diff --git a/thys/Word_Lib/Aligned.thy b/thys/Word_Lib/Aligned.thy
--- a/thys/Word_Lib/Aligned.thy
+++ b/thys/Word_Lib/Aligned.thy
@@ -1,898 +1,894 @@
 (*
- * Copyright 2014, NICTA
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
- * This software may be distributed and modified according to the terms of
- * the BSD 2-Clause license. Note that NO WARRANTY is provided.
- * See "LICENSE_BSD2.txt" for details.
- *
- * @TAG(NICTA_BSD)
+ * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "Word Alignment"
 
 theory Aligned
 imports
   Word_Lib
   HOL_Lemmas
   More_Divides
 begin
 
 
 definition
   is_aligned :: "'a :: len word \<Rightarrow> nat \<Rightarrow> bool" where
   "is_aligned ptr n \<equiv> 2^n dvd unat ptr"
 
 
 lemma is_aligned_mask: "(is_aligned w n) = (w && mask n = 0)"
   unfolding is_aligned_def by (rule and_mask_dvd_nat)
 
 
 lemma is_aligned_to_bl:
   "is_aligned (w :: 'a :: len word) n = (True \<notin> set (drop (size w - n) (to_bl w)))"
   apply (simp add: is_aligned_mask eq_zero_set_bl)
   apply (clarsimp simp: in_set_conv_nth word_size)
   apply (simp add: to_bl_nth word_size cong: conj_cong)
   apply (simp add: diff_diff_less)
   apply safe
    apply (case_tac "n \<le> LENGTH('a)")
     prefer 2
     apply (rule_tac x=i in exI)
     apply clarsimp
    apply (subgoal_tac "\<exists>j < LENGTH('a). j < n \<and> LENGTH('a) - n + j = i")
     apply (erule exE)
     apply (rule_tac x=j in exI)
     apply clarsimp
    apply (thin_tac "w !! t" for t)
    apply (rule_tac x="i + n - LENGTH('a)" in exI)
    apply clarsimp
    apply arith
   apply (rule_tac x="LENGTH('a) - n + i" in exI)
   apply clarsimp
   apply arith
   done
 
 lemma unat_power_lower [simp]:
   assumes nv: "n < LENGTH('a::len)"
   shows "unat ((2::'a::len word) ^ n) = 2 ^ n"
   by (simp add: assms nat_power_eq uint_2p_alt unat_def)
 
 lemma power_overflow:
   "n \<ge> LENGTH('a) \<Longrightarrow> 2 ^ n = (0 :: 'a::len word)"
   by simp
 
 lemma is_alignedI [intro?]:
   fixes x::"'a::len word"
   assumes xv: "x = 2 ^ n * k"
   shows   "is_aligned x n"
 proof cases
   assume nv: "n < LENGTH('a)"
   show ?thesis
     unfolding is_aligned_def
   proof (rule dvdI [where k = "unat k mod 2 ^ (LENGTH('a) - n)"])
     from xv
     have "unat x = (unat (2::word32) ^ n * unat k) mod 2 ^ LENGTH('a)"
       using nv
       by (subst (asm) word_unat.Rep_inject [symmetric], simp,
           subst unat_word_ariths, simp)
 
     also have "\<dots> =  2 ^ n * (unat k mod 2 ^ (LENGTH('a) - n))" using nv
       by (simp add: mult_mod_right power_add [symmetric] add_diff_inverse)
 
     finally show "unat x = 2 ^ n * (unat k mod 2 ^ (LENGTH('a) - n))" .
   qed
 next
   assume "\<not> n < LENGTH('a)"
   with xv
   show ?thesis by (simp add: not_less power_overflow is_aligned_def)
 qed
 
 lemma is_aligned_weaken:
   "\<lbrakk> is_aligned w x; x \<ge> y \<rbrakk> \<Longrightarrow> is_aligned w y"
   unfolding is_aligned_def
   by (erule dvd_trans [rotated]) (simp add: le_imp_power_dvd)
 
 lemma nat_power_less_diff:
   assumes lt: "(2::nat) ^ n * q < 2 ^ m"
   shows "q < 2 ^ (m - n)"
   using lt
 proof (induct n arbitrary: m)
   case 0
   then show ?case by simp
 next
   case (Suc n)
 
   have ih: "\<And>m. 2 ^ n * q < 2 ^ m \<Longrightarrow> q < 2 ^ (m - n)"
     and prem: "2 ^ Suc n * q < 2 ^ m" by fact+
 
   show ?case
   proof (cases m)
     case 0
     then show ?thesis using Suc by simp
   next
     case (Suc m')
     then show ?thesis using prem
       by (simp add: ac_simps ih)
   qed
 qed
 
 lemma is_alignedE_pre:
   fixes w::"'a::len word"
   assumes aligned: "is_aligned w n"
   shows        rl: "\<exists>q. w = 2 ^ n * (of_nat q) \<and> q < 2 ^ (LENGTH('a) - n)"
 proof -
   from aligned obtain q where wv: "unat w = 2 ^ n * q"
     unfolding is_aligned_def ..
 
   show ?thesis
   proof (rule exI, intro conjI)
     show "q < 2 ^ (LENGTH('a) - n)"
     proof (rule nat_power_less_diff)
       have "unat w < 2 ^ size w" unfolding word_size ..
       then have "unat w < 2 ^ LENGTH('a)" by simp
       with wv show "2 ^ n * q < 2 ^ LENGTH('a)" by simp
     qed
 
     have r: "of_nat (2 ^ n) = (2::word32) ^ n"
       by (induct n) simp+
 
     from wv have "of_nat (unat w) = of_nat (2 ^ n * q)" by simp
     then have "w = of_nat (2 ^ n * q)" by (subst word_unat.Rep_inverse [symmetric])
     then show "w = 2 ^ n * (of_nat q)" by (simp add: r)
   qed
 qed
 
 lemma is_alignedE:
   "\<lbrakk>is_aligned (w::'a::len word) n;
     \<And>q. \<lbrakk>w = 2 ^ n * (of_nat q); q < 2 ^ (LENGTH('a) - n)\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
   by (auto dest: is_alignedE_pre)
 
 lemma is_aligned_replicate:
   fixes w::"'a::len word"
   assumes aligned: "is_aligned w n"
   and          nv: "n \<le> LENGTH('a)"
   shows   "to_bl w = (take (LENGTH('a) - n) (to_bl w)) @ replicate n False"
 proof -
   from nv have rl: "\<And>q. q < 2 ^ (LENGTH('a) - n) \<Longrightarrow>
       to_bl (2 ^ n * (of_nat q :: 'a word)) =
       drop n (to_bl (of_nat q :: 'a word)) @ replicate n False"
     by (metis bl_shiftl le_antisym min_def shiftl_t2n wsst_TYs(3))
   show ?thesis using aligned
     by (auto simp: rl elim: is_alignedE)
 qed
 
 lemma is_aligned_drop:
   fixes w::"'a::len word"
   assumes "is_aligned w n" "n \<le> LENGTH('a)"
   shows "drop (LENGTH('a) - n) (to_bl w) = replicate n False"
 proof -
   have "to_bl w = take (LENGTH('a) - n) (to_bl w) @ replicate n False"
     by (rule is_aligned_replicate) fact+
   then have "drop (LENGTH('a) - n) (to_bl w) = drop (LENGTH('a) - n) \<dots>" by simp
   also have "\<dots> = replicate n False" by simp
   finally show ?thesis .
 qed
 
 lemma less_is_drop_replicate:
   fixes x::"'a::len word"
   assumes lt: "x < 2 ^ n"
   shows   "to_bl x = replicate (LENGTH('a) - n) False @ drop (LENGTH('a) - n) (to_bl x)"
   by (metis assms bl_and_mask' less_mask_eq)
 
 lemma is_aligned_add_conv:
   fixes off::"'a::len word"
   assumes aligned: "is_aligned w n"
   and        offv: "off < 2 ^ n"
   shows    "to_bl (w + off) =
    (take (LENGTH('a) - n) (to_bl w)) @ (drop (LENGTH('a) - n) (to_bl off))"
 proof cases
   assume nv: "n \<le> LENGTH('a)"
   show ?thesis
   proof (subst aligned_bl_add_size, simp_all only: word_size)
     show "drop (LENGTH('a) - n) (to_bl w) = replicate n False"
       by (subst is_aligned_replicate [OF aligned nv]) (simp add: word_size)
 
     from offv show "take (LENGTH('a) - n) (to_bl off) =
                     replicate (LENGTH('a) - n) False"
       by (subst less_is_drop_replicate, assumption) simp
   qed fact
 next
   assume "\<not> n \<le> LENGTH('a)"
   with offv show ?thesis by (simp add: power_overflow)
 qed
 
 lemma aligned_add_aligned:
   fixes x::"'a::len word"
   assumes aligned1: "is_aligned x n"
   and     aligned2: "is_aligned y m"
   and           lt: "m \<le> n"
   shows   "is_aligned (x + y) m"
 proof cases
   assume nlt: "n < LENGTH('a)"
   show ?thesis
     unfolding is_aligned_def dvd_def
   proof -
     from aligned2 obtain q2 where yv: "y = 2 ^ m * of_nat q2"
       and q2v: "q2 < 2 ^ (LENGTH('a) - m)"
       by (auto elim: is_alignedE)
 
     from lt obtain k where kv: "m + k = n" by (auto simp: le_iff_add)
     with aligned1 obtain q1 where xv: "x = 2 ^ (m + k) * of_nat q1"
       and q1v: "q1 < 2 ^ (LENGTH('a) - (m + k))"
       by (auto elim: is_alignedE)
 
     have l1: "2 ^ (m + k) * q1 < 2 ^ LENGTH('a)"
       by (rule nat_less_power_trans [OF q1v])
          (subst kv, rule order_less_imp_le [OF nlt])
 
     have l2: "2 ^ m * q2 < 2 ^ LENGTH('a)"
       by (rule nat_less_power_trans [OF q2v],
           rule order_less_imp_le [OF order_le_less_trans])
          fact+
 
     have "x = of_nat (2 ^ (m + k) *  q1)" using xv
       by simp
 
     moreover have "y = of_nat (2 ^ m * q2)" using yv
       by simp
 
     ultimately have upls: "unat x + unat y = 2 ^ m * (2 ^ k * q1 + q2)"
     proof -
       have f1: "unat x = 2 ^ (m + k) * q1"
         by (metis (no_types) \<open>x = of_nat (2 ^ (m + k) * q1)\<close> l1 nat_mod_lem word_unat.inverse_norm
                              zero_less_numeral zero_less_power)
       have "unat y = 2 ^ m * q2"
         by (metis (no_types) \<open>y = of_nat (2 ^ m * q2)\<close> l2 nat_mod_lem word_unat.inverse_norm
                              zero_less_numeral zero_less_power)
       then show ?thesis
         using f1 by (simp add: power_add semiring_normalization_rules(34))
     qed
 
     (* (2 ^ k * q1 + q2) *)
     show "\<exists>d. unat (x + y) = 2 ^ m * d"
     proof (cases "unat x + unat y < 2 ^ LENGTH('a)")
       case True
 
       have "unat (x + y) = unat x + unat y"
         by (subst unat_plus_if', rule if_P) fact
 
       also have "\<dots> = 2 ^ m * (2 ^ k * q1 + q2)" by (rule upls)
       finally show ?thesis ..
     next
       case False
       then have "unat (x + y) = (unat x + unat y) mod 2 ^ LENGTH('a)"
         by (subst unat_word_ariths(1)) simp
 
       also have "\<dots> = (2 ^ m * (2 ^ k * q1 + q2)) mod 2 ^ LENGTH('a)"
         by (subst upls, rule refl)
 
       also
       have "\<dots> = 2 ^ m * ((2 ^ k * q1 +  q2) mod 2 ^ (LENGTH('a) - m))"
       proof -
         have "m \<le> len_of (TYPE('a))"
           by (meson le_trans less_imp_le_nat lt nlt)
         then show ?thesis
           by (metis mult_mod_right ordered_cancel_comm_monoid_diff_class.add_diff_inverse power_add)
       qed
 
       finally show ?thesis ..
     qed
   qed
 next
   assume "\<not> n < LENGTH('a)"
   with assms
   show ?thesis by (simp add: not_less power_overflow is_aligned_mask mask_def)
 qed
 
 corollary aligned_sub_aligned:
   "\<lbrakk>is_aligned (x::'a::len word) n; is_aligned y m; m \<le> n\<rbrakk>
    \<Longrightarrow> is_aligned (x - y) m"
   apply (simp del: add_uminus_conv_diff add:diff_conv_add_uminus)
   apply (erule aligned_add_aligned, simp_all)
   apply (erule is_alignedE)
   apply (rule_tac k="- of_nat q" in is_alignedI)
   apply simp
   done
 
 lemma is_aligned_shift:
   fixes k::"'a::len word"
   shows "is_aligned (k << m) m"
 proof cases
   assume mv: "m < LENGTH('a)"
   from mv obtain q where mq: "m + q = LENGTH('a)" and "0 < q"
     by (auto dest: less_imp_add_positive)
 
   have "(2::nat) ^ m dvd unat (k << m)"
   proof
     have kv: "(unat k div 2 ^ q) * 2 ^ q + unat k mod 2 ^ q = unat k"
       by (rule div_mult_mod_eq)
 
     have "unat (k << m) = unat (2 ^ m * k)" by (simp add: shiftl_t2n)
     also have "\<dots> = (2 ^ m * unat k) mod (2 ^ LENGTH('a))" using mv
       by (subst unat_word_ariths(2))+ simp
     also have "\<dots> = 2 ^ m * (unat k mod 2 ^ q)"
       by (subst mq [symmetric], subst power_add, subst mod_mult2_eq) simp
     finally show "unat (k << m) = 2 ^ m * (unat k mod 2 ^ q)" .
   qed
 
   then show ?thesis by (unfold is_aligned_def)
 next
   assume "\<not> m < LENGTH('a)"
   then show ?thesis
     by (simp add: not_less power_overflow is_aligned_mask mask_def
                   shiftl_zero_size word_size)
 qed
 
 lemma word_mod_by_0: "k mod (0::'a::len word) = k"
   by (simp add: word_arith_nat_mod)
 
 lemma aligned_mod_eq_0:
   fixes p::"'a::len word"
   assumes al: "is_aligned p sz"
   shows   "p mod 2 ^ sz = 0"
 proof cases
   assume szv: "sz < LENGTH('a)"
   with al
   show ?thesis
     unfolding is_aligned_def
     by (simp add: and_mask_dvd_nat p2_gt_0 word_mod_2p_is_mask)
 next
   assume "\<not> sz < LENGTH('a)"
   with al show ?thesis
     by (simp add: not_less power_overflow is_aligned_mask mask_def word_mod_by_0)
 qed
 
 lemma is_aligned_triv: "is_aligned (2 ^ n ::'a::len word) n"
   by (rule is_alignedI [where k = 1], simp)
 
 lemma is_aligned_mult_triv1: "is_aligned (2 ^ n * x  ::'a::len word) n"
   by (rule is_alignedI [OF refl])
 
 lemma is_aligned_mult_triv2: "is_aligned (x * 2 ^ n ::'a::len word) n"
   by (subst mult.commute, simp add: is_aligned_mult_triv1)
 
 lemma word_power_less_0_is_0:
   fixes x :: "'a::len word"
   shows "x < a ^ 0 \<Longrightarrow> x = 0" by simp
 
 lemma nat_add_offset_less:
   fixes x :: nat
   assumes yv: "y < 2 ^ n"
   and     xv: "x < 2 ^ m"
   and     mn: "sz = m + n"
   shows   "x * 2 ^ n + y < 2 ^ sz"
 proof (subst mn)
   from yv obtain qy where "y + qy = 2 ^ n" and "0 < qy"
     by (auto dest: less_imp_add_positive)
 
   have "x * 2 ^ n + y < x * 2 ^ n + 2 ^ n" by simp fact+
   also have "\<dots> = (x + 1) * 2 ^ n" by simp
   also have "\<dots> \<le> 2 ^ (m + n)" using xv
     by (subst power_add) (rule mult_le_mono1, simp)
   finally show "x * 2 ^ n + y < 2 ^ (m + n)" .
 qed
 
 lemma is_aligned_no_wrap:
   fixes off :: "'a::len word"
   fixes ptr :: "'a::len word"
   assumes al: "is_aligned ptr sz"
   and    off: "off < 2 ^ sz"
   shows  "unat ptr + unat off < 2 ^ LENGTH('a)"
 proof -
   have szv: "sz < LENGTH('a)"
   using off p2_gt_0 word_neq_0_conv by fastforce
 
   from al obtain q where ptrq: "ptr = 2 ^ sz * of_nat q" and
     qv: "q < 2 ^ (LENGTH('a) - sz)" by (auto elim: is_alignedE)
 
   show ?thesis
   proof (cases "sz = 0")
     case True
     then show ?thesis using off ptrq qv by clarsimp
   next
     case False
     then have sne: "0 < sz" ..
 
     show ?thesis
     proof -
       have uq: "unat (of_nat q ::'a::len word) = q"
         apply (subst unat_of_nat)
         apply (rule mod_less)
         apply (rule order_less_trans [OF qv])
         apply (rule power_strict_increasing [OF diff_less [OF sne]])
          apply (simp_all)
         done
 
       have uptr: "unat ptr = 2 ^ sz * q"
         apply (subst ptrq)
         apply (subst iffD1 [OF unat_mult_lem])
          apply (subst unat_power_lower [OF szv])
          apply (subst uq)
          apply (rule nat_less_power_trans [OF qv order_less_imp_le [OF szv]])
         apply (subst uq)
         apply (subst unat_power_lower [OF szv])
         apply simp
         done
 
       show "unat ptr + unat off < 2 ^ LENGTH('a)" using szv
         apply (subst uptr)
         apply (subst mult.commute, rule nat_add_offset_less [OF _ qv])
         apply (rule order_less_le_trans [OF unat_mono [OF off] order_eq_refl])
          apply simp_all
         done
     qed
   qed
 qed
 
 lemma is_aligned_no_wrap':
   fixes ptr :: "'a::len word"
   assumes al: "is_aligned ptr sz"
   and    off: "off < 2 ^ sz"
   shows  "ptr \<le> ptr + off"
   by (subst no_plus_overflow_unat_size, subst word_size, rule is_aligned_no_wrap) fact+
 
 lemma is_aligned_no_overflow':
   fixes p :: "'a::len word"
   assumes al: "is_aligned p n"
   shows "p \<le> p + (2 ^ n - 1)"
 proof cases
   assume "n<LENGTH('a)"
   with al
   have "2^n - (1::'a::len word) < 2^n"
     by (simp add: word_less_nat_alt unat_sub_if_size)
   with al
   show ?thesis by (rule is_aligned_no_wrap')
 next
   assume "\<not> n<LENGTH('a)"
   with al
   show ?thesis
   by (simp add: not_less power_overflow is_aligned_mask mask_2pm1)
 qed
 
 lemma is_aligned_no_overflow:
   "is_aligned ptr sz \<Longrightarrow> ptr \<le> ptr + 2^sz - 1"
   by (drule is_aligned_no_overflow') (simp add: field_simps)
 
 lemma replicate_not_True:
   "\<And>n. xs = replicate n False \<Longrightarrow> True \<notin> set xs"
   by (induct xs) auto
 
 lemma is_aligned_replicateI:
   "to_bl p = addr @ replicate n False \<Longrightarrow> is_aligned (p::'a::len word) n"
   apply (simp add: is_aligned_to_bl word_size)
   apply (subgoal_tac "length addr = LENGTH('a) - n")
    apply (simp add: replicate_not_True)
   apply (drule arg_cong [where f=length])
   apply simp
   done
 
 lemma to_bl_2p:
   "n < LENGTH('a) \<Longrightarrow>
    to_bl ((2::'a::len word) ^ n) =
    replicate (LENGTH('a) - Suc n) False @ True # replicate n False"
   apply (subst shiftl_1 [symmetric])
   apply (subst bl_shiftl)
   apply (simp add: to_bl_1 min_def word_size)
   done
 
 lemma map_zip_replicate_False_xor:
   "n = length xs \<Longrightarrow> map (\<lambda>(x, y). x = (\<not> y)) (zip xs (replicate n False)) = xs"
   by (induct xs arbitrary: n, auto)
 
 lemma drop_minus_lem:
   "\<lbrakk> n \<le> length xs; 0 < n; n' = length xs \<rbrakk> \<Longrightarrow> drop (n' - n) xs = rev xs ! (n - 1)  # drop (Suc (n' - n)) xs"
 proof (induct xs arbitrary: n n')
   case Nil then show ?case by simp
 next
   case (Cons y ys)
   from Cons.prems
   show ?case
     apply simp
     apply (cases "n = Suc (length ys)")
      apply (simp add: nth_append)
     apply (simp add: Suc_diff_le Cons.hyps nth_append)
     apply clarsimp
     apply arith
     done
 qed
 
