insert LENGTH('a) {n. bit w n})\<close> for w :: \<open>'a::len word\<close>
 proof (cases \<open>\<exists>n. bit w n\<close>)
   case True
   then obtain n where \<open>bit w n\<close> ..
   from \<open>bit w n\<close> show ?thesis
     apply (simp add: word_ctz_unfold)
     apply (subst Min_eq_length_takeWhile [symmetric])
       apply (auto simp add: bit_imp_le_length)
     apply (subst Min_insert)
       apply auto
     apply (subst min.absorb2)
      apply (subst Min_le_iff)
        apply auto
     apply (meson bit_imp_le_length order_less_le)
     done
 next
   case False
   then have \<open>bit w = bot\<close>
     by auto
   then have \<open>word_ctz w = LENGTH('a)\<close>
     by (simp add: word_ctz_def rev_to_bl_eq bot_fun_def map_replicate_const)
   with \<open>bit w = bot\<close> show ?thesis
     by simp
 qed
 
 lemma word_ctz_le:
   "word_ctz (w :: ('a::len word)) \<le> LENGTH('a)"
   apply (clarsimp simp: word_ctz_def)
   using length_takeWhile_le apply (rule order_trans)
   apply simp
   done
 
 lemma word_ctz_less:
   "w \<noteq> 0 \<Longrightarrow> word_ctz (w :: ('a::len word)) < LENGTH('a)"
   apply (clarsimp simp: word_ctz_def eq_zero_set_bl)
   using length_takeWhile_less apply (rule less_le_trans)
   apply auto
   done
 
 lemma take_bit_word_ctz_eq [simp]:
   \<open>take_bit LENGTH('a) (word_ctz w) = word_ctz w\<close>
   for w :: \<open>'a::len word\<close>
   apply (simp add: take_bit_nat_eq_self_iff word_ctz_def to_bl_unfold)
   using length_takeWhile_le apply (rule le_less_trans)
   apply simp
   done
 
 lemma word_ctz_not_minus_1:
   \<open>word_of_nat (word_ctz (w :: 'a :: len word)) \<noteq> (- 1 :: 'a::len word)\<close> if \<open>1 < LENGTH('a)\<close>
 proof -
   note word_ctz_le
   also from that have \<open>LENGTH('a) < mask LENGTH('a)\<close>
     by (simp add: less_mask)
   finally have \<open>word_ctz w < mask LENGTH('a)\<close> .
   then have \<open>word_of_nat (word_ctz w) < (word_of_nat (mask LENGTH('a)) :: 'a word)\<close>
     by (simp add: of_nat_word_less_iff)
   also have \<open>\<dots> = - 1\<close>
     by (rule bit_word_eqI) (simp add: bit_simps)
   finally show ?thesis
     by simp
 qed
 
 lemma unat_of_nat_ctz_mw:
   "unat (of_nat (word_ctz (w :: 'a :: len word)) :: 'a :: len word) = word_ctz w"
   by (simp add: unsigned_of_nat)
 
 lemma unat_of_nat_ctz_smw:
   "unat (of_nat (word_ctz (w :: 'a :: len word)) :: 'a :: len signed word) = word_ctz w"
   by (simp add: unsigned_of_nat)
 
 definition
   word_log2 :: "'a::len word \<Rightarrow> nat"
 where
   "word_log2 (w::'a::len word) \<equiv> size w - 1 - word_clz w"
 
 (* Bit population count. Equivalent of __builtin_popcount. *)
 definition
   pop_count :: "('a::len) word \<Rightarrow> nat"
 where
   "pop_count w \<equiv> length (filter id (to_bl w))"
 
 (* Sign extension from bit n *)
 definition
   sign_extend :: "nat \<Rightarrow> 'a::len word \<Rightarrow> 'a word"
 where
   "sign_extend n w \<equiv> if bit w n then w OR NOT (mask n) else w AND mask n"
 
 lemma sign_extend_eq_signed_take_bit:
   \<open>sign_extend = signed_take_bit\<close>
 proof (rule ext)+
   fix n and w :: \<open>'a::len word\<close>
   show \<open>sign_extend n w = signed_take_bit n w\<close>
   proof (rule bit_word_eqI)
     fix q
     assume \<open>q < LENGTH('a)\<close>
     then show \<open>bit (sign_extend n w) q \<longleftrightarrow> bit (signed_take_bit n w) q\<close>
       by (auto simp add: bit_signed_take_bit_iff
         sign_extend_def bit_and_iff bit_or_iff bit_not_iff bit_mask_iff not_less
         exp_eq_0_imp_not_bit not_le min_def)
   qed
 qed
 
 definition
   sign_extended :: "nat \<Rightarrow> 'a::len word \<Rightarrow> bool"
 where
   "sign_extended n w \<equiv> \<forall>i. n < i \<longrightarrow> i < size w \<longrightarrow> bit w i = bit w n"
 
 lemma ptr_add_0 [simp]:
   "ptr_add ref 0 = ref "
   unfolding ptr_add_def by simp
 
 lemma pop_count_0[simp]:
   "pop_count 0 = 0"
   by (clarsimp simp:pop_count_def)
 
 lemma pop_count_1[simp]:
   "pop_count 1 = 1"
   by (clarsimp simp:pop_count_def to_bl_1)
 
 lemma pop_count_0_imp_0:
   "(pop_count w = 0) = (w = 0)"
   apply (rule iffI)
    apply (clarsimp simp:pop_count_def)
    apply (subst (asm) filter_empty_conv)
    apply (clarsimp simp:eq_zero_set_bl)
    apply fast
   apply simp
   done
 
 lemma word_log2_zero_eq [simp]:
   \<open>word_log2 0 = 0\<close>
   by (simp add: word_log2_def word_clz_def word_size)
 
 lemma word_log2_unfold:
   \<open>word_log2 w = (if w = 0 then 0 else Max {n. bit w n})\<close>
   for w :: \<open>'a::len word\<close>
 proof (cases \<open>w = 0\<close>)
   case True
   then show ?thesis
     by simp
 next
   case False
   then obtain r where \<open>bit w r\<close>
     by (auto simp add: bit_eq_iff)
   then have \<open>Max {m. bit w m} = LENGTH('a) - Suc (length
     (takeWhile (Not \<circ> bit w) (rev [0..<LENGTH('a)])))\<close>
     by (subst Max_eq_length_takeWhile [of _ \<open>LENGTH('a)\<close>])
       (auto simp add: bit_imp_le_length)
   then have \<open>word_log2 w = Max {x. bit w x}\<close>
     by (simp add: word_log2_def word_clz_def word_size to_bl_unfold rev_map takeWhile_map)
   with \<open>w \<noteq> 0\<close> show ?thesis
     by simp
 qed
 
 lemma word_log2_eqI:
   \<open>word_log2 w = n\<close>
   if \<open>w \<noteq> 0\<close> \<open>bit w n\<close> \<open>\<And>m. bit w m \<Longrightarrow> m \<le> n\<close>
   for w :: \<open>'a::len word\<close>
 proof -
   from \<open>w \<noteq> 0\<close> have \<open>word_log2 w = Max {n. bit w n}\<close>
     by (simp add: word_log2_unfold)
   also have \<open>Max {n. bit w n} = n\<close>
     using that by (auto intro: Max_eqI)
   finally show ?thesis .
 qed
 
 lemma bit_word_log2:
   \<open>bit w (word_log2 w)\<close> if \<open>w \<noteq> 0\<close>
 proof -
   from \<open>w \<noteq> 0\<close> have \<open>\<exists>r. bit w r\<close>
     by (auto intro: bit_eqI)
   then obtain r where \<open>bit w r\<close> ..
   from \<open>w \<noteq> 0\<close> have \<open>word_log2 w = Max {n. bit w n}\<close>
     by (simp add: word_log2_unfold)
   also have \<open>Max {n. bit w n} \<in> {n. bit w n}\<close>
     using \<open>bit w r\<close> by (subst Max_in) auto
   finally show ?thesis
     by simp
 qed
 
 lemma word_log2_maximum:
   \<open>n \<le> word_log2 w\<close> if \<open>bit w n\<close>
 proof -
   have \<open>n \<le> Max {n. bit w n}\<close>
     using that by (auto intro: Max_ge)
   also from that have \<open>w \<noteq> 0\<close>
     by force
   then have \<open>Max {n. bit w n} = word_log2 w\<close>
     by (simp add: word_log2_unfold)
   finally show ?thesis .
 qed
 
 lemma word_log2_nth_same:
   "w \<noteq> 0 \<Longrightarrow> bit w (word_log2 w)"
   by (drule bit_word_log2) simp
 
 lemma word_log2_nth_not_set:
   "\<lbrakk> word_log2 w < i ; i < size w \<rbrakk> \<Longrightarrow> \<not> bit w i"
   using word_log2_maximum [of w i] by auto
 
 lemma word_log2_highest:
   assumes a: "bit w i"
   shows "i \<le> word_log2 w"
   using a by (simp add: word_log2_maximum)
 
 lemma word_log2_max:
   "word_log2 w < size w"
   apply (cases \<open>w = 0\<close>)
    apply (simp_all add: word_size)
   apply (drule bit_word_log2)
   apply (fact bit_imp_le_length)
   done
 
 lemma word_clz_0[simp]:
   "word_clz (0::'a::len word) = LENGTH('a)"
   unfolding word_clz_def by simp
 
 lemma word_clz_minus_one[simp]:
   "word_clz (-1::'a::len word) = 0"
   unfolding word_clz_def by simp
 
 lemma is_aligned_alignUp[simp]:
   "is_aligned (alignUp p n) n"
   by (simp add: alignUp_def is_aligned_mask mask_eq_decr_exp word_bw_assocs)
 
 lemma alignUp_le[simp]:
   "alignUp p n \<le> p + 2 ^ n - 1"
   unfolding alignUp_def by (rule word_and_le2)
 
 lemma alignUp_idem:
   fixes a :: "'a::len word"
   assumes "is_aligned a n" "n < LENGTH('a)"
   shows "alignUp a n = a"
   using assms unfolding alignUp_def
   by (metis add_cancel_right_right add_diff_eq and_mask_eq_iff_le_mask mask_eq_decr_exp mask_out_add_aligned order_refl word_plus_and_or_coroll2)
 
 lemma alignUp_not_aligned_eq:
   fixes a :: "'a :: len word"
   assumes al: "\<not> is_aligned a n"
   and     sz: "n < LENGTH('a)"
   shows   "alignUp a n = (a div 2 ^ n + 1) * 2 ^ n"
 proof -
   from \<open>n < LENGTH('a)\<close> have \<open>(2::int) ^ n < 2 ^ LENGTH('a)\<close>
     by simp
   with take_bit_int_less_exp [of n]
     have *: \<open>take_bit n k < 2 ^ LENGTH('a)\<close> for k :: int
     by (rule less_trans)
   have anz: "a mod 2 ^ n \<noteq> 0"
     by (rule not_aligned_mod_nz) fact+
   then have um: "unat (a mod 2 ^ n - 1) div 2 ^ n = 0"
     apply (transfer fixing: n) using sz
     apply (simp flip: take_bit_eq_mod add: div_eq_0_iff)
     apply (subst take_bit_int_eq_self)
     using *
      apply (auto simp add: diff_less_eq intro: less_imp_le)
     apply (simp add: less_le)
     done
   have "a + 2 ^ n - 1 = (a div 2 ^ n) * 2 ^ n + (a mod 2 ^ n) + 2 ^ n - 1"
     by (simp add: word_mod_div_equality)
   also have "\<dots> = (a mod 2 ^ n - 1) + (a div 2 ^ n + 1) * 2 ^ n"
     by (simp add: field_simps)
   finally show "alignUp a n = (a div 2 ^ n + 1) * 2 ^ n" using sz
     unfolding alignUp_def
     apply (subst mask_eq_decr_exp [symmetric])
     apply (erule ssubst)
     apply (subst neg_mask_is_div)
     apply (simp add: word_arith_nat_div)
     apply (subst unat_word_ariths(1) unat_word_ariths(2))+
     apply (subst uno_simps)
     apply (subst unat_1)
     apply (subst mod_add_right_eq)
     apply simp
     apply (subst power_mod_div)
     apply (subst div_mult_self1)
      apply simp
     apply (subst um)
     apply simp
     apply (subst mod_mod_power)
     apply simp
     apply (subst word_unat_power, subst Abs_fnat_hom_mult)
     apply (subst mult_mod_left)
     apply (subst power_add [symmetric])
     apply simp
     apply (subst Abs_fnat_hom_1)
     apply (subst Abs_fnat_hom_add)
     apply (subst word_unat_power, subst Abs_fnat_hom_mult)
     apply (subst word_unat.Rep_inverse[symmetric], subst Abs_fnat_hom_mult)
     apply simp
     done
 qed
 
 lemma alignUp_ge:
   fixes a :: "'a :: len word"
   assumes sz: "n < LENGTH('a)"
   and nowrap: "alignUp a n \<noteq> 0"
   shows "a \<le> alignUp a n"
 proof (cases "is_aligned a n")
   case True
   then show ?thesis using sz
     by (subst alignUp_idem, simp_all)
 next
   case False
 
   have lt0: "unat a div 2 ^ n < 2 ^ (LENGTH('a) - n)" using sz
     by (metis le_add_diff_inverse2 less_mult_imp_div_less order_less_imp_le power_add unsigned_less)
 
   have"2 ^ n * (unat a div 2 ^ n + 1) \<le> 2 ^ LENGTH('a)" using sz
     by (metis One_nat_def Suc_leI add.right_neutral add_Suc_right lt0 nat_le_power_trans nat_less_le)
   moreover have "2 ^ n * (unat a div 2 ^ n + 1) \<noteq> 2 ^ LENGTH('a)" using nowrap sz
     apply -
     apply (erule contrapos_nn)
     apply (subst alignUp_not_aligned_eq [OF False sz])
     apply (subst unat_arith_simps)
     apply (subst unat_word_ariths)
     apply (subst unat_word_ariths)
     apply simp
     apply (subst mult_mod_left)
     apply (simp add: unat_div field_simps power_add[symmetric] mod_mod_power)
     done
   ultimately have lt: "2 ^ n * (unat a div 2 ^ n + 1) < 2 ^ LENGTH('a)" by simp
 
   have "a = a div 2 ^ n * 2 ^ n + a mod 2 ^ n" by (rule word_mod_div_equality [symmetric])
   also have "\<dots> < (a div 2 ^ n + 1) * 2 ^ n" using sz lt
     apply (simp add: field_simps)
     apply (rule word_add_less_mono1)
     apply (rule word_mod_less_divisor)
     apply (simp add: word_less_nat_alt)
     apply (subst unat_word_ariths)
     apply (simp add: unat_div)
     done
   also have "\<dots> =  alignUp a n"
     by (rule alignUp_not_aligned_eq [symmetric]) fact+
   finally show ?thesis by (rule order_less_imp_le)
 qed
 
 lemma alignUp_le_greater_al:
   fixes x :: "'a :: len word"
   assumes le: "a \<le> x"
   and     sz: "n < LENGTH('a)"
   and     al: "is_aligned x n"
   shows   "alignUp a n \<le> x"
 proof (cases "is_aligned a n")
   case True
   then show ?thesis using sz le by (simp add: alignUp_idem)
 next
   case False
 
   then have anz: "a mod 2 ^ n \<noteq> 0"
     by (rule not_aligned_mod_nz)
 
   from al obtain k where xk: "x = 2 ^ n * of_nat k" and kv: "k < 2 ^ (LENGTH('a) - n)"
     by (auto elim!: is_alignedE)
 
   then have kn: "unat (of_nat k :: 'a word) * unat ((2::'a word) ^ n) < 2 ^ LENGTH('a)"
     using sz
     apply (subst unat_of_nat_eq)
      apply (erule order_less_le_trans)
      apply simp
     apply (subst mult.commute)
     apply simp
     apply (rule nat_less_power_trans)
      apply simp
     apply simp
     done
 
   have au: "alignUp a n = (a div 2 ^ n + 1) * 2 ^ n"
     by (rule alignUp_not_aligned_eq) fact+
   also have "\<dots> \<le> of_nat k * 2 ^ n"
   proof (rule word_mult_le_mono1 [OF inc_le _ kn])
     show "a div 2 ^ n < of_nat k" using kv xk le sz anz
       by (simp add: alignUp_div_helper)
 
     show "(0:: 'a word) < 2 ^ n" using sz by (simp add: p2_gt_0 sz)
   qed
 
   finally show ?thesis using xk by (simp add: field_simps)
 qed
 
 lemma alignUp_is_aligned_nz:
   fixes a :: "'a :: len word"
   assumes al: "is_aligned x n"
   and     sz: "n < LENGTH('a)"
   and     ax: "a \<le> x"
   and     az: "a \<noteq> 0"
   shows   "alignUp (a::'a :: len word) n \<noteq> 0"
 proof (cases "is_aligned a n")
   case True
   then have "alignUp a n = a" using sz by (simp add: alignUp_idem)
   then show ?thesis using az by simp
 next
   case False
   then have anz: "a mod 2 ^ n \<noteq> 0"
     by (rule not_aligned_mod_nz)
 
   {
     assume asm: "alignUp a n = 0"
 
     have lt0: "unat a div 2 ^ n < 2 ^ (LENGTH('a) - n)" using sz
       by (metis le_add_diff_inverse2 less_mult_imp_div_less order_less_imp_le power_add unsigned_less)
 
     have leq: "2 ^ n * (unat a div 2 ^ n + 1) \<le> 2 ^ LENGTH('a)" using sz
       by (metis One_nat_def Suc_leI add.right_neutral add_Suc_right lt0 nat_le_power_trans
                 order_less_imp_le)
 
     from al obtain k where  kv: "k < 2 ^ (LENGTH('a) - n)" and xk: "x = 2 ^ n * of_nat k"
       by (auto elim!: is_alignedE)
 
     then have "a div 2 ^ n < of_nat k" using ax sz anz
       by (rule alignUp_div_helper)
 
     then have r: "unat a div 2 ^ n < k" using sz
       by (simp flip: drop_bit_eq_div unat_drop_bit_eq) (metis leI le_unat_uoi unat_mono)
 
     have "alignUp a n = (a div 2 ^ n + 1) * 2 ^ n"
       by (rule alignUp_not_aligned_eq) fact+
 
     then have "\<dots> = 0" using asm by simp
     then have "2 ^ LENGTH('a) dvd 2 ^ n * (unat a div 2 ^ n + 1)"
       using sz by (simp add: unat_arith_simps ac_simps)
                   (simp add: unat_word_ariths mod_simps mod_eq_0_iff_dvd)
     with leq have "2 ^ n * (unat a div 2 ^ n + 1) = 2 ^ LENGTH('a)"
       by (force elim!: le_SucE)
     then have "unat a div 2 ^ n = 2 ^ LENGTH('a) div 2 ^ n - 1"
       by (metis (no_types, opaque_lifting) Groups.add_ac(2) add.right_neutral
                 add_diff_cancel_left' div_le_dividend div_mult_self4 gr_implies_not0
                 le_neq_implies_less power_eq_0_iff zero_neq_numeral)
     then have "unat a div 2 ^ n = 2 ^ (LENGTH('a) - n) - 1"
       using sz by (simp add: power_sub)
     then have "2 ^ (LENGTH('a) - n) - 1 < k" using r
       by simp
     then have False using kv by simp
   } then show ?thesis by clarsimp
 qed
 
 lemma alignUp_ar_helper:
   fixes a :: "'a :: len word"
   assumes al: "is_aligned x n"
   and     sz: "n < LENGTH('a)"
   and    sub: "{x..x + 2 ^ n - 1} \<subseteq> {a..b}"
   and    anz: "a \<noteq> 0"
   shows "a \<le> alignUp a n \<and> alignUp a n + 2 ^ n - 1 \<le> b"
 proof
   from al have xl: "x \<le> x + 2 ^ n - 1" by (simp add: is_aligned_no_overflow)
 
   from xl sub have ax: "a \<le> x"
     by auto
 
   show "a \<le> alignUp a n"
   proof (rule alignUp_ge)
     show "alignUp a n \<noteq> 0" using al sz ax anz
       by (rule alignUp_is_aligned_nz)
   qed fact+
 
   show "alignUp a n + 2 ^ n - 1 \<le> b"
   proof (rule order_trans)
     from xl show tp: "x + 2 ^ n - 1 \<le> b" using sub
       by auto
 
     from ax have "alignUp a n \<le> x"
       by (rule alignUp_le_greater_al) fact+
     then have "alignUp a n + (2 ^ n - 1) \<le> x + (2 ^ n - 1)"
       using xl al is_aligned_no_overflow' olen_add_eqv word_plus_mcs_3 by blast
     then show "alignUp a n + 2 ^ n - 1 \<le> x + 2 ^ n - 1"
       by (simp add: field_simps)
   qed
 qed
 
 lemma alignUp_def2:
   "alignUp a sz = a + 2 ^ sz - 1 AND NOT (mask sz)"
   by (simp add: alignUp_def flip: mask_eq_decr_exp)
 
 lemma alignUp_def3:
   "alignUp a sz = 2^ sz + (a - 1 AND NOT (mask sz))"
   by (simp add: alignUp_def2 is_aligned_triv field_simps mask_out_add_aligned)
 
 lemma  alignUp_plus:
   "is_aligned w us \<Longrightarrow> alignUp (w + a) us  = w + alignUp a us"
   by (clarsimp simp: alignUp_def2 mask_out_add_aligned field_simps)
 
 lemma alignUp_distance:
   "alignUp (q :: 'a :: len word) sz - q \<le> mask sz"
   by (metis (no_types) add.commute add_diff_cancel_left alignUp_def2 diff_add_cancel
                        mask_2pm1 subtract_mask(2) word_and_le1 word_sub_le_iff)
 
 lemma is_aligned_diff_neg_mask:
   "is_aligned p sz \<Longrightarrow> (p - q AND NOT (mask sz)) = (p - ((alignUp q sz) AND NOT (mask sz)))"
   apply (clarsimp simp only:word_and_le2 diff_conv_add_uminus)
   apply (subst mask_out_add_aligned[symmetric]; simp)
   apply (simp add: eq_neg_iff_add_eq_0)
   apply (subst add.commute)
   apply (simp add: alignUp_distance is_aligned_neg_mask_eq mask_out_add_aligned and_mask_eq_iff_le_mask flip: mask_eq_x_eq_0)
   done
 
 lemma word_clz_max:
   "word_clz w \<le> size (w::'a::len word)"
   unfolding word_clz_def
   by (metis length_takeWhile_le word_size_bl)
 
 lemma word_clz_nonzero_max:
   fixes w :: "'a::len word"
   assumes nz: "w \<noteq> 0"
   shows "word_clz w < size (w::'a::len word)"
 proof -
   {
     assume a: "word_clz w = size (w::'a::len word)"
     hence "length (takeWhile Not (to_bl w)) = length (to_bl w)"
       by (simp add: word_clz_def word_size)
     hence allj: "\<forall>j\<in>set(to_bl w). \<not> j"
       by (metis a length_takeWhile_less less_irrefl_nat word_clz_def)
     hence "to_bl w = replicate (length (to_bl w)) False"
       using eq_zero_set_bl nz by fastforce
     hence "w = 0"
       by (metis to_bl_0 word_bl.Rep_eqD word_bl_Rep')
     with nz have False by simp
   }
   thus ?thesis using word_clz_max
     by (fastforce intro: le_neq_trans)
 qed
 