 lemma drop_minus:
   "\<lbrakk> n < length xs; n' = length xs \<rbrakk> \<Longrightarrow> drop (n' - Suc n) xs = rev xs ! n  # drop (n' - n) xs"
   apply (subst drop_minus_lem)
      apply simp
     apply simp
    apply simp
   apply simp
   apply (cases "length xs", simp)
   apply (simp add: Suc_diff_le)
   done
 
 lemma xor_2p_to_bl:
   fixes x::"'a::len word"
   shows "to_bl (x xor 2^n) =
   (if n < LENGTH('a)
    then take (LENGTH('a)-Suc n) (to_bl x) @ (\<not>rev (to_bl x)!n) # drop (LENGTH('a)-n) (to_bl x)
    else to_bl x)"
 proof -
   have x: "to_bl x = take (LENGTH('a)-Suc n) (to_bl x) @ drop (LENGTH('a)-Suc n) (to_bl x)"
     by simp
 
   show ?thesis
   apply simp
   apply (rule conjI)
    apply (clarsimp simp: word_size)
    apply (simp add: bl_word_xor to_bl_2p)
    apply (subst x)
    apply (subst zip_append)
     apply simp
    apply (simp add: map_zip_replicate_False_xor drop_minus)
   apply (auto simp add: word_size nth_w2p intro!: word_eqI)
   done
 qed
 
 lemma aligned_add_xor:
   assumes al: "is_aligned (x::'a::len word) n'" and le: "n < n'"
   shows "(x + 2^n) xor 2^n = x"
 proof cases
   assume "n' < LENGTH('a)"
   with assms show ?thesis
   apply -
   apply (rule word_bl.Rep_eqD)
   apply (subst xor_2p_to_bl)
   apply simp
   apply (subst is_aligned_add_conv, simp, simp add: word_less_nat_alt)+
   apply (simp add: to_bl_2p nth_append)
   apply (cases "n' = Suc n")
    apply simp
    apply (subst is_aligned_replicate [where n="Suc n", simplified, symmetric]; simp)
   apply (subgoal_tac "\<not> LENGTH('a) - Suc n \<le> LENGTH('a) - n'")
    prefer 2
    apply arith
   apply (subst replicate_Suc [symmetric])
   apply (subst replicate_add [symmetric])
   apply (simp add: is_aligned_replicate [simplified, symmetric])
   done
 next
   assume "\<not> n' < LENGTH('a)"
   with al show ?thesis
     by (simp add: is_aligned_mask mask_def not_less power_overflow)
 qed
 
 lemma is_aligned_0 [simp]:
   "is_aligned p 0"
   by (simp add: is_aligned_def)
 
 lemma is_aligned_replicateD:
   "\<lbrakk> is_aligned (w::'a::len word) n; n \<le> LENGTH('a) \<rbrakk>
      \<Longrightarrow> \<exists>xs. to_bl w = xs @ replicate n False
                \<and> length xs = size w - n"
   apply (subst is_aligned_replicate, assumption+)
   apply (rule exI, rule conjI, rule refl)
   apply (simp add: word_size)
   done
 
 lemma is_aligned_add_mult_multI:
   fixes p :: "'a::len word"
   shows "\<lbrakk>is_aligned p m; n \<le> m; n' = n\<rbrakk> \<Longrightarrow> is_aligned (p + x * 2 ^ n * z) n'"
   apply (erule aligned_add_aligned)
   apply (auto intro: is_alignedI [where k="x*z"])
   done
 
 lemma is_aligned_add_multI:
   fixes p :: "'a::len word"
   shows "\<lbrakk>is_aligned p m; n \<le> m; n' = n\<rbrakk> \<Longrightarrow> is_aligned (p + x * 2 ^ n) n'"
   apply (erule aligned_add_aligned)
   apply (auto intro: is_alignedI [where k="x"])
   done
 
 lemma unat_of_nat_len:
   "x < 2 ^ LENGTH('a) \<Longrightarrow> unat (of_nat x :: 'a::len word) = x"
   by (simp add: word_size unat_of_nat)
 
 lemma is_aligned_no_wrap''':
   fixes ptr :: "'a::len word"
   shows"\<lbrakk> is_aligned ptr sz; sz < LENGTH('a); off < 2 ^ sz \<rbrakk>
          \<Longrightarrow> unat ptr + off < 2 ^ LENGTH('a)"
   apply (drule is_aligned_no_wrap[where off="of_nat off"])
    apply (simp add: word_less_nat_alt)
    apply (erule order_le_less_trans[rotated])
    apply (subst unat_of_nat)
    apply (rule mod_less_eq_dividend)
   apply (subst(asm) unat_of_nat_len)
    apply (erule order_less_trans)
    apply (erule power_strict_increasing)
    apply simp
   apply assumption
   done
 
 lemma is_aligned_get_word_bits:
   fixes p :: "'a::len word"
   shows "\<lbrakk> is_aligned p n; \<lbrakk> is_aligned p n; n < LENGTH('a) \<rbrakk> \<Longrightarrow> P;
            \<lbrakk> p = 0; n \<ge> LENGTH('a) \<rbrakk> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
   apply (cases "n < LENGTH('a)")
    apply simp
   apply simp
   apply (erule meta_mp)
   apply (clarsimp simp: is_aligned_mask mask_def power_add
                         power_overflow)
   done
 
 lemma aligned_small_is_0:
   "\<lbrakk> is_aligned x n; x < 2 ^ n \<rbrakk> \<Longrightarrow> x = 0"
   apply (erule is_aligned_get_word_bits)
    apply (frule is_aligned_add_conv [rotated, where w=0])
     apply (simp add: is_aligned_def)
    apply simp
    apply (drule is_aligned_replicateD)
     apply simp
    apply (clarsimp simp: word_size)
    apply (subst (asm) replicate_add [symmetric])
    apply (drule arg_cong[where f="of_bl :: bool list \<Rightarrow> 'a::len word"])
    apply simp
   apply (simp only: replicate.simps[symmetric, where x=False]
                     drop_replicate)
   done
 
 corollary is_aligned_less_sz:
   "\<lbrakk>is_aligned a sz; a \<noteq> 0\<rbrakk> \<Longrightarrow> \<not> a < 2 ^ sz"
   by (rule notI, drule(1) aligned_small_is_0, erule(1) notE)
 
 lemma aligned_at_least_t2n_diff:
   "\<lbrakk>is_aligned x n; is_aligned y n; x < y\<rbrakk> \<Longrightarrow> x \<le> y - 2 ^ n"
   apply (erule is_aligned_get_word_bits[where p=y])
    apply (rule ccontr)
    apply (clarsimp simp: linorder_not_le)
    apply (subgoal_tac "y - x = 0")
     apply clarsimp
    apply (rule aligned_small_is_0)
     apply (erule(1) aligned_sub_aligned)
     apply simp
    apply unat_arith
   apply simp
   done
 
 lemma word_sub_1_le:
   "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 is_aligned_no_overflow'':
   "\<lbrakk>is_aligned x n; x + 2 ^ n \<noteq> 0\<rbrakk> \<Longrightarrow> x \<le> x + 2 ^ n"
   apply (frule is_aligned_no_overflow')
   apply (erule order_trans)
   apply (simp add: field_simps)
   apply (erule word_sub_1_le)
   done
 
 lemma is_aligned_nth:
   "is_aligned p m = (\<forall>n < m. \<not>p !! n)"
   apply (clarsimp simp: is_aligned_mask bang_eq word_size)
   apply (rule iffI)
    apply clarsimp
    apply (case_tac "n < size p")
     apply (simp add: word_size)
    apply (drule test_bit_size)
    apply simp
   apply clarsimp
   done
 
 lemma range_inter:
   "({a..b} \<inter> {c..d} = {}) = (\<forall>x. \<not>(a \<le> x \<and> x \<le> b \<and> c \<le> x \<and> x \<le> d))"
   by auto
 
 lemma aligned_inter_non_empty:
   "\<lbrakk> {p..p + (2 ^ n - 1)} \<inter> {p..p + 2 ^ m - 1} = {};
      is_aligned p n; is_aligned p m\<rbrakk> \<Longrightarrow> False"
   apply (clarsimp simp only: range_inter)
   apply (erule_tac x=p in allE)
   apply simp
   apply (erule impE)
    apply (erule is_aligned_no_overflow')
   apply (erule notE)
   apply (erule is_aligned_no_overflow)
   done
 
 lemma not_aligned_mod_nz:
   assumes al: "\<not> is_aligned a n"
   shows "a mod 2 ^ n \<noteq> 0"
 proof cases
   assume "n < LENGTH('a)"
   with al
   show ?thesis
     apply (simp add: is_aligned_def dvd_eq_mod_eq_0 word_arith_nat_mod)
     apply (erule of_nat_neq_0)
     apply (rule order_less_trans)
      apply (rule mod_less_divisor)
      apply simp
     apply simp
     done
 next
   assume "\<not> n < LENGTH('a)"
   with al
   show ?thesis
     by (simp add: is_aligned_mask mask_def not_less power_overflow
                   word_less_nat_alt word_mod_by_0)
 qed
 
 lemma nat_add_offset_le:
   fixes x :: nat
   assumes yv: "y \<le> 2 ^ n"
   and     xv: "x < 2 ^ m"
   and     mn: "sz = m + n"
   shows   "x * 2 ^ n + y \<le> 2 ^ sz"
 proof (subst mn)
   from yv obtain qy where "y + qy = 2 ^ n"
     by (auto simp: le_iff_add)
 
   have "x * 2 ^ n + y \<le> x * 2 ^ n + 2 ^ n"
     using yv xv by simp
   also have "\<dots> = (x + 1) * 2 ^ n" by simp
   also have "\<dots> \<le> 2 ^ (m + n)" using xv
     by (subst power_add) (rule mult_le_mono1, simp)
   finally show "x * 2 ^ n + y \<le> 2 ^ (m + n)" .
 qed
 
 lemma is_aligned_no_wrap_le:
   fixes ptr::"'a::len word"
   assumes al: "is_aligned ptr sz"
   and    szv: "sz < LENGTH('a)"
   and    off: "off \<le> 2 ^ sz"
   shows  "unat ptr + off \<le> 2 ^ LENGTH('a)"
 proof -
   from al obtain q where ptrq: "ptr = 2 ^ sz * of_nat q" and
     qv: "q < 2 ^ (LENGTH('a) - sz)" by (auto elim: is_alignedE)
 
   show ?thesis
   proof (cases "sz = 0")
     case True
     then show ?thesis using off ptrq qv
       apply (clarsimp)
       apply (erule le_SucE)
        apply (simp add: unat_of_nat)
       apply (simp add: less_eq_Suc_le [symmetric] unat_of_nat)
       done
   next
     case False
     then have sne: "0 < sz" ..
 
     show ?thesis
     proof -
       have uq: "unat (of_nat q :: 'a word) = q"
         apply (subst unat_of_nat)
         apply (rule mod_less)
         apply (rule order_less_trans [OF qv])
         apply (rule power_strict_increasing [OF diff_less [OF sne]])
         apply simp_all
         done
 
       have uptr: "unat ptr = 2 ^ sz * q"
         apply (subst ptrq)
         apply (subst iffD1 [OF unat_mult_lem])
         apply (subst unat_power_lower [OF szv])
         apply (subst uq)
         apply (rule nat_less_power_trans [OF qv order_less_imp_le [OF szv]])
         apply (subst uq)
         apply (subst unat_power_lower [OF szv])
         apply simp
         done
 
       show "unat ptr + off \<le> 2 ^ LENGTH('a)" using szv
         apply (subst uptr)
         apply (subst mult.commute, rule nat_add_offset_le [OF off qv])
         apply simp
         done
     qed
   qed
 qed
 
 lemma is_aligned_neg_mask:
   "m \<le> n \<Longrightarrow> is_aligned (x && ~~ (mask n)) m"
   by (metis and_not_mask is_aligned_shift is_aligned_weaken)
 
 lemma unat_minus:
   "unat (- (x :: 'a :: len word)) = (if x = 0 then 0 else 2 ^ size x - unat x)"
   using unat_sub_if_size[where x="2 ^ size x" and y=x]
   by (simp add: unat_eq_0 word_size)
 
 lemma is_aligned_minus:
   "is_aligned p n \<Longrightarrow> is_aligned (- p) n"
   apply (clarsimp simp: is_aligned_def unat_minus word_size word_neq_0_conv)
   apply (rule dvd_diff_nat, simp_all)
   apply (rule le_imp_power_dvd)
   apply (fold is_aligned_def)
   apply (erule_tac Q="0<p" in contrapos_pp)
   apply (clarsimp simp add: is_aligned_mask mask_def power_overflow)
   done
 
 lemma add_mask_lower_bits:
   "\<lbrakk>is_aligned (x :: 'a :: len word) n;
     \<forall>n' \<ge> n. n' < LENGTH('a) \<longrightarrow> \<not> p !! n'\<rbrakk> \<Longrightarrow> x + p && ~~ (mask n) = x"
   apply (subst word_plus_and_or_coroll)
    apply (rule word_eqI)
    apply (clarsimp simp: word_size is_aligned_nth)
    apply (erule_tac x=na in allE)+
    apply simp
   apply (rule word_eqI)
   apply (clarsimp simp: word_size is_aligned_nth word_ops_nth_size le_def)
   apply blast
   done
 
 lemma is_aligned_andI1:
   "is_aligned x n \<Longrightarrow> is_aligned (x && y) n"
   by (simp add: is_aligned_nth)
 
 lemma is_aligned_andI2:
   "is_aligned y n \<Longrightarrow> is_aligned (x && y) n"
   by (simp add: is_aligned_nth)
 
 lemma is_aligned_shiftl:
   "is_aligned w (n - m) \<Longrightarrow> is_aligned (w << m) n"
   by (simp add: is_aligned_nth nth_shiftl)
 
 lemma is_aligned_shiftr:
   "is_aligned w (n + m) \<Longrightarrow> is_aligned (w >> m) n"
   by (simp add: is_aligned_nth nth_shiftr)
 
 lemma is_aligned_shiftl_self:
   "is_aligned (p << n) n"
   by (rule is_aligned_shift)
 
 lemma is_aligned_neg_mask_eq:
   "is_aligned p n \<Longrightarrow> p && ~~ (mask n) = p"
   by (metis add.left_neutral is_aligned_mask word_plus_and_or_coroll2)
 
 lemma is_aligned_shiftr_shiftl:
   "is_aligned w n \<Longrightarrow> w >> n << n = w"
   by (metis and_not_mask is_aligned_neg_mask_eq)
 
 lemma aligned_shiftr_mask_shiftl:
   "is_aligned x n \<Longrightarrow> ((x >> n) && mask v) << n = x && mask (v + n)"
   apply (rule word_eqI)
   apply (simp add: word_size nth_shiftl nth_shiftr)
   apply (subgoal_tac "\<forall>m. x !! m \<longrightarrow> m \<ge> n")
    apply auto[1]
   apply (clarsimp simp: is_aligned_mask)
   apply (drule_tac x=m in word_eqD)
   apply (frule test_bit_size)
   apply (simp add: word_size)
   done
 
 lemma mask_zero:
   "is_aligned x a \<Longrightarrow> x && mask a = 0"
   by (metis is_aligned_mask)
 
 lemma is_aligned_neg_mask_eq_concrete:
   "\<lbrakk> is_aligned p n; msk && ~~(mask n) = ~~(mask n) \<rbrakk>
    \<Longrightarrow> p && msk = p"
   by (metis word_bw_assocs(1) word_bw_comms(1) is_aligned_neg_mask_eq)
 
 lemma is_aligned_and_not_zero:
   "\<lbrakk> is_aligned n k; n \<noteq> 0 \<rbrakk> \<Longrightarrow> 2 ^ k \<le> n"
   using is_aligned_less_sz leI by blast
 
 lemma is_aligned_and_2_to_k:
   "(n && 2 ^ k - 1) = 0 \<Longrightarrow> is_aligned (n :: 'a :: len word) k"
   by (simp add: is_aligned_mask mask_def)
 
 lemma is_aligned_power2:
   "b \<le> a \<Longrightarrow> is_aligned (2 ^ a) b"
   by (metis is_aligned_triv is_aligned_weaken)
 
 lemma aligned_sub_aligned':
   "\<lbrakk> is_aligned (a :: 'a :: len word) n; is_aligned b n; n < LENGTH('a) \<rbrakk>
    \<Longrightarrow> is_aligned (a - b) n"
   by (simp add: aligned_sub_aligned)
 
 lemma is_aligned_neg_mask_weaken:
   "\<lbrakk> is_aligned p n; m \<le> n \<rbrakk> \<Longrightarrow> p && ~~(mask m) = p"
    using is_aligned_neg_mask_eq is_aligned_weaken by blast
 
 lemma is_aligned_neg_mask2[simp]:
   "is_aligned (a && ~~(mask n)) n"
   by (simp add: and_not_mask is_aligned_shift)
 
 end
diff --git a/thys/Word_Lib/Enumeration.thy b/thys/Word_Lib/Enumeration.thy
--- a/thys/Word_Lib/Enumeration.thy
+++ b/thys/Word_Lib/Enumeration.thy
@@ -1,329 +1,325 @@
 (*
- * Copyright 2014, NICTA
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
- * This software may be distributed and modified according to the terms of
- * the BSD 2-Clause license. Note that NO WARRANTY is provided.
- * See "LICENSE_BSD2.txt" for details.
- *
- * @TAG(NICTA_BSD)
+ * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "Enumeration extensions and alternative definition"
 
 theory Enumeration
 imports Main
 begin
 
 abbreviation
   "enum \<equiv> enum_class.enum"
 abbreviation
   "enum_all \<equiv> enum_class.enum_all"
 abbreviation
   "enum_ex \<equiv> enum_class.enum_ex"
 
 primrec (nonexhaustive)
   the_index :: "'a list \<Rightarrow> 'a \<Rightarrow> nat"
 where
   "the_index (x # xs) y = (if x = y then 0 else Suc (the_index xs y))"
 
 lemma the_index_bounded:
   "x \<in> set xs \<Longrightarrow> the_index xs x < length xs"
   by (induct xs, clarsimp+)
 
 lemma nth_the_index:
   "x \<in> set xs \<Longrightarrow> xs ! the_index xs x = x"
   by (induct xs, clarsimp+)
 
 lemma distinct_the_index_is_index[simp]:
   "\<lbrakk> distinct xs ; n < length xs \<rbrakk> \<Longrightarrow> the_index xs (xs ! n) = n"
   by (meson nth_eq_iff_index_eq nth_mem nth_the_index the_index_bounded)
 
 lemma the_index_last_distinct:
   "distinct xs \<and> xs \<noteq> [] \<Longrightarrow> the_index xs (last xs) = length xs - 1"
   apply safe
   apply (subgoal_tac "xs ! (length xs - 1) = last xs")
    apply (subgoal_tac "xs ! the_index xs (last xs) = last xs")
     apply (subst nth_eq_iff_index_eq[symmetric])
        apply assumption
       apply (rule the_index_bounded)
       apply simp_all
    apply (rule nth_the_index)
    apply simp
   apply (induct xs, auto)
   done
 
 context enum begin
 
 (* These two are added for historical reasons. *)
 lemmas enum_surj[simp] = enum_UNIV
 declare enum_distinct[simp]
 
 lemma enum_nonempty[simp]: "(enum :: 'a list) \<noteq> []"
   using enum_surj by fastforce
 
 definition
   maxBound :: 'a where
   "maxBound \<equiv> last enum"
 
 definition
   minBound :: 'a where
   "minBound \<equiv> hd enum"
 
 definition
   toEnum :: "nat \<Rightarrow> 'a" where
   "toEnum n \<equiv> if n < length (enum :: 'a list) then enum ! n else the None"
 
 definition
   fromEnum :: "'a \<Rightarrow> nat" where
   "fromEnum x \<equiv> the_index enum x"
 
 
 lemma maxBound_is_length:
   "fromEnum maxBound = length (enum :: 'a list) - 1"
   by (simp add: maxBound_def fromEnum_def the_index_last_distinct)
 
 lemma maxBound_less_length:
   "(x \<le> fromEnum maxBound) = (x < length (enum :: 'a list))"
   unfolding maxBound_is_length by (cases "length enum") auto
 
 lemma maxBound_is_bound [simp]:
   "fromEnum x \<le> fromEnum maxBound"
   unfolding maxBound_less_length
   by (fastforce simp: fromEnum_def intro: the_index_bounded)
 
 lemma to_from_enum [simp]:
   fixes x :: 'a
   shows "toEnum (fromEnum x) = x"
 proof -
   have "x \<in> set enum" by simp
   then show ?thesis by (simp add: toEnum_def fromEnum_def nth_the_index the_index_bounded)
 qed
 
 lemma from_to_enum [simp]:
   "x \<le> fromEnum maxBound \<Longrightarrow> fromEnum (toEnum x) = x"
   unfolding maxBound_less_length by (simp add: toEnum_def fromEnum_def)
 
 lemma map_enum:
   fixes x :: 'a
   shows "map f enum ! fromEnum x = f x"
 proof -
   have "fromEnum x \<le> fromEnum (maxBound :: 'a)"
     by (rule maxBound_is_bound)
   then have "fromEnum x < length (enum::'a list)"
     by (simp add: maxBound_less_length)
   then have "map f enum ! fromEnum x = f (enum ! fromEnum x)" by simp
   also
   have "x \<in> set enum" by simp
   then have "enum ! fromEnum x = x"
     by (simp add: fromEnum_def nth_the_index)
   finally
   show ?thesis .
 qed
 
 definition
   assocs :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'b) list" where
  "assocs f \<equiv> map (\<lambda>x. (x, f x)) enum"
 
 end
 
 (* For historical naming reasons. *)
 lemmas enum_bool = enum_bool_def
 
 lemma fromEnumTrue [simp]: "fromEnum True = 1"
   by (simp add: fromEnum_def enum_bool)
 
 lemma fromEnumFalse [simp]: "fromEnum False = 0"
   by (simp add: fromEnum_def enum_bool)
 
 
 class enum_alt =
   fixes enum_alt :: "nat \<Rightarrow> 'a option"
 
 class enumeration_alt = enum_alt +
   assumes enum_alt_one_bound:
     "enum_alt x = (None :: 'a option) \<Longrightarrow> enum_alt (Suc x) = (None :: 'a option)"
   assumes enum_alt_surj:
     "range enum_alt \<union> {None} = UNIV"
   assumes enum_alt_inj:
     "(enum_alt x :: 'a option) = enum_alt y \<Longrightarrow> (x = y) \<or> (enum_alt x = (None :: 'a option))"
 begin
 
 lemma enum_alt_inj_2:
   assumes "enum_alt x = (enum_alt y :: 'a option)"
           "enum_alt x \<noteq> (None :: 'a option)"
   shows "x = y"
 proof -
   from assms
   have "(x = y) \<or> (enum_alt x = (None :: 'a option))" by (fastforce intro!: enum_alt_inj)
   with assms show ?thesis by clarsimp
 qed
 
 lemma enum_alt_surj_2:
   "\<exists>x. enum_alt x = Some y"
 proof -
   have "Some y \<in> range enum_alt \<union> {None}" by (subst enum_alt_surj) simp
   then have "Some y \<in> range enum_alt" by simp
   then show ?thesis by auto
 qed
 
 end
 
 definition
   alt_from_ord :: "'a list \<Rightarrow> nat \<Rightarrow> 'a option"
 where
   "alt_from_ord L \<equiv> \<lambda>n. if (n < length L) then Some (L ! n) else None"
 
 lemma handy_if_lemma: "((if P then Some A else None) = Some B) = (P \<and> (A = B))"
   by simp
 
 class enumeration_both = enum_alt + enum +
   assumes enum_alt_rel: "enum_alt = alt_from_ord enum"
 
 instance enumeration_both < enumeration_alt
   apply (intro_classes; simp add: enum_alt_rel alt_from_ord_def)
     apply auto[1]
    apply (safe; simp)[1]
    apply (rule rev_image_eqI; simp)
     apply (rule the_index_bounded; simp)
    apply (subst nth_the_index; simp)
   apply (clarsimp simp: handy_if_lemma)
   apply (subst nth_eq_iff_index_eq[symmetric]; simp)
   done
 
 instantiation bool :: enumeration_both
 begin
   definition enum_alt_bool: "enum_alt \<equiv> alt_from_ord [False, True]"
   instance by (intro_classes, simp add: enum_bool_def enum_alt_bool)
 end
 
 definition
   toEnumAlt :: "nat \<Rightarrow> ('a :: enum_alt)" where
  "toEnumAlt n \<equiv> the (enum_alt n)"
 
 definition
   fromEnumAlt :: "('a :: enum_alt) \<Rightarrow> nat" where
  "fromEnumAlt x \<equiv> THE n. enum_alt n = Some x"
 
 definition
   upto_enum :: "('a :: enumeration_alt) \<Rightarrow> 'a \<Rightarrow> 'a list" ("(1[_ .e. _])") where
  "upto_enum n m \<equiv> map toEnumAlt [fromEnumAlt n ..< Suc (fromEnumAlt m)]"
 
 lemma fromEnum_alt_red[simp]:
   "fromEnumAlt = (fromEnum :: ('a :: enumeration_both) \<Rightarrow> nat)"
   apply (rule ext)
   apply (simp add: fromEnumAlt_def fromEnum_def enum_alt_rel alt_from_ord_def)
   apply (rule theI2)
     apply (rule conjI)
      apply (clarify, rule nth_the_index, simp)
     apply (rule the_index_bounded, simp)
    apply auto
   done
 
 lemma toEnum_alt_red[simp]:
   "toEnumAlt = (toEnum :: nat \<Rightarrow> 'a :: enumeration_both)"
   by (rule ext) (simp add: enum_alt_rel alt_from_ord_def toEnum_def toEnumAlt_def)
 
 lemma upto_enum_red:
   "[(n :: ('a :: enumeration_both)) .e. m] = map toEnum [fromEnum n ..< Suc (fromEnum m)]"
   unfolding upto_enum_def by simp
 
 instantiation nat :: enumeration_alt
 begin
   definition enum_alt_nat: "enum_alt \<equiv> Some"
   instance by (intro_classes; simp add: enum_alt_nat UNIV_option_conv)
 end
 
 lemma toEnumAlt_nat[simp]: "toEnumAlt = id"
   by (rule ext) (simp add: toEnumAlt_def enum_alt_nat)
 
 lemma fromEnumAlt_nat[simp]: "fromEnumAlt = id"
   by (rule ext) (simp add: fromEnumAlt_def enum_alt_nat)
 
 lemma upto_enum_nat[simp]: "[n .e. m] = [n ..< Suc m]"
   by (subst upto_enum_def) simp
 
 definition
   zipE1 :: "'a :: enum_alt \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list"
 where
   "zipE1 x L \<equiv> zip (map toEnumAlt [fromEnumAlt x ..< fromEnumAlt x + length L]) L"
 
 definition
   zipE2 :: "'a :: enum_alt \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list"
 where
   "zipE2 x xn L \<equiv> zip (map (\<lambda>n. toEnumAlt (fromEnumAlt x + (fromEnumAlt xn - fromEnumAlt x) * n))
                       [0 ..< length L]) L"
 
 definition
   zipE3 :: "'a list \<Rightarrow> 'b :: enum_alt \<Rightarrow> ('a \<times> 'b) list"
 where
   "zipE3 L x \<equiv> zip L (map toEnumAlt [fromEnumAlt x ..< fromEnumAlt x + length L])"
 
 definition
   zipE4 :: "'a list \<Rightarrow> 'b :: enum_alt \<Rightarrow> 'b \<Rightarrow> ('a \<times> 'b) list"
 where
   "zipE4 L x xn \<equiv> zip L (map (\<lambda>n. toEnumAlt (fromEnumAlt x + (fromEnumAlt xn - fromEnumAlt x) * n))
                          [0 ..< length L])"
 
 
 lemma to_from_enum_alt[simp]:
   "toEnumAlt (fromEnumAlt x) = (x :: 'a :: enumeration_alt)"
 proof -
   have rl: "\<And>a b. a = Some b \<Longrightarrow> the a = b" by simp
   show ?thesis
     unfolding fromEnumAlt_def toEnumAlt_def
     by (rule rl, rule theI') (metis enum_alt_inj enum_alt_surj_2 not_None_eq)
 qed
 
 lemma upto_enum_triv [simp]: "[x .e. x] = [x]"
   unfolding upto_enum_def by simp
 
 lemma toEnum_eq_to_fromEnum_eq:
   fixes v :: "'a :: enum"
   shows "n \<le> fromEnum (maxBound :: 'a) \<Longrightarrow> (toEnum n = v) = (n = fromEnum v)"
   by auto
 
 lemma le_imp_diff_le:
   "(j::nat) \<le> k \<Longrightarrow> j - n \<le> k"
   by simp
 
 lemma fromEnum_upto_nth:
   fixes start :: "'a :: enumeration_both"
   assumes "n < length [start .e. end]"
   shows "fromEnum ([start .e. end] ! n) = fromEnum start + n"
 proof -
   have less_sub: "\<And>m k m' n. \<lbrakk> (n::nat) < m - k ; m \<le> m' \<rbrakk> \<Longrightarrow> n < m' - k" by fastforce
   note upt_Suc[simp del]
   show ?thesis using assms
   by (fastforce simp: upto_enum_red
                 dest: less_sub[where m'="Suc (fromEnum maxBound)"] intro: maxBound_is_bound)
 qed
 
 lemma length_upto_enum_le_maxBound:
   fixes start :: "'a :: enumeration_both"
   shows "length [start .e. end] \<le> Suc (fromEnum (maxBound :: 'a))"
   apply (clarsimp simp add: upto_enum_red split: if_splits)
   apply (rule le_imp_diff_le[OF maxBound_is_bound[of "end"]])
   done
 
 lemma less_length_upto_enum_maxBoundD:
   fixes start :: "'a :: enumeration_both"
   assumes "n < length [start .e. end]"
   shows "n \<le> fromEnum (maxBound :: 'a)"
   using assms
   by (simp add: upto_enum_red less_Suc_eq_le
                 le_trans[OF _ le_imp_diff_le[OF maxBound_is_bound[of "end"]]]
            split: if_splits)
 
 lemma fromEnum_eq_iff:
   "(fromEnum e = fromEnum f) = (e = f)"
 proof -
   have a: "e \<in> set enum" by auto
   have b: "f \<in> set enum" by auto
   from nth_the_index[OF a] nth_the_index[OF b] show ?thesis unfolding fromEnum_def by metis
 qed
 
 lemma maxBound_is_bound':
   "i = fromEnum (e::('a::enum)) \<Longrightarrow> i \<le> fromEnum (maxBound::('a::enum))"
   by clarsimp
 
 end
diff --git a/thys/Word_Lib/HOL_Lemmas.thy b/thys/Word_Lib/HOL_Lemmas.thy
--- a/thys/Word_Lib/HOL_Lemmas.thy
+++ b/thys/Word_Lib/HOL_Lemmas.thy
@@ -1,233 +1,229 @@
 (*
- * Copyright 2014, NICTA
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
- * This software may be distributed and modified according to the terms of
- * the BSD 2-Clause license. Note that NO WARRANTY is provided.
- * See "LICENSE_BSD2.txt" for details.
- *
- * @TAG(NICTA_BSD)
+ * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "Generic Lemmas used in the Word Library"
 
 theory HOL_Lemmas
 imports Main
 begin
 
 definition
   strict_part_mono :: "'a set \<Rightarrow> ('a :: order \<Rightarrow> 'b :: order) \<Rightarrow> bool" where
  "strict_part_mono S f \<equiv> \<forall>A\<in>S. \<forall>B\<in>S. A < B \<longrightarrow> f A < f B"
 
 lemma strict_part_mono_by_steps:
   "strict_part_mono {..n :: nat} f = (n \<noteq> 0 \<longrightarrow> f (n - 1) < f n \<and> strict_part_mono {.. n - 1} f)"
   apply (cases n; simp add: strict_part_mono_def)
   apply (safe; clarsimp)
   apply (case_tac "B = Suc nat"; simp)
   apply (case_tac "A = nat"; clarsimp)
   apply (erule order_less_trans [rotated])
   apply simp
   done
 
 lemma strict_part_mono_singleton[simp]:
   "strict_part_mono {x} f"
   by (simp add: strict_part_mono_def)
 
 lemma strict_part_mono_lt:
   "\<lbrakk> x < f 0; strict_part_mono {.. n :: nat} f \<rbrakk> \<Longrightarrow> \<forall>m \<le> n. x < f m"
   by (metis atMost_iff le_0_eq le_cases neq0_conv order.strict_trans strict_part_mono_def)
 
 lemma strict_part_mono_reverseE:
   "\<lbrakk> f n \<le> f m; strict_part_mono {.. N :: nat} f; n \<le> N \<rbrakk> \<Longrightarrow> n \<le> m"
   by (rule ccontr) (fastforce simp: linorder_not_le strict_part_mono_def)
 
 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)
 
 lemma takeWhile_replicate_id:
   "f x \<Longrightarrow> takeWhile f (replicate len x) = replicate len x"
   by (simp add: takeWhile_replicate)
 
 lemma power_sub:
   fixes a :: nat
   assumes lt: "n \<le> m"
   and     av: "0 < a"
   shows "a ^ (m - n) = a ^ m div a ^ n"
 proof (subst nat_mult_eq_cancel1 [symmetric])
   show "(0::nat) < a ^ n" using av by simp
 next
   from lt obtain q where mv: "n + q = m"
     by (auto simp: le_iff_add)
 
   have "a ^ n * (a ^ m div a ^ n) = a ^ m"
   proof (subst mult.commute)
     have "a ^ m = (a ^ m div a ^ n) * a ^ n + a ^ m mod a ^ n"
       by (rule  div_mult_mod_eq [symmetric])
 
     moreover have "a ^ m mod a ^ n = 0"
       by (subst mod_eq_0_iff_dvd, subst dvd_def, rule exI [where x = "a ^ q"],
       (subst power_add [symmetric] mv)+, rule refl)
 
     ultimately show "(a ^ m div a ^ n) * a ^ n = a ^ m" by simp
   qed
 
   then show "a ^ n * a ^ (m - n) = a ^ n * (a ^ m div a ^ n)" using lt
     by (simp add: power_add [symmetric])
 qed
 
 
 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 nat_less_power_trans:
   fixes n :: nat
   assumes nv: "n < 2 ^ (m - k)"
   and     kv: "k \<le> m"
   shows "2 ^ k * n < 2 ^ m"
 proof (rule order_less_le_trans)
   show "2 ^ k * n < 2 ^ k * 2 ^ (m - k)"
     by (rule mult_less_mono2 [OF nv zero_less_power]) simp
 
   show "(2::nat) ^ k * 2 ^ (m - k) \<le> 2 ^ m" using nv kv
     by (subst power_add [symmetric]) simp
 qed
 
 lemma nat_le_power_trans:
   fixes n :: nat
   shows "\<lbrakk>n \<le> 2 ^ (m - k); k \<le> m\<rbrakk> \<Longrightarrow> 2 ^ k * n \<le> 2 ^ m"
   by (metis le_add_diff_inverse mult_le_mono2 semiring_normalization_rules(26))
 
 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 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 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 if_f:
   "(if a then f b else f c) = f (if a then b else c)"
   by simp
 
 lemma upt_add_eq_append':
   assumes "i \<le> j" and "j \<le> k"
   shows "[i..<k] = [i..<j] @ [j..<k]"
   using assms le_Suc_ex upt_add_eq_append by blast
 
 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_Suc_nth:
   "n < length xs \<Longrightarrow> drop n xs = xs!n # drop (Suc n) xs"
   by (simp add: Cons_nth_drop_Suc)
 
 lemma n_less_equal_power_2 [simp]:
   "n < 2 ^ n"
   by (induct n; simp)
 
 lemma nat_min_simps [simp]:
   "(a::nat) \<le> b \<Longrightarrow> min b a = a"
   "a \<le> b \<Longrightarrow> min a b = a"
   by auto
 
 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 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 length_takeWhile_less:
   "\<exists>x\<in>set xs. \<not> P x \<Longrightarrow> length (takeWhile P xs) < length xs"
   by (induct xs) (auto split: if_splits)
 
 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 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 mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
   apply (cut_tac m = q and n = c in mod_less_divisor)
   apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
   apply (erule_tac P = "%x. lhs < rhs x" for lhs rhs in ssubst)
   apply (simp add: add_mult_distrib2)
   done
 
 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
 
 end
diff --git a/thys/Word_Lib/Hex_Words.thy b/thys/Word_Lib/Hex_Words.thy
--- a/thys/Word_Lib/Hex_Words.thy
+++ b/thys/Word_Lib/Hex_Words.thy
@@ -1,52 +1,48 @@
 (*
- * Copyright 2014, NICTA
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
- * This software may be distributed and modified according to the terms of
- * the BSD 2-Clause license. Note that NO WARRANTY is provided.
- * See "LICENSE_BSD2.txt" for details.
- *
- * @TAG(NICTA_BSD)
+ * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "Print Words in Hex"
 
 theory Hex_Words
 imports
   "HOL-Word.Word"
 begin
 
 text \<open>Print words in hex.\<close>
 
 (* mostly clagged from Num.thy *)
 typed_print_translation  \<open>
 let
   fun dest_num (Const (@{const_syntax Num.Bit0}, _) $ n) = 2 * dest_num n
     | dest_num (Const (@{const_syntax Num.Bit1}, _) $ n) = 2 * dest_num n + 1
     | dest_num (Const (@{const_syntax Num.One}, _)) = 1;
 
   fun dest_bin_hex_str tm =
   let
     val num = dest_num tm;
     val pre = if num < 10 then "" else "0x"
   in
     pre ^ (Int.fmt StringCvt.HEX num)
   end;
 
   fun num_tr' sign ctxt T [n] =
     let
       val k = dest_bin_hex_str n;
       val t' = Syntax.const @{syntax_const "_Numeral"} $
         Syntax.free (sign ^ k);
     in
       case T of
         Type (@{type_name fun}, [_, T' as Type("Word.word",_)]) =>
           if not (Config.get ctxt show_types) andalso can Term.dest_Type T'
           then t'
           else Syntax.const @{syntax_const "_constrain"} $ t' $
                             Syntax_Phases.term_of_typ ctxt T'
       | T' => if T' = dummyT then t' else raise Match
     end;
 in [(@{const_syntax numeral}, num_tr' "")] end
 \<close>
 
 end
diff --git a/thys/Word_Lib/More_Divides.thy b/thys/Word_Lib/More_Divides.thy
--- a/thys/Word_Lib/More_Divides.thy
+++ b/thys/Word_Lib/More_Divides.thy
@@ -1,58 +1,54 @@
 (*
- * Copyright 2014, NICTA
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
- * This software may be distributed and modified according to the terms of
- * the BSD 2-Clause license. Note that NO WARRANTY is provided.
- * See "LICENSE_BSD2.txt" for details.
- *
- * @TAG(NICTA_BSD)
+ * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "More Lemmas on Division"
 
 theory More_Divides
 imports Main
 begin
 
 lemma div_mult_le:
   "(a::nat) div b * b \<le> a"
   by (fact div_times_less_eq_dividend)
 
 lemma diff_mod_le:
   "\<lbrakk> (a::nat) < d; b dvd d \<rbrakk> \<Longrightarrow> a - a mod b \<le> d - b"
   apply(subst minus_mod_eq_mult_div)
   apply(clarsimp simp: dvd_def)
   apply(case_tac "b = 0")
    apply simp
   apply(subgoal_tac "a div b \<le> k - 1")
    prefer 2
    apply(subgoal_tac "a div b < k")
     apply(simp add: less_Suc_eq_le [symmetric])
    apply(subgoal_tac "b * (a div b) < b * ((b * k) div b)")
     apply clarsimp
    apply(subst div_mult_self1_is_m)
     apply arith
    apply(rule le_less_trans)
     apply simp
     apply(subst mult.commute)
     apply(rule div_mult_le)
    apply assumption
   apply clarsimp
   apply(subgoal_tac "b * (a div b) \<le> b * (k - 1)")
    apply(erule le_trans)
    apply(simp add: diff_mult_distrib2)
   apply simp
   done
 
 lemma int_div_same_is_1 [simp]:
   "0 < a \<Longrightarrow> ((a :: int) div b = a) = (b = 1)"
   by (metis div_by_1 abs_ge_zero abs_of_pos int_div_less_self neq_iff
             nonneg1_imp_zdiv_pos_iff zabs_less_one_iff)
 
 declare div_eq_dividend_iff [simp]
 
 lemma int_div_minus_is_minus1 [simp]:
     "a < 0 \<Longrightarrow> ((a :: int) div b = -a) = (b = -1)"
   by (metis div_minus_right equation_minus_iff int_div_same_is_1 neg_0_less_iff_less)
 
 end
diff --git a/thys/Word_Lib/Norm_Words.thy b/thys/Word_Lib/Norm_Words.thy
--- a/thys/Word_Lib/Norm_Words.thy
+++ b/thys/Word_Lib/Norm_Words.thy
@@ -1,116 +1,112 @@
 (*
- * Copyright 2014, NICTA
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
- * This software may be distributed and modified according to the terms of
- * the BSD 2-Clause license. Note that NO WARRANTY is provided.
- * See "LICENSE_BSD2.txt" for details.
- *
- * @TAG(NICTA_BSD)
+ * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "Normalising Word Numerals"
 
 theory Norm_Words
 imports "Signed_Words"
 begin
 
 text \<open>
   Normalise word numerals, including negative ones apart from @{term "-1"}, to the
   interval \<open>[0..2^len_of 'a)\<close>. Only for concrete word lengths.
 \<close>
 
 lemma neg_num_bintr:
   "(- numeral x :: 'a::len word) =
   word_of_int (bintrunc (LENGTH('a)) (-numeral x))"
   by (simp only: word_ubin.Abs_norm word_neg_numeral_alt)
 