 (* Sign extension from bit n. *)
 
 lemma bin_sign_extend_iff [bit_simps]:
   \<open>bit (sign_extend e w) i \<longleftrightarrow> bit w (min e i)\<close>
   if \<open>i < LENGTH('a)\<close> for w :: \<open>'a::len word\<close>
   using that by (simp add: sign_extend_def bit_simps min_def)
 
 lemma sign_extend_bitwise_if:
   "i < size w \<Longrightarrow> bit (sign_extend e w) i \<longleftrightarrow> (if i < e then bit w i else bit w e)"
   by (simp add: word_size bit_simps)
 
 lemma sign_extend_bitwise_if'  [word_eqI_simps]:
   \<open>i < LENGTH('a) \<Longrightarrow> bit (sign_extend e w) i \<longleftrightarrow> (if i < e then bit w i else bit w e)\<close>
   for w :: \<open>'a::len word\<close>
   using sign_extend_bitwise_if [of i w e] by (simp add: word_size)
 
 lemma sign_extend_bitwise_disj:
   "i < size w \<Longrightarrow> bit (sign_extend e w) i \<longleftrightarrow> i \<le> e \<and> bit w i \<or> e \<le> i \<and> bit w e"
   by (auto simp: sign_extend_bitwise_if)
 
 lemma sign_extend_bitwise_cases:
   "i < size w \<Longrightarrow> bit (sign_extend e w) i \<longleftrightarrow> (i \<le> e \<longrightarrow> bit w i) \<and> (e \<le> i \<longrightarrow> bit w e)"
   by (auto simp: sign_extend_bitwise_if)
 
 lemmas sign_extend_bitwise_disj' = sign_extend_bitwise_disj[simplified word_size]
 lemmas sign_extend_bitwise_cases' = sign_extend_bitwise_cases[simplified word_size]
 
 (* Often, it is easier to reason about an operation which does not overwrite
    the bit which determines which mask operation to apply. *)
 lemma sign_extend_def':
   "sign_extend n w = (if bit w n then w OR NOT (mask (Suc n)) else w AND mask (Suc n))"
   by (rule bit_word_eqI) (auto simp add: bit_simps sign_extend_eq_signed_take_bit min_def less_Suc_eq_le)
 
 lemma sign_extended_sign_extend:
   "sign_extended n (sign_extend n w)"
   by (clarsimp simp: sign_extended_def word_size sign_extend_bitwise_if)
 
 lemma sign_extended_iff_sign_extend:
   "sign_extended n w \<longleftrightarrow> sign_extend n w = w"
   apply auto
    apply (auto simp add: bit_eq_iff)
     apply (simp_all add: bit_simps sign_extend_eq_signed_take_bit not_le min_def sign_extended_def word_size split: if_splits)
   using le_imp_less_or_eq apply auto
   done
 
 lemma sign_extended_weaken:
   "sign_extended n w \<Longrightarrow> n \<le> m \<Longrightarrow> sign_extended m w"
   unfolding sign_extended_def by (cases "n < m") auto
 
 lemma sign_extend_sign_extend_eq:
   "sign_extend m (sign_extend n w) = sign_extend (min m n) w"
   by (rule bit_word_eqI) (simp add: sign_extend_eq_signed_take_bit bit_simps)
 
 lemma sign_extended_high_bits:
   "\<lbrakk> sign_extended e p; j < size p; e \<le> i; i < j \<rbrakk> \<Longrightarrow> bit p i = bit p j"
   by (drule (1) sign_extended_weaken; simp add: sign_extended_def)
 
 lemma sign_extend_eq:
   "w AND mask (Suc n) = v AND mask (Suc n) \<Longrightarrow> sign_extend n w = sign_extend n v"
   by (simp flip: take_bit_eq_mask add: sign_extend_eq_signed_take_bit signed_take_bit_eq_iff_take_bit_eq)
 
 lemma sign_extended_add:
   assumes p: "is_aligned p n"
   assumes f: "f < 2 ^ n"
   assumes e: "n \<le> e"
   assumes "sign_extended e p"
   shows "sign_extended e (p + f)"
 proof (cases "e < size p")
   case True
   note and_or = is_aligned_add_or[OF p f]
   have "\<not> bit f e"
     using True e less_2p_is_upper_bits_unset[THEN iffD1, OF f]
     by (fastforce simp: word_size)
   hence i: "bit (p + f) e = bit p e"
     by (simp add: and_or bit_simps)
   have fm: "f AND mask e = f"
     by (fastforce intro: subst[where P="\<lambda>f. f AND mask e = f", OF less_mask_eq[OF f]]
                   simp: mask_twice e)
   show ?thesis
     using assms
      apply (simp add: sign_extended_iff_sign_extend sign_extend_def i)
      apply (simp add: and_or word_bw_comms[of p f])
      apply (clarsimp simp: word_ao_dist fm word_bw_assocs split: if_splits)
     done
 next
   case False thus ?thesis
     by (simp add: sign_extended_def word_size)
 qed
 
 lemma sign_extended_neq_mask:
   "\<lbrakk>sign_extended n ptr; m \<le> n\<rbrakk> \<Longrightarrow> sign_extended n (ptr AND NOT (mask m))"
   by (fastforce simp: sign_extended_def word_size neg_mask_test_bit bit_simps)
 
 definition
   "limited_and (x :: 'a :: len word) y \<longleftrightarrow> (x AND y = x)"
 
 lemma limited_and_eq_0:
   "\<lbrakk> limited_and x z; y AND NOT z = y \<rbrakk> \<Longrightarrow> x AND y = 0"
   unfolding limited_and_def
   apply (subst arg_cong2[where f="(AND)"])
     apply (erule sym)+
   apply (simp(no_asm) add: word_bw_assocs word_bw_comms word_bw_lcs)
   done
 
 lemma limited_and_eq_id:
   "\<lbrakk> limited_and x z; y AND z = z \<rbrakk> \<Longrightarrow> x AND y = x"
   unfolding limited_and_def
   by (erule subst, fastforce simp: word_bw_lcs word_bw_assocs word_bw_comms)
 
 lemma lshift_limited_and:
   "limited_and x z \<Longrightarrow> limited_and (x << n) (z << n)"
   using push_bit_and [of n x z] by (simp add: limited_and_def shiftl_def)
 
 lemma rshift_limited_and:
   "limited_and x z \<Longrightarrow> limited_and (x >> n) (z >> n)"
   using drop_bit_and [of n x z] by (simp add: limited_and_def shiftr_def)
 
 lemmas limited_and_simps1 = limited_and_eq_0 limited_and_eq_id
 
 lemmas is_aligned_limited_and
     = is_aligned_neg_mask_eq[unfolded mask_eq_decr_exp, folded limited_and_def]
 
 lemmas limited_and_simps = limited_and_simps1
        limited_and_simps1[OF is_aligned_limited_and]
        limited_and_simps1[OF lshift_limited_and]
        limited_and_simps1[OF rshift_limited_and]
        limited_and_simps1[OF rshift_limited_and, OF is_aligned_limited_and]
        not_one_eq
 
 definition
   from_bool :: "bool \<Rightarrow> 'a::len word" where
   "from_bool b \<equiv> case b of True \<Rightarrow> of_nat 1
                          | False \<Rightarrow> of_nat 0"
 
 lemma from_bool_eq:
   \<open>from_bool = of_bool\<close>
   by (simp add: fun_eq_iff from_bool_def)
 
 lemma from_bool_0:
   "(from_bool x = 0) = (\<not> x)"
   by (simp add: from_bool_def split: bool.split)
 
 lemma from_bool_eq_if':
   "((if P then 1 else 0) = from_bool Q) = (P = Q)"
   by (cases Q) (simp_all add: from_bool_def)
 
 definition
   to_bool :: "'a::len word \<Rightarrow> bool" where
   "to_bool \<equiv> (\<noteq>) 0"
 
 lemma to_bool_and_1:
   "to_bool (x AND 1) \<longleftrightarrow> bit x 0"
   by (simp add: to_bool_def word_and_1)
 
 lemma to_bool_from_bool [simp]:
   "to_bool (from_bool r) = r"
   unfolding from_bool_def to_bool_def
   by (simp split: bool.splits)
 
 lemma from_bool_neq_0 [simp]:
   "(from_bool b \<noteq> 0) = b"
   by (simp add: from_bool_def split: bool.splits)
 
 lemma from_bool_mask_simp [simp]:
   "(from_bool r :: 'a::len word) AND 1 = from_bool r"
   unfolding from_bool_def
   by (clarsimp split: bool.splits)
 
 lemma from_bool_1 [simp]:
   "(from_bool P = 1) = P"
   by (simp add: from_bool_def split: bool.splits)
 
 lemma ge_0_from_bool [simp]:
   "(0 < from_bool P) = P"
   by (simp add: from_bool_def split: bool.splits)
 
 lemma limited_and_from_bool:
   "limited_and (from_bool b) 1"
   by (simp add: from_bool_def limited_and_def split: bool.split)
 
 lemma to_bool_1 [simp]: "to_bool 1" by (simp add: to_bool_def)
 lemma to_bool_0 [simp]: "\<not>to_bool 0" by (simp add: to_bool_def)
 
 lemma from_bool_eq_if:
   "(from_bool Q = (if P then 1 else 0)) = (P = Q)"
   by (cases Q) (simp_all add: from_bool_def)
 
 lemma to_bool_eq_0:
   "(\<not> to_bool x) = (x = 0)"
   by (simp add: to_bool_def)
 
 lemma to_bool_neq_0:
   "(to_bool x) = (x \<noteq> 0)"
   by (simp add: to_bool_def)
 
 lemma from_bool_all_helper:
   "(\<forall>bool. from_bool bool = val \<longrightarrow> P bool)
       = ((\<exists>bool. from_bool bool = val) \<longrightarrow> P (val \<noteq> 0))"
   by (auto simp: from_bool_0)
 
 lemma fold_eq_0_to_bool:
   "(v = 0) = (\<not> to_bool v)"
   by (simp add: to_bool_def)
 
 lemma from_bool_to_bool_iff:
   "w = from_bool b \<longleftrightarrow> to_bool w = b \<and> (w = 0 \<or> w = 1)"
   by (cases b) (auto simp: from_bool_def to_bool_def)
 
 lemma from_bool_eqI:
   "from_bool x = from_bool y \<Longrightarrow> x = y"
   unfolding from_bool_def
   by (auto split: bool.splits)
 
 lemma neg_mask_in_mask_range:
   "is_aligned ptr bits \<Longrightarrow> (ptr' AND NOT(mask bits) = ptr) = (ptr' \<in> mask_range ptr bits)"
   apply (erule is_aligned_get_word_bits)
    apply (rule iffI)
     apply (drule sym)
     apply (simp add: word_and_le2)
     apply (subst word_plus_and_or_coroll, word_eqI_solve)
     apply (metis bit.disj_ac(2) bit.disj_conj_distrib2 le_word_or2 word_and_max word_or_not)
    apply clarsimp
    apply (smt add.right_neutral eq_iff is_aligned_neg_mask_eq mask_out_add_aligned neg_mask_mono_le
               word_and_not)
   apply (simp add: power_overflow mask_eq_decr_exp)
   done
 
 lemma aligned_offset_in_range:
   "\<lbrakk> is_aligned (x :: 'a :: len word) m; y < 2 ^ m; is_aligned p n; n \<ge> m; n < LENGTH('a) \<rbrakk>
    \<Longrightarrow> (x + y \<in> {p .. p + mask n}) = (x \<in> mask_range p n)"
   apply (subst disjunctive_add)
    apply (simp add: bit_simps)
   apply (erule is_alignedE')
    apply (auto simp add: bit_simps not_le)[1]
    apply (metis less_2p_is_upper_bits_unset)
   apply (simp only: is_aligned_add_or word_ao_dist flip: neg_mask_in_mask_range)
   apply (subgoal_tac \<open>y AND NOT (mask n) = 0\<close>)
    apply simp
   apply (metis (full_types) is_aligned_mask is_aligned_neg_mask less_mask_eq word_bw_comms(1) word_bw_lcs(1))
   done
 
 lemma mask_range_to_bl':
   "\<lbrakk> is_aligned (ptr :: 'a :: len word) bits; bits < LENGTH('a) \<rbrakk>
    \<Longrightarrow> mask_range ptr bits
        = {x. take (LENGTH('a) - bits) (to_bl x) = take (LENGTH('a) - bits) (to_bl ptr)}"
   apply (rule set_eqI, rule iffI)
    apply clarsimp
    apply (subgoal_tac "\<exists>y. x = ptr + y \<and> y < 2 ^ bits")
     apply clarsimp
     apply (subst is_aligned_add_conv)
        apply assumption
       apply simp
     apply simp
    apply (rule_tac x="x - ptr" in exI)
    apply (simp add: add_diff_eq[symmetric])
    apply (simp only: word_less_sub_le[symmetric])
    apply (rule word_diff_ls')
     apply (simp add: field_simps mask_eq_decr_exp)
    apply assumption
   apply simp
   apply (subgoal_tac "\<exists>y. y < 2 ^ bits \<and> to_bl (ptr + y) = to_bl x")
    apply clarsimp
    apply (rule conjI)
     apply (erule(1) is_aligned_no_wrap')
    apply (simp only: add_diff_eq[symmetric] mask_eq_decr_exp)
    apply (rule word_plus_mono_right)
     apply simp
    apply (erule is_aligned_no_wrap')
    apply simp
   apply (rule_tac x="of_bl (drop (LENGTH('a) - bits) (to_bl x))" in exI)
   apply (rule context_conjI)
    apply (rule order_less_le_trans [OF of_bl_length])
     apply simp
    apply simp
   apply (subst is_aligned_add_conv)
      apply assumption
     apply simp
   apply (drule sym)
   apply (simp add: word_rep_drop)
   done
 
 lemma mask_range_to_bl:
   "is_aligned (ptr :: 'a :: len word) bits
    \<Longrightarrow> mask_range ptr bits
         = {x. take (LENGTH('a) - bits) (to_bl x) = take (LENGTH('a) - bits) (to_bl ptr)}"
   apply (erule is_aligned_get_word_bits)
    apply (erule(1) mask_range_to_bl')
   apply (rule set_eqI)
   apply (simp add: power_overflow mask_eq_decr_exp)
   done
 
 lemma aligned_mask_range_cases:
   "\<lbrakk> is_aligned (p :: 'a :: len word) n; is_aligned (p' :: 'a :: len word) n' \<rbrakk>
    \<Longrightarrow> mask_range p n \<inter> mask_range p' n' = {} \<or>
        mask_range p n \<subseteq> mask_range p' n' \<or>
        mask_range p n \<supseteq> mask_range p' n'"
   apply (simp add: mask_range_to_bl)
   apply (rule Meson.disj_comm, rule disjCI)
   apply auto
   apply (subgoal_tac "(\<exists>n''. LENGTH('a) - n = (LENGTH('a) - n') + n'')
                     \<or> (\<exists>n''. LENGTH('a) - n' = (LENGTH('a) - n) + n'')")
    apply (fastforce simp: take_add)
   apply arith
   done
 
 lemma aligned_mask_range_offset_subset:
   assumes al: "is_aligned (ptr :: 'a :: len word) sz" and al': "is_aligned x sz'"
   and szv: "sz' \<le> sz"
   and xsz: "x < 2 ^ sz"
   shows "mask_range (ptr+x) sz' \<subseteq> mask_range ptr sz"
   using al
 proof (rule is_aligned_get_word_bits)
   assume p0: "ptr = 0" and szv': "LENGTH ('a) \<le> sz"
   then have "(2 ::'a word) ^ sz = 0" by simp
   show ?thesis using p0
     by (simp add: \<open>2 ^ sz = 0\<close> mask_eq_decr_exp)
 next
   assume szv': "sz < LENGTH('a)"
 
   hence blah: "2 ^ (sz - sz') < (2 :: nat) ^ LENGTH('a)"
     using szv by auto
   show ?thesis using szv szv'
     apply auto
     using al assms(4) is_aligned_no_wrap' apply blast
     apply (simp only: flip: add_diff_eq add_mask_fold)
     apply (subst add.assoc, rule word_plus_mono_right)
      using al' is_aligned_add_less_t2n xsz
      apply fastforce
     apply (simp add: field_simps szv al is_aligned_no_overflow)
     done
 qed
 
 lemma aligned_mask_ranges_disjoint:
   "\<lbrakk> is_aligned (p :: 'a :: len word) n; is_aligned (p' :: 'a :: len word) n';
      p AND NOT(mask n') \<noteq> p'; p' AND NOT(mask n) \<noteq> p \<rbrakk>
    \<Longrightarrow> mask_range p n \<inter> mask_range p' n' = {}"
   using aligned_mask_range_cases
   by (auto simp: neg_mask_in_mask_range)
 
 lemma aligned_mask_ranges_disjoint2:
   "\<lbrakk> is_aligned p n; is_aligned ptr bits; n \<ge> m; n < size p; m \<le> bits;
      (\<forall>y < 2 ^ (n - m). p + (y << m) \<notin> mask_range ptr bits) \<rbrakk>
    \<Longrightarrow> mask_range p n \<inter> mask_range ptr bits = {}"
   apply safe
   apply (simp only: flip: neg_mask_in_mask_range)
   apply (drule_tac x="x AND mask n >> m" in spec)
   apply (erule notE[OF mp])
    apply (simp flip: take_bit_eq_mask add: shiftr_def drop_bit_take_bit)
    apply transfer
   apply simp
    apply (simp add: word_size and_mask_less_size)
   apply (subst disjunctive_add)
    apply (auto simp add: bit_simps word_size intro!: bit_eqI)
   done
 
 lemma word_clz_sint_upper[simp]:
   "LENGTH('a) \<ge> 3 \<Longrightarrow> sint (of_nat (word_clz (w :: 'a :: len word)) :: 'a sword) \<le> int (LENGTH('a))"
   using word_clz_max [of w]
   apply (simp add: word_size signed_of_nat)
   apply (subst signed_take_bit_int_eq_self)
     apply simp_all
    apply (metis negative_zle of_nat_numeral semiring_1_class.of_nat_power)
   apply (drule small_powers_of_2)
   apply (erule le_less_trans)
   apply simp
   done
 
 lemma word_clz_sint_lower[simp]:
   "LENGTH('a) \<ge> 3
    \<Longrightarrow> - sint (of_nat (word_clz (w :: 'a :: len word)) :: 'a signed word) \<le> int (LENGTH('a))"
   apply (subst sint_eq_uint)
   using word_clz_max [of w]
    apply (simp_all add: word_size unsigned_of_nat)
   apply (rule not_msb_from_less)
   apply (simp add: word_less_nat_alt unsigned_of_nat)
   apply (subst take_bit_nat_eq_self)
    apply (simp add: le_less_trans)
   apply (drule small_powers_of_2)
   apply (erule le_less_trans)
   apply simp
   done
 
 lemma mask_range_subsetD:
   "\<lbrakk> p' \<in> mask_range p n; x' \<in> mask_range p' n'; n' \<le> n; is_aligned p n; is_aligned p' n' \<rbrakk> \<Longrightarrow>
    x' \<in> mask_range p n"
   using aligned_mask_step by fastforce
 
 lemma add_mult_in_mask_range:
   "\<lbrakk> is_aligned (base :: 'a :: len word) n; n < LENGTH('a); bits \<le> n; x < 2 ^ (n - bits) \<rbrakk>
    \<Longrightarrow> base + x * 2^bits \<in> mask_range base n"
   by (simp add: is_aligned_no_wrap' mask_2pm1 nasty_split_lt word_less_power_trans2
                 word_plus_mono_right)
 
 lemma from_to_bool_last_bit:
   "from_bool (to_bool (x AND 1)) = x AND 1"
   by (metis from_bool_to_bool_iff word_and_1)
 
 lemma sint_ctz:
   \<open>0 \<le> sint (of_nat (word_ctz (x :: 'a :: len word)) :: 'a signed word)
      \<and> sint (of_nat (word_ctz x) :: 'a signed word) \<le> int (LENGTH('a))\<close> (is \<open>?P \<and> ?Q\<close>)
   if \<open>LENGTH('a) > 2\<close>
 proof
   have *: \<open>word_ctz x < 2 ^ (LENGTH('a) - Suc 0)\<close>
     using word_ctz_le apply (rule le_less_trans)
     using that small_powers_of_2 [of \<open>LENGTH('a)\<close>] apply simp
     done
   have \<open>int (word_ctz x) div 2 ^ (LENGTH('a) - Suc 0) = 0\<close>
     apply (rule div_pos_pos_trivial)
      apply (simp_all add: *)
     done
   then show ?P by (simp add: signed_of_nat bit_iff_odd)
   show ?Q
     apply (auto simp add: signed_of_nat)
     apply (subst signed_take_bit_int_eq_self)
       apply (auto simp add: word_ctz_le * minus_le_iff [of _ \<open>int (word_ctz x)\<close>])
     apply (rule order.trans [of _ 0])
      apply simp_all
     done
 qed
 
 lemma unat_of_nat_word_log2:
   "LENGTH('a) < 2 ^ LENGTH('b)
    \<Longrightarrow> unat (of_nat (word_log2 (n :: 'a :: len word)) :: 'b :: len word) = word_log2 n"
   by (metis less_trans unat_of_nat_eq word_log2_max word_size)
 
 lemma aligned_mask_diff:
   "\<lbrakk> is_aligned (dest :: 'a :: len word) bits; is_aligned (ptr :: 'a :: len word) sz;
      bits \<le> sz; sz < LENGTH('a); dest < ptr \<rbrakk>
    \<Longrightarrow> mask bits + dest < ptr"
   apply (frule_tac p' = ptr in aligned_mask_range_cases, assumption)
   apply (elim disjE)
     apply (drule_tac is_aligned_no_overflow_mask, simp)+
     apply (simp add: algebra_split_simps word_le_not_less)
    apply (drule is_aligned_no_overflow_mask; fastforce)
   apply (simp add: is_aligned_weaken algebra_split_simps)
   apply (auto simp add: not_le)
   using is_aligned_no_overflow_mask leD apply blast
   apply (meson aligned_add_mask_less_eq is_aligned_weaken le_less_trans)
   done
 