 ML \<open>
   fun is_refl (Const (@{const_name Pure.eq}, _) $ x $ y) = (x = y)
     | is_refl _ = false;
 
   fun signed_dest_wordT (Type (@{type_name word}, [Type (@{type_name signed}, [T])])) = Word_Lib.dest_binT T
     | signed_dest_wordT T = Word_Lib.dest_wordT T
 
   fun typ_size_of t = signed_dest_wordT (type_of (Thm.term_of t));
 
   fun num_len (Const (@{const_name Num.Bit0}, _) $ n) = num_len n + 1
     | num_len (Const (@{const_name Num.Bit1}, _) $ n) = num_len n + 1
     | num_len (Const (@{const_name Num.One}, _)) = 1
     | num_len (Const (@{const_name numeral}, _) $ t) = num_len t
     | num_len (Const (@{const_name uminus}, _) $ t) = num_len t
     | num_len t = raise TERM ("num_len", [t])
 
   fun unsigned_norm is_neg _ ctxt ct =
   (if is_neg orelse num_len (Thm.term_of ct) > typ_size_of ct then let
       val btr = if is_neg
                 then @{thm neg_num_bintr} else @{thm num_abs_bintr}
       val th = [Thm.reflexive ct, mk_eq btr] MRS transitive_thm
 
       (* will work in context of theory Word as well *)
       val ss = simpset_of (@{context} addsimps @{thms bintrunc_numeral})
       val cnv = simplify (put_simpset ss ctxt) th
     in if is_refl (Thm.prop_of cnv) then NONE else SOME cnv end
     else NONE)
   handle TERM ("num_len", _) => NONE
        | TYPE ("dest_binT", _, _) => NONE
 \<close>
 
 simproc_setup
   unsigned_norm ("numeral n::'a::len word") = \<open>unsigned_norm false\<close>
 
 simproc_setup
   unsigned_norm_neg0 ("-numeral (num.Bit0 num)::'a::len word") = \<open>unsigned_norm true\<close>
 
 simproc_setup
   unsigned_norm_neg1 ("-numeral (num.Bit1 num)::'a::len word") = \<open>unsigned_norm true\<close>
 
 declare word_pow_0 [simp]
 
 lemma minus_one_norm:
   "(-1 :: 'a :: len word) = of_nat (2 ^ LENGTH('a) - 1)"
   by (simp add:of_nat_diff)
 
 lemmas minus_one_norm_num =
   minus_one_norm [where 'a="'b::len bit0"] minus_one_norm [where 'a="'b::len0 bit1"]
 
 lemma "f (7 :: 2 word) = f 3" by simp
 
 lemma "f 7 = f (3 :: 2 word)" by simp
 
 lemma "f (-2) = f (21 + 1 :: 3 word)" by simp
 
 lemma "f (-2) = f (13 + 1 :: 'a::len word)"
   apply simp (* does not touch generic word length *)
   oops
 
 lemma "f (-2) = f (0xFFFFFFFE :: 32 word)" by simp
 
 lemma "(-1 :: 2 word) = 3" by simp
 
 lemma "f (-2) = f (0xFFFFFFFE :: 32 signed word)" by simp
 
 
 text \<open>
   We leave @{term "-1"} untouched by default, because it is often useful
   and its normal form can be large.
   To include it in the normalisation, add @{thm [source] minus_one_norm_num}.
   The additional normalisation is restricted to concrete numeral word lengths,
   like the rest.
 \<close>
 context
   notes minus_one_norm_num [simp]
 begin
 
 lemma "f (-1) = f (15 :: 4 word)" by simp
 
 lemma "f (-1) = f (7 :: 3 word)" by simp
 
 lemma "f (-1) = f (0xFFFF :: 16 word)" by simp
 
 lemma "f (-1) = f (0xFFFF + 1 :: 'a::len word)"
   apply simp (* does not touch generic -1 *)
   oops
 
 end
 
 end
diff --git a/thys/Word_Lib/Signed_Words.thy b/thys/Word_Lib/Signed_Words.thy
--- a/thys/Word_Lib/Signed_Words.thy
+++ b/thys/Word_Lib/Signed_Words.thy
@@ -1,44 +1,40 @@
 (*
- * Copyright 2014, NICTA
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
- * This software may be distributed and modified according to the terms of
- * the BSD 2-Clause license. Note that NO WARRANTY is provided.
- * See "LICENSE_BSD2.txt" for details.
- *
- * @TAG(NICTA_BSD)
+ * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "Signed Words"
 
 theory Signed_Words
 imports "HOL-Word.Word"
 begin
 
 text \<open>Signed words as separate (isomorphic) word length class. Useful for tagging words in C.\<close>
 
 typedef ('a::len0) signed = "UNIV :: 'a set" ..
 
 lemma card_signed [simp]: "CARD (('a::len0) signed) = CARD('a)"
   unfolding type_definition.card [OF type_definition_signed]
   by simp
 
 instantiation signed :: (len0) len0
 begin
 
 definition
   len_signed [simp]: "len_of (x::'a::len0 signed itself) = LENGTH('a)"
 
 instance ..
 
 end
 
 instance signed :: (len) len
   by (intro_classes, simp)
 
 type_synonym 'a sword = "'a signed word"
 type_synonym  sword8 =  "8 sword"
 type_synonym sword16 = "16 sword"
 type_synonym sword32 = "32 sword"
 type_synonym sword64 = "64 sword"
 
 end
diff --git a/thys/Word_Lib/WordBitwise_Signed.thy b/thys/Word_Lib/WordBitwise_Signed.thy
--- a/thys/Word_Lib/WordBitwise_Signed.thy
+++ b/thys/Word_Lib/WordBitwise_Signed.thy
@@ -1,33 +1,29 @@
 (*
- * Copyright 2014, NICTA
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
- * This software may be distributed and modified according to the terms of
- * the BSD 2-Clause license. Note that NO WARRANTY is provided.
- * See "LICENSE_BSD2.txt" for details.
- *
- * @TAG(NICTA_BSD)
+ * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "Bitwise tactic for Signed Words"
 
 theory WordBitwise_Signed
 imports
   "HOL-Word.More_Word"
   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"
 
 end
diff --git a/thys/Word_Lib/Word_Enum.thy b/thys/Word_Lib/Word_Enum.thy
--- a/thys/Word_Lib/Word_Enum.thy
+++ b/thys/Word_Lib/Word_Enum.thy
@@ -1,99 +1,95 @@
 (*
- * Copyright 2014, NICTA
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
- * This software may be distributed and modified according to the terms of
- * the BSD 2-Clause license. Note that NO WARRANTY is provided.
- * See "LICENSE_BSD2.txt" for details.
- *
- * @TAG(NICTA_BSD)
+ * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 (* Author: Thomas Sewell *)
 
 section "Enumeration Instances for Words"
 
 theory Word_Enum
 imports Enumeration Word_Lib
 begin
 
 instantiation word :: (len) enum
 begin
 
 definition
   "(enum_class.enum :: ('a :: len) word list) \<equiv> map of_nat [0 ..< 2 ^ LENGTH('a)]"
 
 definition
   "enum_class.enum_all (P :: ('a :: len) word \<Rightarrow> bool) \<longleftrightarrow> Ball UNIV P"
 
 definition
   "enum_class.enum_ex (P :: ('a :: len) word \<Rightarrow> bool) \<longleftrightarrow> Bex UNIV P"
 
 instance
   apply (intro_classes)
      apply (force simp: enum_word_def)
     apply (simp add: distinct_map enum_word_def)
     apply (rule subset_inj_on, rule word_unat.Abs_inj_on)
     apply (clarsimp simp add: unats_def)
    apply (simp add: enum_all_word_def)
   apply (simp add: enum_ex_word_def)
   done
 
 end
 
 lemma fromEnum_unat[simp]: "fromEnum (x :: 'a::len word) = unat x"
 proof -
   have "enum ! the_index enum x = x" by (auto intro: nth_the_index)
   moreover
   have "the_index enum x < length (enum::'a::len word list)" by (auto intro: the_index_bounded)
   moreover
   { fix y assume "of_nat y = x"
     moreover assume "y < 2 ^ LENGTH('a)"
     ultimately have "y = unat x" using of_nat_inverse by fastforce
   }
   ultimately
   show ?thesis by (simp add: fromEnum_def enum_word_def)
 qed
 
-lemma length_word_enum: "length (enum :: ('a :: len) word list) = 2 ^ LENGTH('a)"
+lemma length_word_enum: "length (enum :: 'a :: len word list) = 2 ^ LENGTH('a)"
   by (simp add: enum_word_def)
 
-lemma toEnum_of_nat[simp]: "n < 2 ^ LENGTH('a) \<Longrightarrow> ((toEnum n) :: ('a :: len) word) = of_nat n"
+lemma toEnum_of_nat[simp]: "n < 2 ^ LENGTH('a) \<Longrightarrow> (toEnum n :: 'a :: len word) = of_nat n"
   by (simp add: toEnum_def length_word_enum enum_word_def)
 
 declare of_nat_diff [simp]
 
 instantiation word :: (len) enumeration_both
 begin
 
 definition
   enum_alt_word_def: "enum_alt \<equiv> alt_from_ord (enum :: ('a :: len) word list)"
 
 instance
   by (intro_classes, simp add: enum_alt_word_def)
 
 end
 
 definition
   upto_enum_step :: "('a :: len) word \<Rightarrow> 'a word \<Rightarrow> 'a word \<Rightarrow> 'a word list" ("[_ , _ .e. _]")
 where
   "upto_enum_step a b c \<equiv>
       if c < a then [] else map (\<lambda>x. a + x * (b - a)) [0 .e. (c - a) div (b - a)]"
   (* in the wraparound case, bad things happen. *)
 
 lemma maxBound_word:
   "(maxBound::'a::len word) = -1"
   by (simp add: maxBound_def enum_word_def last_map)
 
 lemma minBound_word:
   "(minBound::'a::len word) = 0"
   by (simp add: minBound_def enum_word_def upt_conv_Cons)
 
 lemma maxBound_max_word:
   "(maxBound::'a::len word) = max_word"
   by (simp add: maxBound_word max_word_minus [symmetric])
 
 lemma leq_maxBound [simp]:
   "(x::'a::len word) \<le> maxBound"
   by (simp add: maxBound_max_word)
 
 end
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
@@ -1,188 +1,184 @@
 (*
- * Copyright 2019, Data61, CSIRO
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
- * This software may be distributed and modified according to the terms of
- * the BSD 2-Clause license. Note that NO WARRANTY is provided.
- * See "LICENSE_BSD2.txt" for details.
- *
- * @TAG(DATA61_BSD)
+ * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "Solving Word Equalities"
 
 theory Word_EqI
   imports
     Word_Next
     "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.
 \<close>
 
 lemma word_or_zero:
   "(a || b = 0) = (a = 0 \<and> b = 0)"
   by (safe; rule word_eqI, drule_tac x=n in word_eqD, simp)
 
 lemma test_bit_over:
   "n \<ge> size (x::'a::len0 word) \<Longrightarrow> (x !! n) = False"
   by (simp add: test_bit_bl word_size)
 
 lemma neg_mask_test_bit:
   "(~~(mask n) :: 'a :: len word) !! m = (n \<le> m \<and> m < LENGTH('a))"
   by (metis not_le nth_mask test_bit_bin word_ops_nth_size word_size)
 
 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 word_power_increasing:
   assumes x: "2 ^ x < (2 ^ y::'a::len word)" "x < LENGTH('a::len)" "y < LENGTH('a::len)"
   shows "x < y" using x
   apply (induct x arbitrary: y)
    apply (case_tac y; simp)
   apply (case_tac y; clarsimp simp: word_2p_mult_inc)
   apply (subst (asm) power_Suc [symmetric])
   apply (subst (asm) p2_eq_0)
   apply simp
   done
 
 lemma upper_bits_unset_is_l2p:
   "n < LENGTH('a) \<Longrightarrow>
    (\<forall>n' \<ge> n. n' < LENGTH('a) \<longrightarrow> \<not> p !! n') = (p < 2 ^ n)" for p :: "'a :: len word"
   apply (cases "Suc 0 < LENGTH('a)")
    prefer 2
    apply (subgoal_tac "LENGTH('a) = 1", auto simp: word_eq_iff)[1]
   apply (rule iffI)
    apply (subst mask_eq_iff_w2p [symmetric])
     apply (clarsimp simp: word_size)
    apply (rule word_eqI, rename_tac n')
    apply (case_tac "n' < n"; simp add: word_size)
   by (meson bang_is_le le_less_trans not_le word_power_increasing)
 
 lemma less_2p_is_upper_bits_unset:
   "p < 2 ^ n \<longleftrightarrow> n < LENGTH('a) \<and> (\<forall>n' \<ge> n. n' < LENGTH('a) \<longrightarrow> \<not> 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 word_le_minus_one_leq:
   "x < y \<Longrightarrow> x \<le> y - 1" for x :: "'a :: len word"
   by (simp add: plus_one_helper)
 
 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 not_greatest_aligned:
   "\<lbrakk> x < y; is_aligned x n; is_aligned y n \<rbrakk> \<Longrightarrow> x + 2 ^ n \<noteq> 0"
   by (metis NOT_mask add_diff_cancel_right' diff_0 is_aligned_neg_mask_eq not_le word_and_le1)
 
 lemma neg_mask_mono_le:
   "x \<le> y \<Longrightarrow> x && ~~(mask n) \<le> y && ~~(mask n)" for x :: "'a :: len word"
 proof (rule ccontr, simp add: linorder_not_le, cases "n < LENGTH('a)")
   case False
   then show "y && ~~(mask n) < x && ~~(mask n) \<Longrightarrow> False"
     by (simp add: mask_def linorder_not_less power_overflow)
 next
   case True
   assume a: "x \<le> y" and b: "y && ~~(mask n) < x && ~~(mask n)"
   have word_bits: "n < LENGTH('a)" by fact
   have "y \<le> (y && ~~(mask n)) + (y && mask n)"
     by (simp add: word_plus_and_or_coroll2 add.commute)
   also have "\<dots> \<le> (y && ~~(mask n)) + 2 ^ n"
     apply (rule word_plus_mono_right)
      apply (rule order_less_imp_le, rule and_mask_less_size)
      apply (simp add: word_size word_bits)
     apply (rule is_aligned_no_overflow'', simp add: is_aligned_neg_mask word_bits)
     apply (rule not_greatest_aligned, rule b; simp add: is_aligned_neg_mask)
     done
   also have "\<dots> \<le> x && ~~(mask n)"
     using b
     apply (subst add.commute)
     apply (rule le_plus)
      apply (rule aligned_at_least_t2n_diff; simp add: is_aligned_neg_mask)
     apply (rule ccontr, simp add: linorder_not_le)
     apply (drule aligned_small_is_0[rotated]; simp add: is_aligned_neg_mask)
     done
   also have "\<dots> \<le> x" by (rule word_and_le2)
   also have "x \<le> y" by fact
   finally
   show "False" using b by simp
 qed
 
 lemma and_neg_mask_eq_iff_not_mask_le:
   "w && ~~(mask n) = ~~(mask n) \<longleftrightarrow> ~~(mask n) \<le> w"
   by (metis eq_iff neg_mask_mono_le word_and_le1 word_and_le2 word_bw_same(1))
 
 lemma le_mask_high_bits:
   "w \<le> mask n \<longleftrightarrow> (\<forall>i \<in> {n ..< size w}. \<not> w !! i)"
   by (auto simp: word_size and_mask_eq_iff_le_mask[symmetric] word_eq_iff)
 
 lemma neg_mask_le_high_bits:
   "~~(mask n) \<le> w \<longleftrightarrow> (\<forall>i \<in> {n ..< size w}. w !! i)"
   by (auto simp: word_size and_neg_mask_eq_iff_not_mask_le[symmetric] word_eq_iff neg_mask_test_bit)
 
 lemma test_bit_conj_lt:
   "(x !! m \<and> m < LENGTH('a)) = x !! m" for x :: "'a :: len word"
   using test_bit_bin by blast
 
 lemma neg_test_bit:
   "(~~ x) !! n = (\<not> 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)
 
 named_theorems word_eqI_simps
 
 lemmas [word_eqI_simps] =
   word_ops_nth_size
   word_size
   word_or_zero
   neg_mask_test_bit
   nth_ucast
   is_aligned_nth
   nth_w2p nth_shiftl
   nth_shiftr
   less_2p_is_upper_bits_unset
   le_mask_high_bits
   neg_mask_le_high_bits
   bang_eq
   neg_test_bit
   is_up
   is_down
 
 lemmas word_eqI_rule = word_eqI[rule_format]
 
 lemma test_bit_lenD:
   "x !! n \<Longrightarrow> n < LENGTH('a) \<and> x !! n" for x :: "'a :: len word"
   by (fastforce dest: test_bit_size simp: word_size)
 
 method word_eqI uses simp simp_del split split_del cong flip =
   ((* reduce conclusion to test_bit: *)
    rule word_eqI_rule,
    (* 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: *)
    (clarsimp simp: word_eqI_simps simp 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): *)
    (clarsimp simp: word_eqI_simps 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)?)
 
 method word_eqI_solve uses simp simp_del split split_del cong flip =
   solves \<open>word_eqI simp: simp simp_del: simp_del split: split split_del: split_del
                    cong: cong simp flip: flip;
           (fastforce dest: test_bit_size simp: word_eqI_simps 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
@@ -1,6241 +1,6257 @@
 (*
- * Copyright 2014, NICTA
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
- * This software may be distributed and modified according to the terms of
- * the BSD 2-Clause license. Note that NO WARRANTY is provided.
- * See "LICENSE_BSD2.txt" for details.
- *
- * @TAG(NICTA_BSD)
+ * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "Lemmas with Generic Word Length"
 
 theory Word_Lemmas
   imports
     Complex_Main
     Word_EqI
     Word_Enum
     "HOL-Library.Sublist"
 begin
 
 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 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 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 (subst (asm) bang_eq) (fastforce simp: nth_ucast word_size intro: word_eqI)
 
 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"
   by (fastforce intro: word_eqI simp: bang_eq nth_ucast word_size)
 
 lemma ucast_0_I:
   "x = 0 \<Longrightarrow> ucast x = 0"
   by simp
 
 text \<open>right-padding a word to a certain length\<close>
 
 definition
   "bl_pad_to bl sz \<equiv> bl @ (replicate (sz - length bl) False)"
 
 lemma bl_pad_to_length:
   assumes lbl: "length bl \<le> sz"
   shows   "length (bl_pad_to bl sz) = sz"
   using lbl by (simp add: bl_pad_to_def)
 
 lemma bl_pad_to_prefix:
   "prefix bl (bl_pad_to bl sz)"
   by (simp add: bl_pad_to_def)
 
 lemma same_length_is_parallel:
   assumes len: "\<forall>y \<in> set as. length y = x"
   shows  "\<forall>x \<in> set as. \<forall>y \<in> set as - {x}. x \<parallel> y"
 proof (rule, rule)
   fix x y
   assume xi: "x \<in> set as" and yi: "y \<in> set as - {x}"
   from len obtain q where len': "\<forall>y \<in> set as. length y = q" ..
 
   show "x \<parallel> y"
   proof (rule not_equal_is_parallel)
     from xi yi show "x \<noteq> y" by auto
     from xi yi len' show "length x = length y" by (auto dest: bspec)
   qed
 qed
 
 text \<open>Lemmas about words\<close>
 
 lemmas and_bang = word_and_nth
 
 lemma of_drop_to_bl:
   "of_bl (drop n (to_bl x)) = (x && mask (size x - n))"
   by (simp add: of_bl_drop word_size_bl)
 
 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 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_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_unat.Rep_eqD)
   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 set_enum_word8_def:
   "(set enum::word8 set) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
                             20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
                             37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
                             54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70,
                             71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87,
                             88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103,
                             104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117,
                             118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131,
                             132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145,
                             146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159,
                             160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173,
                             174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187,
                             188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201,
                             202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215,
                             216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229,
                             230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243,
                             244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255}"
   by eval
 
 lemma set_strip_insert: "\<lbrakk> x \<in> insert a S; x \<noteq> a \<rbrakk> \<Longrightarrow> x \<in> S"
   by simp
 
 lemma word8_exhaust:
   fixes x :: word8
   shows "\<lbrakk>x \<noteq> 0; x \<noteq> 1; x \<noteq> 2; x \<noteq> 3; x \<noteq> 4; x \<noteq> 5; x \<noteq> 6; x \<noteq> 7; x \<noteq> 8; x \<noteq> 9; x \<noteq> 10; x \<noteq> 11; x \<noteq>
           12; x \<noteq> 13; x \<noteq> 14; x \<noteq> 15; x \<noteq> 16; x \<noteq> 17; x \<noteq> 18; x \<noteq> 19; x \<noteq> 20; x \<noteq> 21; x \<noteq> 22; x \<noteq>
           23; x \<noteq> 24; x \<noteq> 25; x \<noteq> 26; x \<noteq> 27; x \<noteq> 28; x \<noteq> 29; x \<noteq> 30; x \<noteq> 31; x \<noteq> 32; x \<noteq> 33; x \<noteq>
           34; x \<noteq> 35; x \<noteq> 36; x \<noteq> 37; x \<noteq> 38; x \<noteq> 39; x \<noteq> 40; x \<noteq> 41; x \<noteq> 42; x \<noteq> 43; x \<noteq> 44; x \<noteq>
           45; x \<noteq> 46; x \<noteq> 47; x \<noteq> 48; x \<noteq> 49; x \<noteq> 50; x \<noteq> 51; x \<noteq> 52; x \<noteq> 53; x \<noteq> 54; x \<noteq> 55; x \<noteq>
           56; x \<noteq> 57; x \<noteq> 58; x \<noteq> 59; x \<noteq> 60; x \<noteq> 61; x \<noteq> 62; x \<noteq> 63; x \<noteq> 64; x \<noteq> 65; x \<noteq> 66; x \<noteq>
           67; x \<noteq> 68; x \<noteq> 69; x \<noteq> 70; x \<noteq> 71; x \<noteq> 72; x \<noteq> 73; x \<noteq> 74; x \<noteq> 75; x \<noteq> 76; x \<noteq> 77; x \<noteq>
           78; x \<noteq> 79; x \<noteq> 80; x \<noteq> 81; x \<noteq> 82; x \<noteq> 83; x \<noteq> 84; x \<noteq> 85; x \<noteq> 86; x \<noteq> 87; x \<noteq> 88; x \<noteq>
           89; x \<noteq> 90; x \<noteq> 91; x \<noteq> 92; x \<noteq> 93; x \<noteq> 94; x \<noteq> 95; x \<noteq> 96; x \<noteq> 97; x \<noteq> 98; x \<noteq> 99; x \<noteq>
           100; x \<noteq> 101; x \<noteq> 102; x \<noteq> 103; x \<noteq> 104; x \<noteq> 105; x \<noteq> 106; x \<noteq> 107; x \<noteq> 108; x \<noteq> 109; x \<noteq>
           110; x \<noteq> 111; x \<noteq> 112; x \<noteq> 113; x \<noteq> 114; x \<noteq> 115; x \<noteq> 116; x \<noteq> 117; x \<noteq> 118; x \<noteq> 119; x \<noteq>
           120; x \<noteq> 121; x \<noteq> 122; x \<noteq> 123; x \<noteq> 124; x \<noteq> 125; x \<noteq> 126; x \<noteq> 127; x \<noteq> 128; x \<noteq> 129; x \<noteq>
           130; x \<noteq> 131; x \<noteq> 132; x \<noteq> 133; x \<noteq> 134; x \<noteq> 135; x \<noteq> 136; x \<noteq> 137; x \<noteq> 138; x \<noteq> 139; x \<noteq>
           140; x \<noteq> 141; x \<noteq> 142; x \<noteq> 143; x \<noteq> 144; x \<noteq> 145; x \<noteq> 146; x \<noteq> 147; x \<noteq> 148; x \<noteq> 149; x \<noteq>
           150; x \<noteq> 151; x \<noteq> 152; x \<noteq> 153; x \<noteq> 154; x \<noteq> 155; x \<noteq> 156; x \<noteq> 157; x \<noteq> 158; x \<noteq> 159; x \<noteq>
           160; x \<noteq> 161; x \<noteq> 162; x \<noteq> 163; x \<noteq> 164; x \<noteq> 165; x \<noteq> 166; x \<noteq> 167; x \<noteq> 168; x \<noteq> 169; x \<noteq>
           170; x \<noteq> 171; x \<noteq> 172; x \<noteq> 173; x \<noteq> 174; x \<noteq> 175; x \<noteq> 176; x \<noteq> 177; x \<noteq> 178; x \<noteq> 179; x \<noteq>
           180; x \<noteq> 181; x \<noteq> 182; x \<noteq> 183; x \<noteq> 184; x \<noteq> 185; x \<noteq> 186; x \<noteq> 187; x \<noteq> 188; x \<noteq> 189; x \<noteq>
           190; x \<noteq> 191; x \<noteq> 192; x \<noteq> 193; x \<noteq> 194; x \<noteq> 195; x \<noteq> 196; x \<noteq> 197; x \<noteq> 198; x \<noteq> 199; x \<noteq>
           200; x \<noteq> 201; x \<noteq> 202; x \<noteq> 203; x \<noteq> 204; x \<noteq> 205; x \<noteq> 206; x \<noteq> 207; x \<noteq> 208; x \<noteq> 209; x \<noteq>
           210; x \<noteq> 211; x \<noteq> 212; x \<noteq> 213; x \<noteq> 214; x \<noteq> 215; x \<noteq> 216; x \<noteq> 217; x \<noteq> 218; x \<noteq> 219; x \<noteq>
           220; x \<noteq> 221; x \<noteq> 222; x \<noteq> 223; x \<noteq> 224; x \<noteq> 225; x \<noteq> 226; x \<noteq> 227; x \<noteq> 228; x \<noteq> 229; x \<noteq>
           230; x \<noteq> 231; x \<noteq> 232; x \<noteq> 233; x \<noteq> 234; x \<noteq> 235; x \<noteq> 236; x \<noteq> 237; x \<noteq> 238; x \<noteq> 239; x \<noteq>
           240; x \<noteq> 241; x \<noteq> 242; x \<noteq> 243; x \<noteq> 244; x \<noteq> 245; x \<noteq> 246; x \<noteq> 247; x \<noteq> 248; x \<noteq> 249; x \<noteq>
           250; x \<noteq> 251; x \<noteq> 252; x \<noteq> 253; x \<noteq> 254; x \<noteq> 255\<rbrakk> \<Longrightarrow> P"
   apply (subgoal_tac "x \<in> set enum", subst (asm) set_enum_word8_def)
     apply (drule set_strip_insert, assumption)+
    apply (erule emptyE)
   apply (subst enum_UNIV, rule UNIV_I)
   done
 
 lemma upto_enum_red':
   assumes lt: "1 \<le> X"
   shows "[(0::'a :: len word) .e. X - 1] =  map of_nat [0 ..< unat X]"
 proof -
   have lt': "unat X < 2 ^ LENGTH('a)"
     by (rule unat_lt2p)
 
   show ?thesis
     apply (subst upto_enum_red)
     apply (simp del: upt.simps)
     apply (subst Suc_unat_diff_1 [OF lt])
     apply (rule map_cong [OF refl])
     apply (rule toEnum_of_nat)
     apply simp
     apply (erule order_less_trans [OF _ lt'])
     done
 qed
 
 lemma upto_enum_red2:
   assumes szv: "sz < LENGTH('a :: len)"
   shows "[(0:: 'a :: len word) .e. 2 ^ sz - 1] =
   map of_nat [0 ..< 2 ^ sz]" using szv
   apply (subst unat_power_lower[OF szv, symmetric])
   apply (rule upto_enum_red')
   apply (subst word_le_nat_alt, simp)
   done
 
 lemma upto_enum_step_red:
   assumes szv: "sz < LENGTH('a)"
   and   usszv: "us \<le> sz"
   shows "[0 :: 'a :: len word , 2 ^ us .e. 2 ^ sz - 1] =
   map (\<lambda>x. of_nat x * 2 ^ us) [0 ..< 2 ^ (sz - us)]" using szv
   unfolding upto_enum_step_def
   apply (subst if_not_P)
    apply (rule leD)
    apply (subst word_le_nat_alt)
    apply (subst unat_minus_one)
     apply simp
    apply simp
   apply simp
   apply (subst upto_enum_red)
   apply (simp del: upt.simps)
   apply (subst Suc_div_unat_helper [where 'a = 'a, OF szv usszv, symmetric])
   apply clarsimp
   apply (subst toEnum_of_nat)
    apply (erule order_less_trans)
    using szv
    apply simp
   apply simp
   done
 
 lemma upto_enum_word:
   "[x .e. y] = map of_nat [unat x ..< Suc (unat y)]"
   apply (subst upto_enum_red)
   apply clarsimp
   apply (subst toEnum_of_nat)
    prefer 2
    apply (rule refl)
   apply (erule disjE, simp)
   apply clarsimp
   apply (erule order_less_trans)
   apply simp
   done
 
 lemma word_upto_Cons_eq:
   "x < y \<Longrightarrow> [x::'a::len word .e. y] = x # [x + 1 .e. y]"
   apply (subst upto_enum_red)
   apply (subst upt_conv_Cons, unat_arith)
   apply (simp only: list.map list.inject upto_enum_red to_from_enum simp_thms)
   apply (rule map_cong[OF _ refl])
   apply (rule arg_cong2[where f = "\<lambda>x y. [x ..< y]"], unat_arith)
   apply (rule refl)
   done
 
 lemma distinct_enum_upto:
   "distinct [(0 :: 'a::len word) .e. b]"
 proof -
   have "\<And>(b::'a word). [0 .e. b] = nths enum {..< Suc (fromEnum b)}"
     apply (subst upto_enum_red)
     apply (subst nths_upt_eq_take)
     apply (subst enum_word_def)
     apply (subst take_map)
     apply (subst take_upt)
      apply (simp only: add_0 fromEnum_unat)
      apply (rule order_trans [OF _ order_eq_refl])
       apply (rule Suc_leI [OF unat_lt2p])
      apply simp
     apply clarsimp
     apply (rule toEnum_of_nat)
     apply (erule order_less_trans [OF _ unat_lt2p])
     done
 
   then show ?thesis
     by (rule ssubst) (rule distinct_nthsI, simp)
 qed
 
 lemma upto_enum_set_conv [simp]:
   fixes a :: "'a :: len word"
   shows "set [a .e. b] = {x. a \<le> x \<and> x \<le> b}"
   apply (subst upto_enum_red)
   apply (subst set_map)
   apply safe
     apply simp
     apply clarsimp
     apply (erule disjE)
      apply simp
      apply (erule iffD2 [OF word_le_nat_alt])
     apply clarsimp
     apply (erule word_unat.Rep_cases [OF unat_le [OF order_less_imp_le]])
     apply simp
     apply (erule iffD2 [OF word_le_nat_alt])
    apply simp
 
    apply clarsimp
    apply (erule disjE)
     apply simp
    apply clarsimp
    apply (rule word_unat.Rep_cases [OF unat_le  [OF order_less_imp_le]])
     apply assumption
    apply simp
    apply (erule order_less_imp_le [OF iffD2 [OF word_less_nat_alt]])
   apply clarsimp
   apply (rule_tac x="fromEnum x" in image_eqI)
    apply clarsimp
   apply clarsimp
   apply (rule conjI)
    apply (subst word_le_nat_alt [symmetric])
    apply simp
   apply safe
    apply (simp add: word_le_nat_alt [symmetric])
   apply (simp add: word_less_nat_alt [symmetric])
   done
 
 lemma upto_enum_less:
   assumes xin: "x \<in> set [(a::'a::len word).e.2 ^ n - 1]"
   and     nv:  "n < LENGTH('a::len)"
   shows   "x < 2 ^ n"
 proof (cases n)
   case 0
   then show ?thesis using xin by simp
 next
   case (Suc m)
   show ?thesis using xin nv by simp
 qed
 
 lemma upto_enum_len_less:
   "\<lbrakk> n \<le> length [a, b .e. c]; n \<noteq> 0 \<rbrakk> \<Longrightarrow> a \<le> c"
   unfolding upto_enum_step_def
   by (simp split: if_split_asm)
 
 lemma length_upto_enum_step:
   fixes x :: "'a :: len word"
   shows "x \<le> z \<Longrightarrow> length [x , y .e. z] = (unat ((z - x) div (y - x))) + 1"
   unfolding upto_enum_step_def
   by (simp add: upto_enum_red)
 
 lemma map_length_unfold_one:
   fixes x :: "'a::len word"
   assumes xv: "Suc (unat x) < 2 ^ LENGTH('a)"
   and     ax: "a < x"
   shows   "map f [a .e. x] = f a # map f [a + 1 .e. x]"
   by (subst word_upto_Cons_eq, auto, fact+)
 
 lemma upto_enum_set_conv2:
   fixes a :: "'a::len word"
   shows "set [a .e. b] = {a .. b}"
   by auto
 
 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 mask_shift:
   "(x && ~~ (mask y)) >> y = x >> y"
   by word_eqI
 
 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 shiftr_div_2n':
   "unat (w >> n) = unat w div 2 ^ n"
   apply (unfold unat_def)
   apply (subst shiftr_div_2n)
   apply (subst nat_div_distrib)
    apply simp
   apply (simp add: nat_power_eq)
   done
 
 lemma shiftl_shiftr_id:
   assumes nv: "n < LENGTH('a)"
   and     xv: "x < 2 ^ (LENGTH('a) - n)"
   shows "x << n >> n = (x::'a::len word)"
   apply (simp add: shiftl_t2n)
   apply (rule word_unat.Rep_eqD)
   apply (subst shiftr_div_2n')
   apply (cases n)
    apply simp
   apply (subst iffD1 [OF unat_mult_lem])+
    apply (subst unat_power_lower[OF nv])
    apply (rule nat_less_power_trans [OF _ order_less_imp_le [OF nv]])
    apply (rule order_less_le_trans [OF unat_mono [OF xv] order_eq_refl])
    apply (rule unat_power_lower)
    apply simp
   apply (subst unat_power_lower[OF nv])
   apply simp
   done
 
 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 (case_tac "size w \<le> n", clarsimp simp: shiftl_zero_size)
      (clarsimp simp: not_le ucast_bl bl_shiftl bang_eq test_bit_of_bl rev_nth nth_append)
 
 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 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 length_upto_enum [simp]:
   fixes a :: "'a :: len word"
   shows "length [a .e. b] = Suc (unat b) - unat a"
   apply (simp add: word_le_nat_alt upto_enum_red)
   apply (clarsimp simp: Suc_diff_le)
   done
 
 lemma length_upto_enum_cases:
   fixes a :: "'a::len word"
   shows "length [a .e. b] = (if a \<le> b then Suc (unat b) - unat a else 0)"
   apply (case_tac "a \<le> b")
    apply (clarsimp)
   apply (clarsimp simp: upto_enum_def)
   apply unat_arith
   done
 
 lemma length_upto_enum_less_one:
   "\<lbrakk>a \<le> b; b \<noteq> 0\<rbrakk>
   \<Longrightarrow> length [a .e. b - 1] = unat (b - a)"
   apply clarsimp
   apply (subst unat_sub[symmetric], assumption)
   apply clarsimp
   done
 
 lemma drop_upto_enum:
   "drop (unat n) [0 .e. m] = [n .e. m]"
   apply (clarsimp simp: upto_enum_def)
   apply (induct m, simp)
   by (metis drop_map drop_upt plus_nat.add_0)
 
 lemma distinct_enum_upto' [simp]:
   "distinct [a::'a::len word .e. b]"
   apply (subst drop_upto_enum [symmetric])
   apply (rule distinct_drop)
   apply (rule distinct_enum_upto)
   done
 
 lemma length_interval:
   "\<lbrakk>set xs = {x. (a::'a::len word) \<le> x \<and> x \<le> b}; distinct xs\<rbrakk>
   \<Longrightarrow> length xs = Suc (unat b) - unat a"
   apply (frule distinct_card)
   apply (subgoal_tac "set xs = set [a .e. b]")
    apply (cut_tac distinct_card [where xs="[a .e. b]"])
     apply (subst (asm) length_upto_enum)
     apply clarsimp
    apply (rule distinct_enum_upto')
   apply simp
   done
 
 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
 
 declare of_nat_power [simp del]
 
 (* TODO: move to word *)
 lemma unat_of_bl_length:
   "unat (of_bl xs :: 'a::len word) < 2 ^ (length xs)"
 proof (cases "length xs < LENGTH('a)")
   case True
   then have "(of_bl xs::'a::len word) < 2 ^ length xs"
     by (simp add: of_bl_length_less)
   with True
   show ?thesis
     by (simp add: word_less_nat_alt word_unat_power unat_of_nat)
 next
   case False
   have "unat (of_bl xs::'a::len word) < 2 ^ LENGTH('a)"
     by (simp split: unat_split)
   also
   from False
   have "LENGTH('a) \<le> length xs" by simp
   then have "2 ^ LENGTH('a) \<le> (2::nat) ^ length xs"
     by (rule power_increasing) simp
   finally
   show ?thesis .
 qed
 
 lemma is_aligned_0'[simp]:
   "is_aligned 0 n"
   by (simp add: is_aligned_def)
 
 lemma p_assoc_help:
   fixes p :: "'a::{ring,power,numeral,one}"
   shows "p + 2^sz - 1 = p + (2^sz - 1)"
   by simp
 
 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_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_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 add: word_unat.Rep_inject [symmetric])
   apply simp
   done
 
 lemma nasty_split_lt:
   "\<lbrakk> (x :: 'a:: len word) < 2 ^ (m - n); n \<le> m; m < LENGTH('a::len) \<rbrakk>
      \<Longrightarrow> x * 2 ^ n + (2 ^ n - 1) \<le> 2 ^ m - 1"
   apply (simp only: add_diff_eq)
   apply (subst mult_1[symmetric], subst distrib_right[symmetric])
   apply (rule word_sub_mono)
      apply (rule order_trans)
       apply (rule word_mult_le_mono1)
         apply (rule inc_le)
         apply assumption
        apply (subst word_neq_0_conv[symmetric])
        apply (rule power_not_zero)
        apply simp
       apply (subst unat_power_lower, simp)+
       apply (subst power_add[symmetric])
       apply (rule power_strict_increasing)
        apply simp
       apply simp
      apply (subst power_add[symmetric])
      apply simp
     apply simp
    apply (rule word_sub_1_le)
    apply (subst mult.commute)
    apply (subst shiftl_t2n[symmetric])
    apply (rule word_shift_nonzero)
      apply (erule inc_le)
     apply simp
    apply (unat_arith)
   apply (drule word_power_less_1)
   apply simp
   done
 
 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
 
 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)
 
 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
   assume szv: "sz < LENGTH('a::len)"
   show ?thesis
   proof (cases "sz = 0")
     case True
     then show ?thesis using kv szv
      by (simp add: unat_of_nat)
   next
     case False
     then have sne: "0 < sz" ..
 
     have uk: "unat (of_nat k :: 'a word) = k"
       apply (subst unat_of_nat)
       apply (simp add: nat_mod_eq less_trans[OF kv] sne)
       done
 
     show ?thesis using szv
       apply (subst iffD1 [OF unat_mult_lem])
       apply (simp add: uk nat_less_power_trans[OF kv order_less_imp_le [OF szv]])+
       done
   qed
 next
   assume "\<not> sz < LENGTH('a)"
   with kv show ?thesis by (simp add: not_less power_overflow)
 qed
 
 lemma aligned_add_offset_no_wrap:
   fixes off :: "('a::len) word"
   and     x :: "'a word"
   assumes al: "is_aligned x sz"
   and   offv: "off < 2 ^ sz"
   shows  "unat x + unat off < 2 ^ LENGTH('a)"
 proof cases
   assume szv: "sz < LENGTH('a)"
   from al obtain k where xv: "x = 2 ^ sz * (of_nat k)"
     and kl: "k < 2 ^ (LENGTH('a) - sz)"
     by (auto elim: is_alignedE)
 
   show ?thesis using szv
     apply (subst xv)
     apply (subst unat_mult_power_lem[OF kl])
     apply (subst mult.commute, rule nat_add_offset_less)
       apply (rule less_le_trans[OF unat_mono[OF offv, simplified]])
       apply (erule eq_imp_le[OF unat_power_lower])
      apply (rule kl)
     apply simp
    done
 next
   assume "\<not> sz < LENGTH('a)"
   with offv show ?thesis by (simp add: not_less power_overflow )
 qed
 
 lemma aligned_add_offset_mod:
   fixes x :: "('a::len) word"
   assumes al: "is_aligned x sz"
   and     kv: "k < 2 ^ sz"
   shows   "(x + k) mod 2 ^ sz = k"
 proof cases
   assume szv: "sz < LENGTH('a)"
 
   have ux: "unat x + unat k < 2 ^ LENGTH('a)"
     by (rule aligned_add_offset_no_wrap) fact+
 
   show ?thesis using al szv
     apply -
     apply (erule is_alignedE)
     apply (subst word_unat.Rep_inject [symmetric])
     apply (subst unat_mod)
     apply (subst iffD1 [OF unat_add_lem], rule ux)
     apply simp
     apply (subst unat_mult_power_lem, assumption+)
     apply (simp)
     apply (rule mod_less[OF less_le_trans[OF unat_mono], OF kv])
     apply (erule eq_imp_le[OF unat_power_lower])
     done
 next
   assume "\<not> sz < LENGTH('a)"
   with al show ?thesis
     by (simp add: not_less power_overflow is_aligned_mask mask_def
                   word_mod_by_0)
 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 aligned_neq_into_no_overlap:
   fixes x :: "'a::len word"
   assumes neq: "x \<noteq> y"
   and     alx: "is_aligned x sz"
   and     aly: "is_aligned y sz"
   shows  "{x .. x + (2 ^ sz - 1)} \<inter> {y .. y + (2 ^ sz - 1)} = {}"
 proof cases
   assume szv: "sz < LENGTH('a)"
   show ?thesis
   proof (rule equals0I, clarsimp)
     fix z
     assume xb: "x \<le> z" and xt: "z \<le> x + (2 ^ sz - 1)"
       and yb: "y \<le> z" and yt: "z \<le> y + (2 ^ sz - 1)"
 
     have rl: "\<And>(p::'a word) k w. \<lbrakk>uint p + uint k < 2 ^ LENGTH('a); w = p + k; w \<le> p + (2 ^ sz - 1) \<rbrakk>
       \<Longrightarrow> k < 2 ^ sz"
       apply -
       apply simp
       apply (subst (asm) add.commute, subst (asm) add.commute, drule word_plus_mcs_4)
       apply (subst add.commute, subst no_plus_overflow_uint_size)
       apply (simp add: word_size_bl)
       apply (erule iffD1 [OF word_less_sub_le[OF szv]])
       done
 
     from xb obtain kx where
       kx: "z = x + kx" and
       kxl: "uint x + uint kx < 2 ^ LENGTH('a)"
       by (clarsimp dest!: word_le_exists')
 
     from yb obtain ky where
       ky: "z = y + ky" and
       kyl: "uint y + uint ky < 2 ^ LENGTH('a)"
       by (clarsimp dest!: word_le_exists')
 
     have "x = y"
     proof -
       have "kx = z mod 2 ^ sz"
       proof (subst kx, rule sym, rule aligned_add_offset_mod)
         show "kx < 2 ^ sz" by (rule rl) fact+
       qed fact+
 
       also have "\<dots> = ky"
       proof (subst ky, rule aligned_add_offset_mod)
         show "ky < 2 ^ sz"
           using kyl ky yt by (rule rl)
       qed fact+
 
       finally have kxky: "kx = ky" .
       moreover have "x + kx = y + ky" by (simp add: kx [symmetric] ky [symmetric])
       ultimately show ?thesis by simp
     qed
 
     then show False using neq by simp
   qed
 next
   assume "\<not> sz < LENGTH('a)"
   with neq alx aly
   have False by (simp add: is_aligned_mask mask_def power_overflow)
   then show ?thesis ..
 qed
 
 lemma less_two_pow_divD:
   "\<lbrakk> (x :: nat) < 2 ^ n div 2 ^ m \<rbrakk>
     \<Longrightarrow> n \<ge> m \<and> (x < 2 ^ (n - m))"
   apply (rule context_conjI)
    apply (rule ccontr)
    apply (simp add: power_strict_increasing)
   apply (simp add: power_sub)
   done
 
 lemma less_two_pow_divI:
   "\<lbrakk> (x :: nat) < 2 ^ (n - m); m \<le> n \<rbrakk> \<Longrightarrow> x < 2 ^ n div 2 ^ m"
   by (simp add: power_sub)
 