+lemma Suc_mask_eq_mask:
+  "\<not>bit a n \<Longrightarrow> a AND mask (Suc n) = a AND mask n" for a::"'a::len word"
+  by (metis sign_extend_def sign_extend_def')
+
+lemma word_less_high_bits:
+  fixes a::"'a::len word"
+  assumes high_bits: "\<forall>i > n. bit a i = bit b i"
+  assumes less: "a AND mask (Suc n) < b AND mask (Suc n)"
+  shows "a < b"
+proof -
+  let ?masked = "\<lambda>x. x AND NOT (mask (Suc n))"
+  from high_bits
+  have "?masked a = ?masked b"
+    by - word_eqI_solve
+  then
+  have "?masked a + (a AND mask (Suc n)) < ?masked b + (b AND mask (Suc n))"
+    by (metis AND_NOT_mask_plus_AND_mask_eq less word_and_le2 word_plus_strict_mono_right)
+  then
+  show ?thesis
+    by (simp add: AND_NOT_mask_plus_AND_mask_eq)
+qed
+
+lemma word_less_bitI:
+  fixes a :: "'a::len word"
+  assumes hi_bits: "\<forall>i > n. bit a i = bit b i"
+  assumes a_bits: "\<not>bit a n"
+  assumes b_bits: "bit b n" "n < LENGTH('a)"
+  shows "a < b"
+proof -
+  from b_bits
+  have "a AND mask n < b AND mask (Suc n)"
+    by (metis bit_mask_iff impossible_bit le2p_bits_unset leI lessI less_Suc_eq_le mask_eq_decr_exp
+              word_and_less' word_ao_nth)
+  with a_bits
+  have "a AND mask (Suc n) < b AND mask (Suc n)"
+    by (simp add: Suc_mask_eq_mask)
+  with hi_bits
+  show ?thesis
+    by (rule word_less_high_bits)
+qed
+
+lemma word_less_bitD:
+  fixes a::"'a::len word"
+  assumes less: "a < b"
+  shows "\<exists>n. (\<forall>i > n. bit a i = bit b i) \<and> \<not>bit a n \<and> bit b n"
+proof -
+  define xs where "xs \<equiv> zip (to_bl a) (to_bl b)"
+  define tk where "tk \<equiv> length (takeWhile (\<lambda>(x,y). x = y) xs)"
+  define  n where  "n \<equiv> LENGTH('a) - Suc tk"
+  have n_less: "n < LENGTH('a)"
+    by (simp add: n_def)
+  moreover
+  { fix i
+    have "\<not> i < LENGTH('a) \<Longrightarrow> bit a i = bit b i"
+      using bit_imp_le_length by blast
+    moreover
+    assume "i > n"
+    with n_less
+    have "i < LENGTH('a) \<Longrightarrow> LENGTH('a) - Suc i < tk"
+      unfolding n_def by arith
+    hence "i < LENGTH('a) \<Longrightarrow> bit a i = bit b i"
+      unfolding n_def tk_def xs_def
+      by (fastforce dest: takeWhile_take_has_property_nth simp: rev_nth simp flip: nth_rev_to_bl)
+    ultimately
+    have "bit a i = bit b i"
+      by blast
+  }
+  note all = this
+  moreover
+  from less
+  have "a \<noteq> b" by simp
+  then
+  obtain i where "to_bl a ! i \<noteq> to_bl b ! i"
+    using nth_equalityI word_bl.Rep_eqD word_rotate.lbl_lbl by blast
+  then
+  have "tk \<noteq> length xs"
+    unfolding tk_def xs_def
+    by (metis length_takeWhile_less list_eq_iff_zip_eq nat_neq_iff word_rotate.lbl_lbl)
+  then
+  have "tk < length xs"
+    using length_takeWhile_le order_le_neq_trans tk_def by blast
+  from nth_length_takeWhile[OF this[unfolded tk_def]]
+  have "fst (xs ! tk) \<noteq> snd (xs ! tk)"
+    by (clarsimp simp: tk_def)
+  with `tk < length xs`
+  have "bit a n \<noteq> bit b n"
+    by (clarsimp simp: xs_def n_def tk_def nth_rev simp flip: nth_rev_to_bl)
+  with less all
+  have "\<not>bit a n \<and> bit b n"
+    by (metis n_less order.asym word_less_bitI)
+  ultimately
+  show ?thesis by blast
+qed
+
+lemma word_less_bit_eq:
+  "(a < b) = (\<exists>n < LENGTH('a). (\<forall>i > n. bit a i = bit b i) \<and> \<not>bit a n \<and> bit b n)" for a::"'a::len word"
+  by (meson bit_imp_le_length word_less_bitD word_less_bitI)
+
 end
 
 end
\ No newline at end of file
diff --git a/thys/Word_Lib/Sgn_Abs.thy b/thys/Word_Lib/Sgn_Abs.thy
new file mode 100644
--- /dev/null
+++ b/thys/Word_Lib/Sgn_Abs.thy
@@ -0,0 +1,131 @@
+(*
+ * Copyright 2023, Proofcraft Pty Ltd
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
+
+theory Sgn_Abs
+  imports Most_significant_bit
+begin
+
+section \<open>@{const sgn} and @{const abs} for @{typ "'a word"}\<close>
+
+subsection \<open>Instances\<close>
+
+text \<open>@{const sgn} on words returns -1, 0, or 1.\<close>
+instantiation word :: (len) sgn
+begin
+
+definition sgn_word :: \<open>'a word \<Rightarrow> 'a word\<close> where
+  \<open>sgn w = (if w = 0 then 0 else if 0 <s w then 1 else -1)\<close>
+
+instance ..
+
+end
+
+
+(* Simplification setup for sgn on numerals *)
+
+lemma word_sgn_0[simp]:
+  "sgn 0 = (0::'a::len word)"
+  by (simp add: sgn_word_def)
+
+lemma word_sgn_1[simp]:
+  "sgn 1 = (1::'a::len word)"
+  by (simp add: sgn_word_def)
+
+lemma word_sgn_max_word[simp]:
+  "sgn (- 1) = (-1::'a::len word)"
+  by (clarsimp simp: sgn_word_def word_sless_alt)
+
+lemmas word_sgn_numeral[simp] = sgn_word_def[where w="numeral w" for w]
+
+
+text \<open>@{const abs} on words is the usual definition.\<close>
+instantiation word :: (len) abs
+begin
+
+definition abs_word :: \<open>'a word \<Rightarrow> 'a word\<close> where
+  \<open>abs w = (if w \<le>s 0 then -w else w)\<close>
+
+instance ..
+
+end
+
+
+(* Simplification setup for abs on numerals *)
+
+lemma word_abs_0[simp]:
+  "\<bar>0\<bar> = (0::'a::len word)"
+  by (simp add: abs_word_def)
+
+lemma word_abs_1[simp]:
+  "\<bar>1\<bar> = (1::'a::len word)"
+  by (simp add: abs_word_def)
+
+lemma word_abs_max_word[simp]:
+  "\<bar>- 1\<bar> = (1::'a::len word)"
+  by (clarsimp simp: abs_word_def word_sle_eq)
+
+lemma word_abs_msb:
+  "\<bar>w\<bar> = (if msb w then -w else w)" for w::"'a::len word"
+  by (simp add: abs_word_def msb_word_iff_sless_0 word_sless_eq)
+
+lemmas word_abs_numeral[simp] = word_abs_msb[where w="numeral w" for w]
+
+
+subsection \<open>Properties\<close>
+
+(* Many of these are from linordered_idom, but need <s instead of < and occasionally
+   an additional assumption on minimum word length, because in "1 word" we have -1 = 1. *)
+
+lemma word_sgn_0_0:
+  "sgn a = 0 \<longleftrightarrow> a = 0" for a::"'a::len word"
+  by (simp add: sgn_word_def)
+
+lemma word_sgn_1_pos:
+  "1 < LENGTH('a) \<Longrightarrow> sgn a = 1 \<longleftrightarrow> 0 <s a" for a::"'a::len word"
+  unfolding sgn_word_def by simp
+
+lemma word_sgn_1_neg:
+  "sgn a = - 1 \<longleftrightarrow> a <s 0"
+  unfolding sgn_word_def
+  using sint_1_cases by force
+
+lemma word_sgn_pos[simp]:
+  "0 <s a \<Longrightarrow> sgn a = 1"
+  by (simp add: sgn_word_def)
+
+lemma word_sgn_neg[simp]:
+  "a <s 0 \<Longrightarrow> sgn a = - 1"
+  by (simp only: word_sgn_1_neg)
+
+lemma word_abs_sgn:
+  "\<bar>k\<bar> = k * sgn k" for k :: "'a::len word"
+  unfolding sgn_word_def abs_word_def
+  by auto
+
+lemma word_sgn_greater[simp]:
+  "0 <s sgn a \<longleftrightarrow> 0 <s a" for a::"'a::len word"
+  by (smt (verit) signed_eq_0_iff sint_1_cases sint_n1 word_sgn_0_0 word_sgn_neg word_sgn_pos
+                  word_sless_alt)
+
+lemma word_sgn_less[simp]:
+  "sgn a <s 0 \<longleftrightarrow> a <s 0" for a::"'a::len word"
+  unfolding sgn_word_def
+  using degenerate_word signed.antisym_conv3 word_sless_alt by force
+
+lemma word_abs_sgn_eq_1[simp]:
+  "a \<noteq> 0 \<Longrightarrow> \<bar>sgn a\<bar> = 1" for a::"'a::len word"
+  unfolding abs_word_def sgn_word_def
+  by (clarsimp simp: word_sle_eq)
+
+lemma word_abs_sgn_eq:
+  "\<bar>sgn a\<bar> = (if a = 0 then 0 else 1)" for a::"'a::len word"
+  by clarsimp
+
+lemma word_sgn_mult_self_eq[simp]:
+  "sgn a * sgn a = of_bool (a \<noteq> 0)" for a::"'a::len word"
+  by (cases "0 <s a"; simp)
+
+end
\ No newline at end of file
diff --git a/thys/Word_Lib/Word_EqI.thy b/thys/Word_Lib/Word_EqI.thy
--- a/thys/Word_Lib/Word_EqI.thy
+++ b/thys/Word_Lib/Word_EqI.thy
@@ -1,81 +1,89 @@
 (*
  * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
  * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "Solving Word Equalities"
 
 theory Word_EqI
   imports
     More_Word
     Aligned
     "HOL-Eisbach.Eisbach_Tools"
 begin
 
 text \<open>
   Some word equalities can be solved by considering the problem bitwise for all
-  @{prop "n < LENGTH('a::len)"}, which is different to running @{text word_bitwise}
-  and expanding into an explicit list of bits.
+  @{prop "n < LENGTH('a::len)"}. This is similar to the existing method @{text word_bitwise}
+  and expanding into an explicit list of bits. The @{text word_bitwise} only works on
+  concrete word lengths, but can treat a wider number of operators (in particular a mix of
+  arithmetic, order, and bit operations). The @{text word_eqI} method below works on words of
+  abstract size (@{typ "'a word"}) and produces smaller, more abstract goals, but does not deal
+  with arithmetic operations.
 \<close>
 
 lemmas le_mask_high_bits_len = le_mask_high_bits[unfolded word_size]
 lemmas neg_mask_le_high_bits_len = neg_mask_le_high_bits[unfolded word_size]
 
 named_theorems word_eqI_simps
 
 lemmas [word_eqI_simps] =
   word_or_zero
   neg_mask_test_bit
   nth_ucast
   less_2p_is_upper_bits_unset
   le_mask_high_bits_len
   neg_mask_le_high_bits_len
   bang_eq
   is_up
   is_down
   is_aligned_nth
   word_size (* keep this, because the goal may contain "n < size x" terms *)
 
 lemmas word_eqI_folds [symmetric] =
   push_bit_eq_mult
   drop_bit_eq_div
   take_bit_eq_mod
 
 (* bit_eqI subsumes the first rule; kept for backwards compatibility *)
 lemmas word_eqI_rules = word_eqI [rule_format, unfolded word_size] bit_eqI
 
 lemma test_bit_lenD:
   "bit x n \<Longrightarrow> n < LENGTH('a) \<and> bit x n" for x :: "'a :: len word"
   by (fastforce dest: test_bit_size simp: word_size)
 
+\<comment> \<open>Method to reduce goals of the form @{prop "P \<Longrightarrow> x = y"} for words of abstract length to
+    reasoning on bits of the words. Leaves open goal if unsolved.\<close>
 method word_eqI uses simp simp_del split split_del cong flip =
   ((* reduce conclusion to test_bit: *)
    rule word_eqI_rules,
    (* fold common patterns into bit form *)
    (simp only: word_eqI_folds)?,
    (* make sure we're in clarsimp normal form: *)
    (clarsimp simp: simp simp del: simp_del simp flip: flip split: split split del: split_del cong: cong)?,
    (* turn x < 2^n assumptions into mask equations: *)
    ((drule less_mask_eq)+)?,
    (* expand and distribute test_bit everywhere: *)
    (simp only: bit_simps word_eqI_simps)?,
    (* clarsimp normal form again, also resolve negated < and \<le> *)
    (clarsimp simp: simp not_less not_le simp del: simp_del simp flip: flip
              split: split split del: split_del cong: cong)?,
    (* add any additional word size constraints to new indices: *)
    ((drule test_bit_lenD)+)?,
    (* try to make progress (can't use +, would loop): *)
    (simp only: bit_simps word_eqI_simps)?,
    (clarsimp simp: simp simp del: simp_del simp flip: flip
              split: split split del: split_del cong: cong)?,
    (* helps sometimes, rarely: *)
    (simp add: simp test_bit_conj_lt del: simp_del flip: flip split: split split del: split_del cong: cong)?)
 
+\<comment> \<open>Method to reduce goals of the form @{prop "P \<Longrightarrow> x = y"} for words of abstract length to
+    reasoning on bits of the words. Fails if goal unsolved, but tries harder than @{method word_eqI}.\<close>
 method word_eqI_solve uses simp simp_del split split_del cong flip dest =
   solves \<open>word_eqI simp: simp simp_del: simp_del split: split split_del: split_del
                    cong: cong simp flip: flip;
           (fastforce dest: dest simp: simp flip: flip
                      simp: simp simp del: simp_del split: split split del: split_del cong: cong)?\<close>
 
 end
diff --git a/thys/Word_Lib/Word_Lemmas.thy b/thys/Word_Lib/Word_Lemmas.thy
--- a/thys/Word_Lib/Word_Lemmas.thy
+++ b/thys/Word_Lib/Word_Lemmas.thy
@@ -1,1888 +1,1915 @@
 (*
  * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
  * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "Lemmas with Generic Word Length"
 
 theory Word_Lemmas
   imports
     Type_Syntax
     Signed_Division_Word
     Signed_Words
     More_Word
     Most_significant_bit
     Enumeration_Word
     Aligned
     Bit_Shifts_Infix_Syntax
+    Boolean_Inequalities
     Word_EqI
 begin
 
 context
   includes bit_operations_syntax
 begin
 
+lemma word_max_le_or:
+  "max x y \<le> x OR y" for x :: "'a::len word"
+  by (simp add: word_bool_le_funs)
+
+lemma word_and_le_min:
+  "x AND y \<le> min x y" for x :: "'a::len word"
+  by (simp add: word_bool_le_funs)
+
+lemma word_not_le_eq:
+  "(NOT x \<le> y) = (NOT y \<le> x)" for x :: "'a::len word"
+  by transfer (auto simp: take_bit_not_eq_mask_diff)
+
+lemma word_not_le_not_eq[simp]:
+  "(NOT y \<le> NOT x) = (x \<le> y)" for x :: "'a::len word"
+  by (subst word_not_le_eq) simp
+
+lemma not_min_eq:
+  "NOT (min x y) = max (NOT x) (NOT y)" for x :: "'a::len word"
+  unfolding min_def max_def
+  by auto
+
+lemma not_max_eq:
+  "NOT (max x y) = min (NOT x) (NOT y)" for x :: "'a::len word"
+  unfolding min_def max_def
+  by auto
+
 lemma ucast_le_ucast_eq:
   fixes x y :: "'a::len word"
   assumes x: "x < 2 ^ n"
   assumes y: "y < 2 ^ n"
   assumes n: "n = LENGTH('b::len)"
   shows "(UCAST('a \<rightarrow> 'b) x \<le> UCAST('a \<rightarrow> 'b) y) = (x \<le> y)"
   apply (rule iffI)
    apply (cases "LENGTH('b) < LENGTH('a)")
     apply (subst less_mask_eq[OF x, symmetric])
     apply (subst less_mask_eq[OF y, symmetric])
     apply (unfold n)
     apply (subst ucast_ucast_mask[symmetric])+
     apply (simp add: ucast_le_ucast)+
   apply (erule ucast_mono_le[OF _ y[unfolded n]])
   done
 
 lemma ucast_zero_is_aligned:
   \<open>is_aligned w n\<close> if \<open>UCAST('a::len \<rightarrow> 'b::len) w = 0\<close> \<open>n \<le> LENGTH('b)\<close>
 proof (rule is_aligned_bitI)
   fix q
   assume \<open>q < n\<close>
   moreover have \<open>bit (UCAST('a::len \<rightarrow> 'b::len) w) q = bit 0 q\<close>
     using that by simp
   with \<open>q < n\<close> \<open>n \<le> LENGTH('b)\<close> show \<open>\<not> bit w q\<close>
     by (simp add: bit_simps)
 qed
 
 lemma unat_ucast_eq_unat_and_mask:
   "unat (UCAST('b::len \<rightarrow> 'a::len) w) = unat (w AND mask LENGTH('a))"
   apply (simp flip: take_bit_eq_mask)
   apply transfer
   apply (simp add: ac_simps)
   done
 