 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  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)
 
 (* shadows the slightly weaker Word.nth_ucast *)
 lemma nth_ucast:
   "(ucast (w::'a::len0 word)::'b::len0 word) !! n =
    (w !! n \<and> n < min LENGTH('a) LENGTH('b))"
   by (simp add: ucast_def test_bit_bin word_ubin.eq_norm nth_bintr word_size)
      (fast elim!: bin_nth_uint_imp)
 
 lemma ucast_less:
   "LENGTH('b) < LENGTH('a) \<Longrightarrow>
    (ucast (x :: 'b :: len word) :: ('a :: len word)) < 2 ^ LENGTH('b)"
   by (meson Word.nth_ucast test_bit_conj_lt le_def upper_bits_unset_is_l2p)
 
 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)
   by word_eqI_solve
 
 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
 
 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 is_aligned_diff:
   fixes m :: "'a::len word"
   assumes alm: "is_aligned m s1"
   and     aln: "is_aligned n s2"
   and    s2wb: "s2 < LENGTH('a)"
   and      nm: "m \<in> {n .. n + (2 ^ s2 - 1)}"
   and    s1s2: "s1 \<le> s2"
   and     s10: "0 < s1" (* Probably can be folded into the proof \<dots> *)
   shows  "\<exists>q. m - n = of_nat q * 2 ^ s1 \<and> q < 2 ^ (s2 - s1)"
 proof -
   have rl: "\<And>m s. \<lbrakk> m < 2 ^ (LENGTH('a) - s); s < LENGTH('a) \<rbrakk> \<Longrightarrow> unat ((2::'a word) ^ s * of_nat m) = 2 ^ s * m"
   proof -
     fix m :: nat and  s
     assume m: "m < 2 ^ (LENGTH('a) - s)" and s: "s < LENGTH('a)"
     then have "unat ((of_nat m) :: 'a word) = m"
       apply (subst unat_of_nat)
       apply (subst mod_less)
        apply (erule order_less_le_trans)
        apply (rule power_increasing)
         apply simp_all
       done
 
     then show "?thesis m s" using s m
       apply (subst iffD1 [OF unat_mult_lem])
       apply (simp add: nat_less_power_trans)+
       done
   qed
   have s1wb: "s1 < LENGTH('a)" using s2wb s1s2 by simp
   from alm obtain mq where mmq: "m = 2 ^ s1 * of_nat mq" and mq: "mq < 2 ^ (LENGTH('a) - s1)"
     by (auto elim: is_alignedE simp: field_simps)
   from aln obtain nq where nnq: "n = 2 ^ s2 * of_nat nq" and nq: "nq < 2 ^ (LENGTH('a) - s2)"
     by (auto elim: is_alignedE simp: field_simps)
   from s1s2 obtain sq where sq: "s2 = s1 + sq" by (auto simp: le_iff_add)
 
   note us1 = rl [OF mq s1wb]
   note us2 = rl [OF nq s2wb]
 
   from nm have "n \<le> m" by clarsimp
   then have "(2::'a word) ^ s2 * of_nat nq \<le> 2 ^ s1 * of_nat mq" using nnq mmq by simp
   then have "2 ^ s2 * nq \<le> 2 ^ s1 * mq" using s1wb s2wb
     by (simp add: word_le_nat_alt us1 us2)
   then have nqmq: "2 ^ sq * nq \<le> mq" using sq by (simp add: power_add)
 
   have "m - n = 2 ^ s1 * of_nat mq - 2 ^ s2 * of_nat nq" using mmq nnq by simp
   also have "\<dots> = 2 ^ s1 * of_nat mq - 2 ^ s1 * 2 ^ sq * of_nat nq" using sq by (simp add: power_add)
   also have "\<dots> = 2 ^ s1 * (of_nat mq - 2 ^ sq * of_nat nq)" by (simp add: field_simps)
   also have "\<dots> = 2 ^ s1 * of_nat (mq - 2 ^ sq * nq)" using s1wb s2wb us1 us2 nqmq
     by (simp add:  word_unat_power)
   finally have mn: "m - n = of_nat (mq - 2 ^ sq * nq) * 2 ^ s1" by simp
   moreover
   from nm have "m - n \<le> 2 ^ s2 - 1"
     by - (rule word_diff_ls', (simp add: field_simps)+)
   then have "(2::'a word) ^ s1 * of_nat (mq - 2 ^ sq * nq) < 2 ^ s2" using mn s2wb by (simp add: field_simps)
   then have "of_nat (mq - 2 ^ sq * nq) < (2::'a word) ^ (s2 - s1)"
   proof (rule word_power_less_diff)
     have mm: "mq - 2 ^ sq * nq < 2 ^ (LENGTH('a) - s1)" using mq by simp
     moreover from s10 have "LENGTH('a) - s1 < LENGTH('a)"
       by (rule diff_less, simp)
     ultimately show "of_nat (mq - 2 ^ sq * nq) < (2::'a word) ^ (LENGTH('a) - s1)"
       apply (simp add: word_less_nat_alt)
       apply (subst unat_of_nat)
       apply (subst mod_less)
        apply (erule order_less_le_trans)
        apply simp+
       done
   qed
   then have "mq - 2 ^ sq * nq < 2 ^ (s2 - s1)" using mq s2wb
     apply (simp add: word_less_nat_alt)
     apply (subst (asm) unat_of_nat)
     apply (subst (asm) mod_less)
     apply (rule order_le_less_trans)
     apply (rule diff_le_self)
     apply (erule order_less_le_trans)
     apply simp
     apply assumption
     done
   ultimately show ?thesis by auto
 qed
 
 lemma word_less_sub_1:
   "x < (y :: 'a :: len word) \<Longrightarrow> x \<le> y - 1"
   apply (erule udvd_minus_le')
    apply (simp add: udvd_def)+
   done
 
 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
 
 lemma word_not_le:
   "(\<not> x \<le> (y :: 'a :: len word)) = (y < x)"
   by fastforce
 
 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 sublist_equal_part:
   "prefix xs ys \<Longrightarrow> take (length xs) ys = xs"
   by (clarsimp simp: prefix_def)
 
 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 prefix_length_less:
   "strict_prefix xs ys \<Longrightarrow> length xs < length ys"
   apply (clarsimp simp: strict_prefix_def)
   apply (frule prefix_length_le)
   apply (rule ccontr, simp)
   apply (clarsimp simp: prefix_def)
   done
 
 lemmas take_less = take_strict_prefix
 
 lemma not_prefix_longer:
   "\<lbrakk> length xs > length ys \<rbrakk> \<Longrightarrow> \<not> prefix xs ys"
   by (clarsimp dest!: prefix_length_le)
 
 lemma of_bl_length:
   "length xs < LENGTH('a) \<Longrightarrow> of_bl xs < (2 :: 'a::len word) ^ length xs"
   by (simp add: of_bl_length_less)
 
 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 (subst unat_of_nat_eq[where x=n, symmetric], simp+)
 
 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: unat_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_def [symmetric])
+  apply (subst unat_def [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 (drule unat_of_nat_eq, simp)
 
 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 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 (simp add: word_unat.norm_eq_iff [symmetric])
 
 lemma map_prefixI:
   "prefix xs ys \<Longrightarrow> prefix (map f xs) (map f ys)"
   by (clarsimp simp: prefix_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
 
 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 list_all2_induct_suffixeq [consumes 1, case_names Nil Cons]:
   assumes lall: "list_all2 Q as bs"
   and     nilr: "P [] []"
   and    consr: "\<And>x xs y ys.
   \<lbrakk>list_all2 Q xs ys; Q x y; P xs ys; suffix (x # xs) as; suffix (y # ys) bs\<rbrakk>
   \<Longrightarrow> P (x # xs) (y # ys)"
   shows  "P as bs"
 proof -
   define as' where "as' == as"
   define bs' where "bs' == bs"
 
   have "suffix as as' \<and> suffix bs bs'" unfolding as'_def bs'_def by simp
   then show ?thesis using lall
   proof (induct rule: list_induct2 [OF list_all2_lengthD [OF lall]])
     case 1 show ?case by 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" using "2.hyps" "2.prems" by (auto dest: suffix_ConsD)
       from "2.prems" show "suffix (x # xs) as" and "suffix (y # ys) bs"
       by (auto simp: as'_def bs'_def)
     qed
   qed
 qed
 
 lemma upto_enum_step_shift:
   "\<lbrakk> is_aligned p n \<rbrakk> \<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 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 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 shiftr_less_t2n':
   "\<lbrakk> x && 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])
   apply word_eqI
   apply (erule_tac x="na + n" in allE)
   apply fastforce
   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 shiftr_not_mask_0:
   "n+m \<ge> LENGTH('a :: len) \<Longrightarrow> ((w::'a::len word) >> n) && ~~ (mask m) = 0"
   apply (simp add: and_not_mask shiftr_less_t2n shiftr_shiftr)
   apply (subgoal_tac "w >> n + m = 0", simp)
   apply (simp add: le_mask_iff[symmetric] mask_def le_def)
   apply (subst (asm) p2_gt_0[symmetric])
   apply (simp add: power_add not_less)
   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])
   apply word_eqI
   apply (erule_tac x="na - n" in allE)
   apply auto
   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 ucast_ucast_mask:
   "(ucast :: 'a :: len word \<Rightarrow> 'b :: len word) (ucast x) = x && mask (len_of TYPE ('a))"
   by word_eqI
 
 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 :: len0) word) = unat x mod 2 ^ (LENGTH('a))"
   apply (simp add: unat_def ucast_def)
   apply (subst word_uint.eq_norm)
   apply (subst nat_mod_distrib)
     apply simp
    apply simp
   apply (subst nat_power_eq)
    apply simp
   apply simp
   done
 
 lemma ucast_less_ucast:
-  "LENGTH('a) < LENGTH('b) \<Longrightarrow>
+  "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 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 sints_subset:
   "m \<le> n \<Longrightarrow> sints m \<subseteq> sints n"
   apply (simp add: sints_num)
   apply clarsimp
   apply (rule conjI)
    apply (erule order_trans[rotated])
    apply simp
   apply (erule order_less_le_trans)
   apply simp
   done
 
 lemma up_scast_inj:
       "\<lbrakk> scast x = (scast y :: 'b :: len word); size x \<le> LENGTH('b) \<rbrakk>
          \<Longrightarrow> x = y"
   apply (simp add: scast_def)
   apply (subst(asm) word_sint.Abs_inject)
     apply (erule subsetD [OF sints_subset])
     apply (simp add: word_size)
    apply (erule subsetD [OF sints_subset])
    apply (simp add: word_size)
   apply simp
   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 nth_bounded:
   "\<lbrakk>(x :: 'a :: len word) !! n; x < 2 ^ m; m \<le> len_of TYPE ('a)\<rbrakk> \<Longrightarrow> n < m"
   apply (frule test_bit_size)
   apply (clarsimp simp: test_bit_bl word_size)
   apply (simp add: nth_rev)
   apply (subst(asm) is_aligned_add_conv[OF is_aligned_0',
                                         simplified add_0_left, rotated])
    apply assumption+
   apply (simp only: to_bl_0)
   apply (simp add: nth_append split: if_split_asm)
   done
 
 lemma is_aligned_add_or:
   "\<lbrakk>is_aligned p n; d < 2 ^ n\<rbrakk> \<Longrightarrow> p + d = p || d"
   by (rule word_plus_and_or_coroll, word_eqI) blast
 
 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 is_aligned_add_less_t2n:
   "\<lbrakk>is_aligned (p::'a::len word) n; d < 2^n; n \<le> m; p < 2^m\<rbrakk> \<Longrightarrow> p + d < 2^m"
   apply (case_tac "m < LENGTH('a)")
    apply (subst mask_eq_iff_w2p[symmetric])
     apply (simp add: word_size)
    apply (simp add: is_aligned_add_or word_ao_dist less_mask_eq)
    apply (subst less_mask_eq)
     apply (erule order_less_le_trans)
     apply (erule(1) two_power_increasing)
    apply simp
   apply (simp add: power_overflow)
   done
 
 
 lemma aligned_offset_non_zero:
   "\<lbrakk> is_aligned x n; y < 2 ^ n; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x + y \<noteq> 0"
   apply (cases "y = 0")
    apply simp
   apply (subst word_neq_0_conv)
   apply (subst gt0_iff_gem1)
   apply (erule is_aligned_get_word_bits)
    apply (subst field_simps[symmetric], subst plus_le_left_cancel_nowrap)
      apply (rule is_aligned_no_wrap')
       apply simp
      apply (rule word_leq_le_minus_one)
       apply simp
      apply assumption
     apply (erule (1) is_aligned_no_wrap')
    apply (simp add: gt0_iff_gem1 [symmetric] word_neq_0_conv)
   apply simp
   done
 
 lemmas mask_inner_mask = mask_eqs(1)
 
 lemma mask_add_aligned:
   "is_aligned p n \<Longrightarrow> (p + q) && mask n = q && mask n"
   apply (simp add: is_aligned_mask)
   apply (subst mask_inner_mask [symmetric])
   apply simp
   done
 
 lemma take_prefix:
   "(take (length xs) ys = xs) = prefix xs ys"
 proof (induct xs arbitrary: ys)
   case Nil then show ?case by simp
 next
   case Cons then show ?case by (cases ys) auto
 qed
 
 lemma cart_singleton_empty:
   "(S \<times> {e} = {}) = (S = {})"
   by blast
 
 lemma word_div_1:
   "(n :: 'a :: len word) div 1 = n"
   by (simp add: word_div_def)
 
 lemma word_minus_one_le:
   "-1 \<le> (x :: 'a :: len word) = (x = -1)"
   apply (insert word_n1_ge[where y=x])
   apply safe
   apply (erule(1) order_antisym)
   done
 
 lemma mask_out_sub_mask:
   "(x && ~~ (mask n)) = x - (x && (mask n))"
   by (simp add: field_simps word_plus_and_or_coroll2)
 
 lemma is_aligned_addD1:
   assumes al1: "is_aligned (x + y) n"
   and     al2: "is_aligned (x::'a::len word) n"
   shows "is_aligned y n"
   using al2
 proof (rule is_aligned_get_word_bits)
   assume "x = 0" then show ?thesis using al1 by simp
 next
   assume nv: "n < LENGTH('a)"
   from al1 obtain q1
     where xy: "x + y = 2 ^ n * of_nat q1" and "q1 < 2 ^ (LENGTH('a) - n)"
     by (rule is_alignedE)
   moreover from al2 obtain q2
     where x: "x = 2 ^ n * of_nat q2" and "q2 < 2 ^ (LENGTH('a) - n)"
     by (rule is_alignedE)
   ultimately have "y = 2 ^ n * (of_nat q1 - of_nat q2)"
     by (simp add: field_simps)
   then show ?thesis using nv by (simp add: is_aligned_mult_triv1)
 qed
 
 lemmas is_aligned_addD2 =
        is_aligned_addD1[OF subst[OF add.commute,
                                  of "%x. is_aligned x n" for n]]
 
 lemma is_aligned_add:
   "\<lbrakk>is_aligned p n; is_aligned q n\<rbrakk> \<Longrightarrow> is_aligned (p + q) n"
   by (simp add: is_aligned_mask mask_add_aligned)
 
 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
 
 lemma shiftl_mask_is_0[simp]:
   "(x << n) && mask n = 0"
   apply (rule iffD1 [OF is_aligned_mask])
   apply (rule is_aligned_shiftl_self)
   done
 
 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_and_helper:
   "(If x v v') && v'' = If x (v && v'') (v' && v'')"
   by (rule if_distrib)
 
 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
 
 lemmas word_unat_Rep_inject1 = word_unat.Rep_inject[where y=1]
 lemmas unat_eq_1 = unat_eq_0 word_unat_Rep_inject1[simplified]
 
 
 lemma rshift_sub_mask_eq:
   "(a >> (size a - b)) && mask b = a >> (size a - b)"
   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_def word_size
                    p2_eq_0[THEN iffD2])
   done
 
 lemma shiftl_shiftr3:
   "b \<le> c \<Longrightarrow> a << b >> c = (a >> c - b) && mask (size a - c)"
   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 && mask m) >> n = (w >> n) && mask (m-n)"
   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 && mask m) << n = (w << n) && mask (m+n)"
   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"
   by (simp add: le_mask_iff shiftl_shiftr3)
 
 lemma and_not_mask_twice:
   "(w && ~~ (mask n)) && ~~ (mask m) = w && ~~ (mask (max m n))"
   apply (simp add: and_not_mask)
   apply (case_tac "n<m";
          simp add: shiftl_shiftr2 shiftl_shiftr1 not_less max_def shiftr_shiftr shiftl_shiftl)
    apply (cut_tac and_mask_shiftr_comm [where w=w and m="size w" and n=m, simplified,symmetric])
    apply (simp add: word_size mask_def)
   apply (cut_tac and_mask_shiftr_comm [where w=w and m="size w" and n=n, simplified,symmetric])
   apply (simp add: word_size mask_def)
   done
 
 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 eq_eqI:
   "a = b \<Longrightarrow> (a = x) = (b = x)"
   by simp
 
 lemma mask_and_mask:
   "mask a && mask b = mask (min a b)"
   by word_eqI
 
 lemma mask_eq_0_eq_x:
   "(x && w = 0) = (x && ~~ w = x)"
   using word_plus_and_or_coroll2[where x=x and w=w]
   by auto
 
 lemma mask_eq_x_eq_0:
   "(x && w = x) = (x && ~~ w = 0)"
   using word_plus_and_or_coroll2[where x=x and w=w]
   by auto
 
 definition
   "limited_and (x :: 'a :: len word) y = (x && y = x)"
 
 lemma limited_and_eq_0:
   "\<lbrakk> limited_and x z; y && ~~ z = y \<rbrakk> \<Longrightarrow> x && y = 0"
   unfolding limited_and_def
   apply (subst arg_cong2[where f="(&&)"])
     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 && z = z \<rbrakk> \<Longrightarrow> x && 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)"
   unfolding limited_and_def
   by (simp add: shiftl_over_and_dist[symmetric])
 
 lemma rshift_limited_and:
   "limited_and x z \<Longrightarrow> limited_and (x >> n) (z >> n)"
   unfolding limited_and_def
   by (simp add: shiftr_over_and_dist[symmetric])
 
 lemmas limited_and_simps1 = limited_and_eq_0 limited_and_eq_id
 
 lemmas is_aligned_limited_and
     = is_aligned_neg_mask_eq[unfolded mask_def, folded limited_and_def]
 
 lemma compl_of_1: "~~ 1 = (-2 :: 'a :: len word)"
   by simp
 
 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]
        compl_of_1 shiftl_shiftr1[unfolded word_size mask_def]
        shiftl_shiftr2[unfolded word_size mask_def]
 
 lemma split_word_eq_on_mask:
   "(x = y) = (x && m = y && m \<and> x && ~~ m = y && ~~ m)"
   by safe word_eqI_solve
 