 lemma le_max_word_ucast_id:
   \<open>UCAST('b \<rightarrow> 'a) (UCAST('a \<rightarrow> 'b) x) = x\<close>
     if \<open>x \<le> UCAST('b::len \<rightarrow> 'a) (- 1)\<close>
     for x :: \<open>'a::len word\<close>
 proof -
   from that have a1: \<open>x \<le> word_of_int (uint (word_of_int (2 ^ LENGTH('b) - 1) :: 'b word))\<close>
     by (simp add: of_int_mask_eq)
   have f2: "((\<exists>i ia. (0::int) \<le> i \<and> \<not> 0 \<le> i + - 1 * ia \<and> i mod ia \<noteq> i) \<or>
             \<not> (0::int) \<le> - 1 + 2 ^ LENGTH('b) \<or> (0::int) \<le> - 1 + 2 ^ LENGTH('b) + - 1 * 2 ^ LENGTH('b) \<or>
             (- (1::int) + 2 ^ LENGTH('b)) mod 2 ^ LENGTH('b) =
               - 1 + 2 ^ LENGTH('b)) = ((\<exists>i ia. (0::int) \<le> i \<and> \<not> 0 \<le> i + - 1 * ia \<and> i mod ia \<noteq> i) \<or>
             \<not> (1::int) \<le> 2 ^ LENGTH('b) \<or>
             2 ^ LENGTH('b) + - (1::int) * ((- 1 + 2 ^ LENGTH('b)) mod 2 ^ LENGTH('b)) = 1)"
     by force
   have f3: "\<forall>i ia. \<not> (0::int) \<le> i \<or> 0 \<le> i + - 1 * ia \<or> i mod ia = i"
     using mod_pos_pos_trivial by force
   have "(1::int) \<le> 2 ^ LENGTH('b)"
     by simp
   then have "2 ^ LENGTH('b) + - (1::int) * ((- 1 + 2 ^ LENGTH('b)) mod 2 ^ len_of TYPE ('b)) = 1"
     using f3 f2 by blast
   then have f4: "- (1::int) + 2 ^ LENGTH('b) = (- 1 + 2 ^ LENGTH('b)) mod 2 ^ LENGTH('b)"
     by linarith
   have f5: "x \<le> word_of_int (uint (word_of_int (- 1 + 2 ^ LENGTH('b))::'b word))"
     using a1 by force
   have f6: "2 ^ LENGTH('b) + - (1::int) = - 1 + 2 ^ LENGTH('b)"
     by force
   have f7: "- (1::int) * 1 = - 1"
     by auto
   have "\<forall>x0 x1. (x1::int) - x0 = x1 + - 1 * x0"
     by force
   then have "x \<le> 2 ^ LENGTH('b) - 1"
     using f7 f6 f5 f4 by (metis uint_word_of_int wi_homs(2) word_arith_wis(8) word_of_int_2p)
   then have \<open>uint x \<le> uint (2 ^ LENGTH('b) - (1 :: 'a word))\<close>
     by (simp add: word_le_def)
   then have \<open>uint x \<le> 2 ^ LENGTH('b) - 1\<close>
     by (simp add: uint_word_ariths)
       (metis \<open>1 \<le> 2 ^ LENGTH('b)\<close> \<open>uint x \<le> uint (2 ^ LENGTH('b) - 1)\<close> linorder_not_less lt2p_lem uint_1 uint_minus_simple_alt uint_power_lower word_le_def zle_diff1_eq)
   then show ?thesis
     apply (simp add: unsigned_ucast_eq take_bit_word_eq_self_iff)
     apply (meson \<open>x \<le> 2 ^ LENGTH('b) - 1\<close> not_le word_less_sub_le)
     done
 qed
 
 lemma uint_shiftr_eq:
   \<open>uint (w >> n) = uint w div 2 ^ n\<close>
   by word_eqI
 
 lemma bit_shiftl_word_iff [bit_simps]:
   \<open>bit (w << m) n \<longleftrightarrow> m \<le> n \<and> n < LENGTH('a) \<and> bit w (n - m)\<close>
   for w :: \<open>'a::len word\<close>
   by (simp add: bit_simps)
 
 lemma bit_shiftr_word_iff:
   \<open>bit (w >> m) n \<longleftrightarrow> bit w (m + n)\<close>
   for w :: \<open>'a::len word\<close>
   by (simp add: bit_simps)
 
 lemma uint_sshiftr_eq:
   \<open>uint (w >>> n) = take_bit LENGTH('a) (sint w div 2 ^  n)\<close>
   for w :: \<open>'a::len word\<close>
   by (word_eqI_solve dest: test_bit_lenD)
 
 lemma sshiftr_n1: "-1 >>> n = -1"
   by (simp add: sshiftr_def)
 
 lemma nth_sshiftr:
   "bit (w >>> m) n = (n < size w \<and> (if n + m \<ge> size w then bit w (size w - 1) else bit w (n + m)))"
   apply (auto simp add: bit_simps word_size ac_simps not_less)
   apply (meson bit_imp_le_length bit_shiftr_word_iff leD)
   done
 
 lemma sshiftr_numeral:
   \<open>(numeral k >>> numeral n :: 'a::len word) =
     word_of_int (signed_take_bit (LENGTH('a) - 1) (numeral k) >> numeral n)\<close>
   using signed_drop_bit_word_numeral [of n k] by (simp add: sshiftr_def shiftr_def)
 
 lemma sshiftr_div_2n: "sint (w >>> n) = sint w div 2 ^ n"
   by word_eqI (cases \<open>n < LENGTH('a)\<close>; fastforce simp: le_diff_conv2)
 
 lemma mask_eq:
   \<open>mask n = (1 << n) - (1 :: 'a::len word)\<close>
   by (simp add: mask_eq_exp_minus_1 shiftl_def)
 
 lemma nth_shiftl': "bit (w << m) n \<longleftrightarrow> n < size w \<and> n >= m \<and> bit w (n - m)"
   for w :: "'a::len word"
   by (simp add: bit_simps word_size ac_simps)
 
 lemmas nth_shiftl = nth_shiftl' [unfolded word_size]
 
 lemma nth_shiftr: "bit (w >> m) n = bit w (n + m)"
   for w :: "'a::len word"
   by (simp add: bit_simps ac_simps)
 
 lemma shiftr_div_2n: "uint (shiftr w n) = uint w div 2 ^ n"
   by (fact uint_shiftr_eq)
 
 lemma shiftl_rev: "shiftl w n = word_reverse (shiftr (word_reverse w) n)"
   by word_eqI_solve
 
 lemma rev_shiftl: "word_reverse w << n = word_reverse (w >> n)"
   by (simp add: shiftl_rev)
 
 lemma shiftr_rev: "w >> n = word_reverse (word_reverse w << n)"
   by (simp add: rev_shiftl)
 
 lemma rev_shiftr: "word_reverse w >> n = word_reverse (w << n)"
   by (simp add: shiftr_rev)
 
 lemmas ucast_up =
   rc1 [simplified rev_shiftr [symmetric] revcast_ucast [symmetric]]
 lemmas ucast_down =
   rc2 [simplified rev_shiftr revcast_ucast [symmetric]]
 
 lemma shiftl_zero_size: "size x \<le> n \<Longrightarrow> x << n = 0"
   for x :: "'a::len word"
   by (simp add: shiftl_def word_size)
 
 lemma shiftl_t2n: "shiftl w n = 2 ^ n * w"
   for w :: "'a::len word"
   by (simp add: shiftl_def push_bit_eq_mult)
 
 lemma word_shift_by_2:
   "x * 4 = (x::'a::len word) << 2"
   by (simp add: shiftl_t2n)
 
 lemma word_shift_by_3:
   "x * 8 = (x::'a::len word) << 3"
   by (simp add: shiftl_t2n)
 
 lemma slice_shiftr: "slice n w = ucast (w >> n)"
   by word_eqI (cases \<open>n \<le> LENGTH('b)\<close>; fastforce simp: ac_simps dest: bit_imp_le_length)
 
 lemma shiftr_zero_size: "size x \<le> n \<Longrightarrow> x >> n = 0"
   for x :: "'a :: len word"
   by word_eqI
 
 lemma shiftr_x_0 [simp]: "x >> 0 = x"
   for x :: "'a::len word"
   by (simp add: shiftr_def)
 
 lemma shiftl_x_0 [simp]: "x << 0 = x"
   for x :: "'a::len word"
   by (simp add: shiftl_def)
 
 lemmas shiftl0 = shiftl_x_0
 
 lemma shiftr_1 [simp]: "(1::'a::len word) >> n = (if n = 0 then 1 else 0)"
   by (simp add: shiftr_def)
 
 lemma and_not_mask:
   "w AND NOT (mask n) = (w >> n) << n"
   for w :: \<open>'a::len word\<close>
   by word_eqI_solve
 
 lemma and_mask:
   "w AND mask n = (w << (size w - n)) >> (size w - n)"
   for w :: \<open>'a::len word\<close>
   by word_eqI_solve
 
 lemma shiftr_div_2n_w: "w >> n = w div (2^n :: 'a :: len word)"
   by (fact shiftr_eq_div)
 
 lemma le_shiftr:
   "u \<le> v \<Longrightarrow> u >> (n :: nat) \<le> (v :: 'a :: len word) >> n"
   apply (unfold shiftr_def)
   apply transfer
   apply (simp add: take_bit_drop_bit)
   apply (simp add: drop_bit_eq_div zdiv_mono1)
   done
 
 lemma le_shiftr':
   "\<lbrakk> u >> n \<le> v >> n ; u >> n \<noteq> v >> n \<rbrakk> \<Longrightarrow> (u::'a::len word) \<le> v"
   by (metis le_cases le_shiftr verit_la_disequality)
 
 lemma shiftr_mask_le:
   "n <= m \<Longrightarrow> mask n >> m = (0 :: 'a::len word)"
   by word_eqI
 
 lemma shiftr_mask [simp]:
   \<open>mask m >> m = (0::'a::len word)\<close>
   by (rule shiftr_mask_le) simp
 
 lemma le_mask_iff:
   "(w \<le> mask n) = (w >> n = 0)"
   for w :: \<open>'a::len word\<close>
   apply safe
    apply (rule word_le_0_iff [THEN iffD1])
    apply (rule xtrans(3))
     apply (erule_tac [2] le_shiftr)
    apply simp
   apply (rule word_leI)
   apply (rename_tac n')
   apply (drule_tac x = "n' - n" in word_eqD)
   apply (simp add : nth_shiftr word_size bit_simps)
   apply (case_tac "n <= n'")
   by auto
 
 lemma and_mask_eq_iff_shiftr_0:
   "(w AND mask n = w) = (w >> n = 0)"
   for w :: \<open>'a::len word\<close>
   by (simp flip: take_bit_eq_mask add: shiftr_def take_bit_eq_self_iff_drop_bit_eq_0)
 
 lemma mask_shiftl_decompose:
   "mask m << n = mask (m + n) AND NOT (mask n :: 'a::len word)"
   by word_eqI_solve
 
 lemma shiftl_over_and_dist:
   fixes a::"'a::len word"
   shows "(a AND b) << c = (a << c) AND (b << c)"
   by (unfold shiftl_def) (fact push_bit_and)
 
 lemma shiftr_over_and_dist:
   fixes a::"'a::len word"
   shows "a AND b >> c = (a >> c) AND (b >> c)"
   by (unfold shiftr_def) (fact drop_bit_and)
 
 lemma sshiftr_over_and_dist:
   fixes a::"'a::len word"
   shows "a AND b >>> c = (a >>> c) AND (b >>> c)"
   by word_eqI
 
 lemma shiftl_over_or_dist:
   fixes a::"'a::len word"
   shows "a OR b << c = (a << c) OR (b << c)"
   by (unfold shiftl_def) (fact push_bit_or)
 
 lemma shiftr_over_or_dist:
   fixes a::"'a::len word"
   shows "a OR b >> c = (a >> c) OR (b >> c)"
   by (unfold shiftr_def) (fact drop_bit_or)
 
 lemma sshiftr_over_or_dist:
   fixes a::"'a::len word"
   shows "a OR b >>> c = (a >>> c) OR (b >>> c)"
   by word_eqI
 
 lemmas shift_over_ao_dists =
   shiftl_over_or_dist shiftr_over_or_dist
   sshiftr_over_or_dist shiftl_over_and_dist
   shiftr_over_and_dist sshiftr_over_and_dist
 
 lemma shiftl_shiftl:
   fixes a::"'a::len word"
   shows "a << b << c = a << (b + c)"
   by (word_eqI_solve simp: add.commute add.left_commute)
 
 lemma shiftr_shiftr:
   fixes a::"'a::len word"
   shows "a >> b >> c = a >> (b + c)"
   by word_eqI (simp add: add.left_commute add.commute)
 
 lemma shiftl_shiftr1:
   fixes a::"'a::len word"
   shows "c \<le> b \<Longrightarrow> a << b >> c = a AND (mask (size a - b)) << (b - c)"
   by word_eqI (auto simp: ac_simps)
 
 lemma shiftl_shiftr2:
   fixes a::"'a::len word"
   shows "b < c \<Longrightarrow> a << b >> c = (a >> (c - b)) AND (mask (size a - c))"
   by word_eqI_solve
 
 lemma shiftr_shiftl1:
   fixes a::"'a::len word"
   shows "c \<le> b \<Longrightarrow> a >> b << c = (a >> (b - c)) AND (NOT (mask c))"
   by word_eqI_solve
 
 lemma shiftr_shiftl2:
   fixes a::"'a::len word"
   shows "b < c \<Longrightarrow> a >> b << c = (a << (c - b)) AND (NOT (mask c))"
   by word_eqI (auto simp: ac_simps)
 
 lemmas multi_shift_simps =
   shiftl_shiftl shiftr_shiftr
   shiftl_shiftr1 shiftl_shiftr2
   shiftr_shiftl1 shiftr_shiftl2
 
 lemma shiftr_mask2:
   "n \<le> LENGTH('a) \<Longrightarrow> (mask n >> m :: ('a :: len) word) = mask (n - m)"
   by word_eqI_solve
 
 lemma word_shiftl_add_distrib:
   fixes x :: "'a :: len word"
   shows "(x + y) << n = (x << n) + (y << n)"
   by (simp add: shiftl_t2n ring_distribs)
 
 lemma mask_shift:
   "(x AND NOT (mask y)) >> y = x >> y"
   for x :: \<open>'a::len word\<close>
   by word_eqI
 
 lemma shiftr_div_2n':
   "unat (w >> n) = unat w div 2 ^ n"
   by word_eqI
 
 lemma shiftl_shiftr_id:
   "\<lbrakk> n < LENGTH('a); x < 2 ^ (LENGTH('a) - n) \<rbrakk> \<Longrightarrow> x << n >> n = (x::'a::len word)" 
   by word_eqI (metis add.commute less_diff_conv)
 
 lemma ucast_shiftl_eq_0:
   fixes w :: "'a :: len word"
   shows "\<lbrakk> n \<ge> LENGTH('b) \<rbrakk> \<Longrightarrow> ucast (w << n) = (0 :: 'b :: len word)"
   by (transfer fixing: n) (simp add: take_bit_push_bit)
 
 lemma word_shift_nonzero:
   "\<lbrakk> (x::'a::len word) \<le> 2 ^ m; m + n < LENGTH('a::len); x \<noteq> 0\<rbrakk>
    \<Longrightarrow> x << n \<noteq> 0"
   apply (simp only: word_neq_0_conv word_less_nat_alt
                     shiftl_t2n mod_0 unat_word_ariths
                     unat_power_lower word_le_nat_alt)
   apply (subst mod_less)
    apply (rule order_le_less_trans)
     apply (erule mult_le_mono2)
    apply (subst power_add[symmetric])
    apply (rule power_strict_increasing)
     apply simp
    apply simp
   apply simp
   done
 
 lemma word_shiftr_lt:
   fixes w :: "'a::len word"
   shows "unat (w >> n) < (2 ^ (LENGTH('a) - n))"
   apply (subst shiftr_div_2n')
   apply transfer
   apply (simp flip: drop_bit_eq_div add: drop_bit_nat_eq drop_bit_take_bit)
   done
 
 lemma shiftr_less_t2n':
   "\<lbrakk> x AND mask (n + m) = x; m < LENGTH('a) \<rbrakk> \<Longrightarrow> x >> n < 2 ^ m" for x :: "'a :: len word"
   apply (simp add: word_size mask_eq_iff_w2p [symmetric] flip: take_bit_eq_mask)
   apply transfer
   apply (simp add: take_bit_drop_bit ac_simps)
   done
 
 lemma shiftr_less_t2n:
   "x < 2 ^ (n + m) \<Longrightarrow> x >> n < 2 ^ m" for x :: "'a :: len word"
   apply (rule shiftr_less_t2n')
    apply (erule less_mask_eq)
   apply (rule ccontr)
   apply (simp add: not_less)
   apply (subst (asm) p2_eq_0[symmetric])
   apply (simp add: power_add)
   done
 
 lemma shiftr_eq_0:
   "n \<ge> LENGTH('a) \<Longrightarrow> ((w::'a::len word) >> n) = 0"
   apply (cut_tac shiftr_less_t2n'[of w n 0], simp)
    apply (simp add: mask_eq_iff)
   apply (simp add: lt2p_lem)
   apply simp
   done
 
 lemma shiftl_less_t2n:
   fixes x :: "'a :: len word"
   shows "\<lbrakk> x < (2 ^ (m - n)); m < LENGTH('a) \<rbrakk> \<Longrightarrow> (x << n) < 2 ^ m"
   apply (simp add: word_size mask_eq_iff_w2p [symmetric] flip: take_bit_eq_mask)
   apply transfer
   apply (simp add: take_bit_push_bit)
   done
 
 lemma shiftl_less_t2n':
   "(x::'a::len word) < 2 ^ m \<Longrightarrow> m+n < LENGTH('a) \<Longrightarrow> x << n < 2 ^ (m + n)"
   by (rule shiftl_less_t2n) simp_all
 
 lemma scast_bit_test [simp]:
   "scast ((1 :: 'a::len signed word) << n) = (1 :: 'a word) << n"
   by word_eqI
 
 lemma signed_shift_guard_to_word:
   \<open>unat x * 2 ^ y < 2 ^ n \<longleftrightarrow> x = 0 \<or> x < 1 << n >> y\<close>
     if \<open>n < LENGTH('a)\<close> \<open>0 < n\<close>
     for x :: \<open>'a::len word\<close>
 proof (cases \<open>x = 0\<close>)
   case True
   then show ?thesis
     by simp
 next
   case False
   then have \<open>unat x \<noteq> 0\<close>
     by (simp add: unat_eq_0)
   then have \<open>unat x \<ge> 1\<close>
     by simp
   show ?thesis
   proof (cases \<open>y < n\<close>)
     case False
     then have \<open>n \<le> y\<close>
       by simp
     then obtain q where \<open>y = n + q\<close>
       using le_Suc_ex by blast
     moreover have \<open>(2 :: nat) ^ n >> n + q \<le> 1\<close>
       by (simp add: drop_bit_eq_div power_add shiftr_def)
     ultimately show ?thesis
       using \<open>x \<noteq> 0\<close> \<open>unat x \<ge> 1\<close> \<open>n < LENGTH('a)\<close>
       by (simp add: power_add not_less word_le_nat_alt unat_drop_bit_eq shiftr_def shiftl_def)
   next
     case True
     with that have \<open>y < LENGTH('a)\<close>
       by simp
     show ?thesis
     proof (cases \<open>2 ^ n = unat x * 2 ^ y\<close>)
       case True
       moreover have \<open>unat x * 2 ^ y < 2 ^ LENGTH('a)\<close>
         using \<open>n < LENGTH('a)\<close> by (simp flip: True)
       moreover have \<open>(word_of_nat (2 ^ n) :: 'a word) = word_of_nat (unat x * 2 ^ y)\<close>
         using True by simp
       then have \<open>2 ^ n = x * 2 ^ y\<close>
         by simp
       ultimately show ?thesis
         using \<open>y < LENGTH('a)\<close>
         by (auto simp add: drop_bit_eq_div word_less_nat_alt unat_div unat_word_ariths
           shiftr_def shiftl_def)
     next
       case False
       with \<open>y < n\<close> have *: \<open>unat x \<noteq> 2 ^ n div 2 ^ y\<close>
         by (auto simp flip: power_sub power_add)
       have \<open>unat x * 2 ^ y < 2 ^ n \<longleftrightarrow> unat x * 2 ^ y \<le> 2 ^ n\<close>
         using False by (simp add: less_le)
       also have \<open>\<dots> \<longleftrightarrow> unat x \<le> 2 ^ n div 2 ^ y\<close>
         by (simp add: less_eq_div_iff_mult_less_eq)
       also have \<open>\<dots> \<longleftrightarrow> unat x < 2 ^ n div 2 ^ y\<close>
         using * by (simp add: less_le)
       finally show ?thesis
       using that \<open>x \<noteq> 0\<close> by (simp flip: push_bit_eq_mult drop_bit_eq_div
         add: shiftr_def shiftl_def unat_drop_bit_eq word_less_iff_unsigned [where ?'a = nat])
     qed
   qed
 qed
 
 lemma shiftr_not_mask_0:
   "n+m \<ge> LENGTH('a :: len) \<Longrightarrow> ((w::'a::len word) >> n) AND NOT (mask m) = 0"
   by word_eqI
 
 lemma shiftl_mask_is_0[simp]:
   "(x << n) AND mask n = 0"
   for x :: \<open>'a::len word\<close>
   by (simp flip: take_bit_eq_mask add: take_bit_push_bit shiftl_def)
 
 lemma rshift_sub_mask_eq:
   "(a >> (size a - b)) AND mask b = a >> (size a - b)"
   for a :: \<open>'a::len word\<close>
   using shiftl_shiftr2[where a=a and b=0 and c="size a - b"]
   apply (cases "b < size a")
    apply simp
   apply (simp add: linorder_not_less mask_eq_decr_exp word_size
                    p2_eq_0[THEN iffD2])
   done
 
 lemma shiftl_shiftr3:
   "b \<le> c \<Longrightarrow> a << b >> c = (a >> c - b) AND mask (size a - c)"
   for a :: \<open>'a::len word\<close>
   apply (cases "b = c")
    apply (simp add: shiftl_shiftr1)
   apply (simp add: shiftl_shiftr2)
   done
 
 lemma and_mask_shiftr_comm:
   "m \<le> size w \<Longrightarrow> (w AND mask m) >> n = (w >> n) AND mask (m-n)"
   for w :: \<open>'a::len word\<close>
   by (simp add: and_mask shiftr_shiftr) (simp add: word_size shiftl_shiftr3)
 
 lemma and_mask_shiftl_comm:
   "m+n \<le> size w \<Longrightarrow> (w AND mask m) << n = (w << n) AND mask (m+n)"
   for w :: \<open>'a::len word\<close>
   by (simp add: and_mask word_size shiftl_shiftl) (simp add: shiftl_shiftr1)
 
 lemma le_mask_shiftl_le_mask: "s = m + n \<Longrightarrow> x \<le> mask n \<Longrightarrow> x << m \<le> mask s"
   for x :: \<open>'a::len word\<close>
   by (simp add: le_mask_iff shiftl_shiftr3)
 
 lemma word_and_1_shiftl:
   "x AND (1 << n) = (if bit x n then (1 << n) else 0)" for x :: "'a :: len word"
   by word_eqI_solve
 
 lemmas word_and_1_shiftls'
     = word_and_1_shiftl[where n=0]
       word_and_1_shiftl[where n=1]
       word_and_1_shiftl[where n=2]
 
 lemmas word_and_1_shiftls = word_and_1_shiftls' [simplified]
 
 lemma word_and_mask_shiftl:
   "x AND (mask n << m) = ((x >> m) AND mask n) << m"
   for x :: \<open>'a::len word\<close>
   by word_eqI_solve
 
 lemma shift_times_fold:
   "(x :: 'a :: len word) * (2 ^ n) << m = x << (m + n)"
   by (simp add: shiftl_t2n ac_simps power_add)
 
 lemma of_bool_nth:
   "of_bool (bit x v) = (x >> v) AND 1"
   for x :: \<open>'a::len word\<close>
   by (simp add: bit_iff_odd_drop_bit word_and_1 shiftr_def)
 
 lemma shiftr_mask_eq:
   "(x >> n) AND mask (size x - n) = x >> n" for x :: "'a :: len word"
   by (word_eqI_solve dest: test_bit_lenD)
 
 lemma shiftr_mask_eq':
   "m = (size x - n) \<Longrightarrow> (x >> n) AND mask m = x >> n" for x :: "'a :: len word"
   by (simp add: shiftr_mask_eq)
 
 lemma and_eq_0_is_nth:
   fixes x :: "'a :: len word"
   shows "y = 1 << n \<Longrightarrow> ((x AND y) = 0) = (\<not> (bit x n))"
   by (simp add: and_exp_eq_0_iff_not_bit shiftl_def)
 
 lemma word_shift_zero:
   "\<lbrakk> x << n = 0; x \<le> 2^m; m + n < LENGTH('a)\<rbrakk> \<Longrightarrow> (x::'a::len word) = 0"
   apply (rule ccontr)
   apply (drule (2) word_shift_nonzero)
   apply simp
   done
 
 lemma mask_shift_and_negate[simp]:"(w AND mask n << m) AND NOT (mask n << m) = 0"
   for w :: \<open>'a::len word\<close>
   by word_eqI
 
 (* The seL4 bitfield generator produces functions containing mask and shift operations, such that
  * invoking two of them consecutively can produce something like the following.
  *)
 lemma bitfield_op_twice:
   "(x AND NOT (mask n << m) OR ((y AND mask n) << m)) AND NOT (mask n << m) = x AND NOT (mask n << m)"
   for x :: \<open>'a::len word\<close>
   by word_eqI_solve
 
 lemma bitfield_op_twice'':
   "\<lbrakk>NOT a = b << c; \<exists>x. b = mask x\<rbrakk> \<Longrightarrow> (x AND a OR (y AND b << c)) AND a = x AND a"
   for a b :: \<open>'a::len word\<close>
   by word_eqI_solve
 
 lemma shiftr1_unfold: "x div 2 = x >> 1"
   by (simp add: drop_bit_eq_div shiftr_def)
 
 lemma shiftr1_is_div_2: "(x::('a::len) word) >> 1 = x div 2"
   by (simp add: drop_bit_eq_div shiftr_def)
 
 lemma shiftl1_is_mult: "(x << 1) = (x :: 'a::len word) * 2"
   by (metis One_nat_def mult_2 mult_2_right one_add_one
         power_0 power_Suc shiftl_t2n)
 
 lemma shiftr1_lt:"x \<noteq> 0 \<Longrightarrow> (x::('a::len) word) >> 1 < x"
   apply (subst shiftr1_is_div_2)
   apply (rule div_less_dividend_word)
    apply simp+
   done
 
 lemma shiftr1_0_or_1:"(x::('a::len) word) >> 1 = 0 \<Longrightarrow> x = 0 \<or> x = 1"
   apply (subst (asm) shiftr1_is_div_2)
   apply (drule word_less_div)
   apply (case_tac "LENGTH('a) = 1")
    apply (simp add:degenerate_word)
   apply (erule disjE)
    apply (subgoal_tac "(2::'a word) \<noteq> 0")
     apply simp
    apply (rule not_degenerate_imp_2_neq_0)
    apply (subgoal_tac "LENGTH('a) \<noteq> 0")
     apply arith
    apply simp
   apply (rule x_less_2_0_1', simp+)
   done
 
 lemma shiftr1_irrelevant_lsb: "bit (x::('a::len) word) 0 \<or> x >> 1 = (x + 1) >> 1"
   by (auto simp add: bit_0 shiftr_def drop_bit_Suc ac_simps elim: evenE)
 
 lemma shiftr1_0_imp_only_lsb:"((x::('a::len) word) + 1) >> 1 = 0 \<Longrightarrow> x = 0 \<or> x + 1 = 0"
   by (metis One_nat_def shiftr1_0_or_1 word_less_1 word_overflow)
 
 lemma shiftr1_irrelevant_lsb': "\<not> (bit (x::('a::len) word) 0) \<Longrightarrow> x >> 1 = (x + 1) >> 1"
   using shiftr1_irrelevant_lsb [of x] by simp
 
 (* Perhaps this one should be a simp lemma, but it seems a little dangerous. *)
 lemma cast_chunk_assemble_id:
   "\<lbrakk>n = LENGTH('a::len); m = LENGTH('b::len); n * 2 = m\<rbrakk> \<Longrightarrow>
   (((ucast ((ucast (x::'b word))::'a word))::'b word) OR (((ucast ((ucast (x >> n))::'a word))::'b word) << n)) = x"
   apply (subgoal_tac "((ucast ((ucast (x >> n))::'a word))::'b word) = x >> n")
    apply clarsimp
    apply (subst and_not_mask[symmetric])
    apply (subst ucast_ucast_mask)
    apply (subst word_ao_dist2[symmetric])
    apply clarsimp
   apply (rule ucast_ucast_len)
   apply (rule shiftr_less_t2n')
    apply (subst and_mask_eq_iff_le_mask)
    apply (simp_all add: mask_eq_decr_exp flip: mult_2_right)
   apply (metis add_diff_cancel_left' len_gt_0 mult_2_right zero_less_diff)
   done
 
 lemma cast_chunk_scast_assemble_id:
   "\<lbrakk>n = LENGTH('a::len); m = LENGTH('b::len); n * 2 = m\<rbrakk> \<Longrightarrow>
   (((ucast ((scast (x::'b word))::'a word))::'b word) OR
    (((ucast ((scast (x >> n))::'a word))::'b word) << n)) = x"
   apply (subgoal_tac "((scast x)::'a word) = ((ucast x)::'a word)")
    apply (subgoal_tac "((scast (x >> n))::'a word) = ((ucast (x >> n))::'a word)")
     apply (simp add:cast_chunk_assemble_id)
    apply (subst down_cast_same[symmetric], subst is_down, arith, simp)+
   done
 
 lemma unat_shiftr_less_t2n:
   fixes x :: "'a :: len word"
   shows "unat x < 2 ^ (n + m) \<Longrightarrow> unat (x >> n) < 2 ^ m"
   by (simp add: shiftr_div_2n' power_add mult.commute less_mult_imp_div_less)
 
 lemma ucast_less_shiftl_helper:
   "\<lbrakk> LENGTH('b) + 2 < LENGTH('a); 2 ^ (LENGTH('b) + 2) \<le> n\<rbrakk>
     \<Longrightarrow> (ucast (x :: 'b::len word) << 2) < (n :: 'a::len word)"
   apply (erule order_less_le_trans[rotated])
   using ucast_less[where x=x and 'a='a]
   apply (simp only: shiftl_t2n field_simps)
   apply (rule word_less_power_trans2; simp)
   done
 
 (* negating a mask which has been shifted to the very left *)
 lemma NOT_mask_shifted_lenword:
   "NOT (mask len << (LENGTH('a) - len) ::'a::len word) = mask (LENGTH('a) - len)"
   by word_eqI_solve
 
 (* Comparisons between different word sizes. *)
 
 lemma shiftr_less:
   "(w::'a::len word) < k \<Longrightarrow> w >> n < k"
   by (metis div_le_dividend le_less_trans shiftr_div_2n' unat_arith_simps(2))
 
 lemma word_and_notzeroD:
   "w AND w' \<noteq> 0 \<Longrightarrow> w \<noteq> 0 \<and> w' \<noteq> 0"
   by auto
 
 lemma shiftr_le_0:
   "unat (w::'a::len word) < 2 ^ n \<Longrightarrow> w >> n = (0::'a::len word)"
   by (auto simp add: take_bit_word_eq_self_iff word_less_nat_alt shiftr_def
     simp flip: take_bit_eq_self_iff_drop_bit_eq_0 intro: ccontr)
 
 lemma of_nat_shiftl:
   "(of_nat x << n) = (of_nat (x * 2 ^ n) :: ('a::len) word)"
 proof -
   have "(of_nat x::'a word) << n = of_nat (2 ^ n) * of_nat x"
     using shiftl_t2n by (metis word_unat_power)
   thus ?thesis by simp
 qed
 
 lemma shiftl_1_not_0:
   "n < LENGTH('a) \<Longrightarrow> (1::'a::len word) << n \<noteq> 0"
   by (simp add: shiftl_t2n)
 
 (* continue sorting out from here *)
 
 (* usually: x,y = (len_of TYPE ('a)) *)
 lemma bitmagic_zeroLast_leq_or1Last:
   "(a::('a::len) word) AND (mask len << x - len) \<le> a OR mask (y - len)"
   by (meson le_word_or2 order_trans word_and_le2)
 
 lemma zero_base_lsb_imp_set_eq_as_bit_operation:
   fixes base ::"'a::len word"
   assumes valid_prefix: "mask (LENGTH('a) - len) AND base = 0"
   shows "(base = NOT (mask (LENGTH('a) - len)) AND a) \<longleftrightarrow>
          (a \<in> {base .. base OR mask (LENGTH('a) - len)})"
 proof
   have helper3: "x OR y = x OR y AND NOT x" for x y ::"'a::len word" by (simp add: word_oa_dist2)
   from assms show "base = NOT (mask (LENGTH('a) - len)) AND a \<Longrightarrow>
                     a \<in> {base..base OR mask (LENGTH('a) - len)}"
     apply(simp add: word_and_le1)
     apply(metis helper3 le_word_or2 word_bw_comms(1) word_bw_comms(2))
   done
 next
   assume "a \<in> {base..base OR mask (LENGTH('a) - len)}"
   hence a: "base \<le> a \<and> a \<le> base OR mask (LENGTH('a) - len)" by simp
   show "base = NOT (mask (LENGTH('a) - len)) AND a"
   proof -
     have f2: "\<forall>x\<^sub>0. base AND NOT (mask x\<^sub>0) \<le> a AND NOT (mask x\<^sub>0)"
       using a neg_mask_mono_le by blast
     have f3: "\<forall>x\<^sub>0. a AND NOT (mask x\<^sub>0) \<le> (base OR mask (LENGTH('a) - len)) AND NOT (mask x\<^sub>0)"
       using a neg_mask_mono_le by blast
     have f4: "base = base AND NOT (mask (LENGTH('a) - len))"
       using valid_prefix by (metis mask_eq_0_eq_x word_bw_comms(1))
     hence f5: "\<forall>x\<^sub>6. (base OR x\<^sub>6) AND NOT (mask (LENGTH('a) - len)) =
                       base OR x\<^sub>6 AND NOT (mask (LENGTH('a) - len))"
       using word_ao_dist by (metis)
     have f6: "\<forall>x\<^sub>2 x\<^sub>3. a AND NOT (mask x\<^sub>2) \<le> x\<^sub>3 \<or>
                       \<not> (base OR mask (LENGTH('a) - len)) AND NOT (mask x\<^sub>2) \<le> x\<^sub>3"
       using f3 dual_order.trans by auto
     have "base = (base OR mask (LENGTH('a) - len)) AND NOT (mask (LENGTH('a) - len))"
       using f5 by auto
     hence "base = a AND NOT (mask (LENGTH('a) - len))"
       using f2 f4 f6 by (metis eq_iff)
     thus "base = NOT (mask (LENGTH('a) - len)) AND a"
       by (metis word_bw_comms(1))
   qed
 qed
 
 lemma of_nat_eq_signed_scast:
   "(of_nat x = (y :: ('a::len) signed word))
    = (of_nat x = (scast y :: 'a word))"
   by (metis scast_of_nat scast_scast_id(2))
 
 lemma word_aligned_add_no_wrap_bounded:
   "\<lbrakk> w + 2^n \<le> x; w + 2^n \<noteq> 0; is_aligned w n \<rbrakk> \<Longrightarrow> (w::'a::len word) < x"
   by (blast dest: is_aligned_no_overflow le_less_trans word_leq_le_minus_one)
 
 lemma mask_Suc:
   "mask (Suc n) = (2 :: 'a::len word) ^ n + mask n"
   by (simp add: mask_eq_decr_exp)
 
 lemma mask_mono:
   "sz' \<le> sz \<Longrightarrow> mask sz' \<le> (mask sz :: 'a::len word)"
   by (simp add: le_mask_iff shiftr_mask_le)
 
 lemma aligned_mask_disjoint:
   "\<lbrakk> is_aligned (a :: 'a :: len word) n; b \<le> mask n \<rbrakk> \<Longrightarrow> a AND b = 0"
   by (metis and_zero_eq is_aligned_mask le_mask_imp_and_mask word_bw_lcs(1))
 
 lemma word_and_or_mask_aligned:
   "\<lbrakk> is_aligned a n; b \<le> mask n \<rbrakk> \<Longrightarrow> a + b = a OR b"
   by (simp add: aligned_mask_disjoint word_plus_and_or_coroll)
 
 lemma word_and_or_mask_aligned2:
   \<open>is_aligned b n \<Longrightarrow> a \<le> mask n \<Longrightarrow> a + b = a OR b\<close>
   using word_and_or_mask_aligned [of b n a] by (simp add: ac_simps)
 
 lemma is_aligned_ucastI:
   "is_aligned w n \<Longrightarrow> is_aligned (ucast w) n"
   by (simp add: bit_ucast_iff is_aligned_nth)
 
 lemma ucast_le_maskI:
   "a \<le> mask n \<Longrightarrow> UCAST('a::len \<rightarrow> 'b::len) a \<le> mask n"
   by (metis and_mask_eq_iff_le_mask ucast_and_mask)
 
 lemma ucast_add_mask_aligned:
   "\<lbrakk> a \<le> mask n; is_aligned b n \<rbrakk> \<Longrightarrow> UCAST ('a::len \<rightarrow> 'b::len) (a + b) = ucast a + ucast b"
   by (metis add.commute is_aligned_ucastI ucast_le_maskI ucast_or_distrib word_and_or_mask_aligned)
 
 lemma ucast_shiftl:
   "LENGTH('b) \<le> LENGTH ('a) \<Longrightarrow> UCAST ('a::len \<rightarrow> 'b::len) x << n = ucast (x << n)"
   by word_eqI_solve
 
 lemma ucast_leq_mask:
   "LENGTH('a) \<le> n \<Longrightarrow> ucast (x::'a::len word) \<le> mask n"
   apply (simp add: less_eq_mask_iff_take_bit_eq_self)
   apply transfer
   apply (simp add: ac_simps)
   done
 
 lemma shiftl_inj:
   \<open>x = y\<close>
     if \<open>x << n = y << n\<close> \<open>x \<le> mask (LENGTH('a) - n)\<close> \<open>y \<le> mask (LENGTH('a) - n)\<close>
     for x y :: \<open>'a::len word\<close>
 proof (cases \<open>n < LENGTH('a)\<close>)
   case False
   with that show ?thesis
     by simp
 next
   case True
   moreover from that have \<open>take_bit (LENGTH('a) - n) x = x\<close> \<open>take_bit (LENGTH('a) - n) y = y\<close>
     by (simp_all add: less_eq_mask_iff_take_bit_eq_self)
   ultimately show ?thesis
     using \<open>x << n = y << n\<close> by (metis diff_less gr_implies_not0 linorder_cases linorder_not_le shiftl_shiftr_id shiftl_x_0 take_bit_word_eq_self_iff)
 qed
 
 lemma distinct_word_add_ucast_shift_inj:
   \<open>p' = p \<and> off' = off\<close>
   if *: \<open>p + (UCAST('a::len \<rightarrow> 'b::len) off << n) = p' + (ucast off' << n)\<close>
     and \<open>is_aligned p n'\<close> \<open>is_aligned p' n'\<close> \<open>n' = n + LENGTH('a)\<close> \<open>n' < LENGTH('b)\<close>
 proof -
   from \<open>n' = n + LENGTH('a)\<close>
   have [simp]: \<open>n' - n = LENGTH('a)\<close> \<open>n + LENGTH('a) = n'\<close>
     by simp_all
   from \<open>is_aligned p n'\<close> obtain q
     where p: \<open>p = push_bit n' (word_of_nat q)\<close> \<open>q < 2 ^ (LENGTH('b) - n')\<close>
     by (rule is_alignedE')
   from \<open>is_aligned p' n'\<close> obtain q'
     where p': \<open>p' = push_bit n' (word_of_nat q')\<close> \<open>q' < 2 ^ (LENGTH('b) - n')\<close>
     by (rule is_alignedE')
   define m :: nat where \<open>m = unat off\<close>
   then have off: \<open>off = word_of_nat m\<close>
     by simp
   define m' :: nat where \<open>m' = unat off'\<close>
   then have off': \<open>off' = word_of_nat m'\<close>
     by simp
   have \<open>push_bit n' q + take_bit n' (push_bit n m) < 2 ^ LENGTH('b)\<close>
     by (metis id_apply is_aligned_no_wrap''' of_nat_eq_id of_nat_push_bit p(1) p(2) take_bit_nat_eq_self_iff take_bit_nat_less_exp take_bit_push_bit that(2) that(5) unsigned_of_nat)
   moreover have \<open>push_bit n' q' + take_bit n' (push_bit n m') < 2 ^ LENGTH('b)\<close>
     by (metis \<open>n' - n = LENGTH('a)\<close> id_apply is_aligned_no_wrap''' m'_def of_nat_eq_id of_nat_push_bit off' p'(1) p'(2) take_bit_nat_eq_self_iff take_bit_push_bit that(3) that(5) unsigned_of_nat)
   ultimately have \<open>push_bit n' q + take_bit n' (push_bit n m) = push_bit n' q' + take_bit n' (push_bit n m')\<close>
     using * by (simp add: p p' off off' push_bit_of_nat push_bit_take_bit word_of_nat_inj unsigned_of_nat shiftl_def flip: of_nat_add)
   then have \<open>int (push_bit n' q + take_bit n' (push_bit n m))
     = int (push_bit n' q' + take_bit n' (push_bit n m'))\<close>
     by simp
   then have \<open>concat_bit n' (int (push_bit n m)) (int q)
     = concat_bit n' (int (push_bit n m')) (int q')\<close>
     by (simp add: of_nat_push_bit of_nat_take_bit concat_bit_eq)
   then show ?thesis
     by (simp add: p p' off off' take_bit_of_nat take_bit_push_bit word_of_nat_eq_iff concat_bit_eq_iff)
       (simp add: push_bit_eq_mult)
 qed
 
 lemma word_upto_Nil:
   "y < x \<Longrightarrow> [x .e. y ::'a::len word] = []"
   by (simp add: upto_enum_red not_le word_less_nat_alt)
 
 lemma word_enum_decomp_elem:
   assumes "[x .e. (y ::'a::len word)] = as @ a # bs"
   shows "x \<le> a \<and> a \<le> y"
 proof -
   have "set as \<subseteq> set [x .e. y] \<and> a \<in> set [x .e. y]"
     using assms by (auto dest: arg_cong[where f=set])
   then show ?thesis by auto
 qed
 
 lemma word_enum_prefix:
   "[x .e. (y ::'a::len word)] = as @ a # bs \<Longrightarrow> as = (if x < a then [x .e. a - 1] else [])"
   apply (induct as arbitrary: x; clarsimp)
    apply (case_tac "x < y")
     prefer 2
     apply (case_tac "x = y", simp)
     apply (simp add: not_less)
     apply (drule (1) dual_order.not_eq_order_implies_strict)
     apply (simp add: word_upto_Nil)
    apply (simp add: word_upto_Cons_eq)
   apply (case_tac "x < y")
    prefer 2
    apply (case_tac "x = y", simp)
    apply (simp add: not_less)
    apply (drule (1) dual_order.not_eq_order_implies_strict)
    apply (simp add: word_upto_Nil)
   apply (clarsimp simp: word_upto_Cons_eq)
   apply (frule word_enum_decomp_elem)
   apply clarsimp
   apply (rule conjI)
    prefer 2
    apply (subst word_Suc_le[symmetric]; clarsimp)
   apply (drule meta_spec)
   apply (drule (1) meta_mp)
   apply clarsimp
   apply (rule conjI; clarsimp)
   apply (subst (2) word_upto_Cons_eq)
    apply unat_arith
   apply simp
   done
 