 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 word_and_1_shiftl:
   "x && (1 << n) = (if 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 && (mask n << m) = ((x >> m) && mask n) << m"
   by word_eqI_solve
 
 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 word_FF_is_mask:
   "0xFF = mask 8"
   by (simp add: mask_def)
 
 lemma word_1FF_is_mask:
   "0x1FF = mask 9"
   by (simp add: mask_def)
 
 lemma ucast_of_nat_small:
   "x < 2 ^ LENGTH('a) \<Longrightarrow> ucast (of_nat x :: 'a :: len word) = (of_nat x :: 'b :: len word)"
   apply (rule sym, subst word_unat.inverse_norm)
   apply (simp add: ucast_def word_of_int[symmetric]
                    of_nat_nat[symmetric] unat_def[symmetric])
   apply (simp add: unat_of_nat)
   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 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 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 mod_mod_power:
   fixes k :: nat
   shows "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ (min m n)"
 proof (cases "m \<le> n")
   case True
 
   then have "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ m"
     apply -
     apply (subst mod_less [where n = "2 ^ n"])
     apply (rule order_less_le_trans [OF mod_less_divisor])
     apply simp+
     done
   also have "\<dots> = k mod  2 ^ (min m n)" using True by simp
   finally show ?thesis .
 next
   case False
   then have "n < m" by simp
   then obtain d where md: "m = n + d"
     by (auto dest: less_imp_add_positive)
   then have "k mod 2 ^ m = 2 ^ n * (k div 2 ^ n mod 2 ^ d) + k mod 2 ^ n"
     by (simp add: mod_mult2_eq power_add)
   then have "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ n"
     by (simp add: mod_add_left_eq)
   then show ?thesis using False
     by simp
 qed
 
 lemma word_div_less:
   "m < n \<Longrightarrow> m div n = 0" for m :: "'a :: len word"
   by (simp add: unat_mono word_arith_nat_defs(6))
 
 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 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 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 power_mod_div:
   fixes x :: "nat"
   shows "x mod 2 ^ n div 2 ^ m = x div 2 ^ m mod 2 ^ (n - m)" (is "?LHS = ?RHS")
 proof (cases "n \<le> m")
   case True
   then have "?LHS = 0"
     apply -
     apply (rule div_less)
     apply (rule order_less_le_trans [OF mod_less_divisor]; simp)
     done
   also have "\<dots> = ?RHS" using True
     by simp
   finally show ?thesis .
 next
   case False
   then have lt: "m < n" by simp
   then obtain q where nv: "n = m + q" and "0 < q"
     by (auto dest: less_imp_Suc_add)
 
   then have "x mod 2 ^ n = 2 ^ m * (x div 2 ^ m mod 2 ^ q) + x mod 2 ^ m"
     by (simp add: power_add mod_mult2_eq)
 
   then have "?LHS = x div 2 ^ m mod 2 ^ q"
     by (simp add: div_add1_eq)
 
   also have "\<dots> = ?RHS" using nv
     by simp
 
   finally show ?thesis .
 qed
 
 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 (simp add: ucast_def word_of_int_nat unat_def)
 
 lemmas if_fun_split = if_apply_def2
 
 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 overflow_plus_one_self:
   "(1 + p \<le> p) = (p = (-1 :: 'a :: len word))"
   apply (safe, simp_all)
   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)
   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 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 auto
 
 lemma emptyE2:
   "\<lbrakk> S = {}; x \<in> S \<rbrakk> \<Longrightarrow> P"
   by simp
 
 lemma mod_div_equality_div_eq:
   "a div b * b = (a - (a mod b) :: int)"
   by (simp add: field_simps)
 
 lemma zmod_helper:
   "n mod m = k \<Longrightarrow> ((n :: int) + a) mod m = (k + a) mod m"
   by (metis add.commute mod_add_right_eq)
 
 lemma int_div_sub_1:
   "\<lbrakk> m \<ge> 1 \<rbrakk> \<Longrightarrow> (n - (1 :: int)) div m = (if m dvd n then (n div m) - 1 else n div m)"
   apply (subgoal_tac "m = 0 \<or> (n - (1 :: int)) div m = (if m dvd n then (n div m) - 1 else n div m)")
    apply fastforce
   apply (subst mult_cancel_right[symmetric])
   apply (simp only: left_diff_distrib split: if_split)
   apply (simp only: mod_div_equality_div_eq)
   apply (clarsimp simp: field_simps)
   apply (clarsimp simp: dvd_eq_mod_eq_0)
   apply (cases "m = 1")
    apply simp
   apply (subst mod_diff_eq[symmetric], simp add: zmod_minus1)
   apply clarsimp
   apply (subst diff_add_cancel[where b=1, symmetric])
   apply (subst mod_add_eq[symmetric])
   apply (simp add: field_simps)
   apply (rule mod_pos_pos_trivial)
    apply (subst add_0_right[where a=0, symmetric])
    apply (rule add_mono)
     apply simp
    apply simp
   apply (cases "(n - 1) mod m = m - 1")
    apply (drule zmod_helper[where a=1])
    apply simp
   apply (subgoal_tac "1 + (n - 1) mod m \<le> m")
    apply simp
   apply (subst field_simps, rule zless_imp_add1_zle)
   apply simp
   done
 
 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"
   apply (simp add: image_def)
   apply (erule rev_bexI)
   apply (rule arg_cong[where f="ptr_add ptr"])
   apply simp
   done
 
 lemma shift_times_fold:
   "(x :: 'a :: len word) * (2 ^ n) << m = x << (m + n)"
   by (simp add: shiftl_t2n ac_simps power_add)
 
 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 replicate_minus:
   "k < n \<Longrightarrow> replicate n False = replicate (n - k) False @ replicate k False"
   by (subst replicate_add [symmetric]) simp
 
 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 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 add: word_less_nat_alt)
     apply (subst unat_of_nat_eq)
      apply (erule order_less_trans)
      apply simp+
     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 enum_word_div:
   fixes v :: "'a :: len word" shows
   "\<exists>xs ys. enum = xs @ [v] @ ys
              \<and> (\<forall>x \<in> set xs. x < v)
              \<and> (\<forall>y \<in> set ys. v < y)"
   apply (simp only: enum_word_def)
   apply (subst upt_add_eq_append'[where j="unat v"])
     apply simp
    apply (rule order_less_imp_le, simp)
   apply (simp add: upt_conv_Cons)
   apply (intro exI conjI)
     apply fastforce
    apply clarsimp
    apply (drule of_nat_mono_maybe[rotated, where 'a='a])
     apply simp
    apply simp
   apply (clarsimp simp: Suc_le_eq)
   apply (drule of_nat_mono_maybe[rotated, where 'a='a])
    apply simp
   apply simp
   done
 
 lemma of_bool_nth:
   "of_bool (x !! v) = (x >> v) && 1"
   by (word_eqI cong: rev_conj_cong)
 
 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 (induct x)
    apply clarsimp+
   by (metis Suc_eq_plus1 add_lessD1 less_irrefl one_add_one unatSuc word_less_nat_alt)
 
 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)
   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 && 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 shiftr_mask_eq:
   "(x >> n) && mask (size x - n) = x >> n" for x :: "'a :: len word"
   by word_eqI_solve
 
 lemma shiftr_mask_eq':
   "m = (size x - n) \<Longrightarrow> (x >> n) && mask m = x >> n" for x :: "'a :: len word"
   by (simp add: shiftr_mask_eq)
 
 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)
 
 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 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 (simp add: word_unat.norm_eq_iff [symmetric])
   done
 
 lemma mask_AND_NOT_mask:
   "(w && ~~ (mask n)) && mask n = 0"
   by word_eqI
 
 lemma AND_NOT_mask_plus_AND_mask_eq:
   "(w && ~~ (mask n)) + (w && mask n) = w"
   by (subst word_plus_and_or_coroll; word_eqI_solve)
 
 lemma mask_eqI:
   fixes x :: "'a :: len word"
   assumes m1: "x && mask n = y && mask n"
   and     m2: "x && ~~ (mask n) = y && ~~ (mask n)"
   shows "x = y"
 proof (subst bang_eq, rule allI)
   fix m
 
   show "x !! m = y !! m"
   proof (cases "m < n")
     case True
     then have "x !! m = ((x && mask n) !! m)"
       by (simp add: word_size test_bit_conj_lt)
     also have "\<dots> = ((y && mask n) !! m)" using m1 by simp
     also have "\<dots> = y !! m" using True
       by (simp add: word_size test_bit_conj_lt)
     finally show ?thesis .
   next
     case False
     then have "x !! m = ((x && ~~ (mask n)) !! m)"
       by (simp add: neg_mask_test_bit test_bit_conj_lt)
     also have "\<dots> = ((y && ~~ (mask n)) !! m)" using m2 by simp
     also have "\<dots> = y !! m" using False
       by (simp add: neg_mask_test_bit test_bit_conj_lt)
     finally show ?thesis .
   qed
 qed
 
 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 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 plus_minus_one_rewrite:
   "v + (- 1 :: ('a :: {ring, one, uminus})) \<equiv> v - 1"
   by (simp add: field_simps)
 
 lemma power_minus_is_div:
   "b \<le> a \<Longrightarrow> (2 :: nat) ^ (a - b) = 2 ^ a div 2 ^ b"
   apply (induct a arbitrary: b)
    apply simp
   apply (erule le_SucE)
    apply (clarsimp simp:Suc_diff_le le_iff_add power_add)
   apply simp
   done
 
 lemma two_pow_div_gt_le:
   "v < 2 ^ n div (2 ^ m :: nat) \<Longrightarrow> m \<le> n"
   by (clarsimp dest!: less_two_pow_divD)
 
 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_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: unat_of_nat)
   done
 
 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
   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 and_eq_0_is_nth:
   fixes x :: "'a :: len word"
   shows "y = 1 << n \<Longrightarrow> ((x && y) = 0) = (\<not> (x !! n))"
   apply safe
    apply (drule_tac u="(x && (1 << n))" and x=n in word_eqD)
    apply (simp add: nth_w2p)
    apply (simp add: test_bit_bin)
   apply word_eqI
   done
 
 lemmas arg_cong_Not = arg_cong [where f=Not]
 lemmas and_neq_0_is_nth = arg_cong_Not [OF and_eq_0_is_nth, simplified]
 
 lemma nth_is_and_neq_0:
   "(x::'a::len word) !! n = (x && 2 ^ n \<noteq> 0)"
   by (subst and_neq_0_is_nth; rule refl)
 
 lemma mask_Suc_0 : "mask (Suc 0) = 1"
   by (simp add: mask_def)
 
 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 (rule word_unat.Rep_eqD)
   apply (simp add: unat_ucast unat_word_ariths mod_mod_power
                    min.absorb2 unat_of_nat)
   apply (subst mod_add_left_eq[symmetric])
   apply (simp add: mod_mod_power min.absorb2)
   apply (subst mod_add_right_eq)
   apply simp
   done
 
 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 bool_mask':
   fixes x :: "'a :: len word"
   shows "2 < LENGTH('a) \<Longrightarrow> (0 < x && 1) = (x && 1 = 1)"
   apply (rule iffI)
    prefer 2
    apply simp
   apply (subgoal_tac "x && mask 1 < 2^1")
    prefer 2
    apply (rule and_mask_less_size)
    apply (simp add: word_size)
   apply (simp add: mask_def)
   apply (drule word_less_cases [where y=2])
   apply (erule disjE, simp)
   apply simp
   done
 
 lemma sint_eq_uint:
   "\<not> msb x \<Longrightarrow> sint x = uint x"
   apply (rule word_uint.Abs_eqD, subst word_sint.Rep_inverse)
     apply simp_all
   apply (cut_tac x=x in word_sint.Rep)
   apply (clarsimp simp add: uints_num sints_num)
   apply (rule conjI)
    apply (rule ccontr)
    apply (simp add: linorder_not_le word_msb_sint[symmetric])
   apply (erule order_less_le_trans)
   apply simp
   done
 
 lemma scast_eq_ucast:
   "\<not> msb x \<Longrightarrow> scast x = ucast x"
   by (simp add: scast_def ucast_def sint_eq_uint)
 
 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 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 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
 
 lemma word_0_sle_from_less[unfolded word_size]:
   "\<lbrakk> x < 2 ^ (size x - 1) \<rbrakk>  \<Longrightarrow> 0 <=s x"
   apply (clarsimp simp: word_sle_msb_le)
   apply (simp add: word_msb_nth)
   apply (subst (asm) word_test_bit_def [symmetric])
   apply (drule less_mask_eq)
   apply (drule_tac x="size x - 1" in word_eqD)
   apply (simp add: word_size)
   done
 
 lemma not_msb_from_less:
   "(v :: 'a word) < 2 ^ (LENGTH('a :: len) - 1) \<Longrightarrow> \<not> msb v"
   apply (clarsimp simp add: msb_nth)
   apply (drule less_mask_eq)
   apply (drule word_eqD, drule(1) iffD2)
   apply simp
   done
 
 lemma distinct_lemma: "f x \<noteq> f y \<Longrightarrow> x \<noteq> y" by auto
 
 
 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"
   by unat_arith
 
 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)
 
 (*
  * Basic signed arithemetic properties.
  *)
 
 lemma sint_minus1 [simp]: "(sint x = -1) = (x = -1)"
   by (metis sint_n1 word_sint.Rep_inverse')
 
 lemma sint_0 [simp]: "(sint x = 0) = (x = 0)"
   by (metis sint_0 word_sint.Rep_inverse')
 
 (* 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:
   "\<lbrakk> \<lbrakk> len_of TYPE ('a::len) = 1; (a::'a word) = 0; sint a = 0 \<rbrakk> \<Longrightarrow> P;
      \<lbrakk> len_of TYPE ('a) = 1; a = 1; sint (1 :: 'a word) = -1 \<rbrakk> \<Longrightarrow> P;
       \<lbrakk> len_of TYPE ('a) > 1; sint (1 :: 'a word) = 1 \<rbrakk> \<Longrightarrow> P \<rbrakk>
                 \<Longrightarrow> P"
    apply atomize_elim
   apply (case_tac "len_of TYPE ('a) = 1")
    apply clarsimp
    apply (subgoal_tac "(UNIV :: 'a word set) = {0, 1}")
     apply (metis UNIV_I insert_iff singletonE)
    apply (subst word_unat.univ)
    apply (clarsimp simp: unats_def image_def)
    apply (rule set_eqI, rule iffI)
     apply clarsimp
     apply (metis One_nat_def less_2_cases of_nat_1 semiring_1_class.of_nat_0)
    apply clarsimp
    apply (metis Abs_fnat_hom_0 Suc_1 lessI of_nat_1 zero_less_Suc)
   apply clarsimp
   apply (metis One_nat_def arith_is_1 le_def len_gt_0)
   done
 
 lemma sint_int_min:
   "sint (- (2 ^ (LENGTH('a) - Suc 0)) :: ('a::len) word) = - (2 ^ (LENGTH('a) - Suc 0))"
   apply (subst word_sint.Abs_inverse' [where r="- (2 ^ (LENGTH('a) - Suc 0))"])
     apply (clarsimp simp: sints_num)
    apply (clarsimp simp: wi_hom_syms word_of_int_2p)
   apply clarsimp
   done
 
 lemma sint_int_max_plus_1:
   "sint (2 ^ (LENGTH('a) - Suc 0) :: ('a::len) word) = - (2 ^ (LENGTH('a) - Suc 0))"
   apply (subst word_of_int_2p [symmetric])
   apply (subst int_word_sint)
   apply clarsimp
   apply (metis Suc_pred int_word_uint len_gt_0 power_Suc uint_eq_0 word_of_int_2p word_pow_0)
   done
 
 lemma sbintrunc_eq_in_range:
   "(sbintrunc n x = x) = (x \<in> range (sbintrunc n))"
   "(x = sbintrunc n x) = (x \<in> range (sbintrunc n))"
   apply (simp_all add: image_def)
   apply (metis sbintrunc_sbintrunc)+
   done
 
 lemma sbintrunc_If:
   "- 3 * (2 ^ n) \<le> x \<and> x < 3 * (2 ^ n)
     \<Longrightarrow> sbintrunc n x = (if x < - (2 ^ n) then x + 2 * (2 ^ n)
         else if x \<ge> 2 ^ n then x - 2 * (2 ^ n) else x)"
   apply (simp add: no_sbintr_alt2, safe)
    apply (simp add: mod_pos_geq)
   apply (subst mod_add_self1[symmetric], simp)
   done
 
 lemma signed_arith_eq_checks_to_ord:
   "(sint a + sint b = sint (a + b ))
     = ((a <=s a + b) = (0 <=s b))"
   "(sint a - sint b = sint (a - b ))
     = ((0 <=s a - b) = (b <=s a))"
   "(- sint a = sint (- a)) = (0 <=s (- a) = (a <=s 0))"
   using sint_range'[where x=a] sint_range'[where x=b]
   by (simp_all add: sint_word_ariths word_sle_def word_sless_alt sbintrunc_If)
 
 (* Basic proofs that signed word div/mod operations are
  * truncations of their integer counterparts. *)
 
 lemma signed_div_arith:
     "sint ((a::('a::len) word) sdiv b) = sbintrunc (LENGTH('a) - 1) (sint a sdiv sint b)"
   apply (subst word_sbin.norm_Rep [symmetric])
   apply (subst bin_sbin_eq_iff' [symmetric])
    apply simp
   apply (subst uint_sint [symmetric])
   apply (clarsimp simp: sdiv_int_def sdiv_word_def)
   apply (metis word_ubin.eq_norm)
   done
 
 lemma signed_mod_arith:
     "sint ((a::('a::len) word) smod b) = sbintrunc (LENGTH('a) - 1) (sint a smod sint b)"
   apply (subst word_sbin.norm_Rep [symmetric])
   apply (subst bin_sbin_eq_iff' [symmetric])
    apply simp
   apply (subst uint_sint [symmetric])
   apply (clarsimp simp: smod_int_def smod_word_def)
   apply (metis word_ubin.eq_norm)
   done
 
 (* Signed word arithmetic overflow constraints. *)
 
 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
                     sbintrunc_eq_in_range range_sbintrunc)
 
 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 signed_mult_eq_checks_double_size:
   assumes mult_le: "(2 ^ (len_of TYPE ('a) - 1) + 1) ^ 2 \<le> (2 :: int) ^ (len_of TYPE ('b) - 1)"
            and le: "2 ^ (LENGTH('a) - 1) \<le> (2 :: int) ^ (len_of TYPE ('b) - 1)"
   shows "(sint (a :: 'a :: len word) * sint b = sint (a * b))
        = (scast a * scast b = (scast (a * b) :: 'b :: len word))"
 proof -
   have P: "sbintrunc (size a - 1) (sint a * sint b) \<in> range (sbintrunc (size a - 1))"
     by simp
 
   have abs: "!! x :: 'a word. abs (sint x) < 2 ^ (size a - 1) + 1"
     apply (cut_tac x=x in sint_range')
     apply (simp add: abs_le_iff word_size)
     done
   have abs_ab: "abs (sint a * sint b) < 2 ^ (LENGTH('b) - 1)"
     using abs_mult_less[OF abs[where x=a] abs[where x=b]] mult_le
     by (simp add: abs_mult power2_eq_square word_size)
   show ?thesis
     using P[unfolded range_sbintrunc] abs_ab le
     apply (simp add: sint_word_ariths scast_def)
     apply (simp add: wi_hom_mult)
     apply (subst word_sint.Abs_inject, simp_all)
      apply (simp add: sints_def range_sbintrunc abs_less_iff)
     apply clarsimp
     apply (simp add: sints_def range_sbintrunc word_size)
     apply (auto elim: order_less_le_trans order_trans[rotated])
     done
 qed
 