 lemma word_enum_decomp_set:
   "[x .e. (y ::'a::len word)] = as @ a # bs \<Longrightarrow> a \<notin> set as"
   by (metis distinct_append distinct_enum_upto' not_distinct_conv_prefix)
 
 lemma word_enum_decomp:
   assumes "[x .e. (y ::'a::len word)] = as @ a # bs"
   shows "x \<le> a \<and> a \<le> y \<and> a \<notin> set as \<and> (\<forall>z \<in> set as. x \<le> z \<and> z \<le> y)"
 proof -
   from assms
   have "set as \<subseteq> set [x .e. y] \<and> a \<in> set [x .e. y]"
     by (auto dest: arg_cong[where f=set])
   with word_enum_decomp_set[OF assms]
   show ?thesis by auto
 qed
 
 lemma of_nat_unat_le_mask_ucast:
   "\<lbrakk>of_nat (unat t) = w; t \<le> mask LENGTH('a)\<rbrakk> \<Longrightarrow> t = UCAST('a::len \<rightarrow> 'b::len) w"
   by (clarsimp simp: ucast_nat_def ucast_ucast_mask simp flip: and_mask_eq_iff_le_mask)
 
 lemma less_diff_gt0:
   "a < b \<Longrightarrow> (0 :: 'a :: len word) < b - a"
   by unat_arith
 
 lemma unat_plus_gt:
   "unat ((a :: 'a :: len word) + b) \<le> unat a + unat b"
   by (clarsimp simp: unat_plus_if_size)
 
 lemma const_less:
   "\<lbrakk> (a :: 'a :: len word) - 1 < b; a \<noteq> b \<rbrakk> \<Longrightarrow> a < b"
   by (metis less_1_simp word_le_less_eq)
 
 lemma add_mult_aligned_neg_mask:
   \<open>(x + y * m) AND NOT(mask n) = (x AND NOT(mask n)) + y * m\<close>
   if \<open>m AND (2 ^ n - 1) = 0\<close>
   for x y m :: \<open>'a::len word\<close>
   by (metis (no_types, opaque_lifting)
             add.assoc add.commute add.right_neutral add_uminus_conv_diff
             mask_eq_decr_exp mask_eqs(2) mask_eqs(6) mult.commute mult_zero_left
             subtract_mask(1) that)
 
 lemma unat_of_nat_minus_1:
   "\<lbrakk> n < 2 ^ LENGTH('a); n \<noteq> 0 \<rbrakk> \<Longrightarrow> unat ((of_nat n:: 'a :: len word) - 1) = n - 1"
   by (simp add: of_nat_diff unat_eq_of_nat)
 
 lemma word_eq_zeroI:
   "a \<le> a - 1 \<Longrightarrow> a = 0" for a :: "'a :: len word"
   by (simp add: word_must_wrap)
 
 lemma word_add_format:
   "(-1 :: 'a :: len  word) + b + c = b + (c - 1)"
   by simp
 
 lemma upto_enum_word_nth:
   "\<lbrakk> i \<le> j; k \<le> unat (j - i) \<rbrakk> \<Longrightarrow> [i .e. j] ! k = i + of_nat k"
   apply (clarsimp simp: upto_enum_def nth_append)
   apply (clarsimp simp: word_le_nat_alt[symmetric])
   apply (rule conjI, clarsimp)
    apply (subst toEnum_of_nat, unat_arith)
    apply unat_arith
   apply (clarsimp simp: not_less unat_sub[symmetric])
   apply unat_arith
   done
 
 lemma upto_enum_step_nth:
   "\<lbrakk> a \<le> c; n \<le> unat ((c - a) div (b - a)) \<rbrakk>
    \<Longrightarrow> [a, b .e. c] ! n = a + of_nat n * (b - a)"
   by (clarsimp simp: upto_enum_step_def not_less[symmetric] upto_enum_word_nth)
 
 lemma upto_enum_inc_1_len:
   "a < - 1 \<Longrightarrow> [(0 :: 'a :: len word) .e. 1 + a] = [0 .e. a] @ [1 + a]"
   apply (simp add: upto_enum_word)
   apply (subgoal_tac "unat (1+a) = 1 + unat a")
    apply simp
   apply (subst unat_plus_simple[THEN iffD1])
    apply (metis add.commute no_plus_overflow_neg olen_add_eqv)
   apply unat_arith
   done
 
 lemma neg_mask_add:
   "y AND mask n = 0 \<Longrightarrow> x + y AND NOT(mask n) = (x AND NOT(mask n)) + y"
   for x y :: \<open>'a::len word\<close>
   by (clarsimp simp: mask_out_sub_mask mask_eqs(7)[symmetric] mask_twice)
 
 lemma shiftr_shiftl_shiftr[simp]:
   "(x :: 'a :: len word) >> a << a >> a = x >> a"
   by (word_eqI_solve dest: bit_imp_le_length)
 
 lemma add_right_shift:
   "\<lbrakk> x AND mask n = 0; y AND mask n = 0; x \<le> x + y \<rbrakk>
    \<Longrightarrow> (x + y :: ('a :: len) word) >> n = (x >> n) + (y >> n)"
   apply (simp add: no_olen_add_nat is_aligned_mask[symmetric])
   apply (simp add: unat_arith_simps shiftr_div_2n' split del: if_split)
   apply (subst if_P)
    apply (erule order_le_less_trans[rotated])
    apply (simp add: add_mono)
   apply (simp add: shiftr_div_2n' is_aligned_iff_dvd_nat)
   done
 
 lemma sub_right_shift:
   "\<lbrakk> x AND mask n = 0; y AND mask n = 0; y \<le> x \<rbrakk>
    \<Longrightarrow> (x - y) >> n = (x >> n :: 'a :: len word) - (y >> n)"
   using add_right_shift[where x="x - y" and y=y and n=n]
   by (simp add: aligned_sub_aligned is_aligned_mask[symmetric] word_sub_le)
 
 lemma and_and_mask_simple:
   "y AND mask n = mask n \<Longrightarrow> (x AND y) AND mask n = x AND mask n"
   by (simp add: ac_simps)
 
 lemma and_and_mask_simple_not:
   "y AND mask n = 0 \<Longrightarrow> (x AND y) AND mask n = 0"
   by (simp add: ac_simps)
 
 lemma word_and_le':
   "b \<le> c \<Longrightarrow> (a :: 'a :: len word) AND b \<le> c"
   by (metis word_and_le1 order_trans)
 
 lemma word_and_less':
   "b < c \<Longrightarrow> (a :: 'a :: len word) AND b < c"
   by transfer simp
 
 lemma shiftr_w2p:
   "x < LENGTH('a) \<Longrightarrow> 2 ^ x = (2 ^ (LENGTH('a) - 1) >> (LENGTH('a) - 1 - x) :: 'a :: len word)"
   by word_eqI_solve
 
 lemma t2p_shiftr:
   "\<lbrakk> b \<le> a; a < LENGTH('a) \<rbrakk> \<Longrightarrow> (2 :: 'a :: len word) ^ a >> b = 2 ^ (a - b)"
   by word_eqI_solve
 
 lemma scast_1[simp]:
   "scast (1 :: 'a :: len signed word) = (1 :: 'a word)"
   by simp
 
 lemma unsigned_uminus1 [simp]:
   \<open>(unsigned (-1::'b::len word)::'c::len word) = mask LENGTH('b)\<close>
   by (fact unsigned_minus_1_eq_mask)
 
 lemma ucast_ucast_mask_eq:
   "\<lbrakk> UCAST('a::len \<rightarrow> 'b::len) x = y; x AND mask LENGTH('b) = x \<rbrakk> \<Longrightarrow> x = ucast y"
   by (drule sym) (simp flip: take_bit_eq_mask add: unsigned_ucast_eq)
 
 lemma ucast_up_eq:
   "\<lbrakk> ucast x = (ucast y::'b::len word); LENGTH('a) \<le> LENGTH ('b) \<rbrakk>
    \<Longrightarrow> ucast x = (ucast y::'a::len word)"
   by (simp add: word_eq_iff bit_simps)
 
 lemma ucast_up_neq:
   "\<lbrakk> ucast x \<noteq> (ucast y::'b::len word); LENGTH('b) \<le> LENGTH ('a) \<rbrakk>
    \<Longrightarrow> ucast x \<noteq> (ucast y::'a::len word)"
   by (fastforce dest: ucast_up_eq)
 
 lemma mask_AND_less_0:
   "\<lbrakk> x AND mask n = 0; m \<le> n \<rbrakk> \<Longrightarrow> x AND mask m = 0"
   for x :: \<open>'a::len word\<close>
   by (metis mask_twice2 word_and_notzeroD)
 
 lemma mask_len_id [simp]:
   "(x :: 'a :: len word) AND mask LENGTH('a) = x"
   using uint_lt2p [of x] by (simp add: mask_eq_iff)
 
 lemma scast_ucast_down_same:
   "LENGTH('b) \<le> LENGTH('a) \<Longrightarrow> SCAST('a \<rightarrow> 'b) = UCAST('a::len \<rightarrow> 'b::len)"
   by (simp add: down_cast_same is_down)
 
 lemma word_aligned_0_sum:
   "\<lbrakk> a + b = 0; is_aligned (a :: 'a :: len word) n; b \<le> mask n; n < LENGTH('a) \<rbrakk>
    \<Longrightarrow> a = 0 \<and> b = 0"
   by (simp add: word_plus_and_or_coroll aligned_mask_disjoint word_or_zero)
 
 lemma mask_eq1_nochoice:
   "\<lbrakk> LENGTH('a) > 1; (x :: 'a :: len word) AND 1 = x \<rbrakk> \<Longrightarrow> x = 0 \<or> x = 1"
   by (metis word_and_1)
 
 lemma shiftr_and_eq_shiftl:
   "(w >> n) AND x = y \<Longrightarrow> w AND (x << n) = (y << n)" for y :: "'a:: len word"
   apply (drule sym)
   apply simp
   apply (rule bit_word_eqI)
   apply (auto simp add: bit_simps)
   done
 
 lemma add_mask_lower_bits':
   "\<lbrakk> len = LENGTH('a); is_aligned (x :: 'a :: len word) n;
      \<forall>n' \<ge> n. n' < len \<longrightarrow> \<not> bit p n' \<rbrakk>
    \<Longrightarrow> x + p AND NOT(mask n) = x"
   using add_mask_lower_bits by auto
 
 lemma leq_mask_shift:
   "(x :: 'a :: len word) \<le> mask (low_bits + high_bits) \<Longrightarrow> (x >> low_bits) \<le> mask high_bits"
   by (simp add: le_mask_iff shiftr_shiftr ac_simps)
 
 lemma ucast_ucast_eq_mask_shift:
   "(x :: 'a :: len word) \<le> mask (low_bits + LENGTH('b))
    \<Longrightarrow> ucast((ucast (x >> low_bits)) :: 'b :: len word) = x >> low_bits"
   by (meson and_mask_eq_iff_le_mask eq_ucast_ucast_eq not_le_imp_less shiftr_less_t2n'
             ucast_ucast_len)
 
 lemma const_le_unat:
   "\<lbrakk> b < 2 ^ LENGTH('a); of_nat b \<le> a \<rbrakk> \<Longrightarrow> b \<le> unat (a :: 'a :: len word)"
   by (simp add: word_le_nat_alt unsigned_of_nat take_bit_nat_eq_self)
 
 lemma upt_enum_offset_trivial:
   "\<lbrakk> x < 2 ^ LENGTH('a) - 1 ; n \<le> unat x \<rbrakk>
    \<Longrightarrow> ([(0 :: 'a :: len word) .e. x] ! n) = of_nat n"
   apply (induct x arbitrary: n)
     apply simp
   by (simp add: upto_enum_word_nth)
 
 lemma word_le_mask_out_plus_2sz:
   "x \<le> (x AND NOT(mask sz)) + 2 ^ sz - 1"
   for x :: \<open>'a::len word\<close>
   by (metis add_diff_eq word_neg_and_le)
 
 lemma ucast_add:
   "ucast (a + (b :: 'a :: len word)) = ucast a + (ucast b :: ('a signed word))"
   by transfer (simp add: take_bit_add)
 
 lemma ucast_minus:
   "ucast (a - (b :: 'a :: len word)) = ucast a - (ucast b :: ('a signed word))"
   apply (insert ucast_add[where a=a and b="-b"])
   apply (metis (no_types, opaque_lifting) add_diff_eq diff_add_cancel ucast_add)
   done
 
 lemma scast_ucast_add_one [simp]:
   "scast (ucast (x :: 'a::len word) + (1 :: 'a signed word)) = x + 1"
   apply (subst ucast_1[symmetric])
   apply (subst ucast_add[symmetric])
   apply clarsimp
   done
 
 lemma word_and_le_plus_one:
   "a > 0 \<Longrightarrow> (x :: 'a :: len word) AND (a - 1) < a"
   by (simp add: gt0_iff_gem1 word_and_less')
 
 lemma unat_of_ucast_then_shift_eq_unat_of_shift[simp]:
   "LENGTH('b) \<ge> LENGTH('a)
    \<Longrightarrow> unat ((ucast (x :: 'a :: len word) :: 'b :: len word) >> n) = unat (x >> n)"
   by (simp add: shiftr_div_2n' unat_ucast_up_simp)
 
 lemma unat_of_ucast_then_mask_eq_unat_of_mask[simp]:
   "LENGTH('b) \<ge> LENGTH('a)
    \<Longrightarrow> unat ((ucast (x :: 'a :: len word) :: 'b :: len word) AND mask m) = unat (x AND mask m)"
   by (metis ucast_and_mask unat_ucast_up_simp)
 
 lemma shiftr_less_t2n3:
   "\<lbrakk> (2 :: 'a word) ^ (n + m) = 0; m < LENGTH('a) \<rbrakk>
    \<Longrightarrow> (x :: 'a :: len word) >> n < 2 ^ m"
   by (fastforce intro: shiftr_less_t2n' simp: mask_eq_decr_exp power_overflow)
 
 lemma unat_shiftr_le_bound:
   "\<lbrakk> 2 ^ (LENGTH('a :: len) - n) - 1 \<le> bnd; 0 < n \<rbrakk>
    \<Longrightarrow> unat ((x :: 'a word) >> n) \<le> bnd"
   apply transfer
   apply (simp add: take_bit_drop_bit)
   apply (simp add: drop_bit_take_bit)
   apply (rule order_trans)
    defer
    apply assumption
   apply (simp add: nat_le_iff of_nat_diff)
   done
 
 lemma shiftr_eqD:
   "\<lbrakk> x >> n = y >> n; is_aligned x n; is_aligned y n \<rbrakk>
    \<Longrightarrow> x = y"
   by (metis is_aligned_shiftr_shiftl)
 
 lemma word_shiftr_shiftl_shiftr_eq_shiftr:
   "a \<ge> b \<Longrightarrow> (x :: 'a :: len word) >> a << b >> b = x >> a"
   apply (rule bit_word_eqI)
   apply (auto simp add: bit_simps dest: bit_imp_le_length)
   done
 
 lemma of_int_uint_ucast:
    "of_int (uint (x :: 'a::len word)) = (ucast x :: 'b::len word)"
   by (fact Word.of_int_uint)
 
 lemma mod_mask_drop:
   "\<lbrakk> m = 2 ^ n; 0 < m; mask n AND msk = mask n \<rbrakk>
    \<Longrightarrow> (x mod m) AND msk = x mod m"
   for x :: \<open>'a::len word\<close>
   by (simp add: word_mod_2p_is_mask word_bw_assocs)
 
 lemma mask_eq_ucast_eq:
   "\<lbrakk> x AND mask LENGTH('a) = (x :: ('c :: len word));
      LENGTH('a) \<le> LENGTH('b)\<rbrakk>
     \<Longrightarrow> ucast (ucast x :: ('a :: len word)) = (ucast x :: ('b :: len word))"
   by (metis ucast_and_mask ucast_id ucast_ucast_mask ucast_up_eq)
 
 lemma of_nat_less_t2n:
   "of_nat i < (2 :: ('a :: len) word) ^ n \<Longrightarrow> n < LENGTH('a) \<and> unat (of_nat i :: 'a word) < 2 ^ n"
   by (metis order_less_trans p2_gt_0 unat_less_power word_neq_0_conv)
 
 lemma two_power_increasing_less_1:
   "\<lbrakk> n \<le> m; m \<le> LENGTH('a) \<rbrakk> \<Longrightarrow> (2 :: 'a :: len word) ^ n - 1 \<le> 2 ^ m - 1"
   by (metis diff_diff_cancel le_m1_iff_lt less_imp_diff_less p2_gt_0 two_power_increasing
             word_1_le_power word_le_minus_mono_left word_less_sub_1)
 
 lemma word_sub_mono4:
   "\<lbrakk> y + x \<le> z + x; y \<le> y + x; z \<le> z + x \<rbrakk> \<Longrightarrow> y \<le> z" for y :: "'a :: len word"
   by (simp add: word_add_le_iff2)
 
 lemma eq_or_less_helperD:
   "\<lbrakk> n = unat (2 ^ m - 1 :: 'a :: len word) \<or> n < unat (2 ^ m - 1 :: 'a word); m < LENGTH('a) \<rbrakk>
    \<Longrightarrow> n < 2 ^ m"
   by (meson le_less_trans nat_less_le unat_less_power word_power_less_1)
 
 lemma mask_sub:
   "n \<le> m \<Longrightarrow> mask m - mask n = mask m AND NOT(mask n :: 'a::len word)"
   by (metis (full_types) and_mask_eq_iff_shiftr_0 mask_out_sub_mask shiftr_mask_le word_bw_comms(1))
 
 lemma neg_mask_diff_bound:
   "sz'\<le> sz \<Longrightarrow> (ptr AND NOT(mask sz')) - (ptr AND NOT(mask sz)) \<le> 2 ^ sz - 2 ^ sz'"
   (is "_ \<Longrightarrow> ?lhs \<le> ?rhs")
   for ptr :: \<open>'a::len word\<close>
 proof -
   assume lt: "sz' \<le> sz"
   hence "?lhs = ptr AND (mask sz AND NOT(mask sz'))"
     by (metis add_diff_cancel_left' multiple_mask_trivia)
   also have "\<dots> \<le> ?rhs" using lt
     by (metis (mono_tags) add_diff_eq diff_eq_eq eq_iff mask_2pm1 mask_sub word_and_le')
   finally show ?thesis by simp
 qed
 
 lemma mask_out_eq_0:
   "\<lbrakk> idx < 2 ^ sz; sz < LENGTH('a) \<rbrakk> \<Longrightarrow> (of_nat idx :: 'a :: len word) AND NOT(mask sz) = 0"
   by (simp add: of_nat_power less_mask_eq mask_eq_0_eq_x)
 
 lemma is_aligned_neg_mask_eq':
   "is_aligned ptr sz = (ptr AND NOT(mask sz) = ptr)"
   using is_aligned_mask mask_eq_0_eq_x by blast
 
 lemma neg_mask_mask_unat:
   "sz < LENGTH('a)
    \<Longrightarrow> unat ((ptr :: 'a :: len word) AND NOT(mask sz)) + unat (ptr AND mask sz) = unat ptr"
   by (metis AND_NOT_mask_plus_AND_mask_eq unat_plus_simple word_and_le2)
 
 lemma unat_pow_le_intro:
   "LENGTH('a) \<le> n \<Longrightarrow> unat (x :: 'a :: len word) < 2 ^ n"
   by (metis lt2p_lem not_le of_nat_le_iff of_nat_numeral semiring_1_class.of_nat_power uint_nat)
 
 lemma unat_shiftl_less_t2n:
   \<open>unat (x << n) < 2 ^ m\<close>
   if \<open>unat (x :: 'a :: len word) < 2 ^ (m - n)\<close> \<open>m < LENGTH('a)\<close>
 proof (cases \<open>n \<le> m\<close>)
   case False
   with that show ?thesis
     apply (transfer fixing: m n)
     apply (simp add: not_le take_bit_push_bit)
     apply (metis diff_le_self order_le_less_trans push_bit_of_0 take_bit_0 take_bit_int_eq_self
       take_bit_int_less_exp take_bit_nonnegative take_bit_tightened)
     done
 next
   case True
   moreover define q r where \<open>q = m - n\<close> and \<open>r = LENGTH('a) - n - q\<close>
   ultimately have \<open>m - n = q\<close> \<open>m = n + q\<close> \<open>LENGTH('a) = r + q + n\<close>
     using that by simp_all
   with that show ?thesis
     apply (transfer fixing: m n q r)
     apply (simp add: not_le take_bit_push_bit)
     apply (simp add: push_bit_eq_mult power_add)
     using take_bit_tightened_less_eq_int [of \<open>r + q\<close> \<open>r + q + n\<close>]
     apply (rule le_less_trans)
      apply simp_all
     done
 qed
 
 lemma unat_is_aligned_add:
   "\<lbrakk> is_aligned p n; unat d < 2 ^ n \<rbrakk>
    \<Longrightarrow> unat (p + d AND mask n) = unat d \<and> unat (p + d AND NOT(mask n)) = unat p"
   by (metis add.right_neutral and_mask_eq_iff_le_mask and_not_mask le_mask_iff mask_add_aligned
             mask_out_add_aligned mult_zero_right shiftl_t2n shiftr_le_0)
 
 lemma unat_shiftr_shiftl_mask_zero:
   "\<lbrakk> c + a \<ge> LENGTH('a) + b ; c < LENGTH('a) \<rbrakk>
    \<Longrightarrow> unat (((q :: 'a :: len word) >> a << b) AND NOT(mask c)) = 0"
   by (fastforce intro: unat_is_aligned_add[where p=0 and n=c, simplified, THEN conjunct2]
                        unat_shiftl_less_t2n unat_shiftr_less_t2n unat_pow_le_intro)
 
 lemmas of_nat_ucast = ucast_of_nat[symmetric]
 
 lemma shift_then_mask_eq_shift_low_bits:
   "x \<le> mask (low_bits + high_bits) \<Longrightarrow> (x >> low_bits) AND mask high_bits = x >> low_bits"
   for x :: \<open>'a::len word\<close>
   by (simp add: leq_mask_shift le_mask_imp_and_mask)
 
 lemma leq_low_bits_iff_zero:
   "\<lbrakk> x \<le> mask (low bits + high bits); x >> low_bits = 0 \<rbrakk> \<Longrightarrow> (x AND mask low_bits = 0) = (x = 0)"
   for x :: \<open>'a::len word\<close>
   using and_mask_eq_iff_shiftr_0 by force
 
 lemma unat_less_iff:
   "\<lbrakk> unat (a :: 'a :: len word) = b; c < 2 ^ LENGTH('a) \<rbrakk> \<Longrightarrow> (a < of_nat c) = (b < c)"
   using unat_ucast_less_no_overflow_simp by blast
 
 lemma is_aligned_no_overflow3:
  "\<lbrakk> is_aligned (a :: 'a :: len word) n; n < LENGTH('a); b < 2 ^ n; c \<le> 2 ^ n; b < c \<rbrakk>
   \<Longrightarrow> a + b \<le> a + (c - 1)"
   by (meson is_aligned_no_wrap' le_m1_iff_lt not_le word_less_sub_1 word_plus_mono_right)
 
 lemma mask_add_aligned_right:
   "is_aligned p n \<Longrightarrow> (q + p) AND mask n = q AND mask n"
   by (simp add: mask_add_aligned add.commute)
 
 lemma leq_high_bits_shiftr_low_bits_leq_bits_mask:
   "x \<le> mask high_bits \<Longrightarrow> (x :: 'a :: len word) << low_bits \<le> mask (low_bits + high_bits)"
   by (metis le_mask_shiftl_le_mask)
 
 lemma word_two_power_neg_ineq:
   "2 ^ m \<noteq> (0 :: 'a word) \<Longrightarrow> 2 ^ n \<le> - (2 ^ m :: 'a :: len word)"
   apply (cases "n < LENGTH('a)"; simp add: power_overflow)
   apply (cases "m < LENGTH('a)"; simp add: power_overflow)
   apply (simp add: word_le_nat_alt unat_minus word_size)
   apply (cases "LENGTH('a)"; simp)
   apply (simp add: less_Suc_eq_le)
   apply (drule power_increasing[where a=2 and n=n] power_increasing[where a=2 and n=m], simp)+
   apply (drule(1) add_le_mono)
   apply simp
   done
 
 lemma unat_shiftl_absorb:
   "\<lbrakk> x \<le> 2 ^ p; p + k < LENGTH('a) \<rbrakk> \<Longrightarrow> unat (x :: 'a :: len word) * 2 ^ k = unat (x * 2 ^ k)"
   by (smt add_diff_cancel_right' add_lessD1 le_add2 le_less_trans mult.commute nat_le_power_trans
           unat_lt2p unat_mult_lem unat_power_lower word_le_nat_alt)
 
 lemma word_plus_mono_right_split:
   "\<lbrakk> unat ((x :: 'a :: len word) AND mask sz) + unat z < 2 ^ sz; sz < LENGTH('a) \<rbrakk>
    \<Longrightarrow> x \<le> x + z"
   apply (subgoal_tac "(x AND NOT(mask sz)) + (x AND mask sz) \<le> (x AND NOT(mask sz)) + ((x AND mask sz) + z)")
    apply (simp add:word_plus_and_or_coroll2 field_simps)
   apply (rule word_plus_mono_right)
    apply (simp add: less_le_trans no_olen_add_nat)
   using of_nat_power is_aligned_no_wrap' by force
 
 lemma mul_not_mask_eq_neg_shiftl:
   "NOT(mask n :: 'a::len word) = -1 << n"
   by (simp add: NOT_mask shiftl_t2n)
 
 lemma shiftr_mul_not_mask_eq_and_not_mask:
   "(x >> n) * NOT(mask n) = - (x AND NOT(mask n))"
   for x :: \<open>'a::len word\<close>
   by (metis NOT_mask and_not_mask mult_minus_left semiring_normalization_rules(7) shiftl_t2n)
 
 lemma mask_eq_n1_shiftr:
   "n \<le> LENGTH('a) \<Longrightarrow> (mask n :: 'a :: len word) = -1 >> (LENGTH('a) - n)"
   by (metis diff_diff_cancel eq_refl mask_full shiftr_mask2)
 
 lemma is_aligned_mask_out_add_eq:
   "is_aligned p n \<Longrightarrow> (p + x) AND NOT(mask n) = p + (x AND NOT(mask n))"
   by (simp add: mask_out_sub_mask mask_add_aligned)
 
 lemmas is_aligned_mask_out_add_eq_sub
     = is_aligned_mask_out_add_eq[where x="a - b" for a b, simplified field_simps]
 
 lemma aligned_bump_down:
   "is_aligned x n \<Longrightarrow> (x - 1) AND NOT(mask n) = x - 2 ^ n"
   by (drule is_aligned_mask_out_add_eq[where x="-1"]) (simp add: NOT_mask)
 
 lemma unat_2tp_if:
   "unat (2 ^ n :: ('a :: len) word) = (if n < LENGTH ('a) then 2 ^ n else 0)"
   by (split if_split, simp_all add: power_overflow)
 
 lemma mask_of_mask:
   "mask (n::nat) AND mask (m::nat) = (mask (min m n) :: 'a::len word)"
   by word_eqI_solve
 
 lemma unat_signed_ucast_less_ucast:
   "LENGTH('a) \<le> LENGTH('b) \<Longrightarrow> unat (ucast (x :: 'a :: len word) :: 'b :: len signed word) = unat x"
   by (simp add: unat_ucast_up_simp)
 
 lemma toEnum_of_ucast:
   "LENGTH('b) \<le> LENGTH('a) \<Longrightarrow>
    (toEnum (unat (b::'b :: len word))::'a :: len word) = of_nat (unat b)"
   by (simp add: unat_pow_le_intro)
 
 lemma plus_mask_AND_NOT_mask_eq:
   "x AND NOT(mask n) = x \<Longrightarrow> (x + mask n) AND NOT(mask n) = x" for x::\<open>'a::len word\<close>
   apply (subst word_plus_and_or_coroll; word_eqI; fastforce?)
   apply (erule allE, drule (1) iffD2)
   apply clarsimp
   done
 
 lemmas unat_ucast_mask = unat_ucast_eq_unat_and_mask[where w=a for a]
 
 lemma t2n_mask_eq_if:
   "2 ^ n AND mask m = (if n < m then 2 ^ n else (0 :: 'a::len word))"
   by word_eqI_solve
 
 lemma unat_ucast_le:
   "unat (ucast (x :: 'a :: len word) :: 'b :: len word) \<le> unat x"
   by (simp add: ucast_nat_def word_unat_less_le)
 
 lemma ucast_le_up_down_iff:
   "\<lbrakk> LENGTH('a) \<le> LENGTH('b); (x :: 'b :: len word) \<le> ucast (- 1 :: 'a :: len word) \<rbrakk>
    \<Longrightarrow> (ucast x \<le> (y :: 'a word)) = (x \<le> ucast y)"
   using le_max_word_ucast_id ucast_le_ucast by metis
 
 lemma ucast_ucast_mask_shift:
   "a \<le> LENGTH('a) + b
    \<Longrightarrow> ucast (ucast (p AND mask a >> b) :: 'a :: len word) = p AND mask a >> b"
   by (metis add.commute le_mask_iff shiftr_mask_le ucast_ucast_eq_mask_shift word_and_le')
 
 lemma unat_ucast_mask_shift:
   "a \<le> LENGTH('a) + b
    \<Longrightarrow> unat (ucast (p AND mask a >> b) :: 'a :: len word) = unat (p AND mask a >> b)"
   by (metis linear ucast_ucast_mask_shift unat_ucast_up_simp)
 
 lemma mask_overlap_zero:
   "a \<le> b \<Longrightarrow> (p AND mask a) AND NOT(mask b) = 0"
   for p :: \<open>'a::len word\<close>
   by (metis NOT_mask_AND_mask mask_lower_twice2 max_def)
 
 lemma mask_shifl_overlap_zero:
   "a + c \<le> b \<Longrightarrow> (p AND mask a << c) AND NOT(mask b) = 0"
   for p :: \<open>'a::len word\<close>
   by (metis and_mask_0_iff_le_mask mask_mono mask_shiftl_decompose order_trans shiftl_over_and_dist word_and_le' word_and_le2)
 
 lemma mask_overlap_zero':
   "a \<ge> b \<Longrightarrow> (p AND NOT(mask a)) AND mask b = 0"
   for p :: \<open>'a::len word\<close>
   using mask_AND_NOT_mask mask_AND_less_0 by blast
 
 lemma mask_rshift_mult_eq_rshift_lshift:
   "((a :: 'a :: len word) >> b) * (1 << c) = (a >> b << c)"
   by (simp add: shiftl_t2n)
 
 lemma shift_alignment:
   "a \<ge> b \<Longrightarrow> is_aligned (p >> a << a) b"
   using is_aligned_shift is_aligned_weaken by blast
 
 lemma mask_split_sum_twice:
   "a \<ge> b \<Longrightarrow> (p AND NOT(mask a)) + ((p AND mask a) AND NOT(mask b)) + (p AND mask b) = p"
   for p :: \<open>'a::len word\<close>
   by (simp add: add.commute multiple_mask_trivia word_bw_comms(1) word_bw_lcs(1) word_plus_and_or_coroll2)
 
 lemma mask_shift_eq_mask_mask:
   "(p AND mask a >> b << b) = (p AND mask a) AND NOT(mask b)"
   for p :: \<open>'a::len word\<close>
   by (simp add: and_not_mask)
 
 lemma mask_shift_sum:
   "\<lbrakk> a \<ge> b; unat n = unat (p AND mask b) \<rbrakk>
    \<Longrightarrow> (p AND NOT(mask a)) + (p AND mask a >> b) * (1 << b) + n = (p :: 'a :: len word)"
   apply (simp add: shiftl_def shiftr_def flip: push_bit_eq_mult take_bit_eq_mask word_unat_eq_iff)
   apply (subst disjunctive_add)
    apply (auto simp add: bit_simps)
   apply (subst disjunctive_add)
    apply (auto simp add: bit_simps)
     apply (rule bit_word_eqI)
   apply (auto simp add: bit_simps)
   done
 
 lemma is_up_compose:
   "\<lbrakk> is_up uc; is_up uc' \<rbrakk> \<Longrightarrow> is_up (uc' \<circ> uc)"
   unfolding is_up_def by (simp add: Word.target_size Word.source_size)
 
 lemma of_int_sint_scast:
   "of_int (sint (x :: 'a :: len word)) = (scast x :: 'b :: len word)"
   by (fact Word.of_int_sint)
 
 lemma scast_of_nat_to_signed [simp]:
   "scast (of_nat x :: 'a :: len word) = (of_nat x :: 'a signed word)"
   by (rule bit_word_eqI) (simp add: bit_simps)
 
 lemma scast_of_nat_signed_to_unsigned_add:
   "scast (of_nat x + of_nat y :: 'a :: len signed word) = (of_nat x + of_nat y :: 'a :: len word)"
   by (metis of_nat_add scast_of_nat)
 
 lemma scast_of_nat_unsigned_to_signed_add:
   "(scast (of_nat x + of_nat y :: 'a :: len word)) = (of_nat x + of_nat y :: 'a :: len signed word)"
   by (metis Abs_fnat_hom_add scast_of_nat_to_signed)
 
 lemma and_mask_cases:
   fixes x :: "'a :: len word"
   assumes len: "n < LENGTH('a)"
   shows "x AND mask n \<in> of_nat ` set [0 ..< 2 ^ n]"
   apply (simp flip: take_bit_eq_mask)
   apply (rule image_eqI [of _ _ \<open>unat (take_bit n x)\<close>])
   using len apply simp_all
   apply transfer
   apply simp
   done
 
 lemma sint_eq_uint_2pl:
   "\<lbrakk> (a :: 'a :: len word) < 2 ^ (LENGTH('a) - 1) \<rbrakk>
    \<Longrightarrow> sint a = uint a"
   by (simp add: not_msb_from_less sint_eq_uint word_2p_lem word_size)
 
 lemma pow_sub_less:
   "\<lbrakk> a + b \<le> LENGTH('a); unat (x :: 'a :: len word) = 2 ^ a \<rbrakk>
    \<Longrightarrow> unat (x * 2 ^ b - 1) < 2 ^ (a + b)"
   by (smt (z3) eq_or_less_helperD le_add2 le_eq_less_or_eq le_trans power_add unat_mult_lem unat_pow_le_intro unat_power_lower word_eq_unatI)
 
 lemma sle_le_2pl:
   "\<lbrakk> (b :: 'a :: len word) < 2 ^ (LENGTH('a) - 1); a \<le> b \<rbrakk> \<Longrightarrow> a <=s b"
   by (simp add: not_msb_from_less word_sle_msb_le)
 
 lemma sless_less_2pl:
   "\<lbrakk> (b :: 'a :: len word) < 2 ^ (LENGTH('a) - 1); a < b \<rbrakk> \<Longrightarrow> a <s b"
   using not_msb_from_less word_sless_msb_less by blast
 
 lemma and_mask2:
   "w << n >> n = w AND mask (size w - n)"
   for w :: \<open>'a::len word\<close>
   by (rule bit_word_eqI) (auto simp add: bit_simps word_size)
 
 lemma aligned_sub_aligned_simple:
   "\<lbrakk> is_aligned a n; is_aligned b n \<rbrakk> \<Longrightarrow> is_aligned (a - b) n"
   by (simp add: aligned_sub_aligned)
 
 lemma minus_one_shift:
   "- (1 << n) = (-1 << n :: 'a::len word)"
   by (simp add: shiftl_def minus_exp_eq_not_mask)
 
 lemma ucast_eq_mask:
   "(UCAST('a::len \<rightarrow> 'b::len) x = UCAST('a \<rightarrow> 'b) y) =
    (x AND mask LENGTH('b) = y AND mask LENGTH('b))"
   by transfer (simp flip: take_bit_eq_mask add: ac_simps)
 
 context
   fixes w :: "'a::len word"
 begin
 
 private lemma sbintrunc_uint_ucast:
   "Suc n = LENGTH('b::len) \<Longrightarrow> signed_take_bit n (uint (ucast w :: 'b word)) = signed_take_bit n (uint w)"
   by word_eqI
 
 private lemma test_bit_sbintrunc:
   assumes "i < LENGTH('a)"
   shows "bit (word_of_int (signed_take_bit n (uint w)) :: 'a word) i
            = (if n < i then bit w n else bit w i)"
   using assms by (simp add: bit_simps)
 
 private lemma test_bit_sbintrunc_ucast:
   assumes len_a: "i < LENGTH('a)"
   shows "bit (word_of_int (signed_take_bit (LENGTH('b) - 1) (uint (ucast w :: 'b word))) :: 'a word) i
           = (if LENGTH('b::len) \<le> i then bit w (LENGTH('b) - 1) else bit w i)"
   using len_a by (auto simp add: sbintrunc_uint_ucast bit_simps)
 
 lemma scast_ucast_high_bits:
   \<open>scast (ucast w :: 'b::len word) = w
      \<longleftrightarrow> (\<forall> i \<in> {LENGTH('b) ..< size w}. bit w i = bit w (LENGTH('b) - 1))\<close>
 proof (cases \<open>LENGTH('a) \<le> LENGTH('b)\<close>)
   case True
   moreover define m where \<open>m = LENGTH('b) - LENGTH('a)\<close>
   ultimately have \<open>LENGTH('b) = m + LENGTH('a)\<close>
     by simp
   then show ?thesis
     by (simp add: signed_ucast_eq word_size) word_eqI
 next
   case False
   define q where \<open>q = LENGTH('b) - 1\<close>
   then have \<open>LENGTH('b) = Suc q\<close>
     by simp
   moreover define m where \<open>m = Suc LENGTH('a) - LENGTH('b)\<close>
   with False \<open>LENGTH('b) = Suc q\<close> have \<open>LENGTH('a) = m + q\<close>
     by (simp add: not_le)
   ultimately show ?thesis
     apply (simp add: signed_ucast_eq word_size)
     apply (transfer fixing: m q)
     apply (simp add: signed_take_bit_take_bit)
     apply (rule impI)
     apply (subst bit_eq_iff)
     apply (simp add: bit_take_bit_iff bit_signed_take_bit_iff min_def)
     apply (auto simp add: Suc_le_eq)
     using less_imp_le_nat apply blast
     using less_imp_le_nat apply blast
     done
 qed
 
 lemma scast_ucast_mask_compare:
   "scast (ucast w :: 'b::len word) = w
    \<longleftrightarrow> (w \<le> mask (LENGTH('b) - 1) \<or> NOT(mask (LENGTH('b) - 1)) \<le> w)"
   apply (clarsimp simp: le_mask_high_bits neg_mask_le_high_bits scast_ucast_high_bits word_size)
   apply (rule iffI; clarsimp)
    apply (rename_tac i j; case_tac "i = LENGTH('b) - 1"; case_tac "j = LENGTH('b) - 1")
   by auto
 
 lemma ucast_less_shiftl_helper':
   "\<lbrakk> LENGTH('b) + (a::nat) < LENGTH('a); 2 ^ (LENGTH('b) + a) \<le> n\<rbrakk>
    \<Longrightarrow> (ucast (x :: 'b::len word) << a) < (n :: 'a::len word)"
   apply (erule order_less_le_trans[rotated])
   using ucast_less[where x=x and 'a='a]
   apply (simp only: shiftl_t2n field_simps)
   apply (rule word_less_power_trans2; simp)
   done
 
 end
 
 lemma ucast_ucast_mask2:
   "is_down (UCAST ('a \<rightarrow> 'b)) \<Longrightarrow>
    UCAST ('b::len \<rightarrow> 'c::len) (UCAST ('a::len \<rightarrow> 'b::len) x) = UCAST ('a \<rightarrow> 'c) (x AND mask LENGTH('b))"
   by word_eqI_solve
 
 lemma ucast_NOT:
   "ucast (NOT x) = NOT(ucast x) AND mask (LENGTH('a))" for x::"'a::len word"
   by word_eqI_solve
 
 lemma ucast_NOT_down:
   "is_down UCAST('a::len \<rightarrow> 'b::len) \<Longrightarrow> UCAST('a \<rightarrow> 'b) (NOT x) = NOT(UCAST('a \<rightarrow> 'b) x)"
   by word_eqI
 
 lemma upto_enum_step_shift:
   "is_aligned p n \<Longrightarrow> ([p , p + 2 ^ m .e. p + 2 ^ n - 1]) = map ((+) p) [0, 2 ^ m .e. 2 ^ n - 1]"
   apply (erule is_aligned_get_word_bits)
    prefer 2
    apply (simp add: map_idI)
   apply (clarsimp simp: upto_enum_step_def)
   apply (frule is_aligned_no_overflow)
   apply (simp add: linorder_not_le [symmetric])
   done
 
 lemma upto_enum_step_shift_red:
   "\<lbrakk> is_aligned p sz; sz < LENGTH('a); us \<le> sz \<rbrakk>
      \<Longrightarrow> [p :: 'a :: len word, p + 2 ^ us .e. p + 2 ^ sz - 1]
           = map (\<lambda>x. p + of_nat x * 2 ^ us) [0 ..< 2 ^ (sz - us)]"
   apply (subst upto_enum_step_shift, assumption)
   apply (simp add: upto_enum_step_red)
   done
 
 lemma upto_enum_step_subset:
   "set [x, y .e. z] \<subseteq> {x .. z}"
   apply (clarsimp simp: upto_enum_step_def linorder_not_less)
   apply (drule div_to_mult_word_lt)
   apply (rule conjI)
    apply (erule word_random[rotated])
    apply simp
   apply (rule order_trans)
    apply (erule word_plus_mono_right)
    apply simp
   apply simp
   done
 
 lemma ucast_distrib:
   fixes M :: "'a::len word \<Rightarrow> 'a::len word \<Rightarrow> 'a::len word"
   fixes M' :: "'b::len word \<Rightarrow> 'b::len word \<Rightarrow> 'b::len word"
   fixes L :: "int \<Rightarrow> int \<Rightarrow> int"
   assumes lift_M: "\<And>x y. uint (M x y) = L (uint x) (uint y)  mod 2 ^ LENGTH('a)"
   assumes lift_M': "\<And>x y. uint (M' x y) = L (uint x) (uint y)  mod 2 ^ LENGTH('b)"
   assumes distrib: "\<And>x y. (L (x mod (2 ^ LENGTH('b))) (y mod (2 ^ LENGTH('b)))) mod (2 ^ LENGTH('b))
                                = (L x y) mod (2 ^ LENGTH('b))"
   assumes is_down: "is_down (ucast :: 'a word \<Rightarrow> 'b word)"
   shows "ucast (M a b) = M' (ucast a) (ucast b)"
   apply (simp only: ucast_eq)
   apply (subst lift_M)
   apply (subst of_int_uint [symmetric], subst lift_M')
   apply (metis local.distrib local.is_down take_bit_eq_mod ucast_down_wi uint_word_of_int_eq word_of_int_uint)
   done
 
 lemma ucast_down_add:
     "is_down (ucast:: 'a word \<Rightarrow> 'b word) \<Longrightarrow>  ucast ((a :: 'a::len word) + b) = (ucast a + ucast b :: 'b::len word)"
   by (rule ucast_distrib [where L="(+)"], (clarsimp simp: uint_word_ariths)+, presburger, simp)
 
 lemma ucast_down_minus:
     "is_down (ucast:: 'a word \<Rightarrow> 'b word) \<Longrightarrow>  ucast ((a :: 'a::len word) - b) = (ucast a - ucast b :: 'b::len word)"
   apply (rule ucast_distrib [where L="(-)"], (clarsimp simp: uint_word_ariths)+)
   apply (metis mod_diff_left_eq mod_diff_right_eq)
   apply simp
   done
 
 lemma ucast_down_mult:
     "is_down (ucast:: 'a word \<Rightarrow> 'b word) \<Longrightarrow>  ucast ((a :: 'a::len word) * b) = (ucast a * ucast b :: 'b::len word)"
   apply (rule ucast_distrib [where L="(*)"], (clarsimp simp: uint_word_ariths)+)
   apply (metis mod_mult_eq)
   apply simp
   done
 
 lemma scast_distrib:
   fixes M :: "'a::len word \<Rightarrow> 'a::len word \<Rightarrow> 'a::len word"
   fixes M' :: "'b::len word \<Rightarrow> 'b::len word \<Rightarrow> 'b::len word"
   fixes L :: "int \<Rightarrow> int \<Rightarrow> int"
   assumes lift_M: "\<And>x y. uint (M x y) = L (uint x) (uint y)  mod 2 ^ LENGTH('a)"
   assumes lift_M': "\<And>x y. uint (M' x y) = L (uint x) (uint y)  mod 2 ^ LENGTH('b)"
   assumes distrib: "\<And>x y. (L (x mod (2 ^ LENGTH('b))) (y mod (2 ^ LENGTH('b)))) mod (2 ^ LENGTH('b))
                                = (L x y) mod (2 ^ LENGTH('b))"
   assumes is_down: "is_down (scast :: 'a word \<Rightarrow> 'b word)"
   shows "scast (M a b) = M' (scast a) (scast b)"
   apply (subst (1 2 3) down_cast_same [symmetric])
    apply (insert is_down)
    apply (clarsimp simp: is_down_def target_size source_size is_down)
   apply (rule ucast_distrib [where L=L, OF lift_M lift_M' distrib])
   apply (insert is_down)
   apply (clarsimp simp: is_down_def target_size source_size is_down)
   done
 
 lemma scast_down_add:
     "is_down (scast:: 'a word \<Rightarrow> 'b word) \<Longrightarrow>  scast ((a :: 'a::len word) + b) = (scast a + scast b :: 'b::len word)"
   by (rule scast_distrib [where L="(+)"], (clarsimp simp: uint_word_ariths)+, presburger, simp)
 
 lemma scast_down_minus:
     "is_down (scast:: 'a word \<Rightarrow> 'b word) \<Longrightarrow>  scast ((a :: 'a::len word) - b) = (scast a - scast b :: 'b::len word)"
   apply (rule scast_distrib [where L="(-)"], (clarsimp simp: uint_word_ariths)+)
   apply (metis mod_diff_left_eq mod_diff_right_eq)
   apply simp
   done
 
 lemma scast_down_mult:
     "is_down (scast:: 'a word \<Rightarrow> 'b word) \<Longrightarrow>  scast ((a :: 'a::len word) * b) = (scast a * scast b :: 'b::len word)"
   apply (rule scast_distrib [where L="(*)"], (clarsimp simp: uint_word_ariths)+)
   apply (metis mod_mult_eq)
   apply simp
   done
 
 lemma scast_ucast_1:
   "\<lbrakk> is_down (ucast :: 'a word \<Rightarrow> 'b word); is_down (ucast :: 'b word \<Rightarrow> 'c word) \<rbrakk> \<Longrightarrow>
          (scast (ucast (a :: 'a::len word) :: 'b::len word) :: 'c::len word) = ucast a"
   by (metis down_cast_same ucast_eq ucast_down_wi)
 
 lemma scast_ucast_3:
   "\<lbrakk> is_down (ucast :: 'a word \<Rightarrow> 'c word); is_down (ucast :: 'b word \<Rightarrow> 'c word) \<rbrakk> \<Longrightarrow>
          (scast (ucast (a :: 'a::len word) :: 'b::len word) :: 'c::len word) = ucast a"
   by (metis down_cast_same ucast_eq ucast_down_wi)
 
 lemma scast_ucast_4:
   "\<lbrakk> is_up (ucast :: 'a word \<Rightarrow> 'b word); is_down (ucast :: 'b word \<Rightarrow> 'c word) \<rbrakk> \<Longrightarrow>
          (scast (ucast (a :: 'a::len word) :: 'b::len word) :: 'c::len word) = ucast a"
   by (metis down_cast_same ucast_eq ucast_down_wi)
 
 lemma scast_scast_b:
   "\<lbrakk> is_up (scast :: 'a word \<Rightarrow> 'b word) \<rbrakk> \<Longrightarrow>
      (scast (scast (a :: 'a::len word) :: 'b::len word) :: 'c::len word) = scast a"
   by (metis scast_eq sint_up_scast)
 
 lemma ucast_scast_1:
   "\<lbrakk> is_down (scast :: 'a word \<Rightarrow> 'b word); is_down (ucast :: 'b word \<Rightarrow> 'c word) \<rbrakk> \<Longrightarrow>
             (ucast (scast (a :: 'a::len word) :: 'b::len word) :: 'c::len word) = scast a"
   by (metis scast_eq ucast_down_wi)
 
 lemma ucast_scast_3:
   "\<lbrakk> is_down (scast :: 'a word \<Rightarrow> 'c word); is_down (ucast :: 'b word \<Rightarrow> 'c word) \<rbrakk> \<Longrightarrow>
      (ucast (scast (a :: 'a::len word) :: 'b::len word) :: 'c::len word) = scast a"
   by (metis scast_eq ucast_down_wi)
 
 lemma ucast_scast_4:
   "\<lbrakk> is_up (scast :: 'a word \<Rightarrow> 'b word); is_down (ucast :: 'b word \<Rightarrow> 'c word) \<rbrakk> \<Longrightarrow>
      (ucast (scast (a :: 'a::len word) :: 'b::len word) :: 'c::len word) = scast a"
   by (metis down_cast_same scast_eq sint_up_scast)
 
 lemma ucast_ucast_a:
   "\<lbrakk> is_down (ucast :: 'b word \<Rightarrow> 'c word) \<rbrakk> \<Longrightarrow>
         (ucast (ucast (a :: 'a::len word) :: 'b::len word) :: 'c::len word) = ucast a"
   by (metis down_cast_same ucast_eq ucast_down_wi)
 
 lemma ucast_ucast_b:
   "\<lbrakk> is_up (ucast :: 'a word \<Rightarrow> 'b word) \<rbrakk> \<Longrightarrow>
      (ucast (ucast (a :: 'a::len word) :: 'b::len word) :: 'c::len word) = ucast a"
   by (metis ucast_up_ucast)
 
 lemma scast_scast_a:
   "\<lbrakk> is_down (scast :: 'b word \<Rightarrow> 'c word) \<rbrakk> \<Longrightarrow>
             (scast (scast (a :: 'a::len word) :: 'b::len word) :: 'c::len word) = scast a"
   apply (simp only: scast_eq)
   apply (metis down_cast_same is_up_down scast_eq ucast_down_wi)
   done
 
 lemma scast_down_wi [OF refl]:
   "uc = scast \<Longrightarrow> is_down uc \<Longrightarrow> uc (word_of_int x) = word_of_int x"
   by (metis down_cast_same is_up_down ucast_down_wi)
 
 lemmas cast_simps =
   is_down is_up
   scast_down_add scast_down_minus scast_down_mult
   ucast_down_add ucast_down_minus ucast_down_mult
   scast_ucast_1 scast_ucast_3 scast_ucast_4
   ucast_scast_1 ucast_scast_3 ucast_scast_4
   ucast_ucast_a ucast_ucast_b
   scast_scast_a scast_scast_b
   ucast_down_wi scast_down_wi
   ucast_of_nat scast_of_nat
   uint_up_ucast sint_up_scast
   up_scast_surj up_ucast_surj
 
 lemma sdiv_word_max:
     "(sint (a :: ('a::len) word) sdiv sint (b :: ('a::len) word) < (2 ^ (size a - 1))) =
           ((a \<noteq> - (2 ^ (size a - 1)) \<or> (b \<noteq> -1)))"
     (is "?lhs = (\<not> ?a_int_min \<or> \<not> ?b_minus1)")
 proof (rule classical)
   assume not_thesis: "\<not> ?thesis"
 
   have not_zero: "b \<noteq> 0"
     using not_thesis
     by (clarsimp)
 
   let ?range = \<open>{- (2 ^ (size a - 1))..<2 ^ (size a - 1)} :: int set\<close>
 
   have result_range: "sint a sdiv sint b \<in> ?range \<union> {2 ^ (size a - 1)}"
     using sdiv_word_min [of a b] sdiv_word_max [of a b] by auto
 
   have result_range_overflow: "(sint a sdiv sint b = 2 ^ (size a - 1)) = (?a_int_min \<and> ?b_minus1)"
     apply (rule iffI [rotated])
      apply (clarsimp simp: signed_divide_int_def sgn_if word_size sint_int_min)
     apply (rule classical)
     apply (case_tac "?a_int_min")
      apply (clarsimp simp: word_size sint_int_min)
      apply (metis diff_0_right
               int_sdiv_negated_is_minus1 minus_diff_eq minus_int_code(2)
               power_eq_0_iff sint_minus1 zero_neq_numeral)
     apply (subgoal_tac "abs (sint a) < 2 ^ (size a - 1)")
      apply (insert sdiv_int_range [where a="sint a" and b="sint b"])[1]
      apply (clarsimp simp: word_size)
     apply (insert sdiv_int_range [where a="sint a" and b="sint b"])[1]
     apply auto
     apply (cases \<open>size a\<close>)
      apply simp_all
     apply (smt (z3) One_nat_def diff_Suc_1 signed_word_eqI sint_int_min sint_range_size wsst_TYs(3))
     done
 
   have result_range_simple: "(sint a sdiv sint b \<in> ?range) \<Longrightarrow> ?thesis"
     apply (insert sdiv_int_range [where a="sint a" and b="sint b"])
     apply (clarsimp simp: word_size sint_int_min)
     done
 
   show ?thesis
     apply (rule UnE [OF result_range result_range_simple])
      apply simp
     apply (clarsimp simp: word_size)
     using result_range_overflow
     apply (clarsimp simp: word_size)
     done
 qed
 
 lemmas sdiv_word_min' = sdiv_word_min [simplified word_size, simplified]
 lemmas sdiv_word_max' = sdiv_word_max [simplified word_size, simplified]
 
 lemma signed_arith_ineq_checks_to_eq:
   "((- (2 ^ (size a - 1)) \<le> (sint a + sint b)) \<and> (sint a + sint b \<le> (2 ^ (size a - 1) - 1)))
     = (sint a + sint b = sint (a + b ))"
   "((- (2 ^ (size a - 1)) \<le> (sint a - sint b)) \<and> (sint a - sint b \<le> (2 ^ (size a - 1) - 1)))
     = (sint a - sint b = sint (a - b))"
   "((- (2 ^ (size a - 1)) \<le> (- sint a)) \<and> (- sint a) \<le> (2 ^ (size a - 1) - 1))
     = ((- sint a) = sint (- a))"
   "((- (2 ^ (size a - 1)) \<le> (sint a * sint b)) \<and> (sint a * sint b \<le> (2 ^ (size a - 1) - 1)))
     = (sint a * sint b = sint (a * b))"
   "((- (2 ^ (size a - 1)) \<le> (sint a sdiv sint b)) \<and> (sint a sdiv sint b \<le> (2 ^ (size a - 1) - 1)))
     = (sint a sdiv sint b = sint (a sdiv b))"
   "((- (2 ^ (size a - 1)) \<le> (sint a smod sint b)) \<and> (sint a smod sint b \<le> (2 ^ (size a - 1) - 1)))
     = (sint a smod sint b = sint (a smod b))"
   by (auto simp: sint_word_ariths word_size signed_div_arith signed_mod_arith signed_take_bit_int_eq_self_iff intro: sym dest: sym)
 
 lemma signed_arith_sint:
   "((- (2 ^ (size a - 1)) \<le> (sint a + sint b)) \<and> (sint a + sint b \<le> (2 ^ (size a - 1) - 1)))
     \<Longrightarrow> sint (a + b) = (sint a + sint b)"
   "((- (2 ^ (size a - 1)) \<le> (sint a - sint b)) \<and> (sint a - sint b \<le> (2 ^ (size a - 1) - 1)))
     \<Longrightarrow> sint (a - b) = (sint a - sint b)"
   "((- (2 ^ (size a - 1)) \<le> (- sint a)) \<and> (- sint a) \<le> (2 ^ (size a - 1) - 1))
     \<Longrightarrow> sint (- a) = (- sint a)"
   "((- (2 ^ (size a - 1)) \<le> (sint a * sint b)) \<and> (sint a * sint b \<le> (2 ^ (size a - 1) - 1)))
     \<Longrightarrow> sint (a * b) = (sint a * sint b)"
   "((- (2 ^ (size a - 1)) \<le> (sint a sdiv sint b)) \<and> (sint a sdiv sint b \<le> (2 ^ (size a - 1) - 1)))
     \<Longrightarrow> sint (a sdiv b) = (sint a sdiv sint b)"
   "((- (2 ^ (size a - 1)) \<le> (sint a smod sint b)) \<and> (sint a smod sint b \<le> (2 ^ (size a - 1) - 1)))
     \<Longrightarrow> sint (a smod b) = (sint a smod sint b)"
   by (subst (asm) signed_arith_ineq_checks_to_eq; simp)+
 
 lemma nasty_split_lt:
   \<open>x * 2 ^ n + (2 ^ n - 1) \<le> 2 ^ m - 1\<close>
     if \<open>x < 2 ^ (m - n)\<close> \<open>n \<le> m\<close> \<open>m < LENGTH('a::len)\<close>
     for x :: \<open>'a::len word\<close>
 proof -
   define q where \<open>q = m - n\<close>
   with \<open>n \<le> m\<close> have \<open>m = q + n\<close>
     by simp
   with \<open>x < 2 ^ (m - n)\<close> have *: \<open>i < q\<close> if \<open>bit x i\<close> for i
     using that by simp (metis bit_take_bit_iff take_bit_word_eq_self_iff)
   from \<open>m = q + n\<close> have \<open>push_bit n x OR mask n \<le> mask m\<close>
     by (auto simp add: le_mask_high_bits word_size bit_simps dest!: *)
   then have \<open>push_bit n x + mask n \<le> mask m\<close>
     by (simp add: disjunctive_add bit_simps)
   then show ?thesis
     by (simp add: mask_eq_exp_minus_1 push_bit_eq_mult)
 qed
 
 lemma nasty_split_less:
   "\<lbrakk>m \<le> n; n \<le> nm; nm < LENGTH('a::len); x < 2 ^ (nm - n)\<rbrakk>
    \<Longrightarrow> (x :: 'a word) * 2 ^ n + (2 ^ m - 1) < 2 ^ nm"
   apply (simp only: word_less_sub_le[symmetric])
   apply (rule order_trans [OF _ nasty_split_lt])
      apply (rule word_plus_mono_right)
       apply (rule word_sub_mono)
          apply (simp add: word_le_nat_alt)
         apply simp
        apply (simp add: word_sub_1_le[OF power_not_zero])
       apply (simp add: word_sub_1_le[OF power_not_zero])
      apply (rule is_aligned_no_wrap')
       apply (rule is_aligned_mult_triv2)
      apply simp
     apply (erule order_le_less_trans, simp)
    apply simp+
   done
 
 end
 
 end
diff --git a/thys/Word_Lib/Word_Lib_Sumo.thy b/thys/Word_Lib/Word_Lib_Sumo.thy
--- a/thys/Word_Lib/Word_Lib_Sumo.thy
+++ b/thys/Word_Lib/Word_Lib_Sumo.thy
@@ -1,137 +1,132 @@
 (*
  * Copyright Florian Haftmann
  *
  * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section \<open>Ancient comprehensive Word Library\<close>
 
 theory Word_Lib_Sumo
 imports
   "HOL-Library.Word"
   Aligned
   Bit_Comprehension
   Bit_Comprehension_Int
   Bit_Shifts_Infix_Syntax
   Bits_Int
   Bitwise_Signed
   Bitwise
   Enumeration_Word
   Generic_set_bit
   Hex_Words
   Least_significant_bit
   More_Arithmetic
   More_Divides
   More_Sublist
   Even_More_List
   More_Misc
   Strict_part_mono
   Legacy_Aliases
   Most_significant_bit
   Next_and_Prev
   Norm_Words
   Reversed_Bit_Lists
   Rsplit
   Signed_Words
   Syntax_Bundles
+  Sgn_Abs
   Typedef_Morphisms
   Type_Syntax
   Word_EqI
   Word_Lemmas
   Word_8
   Word_16
   Word_32
   Word_Syntax
   Signed_Division_Word
   Singleton_Bit_Shifts
   More_Word_Operations
   Many_More
 begin
 
 unbundle bit_operations_syntax
 unbundle bit_projection_infix_syntax
 
 declare word_induct2[induct type]
 declare word_nat_cases[cases type]
 
 declare signed_take_bit_Suc [simp]
 
 (* these generate take_bit terms, which we often don't want for concrete lengths *)
 lemmas of_int_and_nat = unsigned_of_nat unsigned_of_int signed_of_int signed_of_nat
 
 bundle no_take_bit
 begin
 declare of_int_and_nat[simp del]
 end
 
 lemmas bshiftr1_def = bshiftr1_eq
 lemmas is_down_def = is_down_eq
 lemmas is_up_def = is_up_eq
 lemmas mask_def = mask_eq
 lemmas scast_def = scast_eq
 lemmas shiftl1_def = shiftl1_eq
 lemmas shiftr1_def = shiftr1_eq
 lemmas sshiftr1_def = sshiftr1_eq
 lemmas sshiftr_def = sshiftr_eq_funpow_sshiftr1
 lemmas to_bl_def = to_bl_eq
 lemmas ucast_def = ucast_eq
 lemmas unat_def = unat_eq_nat_uint
 lemmas word_cat_def = word_cat_eq
 lemmas word_reverse_def = word_reverse_eq_of_bl_rev_to_bl
 lemmas word_roti_def = word_roti_eq_word_rotr_word_rotl
 lemmas word_rotl_def = word_rotl_eq
 lemmas word_rotr_def = word_rotr_eq
 lemmas word_sle_def = word_sle_eq
 lemmas word_sless_def = word_sless_eq
 
 lemmas uint_0 = uint_nonnegative
 lemmas uint_lt = uint_bounded
 lemmas uint_mod_same = uint_idem
 lemmas of_nth_def = word_set_bits_def
 
 lemmas of_nat_word_eq_iff = word_of_nat_eq_iff
 lemmas of_nat_word_eq_0_iff = word_of_nat_eq_0_iff
 lemmas of_int_word_eq_iff = word_of_int_eq_iff
 lemmas of_int_word_eq_0_iff = word_of_int_eq_0_iff
 
 lemmas word_next_def = word_next_unfold
 
 lemmas word_prev_def = word_prev_unfold
 
 lemmas is_aligned_def = is_aligned_iff_dvd_nat
 
 lemmas word_and_max_simps =
   word8_and_max_simp
   word16_and_max_simp
   word32_and_max_simp
 
 lemma distinct_lemma: "f x \<noteq> f y \<Longrightarrow> x \<noteq> y" by auto
 
 lemmas and_bang = word_and_nth
 
 lemmas sdiv_int_def = signed_divide_int_def
 lemmas smod_int_def = signed_modulo_int_def
 
 (* shortcut for some specific lengths *)
 lemma word_fixed_sint_1[simp]:
   "sint (1::8 word) = 1"
   "sint (1::16 word) = 1"
   "sint (1::32 word) = 1"
   "sint (1::64 word) = 1"
   by (auto simp: sint_word_ariths)
 
 declare of_nat_diff [simp]
 
 (* Haskellish names/syntax *)
 notation (input)
   bit ("testBit")
 
 lemmas cast_simps = cast_simps ucast_down_bl
 
-(* shadows the slightly weaker Word.nth_ucast *)
-lemma nth_ucast:
-  "(ucast (w::'a::len word)::'b::len word) !! n =
-   (w !! n \<and> n < min LENGTH('a) LENGTH('b))"
-  by (auto simp add: bit_simps not_le dest: bit_imp_le_length)
-
 end