 (* Properties about signed division. *)
 
 lemma int_sdiv_simps [simp]:
     "(a :: int) sdiv 1 = a"
     "(a :: int) sdiv 0 = 0"
     "(a :: int) sdiv -1 = -a"
   apply (auto simp: sdiv_int_def sgn_if)
   done
 
 lemma sgn_div_eq_sgn_mult:
     "a div b \<noteq> 0 \<Longrightarrow> sgn ((a :: int) div b) = sgn (a * b)"
   apply (clarsimp simp: sgn_if zero_le_mult_iff neg_imp_zdiv_nonneg_iff not_less)
   apply (metis less_le mult_le_0_iff neg_imp_zdiv_neg_iff not_less pos_imp_zdiv_neg_iff zdiv_eq_0_iff)
   done
 
 lemma sgn_sdiv_eq_sgn_mult:
   "a sdiv b \<noteq> 0 \<Longrightarrow> sgn ((a :: int) sdiv b) = sgn (a * b)"
   by (auto simp: sdiv_int_def sgn_div_eq_sgn_mult sgn_mult)
 
 lemma int_sdiv_same_is_1 [simp]:
     "a \<noteq> 0 \<Longrightarrow> ((a :: int) sdiv b = a) = (b = 1)"
   apply (rule iffI)
    apply (clarsimp simp: sdiv_int_def)
    apply (subgoal_tac "b > 0")
     apply (case_tac "a > 0")
      apply (clarsimp simp: sgn_if)
     apply (clarsimp simp: algebra_split_simps not_less)
     apply (metis int_div_same_is_1 le_neq_trans minus_minus neg_0_le_iff_le neg_equal_0_iff_equal)
    apply (case_tac "a > 0")
     apply (case_tac "b = 0")
      apply clarsimp
     apply (rule classical)
     apply (clarsimp simp: sgn_mult not_less)
     apply (metis le_less neg_0_less_iff_less not_less_iff_gr_or_eq pos_imp_zdiv_neg_iff)
    apply (rule classical)
    apply (clarsimp simp: algebra_split_simps sgn_mult not_less sgn_if split: if_splits)
    apply (metis antisym less_le neg_imp_zdiv_nonneg_iff)
   apply (clarsimp simp: sdiv_int_def sgn_if)
   done
 
 lemma int_sdiv_negated_is_minus1 [simp]:
     "a \<noteq> 0 \<Longrightarrow> ((a :: int) sdiv b = - a) = (b = -1)"
   apply (clarsimp simp: sdiv_int_def)
   apply (rule iffI)
    apply (subgoal_tac "b < 0")
     apply (case_tac "a > 0")
      apply (clarsimp simp: sgn_if algebra_split_simps not_less)
     apply (case_tac "sgn (a * b) = -1")
      apply (clarsimp simp: not_less algebra_split_simps)
     apply (clarsimp simp: algebra_split_simps not_less)
    apply (rule classical)
    apply (case_tac "b = 0")
     apply (clarsimp simp: not_less sgn_mult)
    apply (case_tac "a > 0")
     apply (clarsimp simp: not_less sgn_mult)
     apply (metis less_le neg_less_0_iff_less not_less_iff_gr_or_eq pos_imp_zdiv_neg_iff)
    apply (clarsimp simp: not_less sgn_mult)
    apply (metis antisym_conv div_minus_right neg_imp_zdiv_nonneg_iff neg_le_0_iff_le not_less)
   apply (clarsimp simp: sgn_if)
   done
 
 lemma sdiv_int_range:
     "(a :: int) sdiv b \<in> { - (abs a) .. (abs a) }"
   apply (unfold sdiv_int_def)
   apply (subgoal_tac "(abs a) div (abs b) \<le> (abs a)")
    apply (clarsimp simp: sgn_if)
    apply (meson abs_ge_zero neg_le_0_iff_le nonneg_mod_div order_trans)
   apply (metis abs_eq_0 abs_ge_zero div_by_0 zdiv_le_dividend zero_less_abs_iff)
   done
 
 lemma word_sdiv_div1 [simp]:
     "(a :: ('a::len) word) sdiv 1 = a"
   apply (rule sint_1_cases [where a=a])
     apply (clarsimp simp: sdiv_word_def sdiv_int_def)
    apply (clarsimp simp: sdiv_word_def sdiv_int_def simp del: sint_minus1)
   apply (clarsimp simp: sdiv_word_def)
   done
 
 lemma sdiv_int_div_0 [simp]:
   "(x :: int) sdiv 0 = 0"
   by (clarsimp simp: sdiv_int_def)
 
 lemma sdiv_int_0_div [simp]:
   "0 sdiv (x :: int) = 0"
   by (clarsimp simp: sdiv_int_def)
 
 lemma word_sdiv_div0 [simp]:
     "(a :: ('a::len) word) sdiv 0 = 0"
   apply (auto simp: sdiv_word_def sdiv_int_def sgn_if)
   done
 
 lemma word_sdiv_div_minus1 [simp]:
     "(a :: ('a::len) word) sdiv -1 = -a"
   apply (auto simp: sdiv_word_def sdiv_int_def sgn_if)
   apply (metis wi_hom_neg word_sint.Rep_inverse')
   done
 
 lemmas word_sdiv_0 = word_sdiv_div0
 
 lemma sdiv_word_min:
     "- (2 ^ (size a - 1)) \<le> sint (a :: ('a::len) word) sdiv sint (b :: ('a::len) word)"
   apply (clarsimp simp: word_size)
   apply (cut_tac sint_range' [where x=a])
   apply (cut_tac sint_range' [where x=b])
   apply clarsimp
   apply (insert sdiv_int_range [where a="sint a" and b="sint b"])
   apply (clarsimp simp: max_def abs_if split: if_split_asm)
   done
 
 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)
 
   have result_range: "sint a sdiv sint b \<in> (sints (size a)) \<union> {2 ^ (size a - 1)}"
     apply (cut_tac sdiv_int_range [where a="sint a" and b="sint b"])
     apply (erule rev_subsetD)
     using sint_range' [where x=a]  sint_range' [where x=b]
     apply (auto simp: max_def abs_if word_size sints_num)
     done
 
   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: sdiv_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 (insert word_sint.Rep [where x="a"])[1]
     apply (clarsimp simp: minus_le_iff word_size abs_if sints_num split: if_split_asm)
     apply (metis minus_minus sint_int_min word_sint.Rep_inject)
     done
 
   have result_range_simple: "(sint a sdiv sint b \<in> (sints (size a))) \<Longrightarrow> ?thesis"
     apply (insert sdiv_int_range [where a="sint a" and b="sint b"])
     apply (clarsimp simp: word_size sints_num 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]
 
 lemmas word_sdiv_numerals_lhs = sdiv_word_def[where a="numeral x" for x]
     sdiv_word_def[where a=0] sdiv_word_def[where a=1]
 
 lemmas word_sdiv_numerals = word_sdiv_numerals_lhs[where b="numeral y" for y]
     word_sdiv_numerals_lhs[where b=0] word_sdiv_numerals_lhs[where b=1]
 
 (*
  * Signed modulo properties.
  *)
 
 lemma smod_int_alt_def:
      "(a::int) smod b = sgn (a) * (abs a mod abs b)"
   apply (clarsimp simp: smod_int_def sdiv_int_def)
   apply (clarsimp simp: minus_div_mult_eq_mod [symmetric] abs_sgn sgn_mult sgn_if algebra_split_simps)
   done
 
 lemma smod_int_range:
   "b \<noteq> 0 \<Longrightarrow> (a::int) smod b \<in> { - abs b + 1 .. abs b - 1 }"
   apply (case_tac  "b > 0")
    apply (insert pos_mod_conj [where a=a and b=b])[1]
    apply (insert pos_mod_conj [where a="-a" and b=b])[1]
    apply (auto simp: smod_int_alt_def algebra_simps sgn_if
               abs_if not_less add1_zle_eq [simplified add.commute])[1]
     apply (metis add_nonneg_nonneg int_one_le_iff_zero_less le_less less_add_same_cancel2 not_le pos_mod_conj)
   apply (metis (full_types) add.inverse_inverse eucl_rel_int eucl_rel_int_iff le_less_trans neg_0_le_iff_le)
   apply (insert neg_mod_conj [where a=a and b="b"])[1]
   apply (insert neg_mod_conj [where a="-a" and b="b"])[1]
   apply (clarsimp simp: smod_int_alt_def algebra_simps sgn_if
             abs_if not_less add1_zle_eq [simplified add.commute])
   apply (metis neg_0_less_iff_less neg_mod_conj not_le not_less_iff_gr_or_eq order_trans pos_mod_conj)
   done
 
 lemma smod_int_compares:
    "\<lbrakk> 0 \<le> a; 0 < b \<rbrakk> \<Longrightarrow> (a :: int) smod b < b"
    "\<lbrakk> 0 \<le> a; 0 < b \<rbrakk> \<Longrightarrow> 0 \<le> (a :: int) smod b"
    "\<lbrakk> a \<le> 0; 0 < b \<rbrakk> \<Longrightarrow> -b < (a :: int) smod b"
    "\<lbrakk> a \<le> 0; 0 < b \<rbrakk> \<Longrightarrow> (a :: int) smod b \<le> 0"
    "\<lbrakk> 0 \<le> a; b < 0 \<rbrakk> \<Longrightarrow> (a :: int) smod b < - b"
    "\<lbrakk> 0 \<le> a; b < 0 \<rbrakk> \<Longrightarrow> 0 \<le> (a :: int) smod b"
    "\<lbrakk> a \<le> 0; b < 0 \<rbrakk> \<Longrightarrow> (a :: int) smod b \<le> 0"
    "\<lbrakk> a \<le> 0; b < 0 \<rbrakk> \<Longrightarrow> b \<le> (a :: int) smod b"
   apply (insert smod_int_range [where a=a and b=b])
   apply (auto simp: add1_zle_eq smod_int_alt_def sgn_if)
   done
 
 lemma smod_int_mod_0 [simp]:
   "x smod (0 :: int) = x"
   by (clarsimp simp: smod_int_def)
 
 lemma smod_int_0_mod [simp]:
   "0 smod (x :: int) = 0"
   by (clarsimp simp: smod_int_alt_def)
 
 lemma smod_word_mod_0 [simp]:
   "x smod (0 :: ('a::len) word) = x"
   by (clarsimp simp: smod_word_def)
 
 lemma smod_word_0_mod [simp]:
   "0 smod (x :: ('a::len) word) = 0"
   by (clarsimp simp: smod_word_def)
 
 lemma smod_word_max:
     "sint (a::'a word) smod sint (b::'a word) < 2 ^ (LENGTH('a::len) - Suc 0)"
   apply (case_tac "b = 0")
    apply (insert word_sint.Rep [where x=a, simplified sints_num])[1]
    apply (clarsimp)
   apply (insert word_sint.Rep [where x="b", simplified sints_num])[1]
   apply (insert smod_int_range [where a="sint a" and b="sint b"])
   apply (clarsimp simp: abs_if split: if_split_asm)
   done
 
 lemma smod_word_min:
     "- (2 ^ (LENGTH('a::len) - Suc 0)) \<le> sint (a::'a word) smod sint (b::'a word)"
   apply (case_tac "b = 0")
    apply (insert word_sint.Rep [where x=a, simplified sints_num])[1]
    apply clarsimp
   apply (insert word_sint.Rep [where x=b, simplified sints_num])[1]
   apply (insert smod_int_range [where a="sint a" and b="sint b"])
   apply (clarsimp simp: abs_if add1_zle_eq split: if_split_asm)
   done
 
 lemma smod_word_alt_def:
   "(a :: ('a::len) word) smod b = a - (a sdiv b) * b"
   apply (case_tac "a \<noteq> - (2 ^ (LENGTH('a) - 1)) \<or> b \<noteq> -1")
    apply (clarsimp simp: smod_word_def sdiv_word_def smod_int_def
              minus_word.abs_eq [symmetric] times_word.abs_eq [symmetric])
   apply (clarsimp simp: smod_word_def smod_int_def)
   done
 
 lemmas word_smod_numerals_lhs = smod_word_def[where a="numeral x" for x]
     smod_word_def[where a=0] smod_word_def[where a=1]
 
 lemmas word_smod_numerals = word_smod_numerals_lhs[where b="numeral y" for y]
     word_smod_numerals_lhs[where b=0] word_smod_numerals_lhs[where b=1]
 
 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"
   apply (clarsimp simp: word_of_int int_word_sint)
   apply (subst int_mod_eq')
     apply simp
    apply (subst (2) power_minus_simp)
     apply clarsimp
    apply clarsimp
   apply clarsimp
   done
 
 lemma of_int_sint [simp]:
     "of_int (sint a) = a"
   apply (insert word_sint.Rep [where x=a])
   apply (clarsimp simp: word_of_int)
   done
 
 lemma nth_w2p_scast [simp]:
   "((scast ((2::'a::len signed word) ^ n) :: 'a word) !! m)
          \<longleftrightarrow> ((((2::'a::len  word) ^ n) :: 'a word) !! m)"
   apply (subst nth_w2p)
   apply (case_tac "n \<ge> LENGTH('a)")
    apply (subst power_overflow, simp)
    apply clarsimp
   apply (metis nth_w2p scast_def test_bit_conj_lt
                len_signed nth_word_of_int word_sint.Rep_inverse)
   done
 
 lemma scast_2_power [simp]: "scast ((2 :: 'a::len signed word) ^ x) = ((2 :: 'a word) ^ x)"
   by (clarsimp simp: word_eq_iff)
 
 lemma scast_bit_test [simp]:
     "scast ((1 :: 'a::len signed word) << n) = (1 :: 'a word) << n"
   by (clarsimp simp: word_eq_iff)
 
 lemma ucast_nat_def':
   "of_nat (unat x) = (ucast :: 'a :: len word \<Rightarrow> ('b :: len) signed word) x"
   by (simp add: ucast_def word_of_int_nat unat_def)
 
 lemma mod_mod_power_int:
   fixes k :: int
   shows "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ (min m n)"
   by (metis bintrunc_bintrunc_min bintrunc_mod2p min.commute)
 
 (* Normalise combinations of scast and ucast. *)
 
 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 (clarsimp simp: word_of_int ucast_def)
   apply (subst lift_M)
   apply (subst of_int_uint [symmetric], subst lift_M')
   apply (subst (1 2) int_word_uint)
   apply (subst word_of_int)
   apply (subst word.abs_eq_iff)
   apply (subst (1 2) bintrunc_mod2p)
   apply (insert is_down)
   apply (unfold is_down_def)
   apply (clarsimp simp: target_size source_size)
   apply (clarsimp simp: mod_mod_power_int min_def)
   apply (rule distrib [symmetric])
   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_def 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_def 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_def 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_def 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_def 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_def 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_def 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_def 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 (clarsimp simp: scast_def)
   apply (metis down_cast_same is_up_down scast_def 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_bl
   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 smod_mod_positive:
     "\<lbrakk> 0 \<le> (a :: int); 0 \<le> b \<rbrakk> \<Longrightarrow> a smod b = a mod b"
   by (clarsimp simp: smod_int_alt_def zsgn_def)
 
 lemma nat_mult_power_less_eq:
   "b > 0 \<Longrightarrow> (a * b ^ n < (b :: nat) ^ m) = (a < b ^ (m - n))"
   using mult_less_cancel2[where m = a and k = "b ^ n" and n="b ^ (m - n)"]
         mult_less_cancel2[where m="a * b ^ (n - m)" and k="b ^ m" and n=1]
   apply (simp only: power_add[symmetric] nat_minus_add_max)
   apply (simp only: power_add[symmetric] nat_minus_add_max ac_simps)
   apply (simp add: max_def split: if_split_asm)
   done
 
 lemma signed_shift_guard_to_word:
   "\<lbrakk> n < len_of TYPE ('a); n > 0 \<rbrakk>
     \<Longrightarrow> (unat (x :: 'a :: len word) * 2 ^ y < 2 ^ n)
     = (x = 0 \<or> x < (1 << n >> y))"
   apply (simp only: nat_mult_power_less_eq)
   apply (cases "y \<le> n")
    apply (simp only: shiftl_shiftr1)
    apply (subst less_mask_eq)
     apply (simp add: word_less_nat_alt word_size)
     apply (rule order_less_le_trans[rotated], rule power_increasing[where n=1])
       apply simp
      apply simp
     apply simp
    apply (simp add: nat_mult_power_less_eq word_less_nat_alt word_size)
    apply auto[1]
   apply (simp only: shiftl_shiftr2, simp add: unat_eq_0)
   done
 
 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 (subst sint_eq_uint)
    apply (clarsimp simp: msb_nth nth_ucast is_down)
    apply (metis Suc_leI Suc_pred len_gt_0)
   apply (clarsimp simp: uint_up_ucast is_up is_down)
   done
 
 lemma word_less_nowrapI':
   "(x :: 'a :: len0 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"
   by (clarsimp simp: mask_def)
 
 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_def unat_def
   apply (subst int_word_uint)
   apply (subst mod_pos_pos_trivial)
     apply simp
    apply (rule lt2p_lem)
    apply (clarsimp simp: is_up)
   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 add: ucast_nat_def [symmetric])
   apply (rule of_nat_mono_maybe)
   apply (rule unat_less_helper)
   apply (simp add: Power.of_nat_power)
   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 add: ucast_nat_def [symmetric])
   apply (subst of_nat_mono_maybe_le[symmetric])
     apply (rule unat_less_helper)
     apply (simp add: Power.of_nat_power)
    apply (rule unat_less_helper)
    apply (erule le_less_trans)
    apply (simp add: Power.of_nat_power)
   apply (simp 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('a \<rightarrow> 'b) x \<le> UCAST('a \<rightarrow> 'b) y"
   by (auto simp: word_less_nat_alt intro: ucast_mono_le)
 
 lemma zero_sle_ucast_up:
   "\<not> is_down (ucast :: 'a word \<Rightarrow> 'b signed word) \<Longrightarrow>
           (0 <=s ((ucast (b::('a::len) word)) :: ('b::len) signed word))"
   apply (subgoal_tac "\<not> msb (ucast b :: 'b signed word)")
    apply (clarsimp simp: word_sle_msb_le)
   apply (clarsimp simp: is_down not_le msb_nth nth_ucast)
   apply (subst (asm) test_bit_conj_lt [symmetric])
   apply clarsimp
   apply arith
   done
 
 lemma word_le_ucast_sless:
   "\<lbrakk> x \<le> y; y \<noteq> -1; LENGTH('a) < LENGTH('b) \<rbrakk> \<Longrightarrow>
     UCAST (('a :: len) \<rightarrow> ('b :: len) signed) x <s ucast (y + 1)"
   by (clarsimp simp: word_le_make_less word_sless_alt sint_ucast_eq_uint is_down word_less_def)
 
 lemma msb_ucast_eq:
     "LENGTH('a) = LENGTH('b) \<Longrightarrow>
          msb (ucast x :: ('a::len) word) = msb (x :: ('b::len) word)"
   apply (clarsimp simp: word_msb_alt)
   apply (subst ucast_down_drop [where n=0])
    apply (clarsimp simp: source_size_def target_size_def word_size)
   apply clarsimp
   done
 
 lemma msb_big:
      "msb (a :: ('a::len) word) = (a \<ge> 2 ^ (LENGTH('a)  - Suc 0))"
   apply (rule iffI)
    apply (clarsimp simp: msb_nth)
    apply (drule bang_is_le)
    apply simp
   apply (rule ccontr)
   apply (subgoal_tac "a = a && mask (LENGTH('a) - Suc 0)")
    apply (cut_tac and_mask_less' [where w=a and n="LENGTH('a) - Suc 0"])
     apply (clarsimp simp: word_not_le [symmetric])
    apply clarsimp
   apply (rule sym, subst and_mask_eq_iff_shiftr_0)
   apply (clarsimp simp: msb_shift)
   done
 
 lemma zero_sle_ucast:
   "(0 <=s ((ucast (b::('a::len) word)) :: ('a::len) signed word))
                 = (uint b < 2 ^ (LENGTH('a) - 1))"
   apply (case_tac "msb b")
    apply (clarsimp simp: word_sle_msb_le not_less msb_ucast_eq del: notI)
    apply (clarsimp simp: msb_big word_le_def uint_2p_alt)
   apply (clarsimp simp: word_sle_msb_le not_less msb_ucast_eq del: notI)
   apply (clarsimp simp: msb_big word_le_def uint_2p_alt)
   done
 
 
 (* to_bool / from_bool. *)
 
 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_0:
   "(from_bool x = 0) = (\<not> x)"
   by (simp add: from_bool_def split: bool.split)
 
 definition
   to_bool :: "'a::len word \<Rightarrow> bool" where
   "to_bool \<equiv> (\<noteq>) 0"
 
 lemma to_bool_and_1:
   "to_bool (x && 1) = (x !! 0)"
   apply (simp add: to_bool_def)
   apply (rule iffI)
    apply (rule classical, erule notE, word_eqI)
    apply (case_tac n; simp)
   apply (rule notI, drule word_eqD[where x=0])
   apply simp
   done
 
 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) && 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 (simp add: case_bool_If from_bool_def split: if_split)
 
 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 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 word_rsplit_upt:
   "\<lbrakk> size x = LENGTH('a :: len) * n; n \<noteq> 0 \<rbrakk>
     \<Longrightarrow> word_rsplit x = map (\<lambda>i. ucast (x >> i * len_of TYPE ('a)) :: 'a word) (rev [0 ..< n])"
   apply (subgoal_tac "length (word_rsplit x :: 'a word list) = n")
    apply (rule nth_equalityI, simp)
    apply (intro allI word_eqI impI)
    apply (simp add: test_bit_rsplit_alt word_size)
    apply (simp add: nth_ucast nth_shiftr nth_rev field_simps)
   apply (simp add: length_word_rsplit_exp_size)
   apply (metis mult.commute given_quot_alt word_size word_size_gt_0)
   done
 
 lemma aligned_shift:
   "\<lbrakk>x < 2 ^ n; is_aligned (y :: 'a :: len word) n;n \<le> LENGTH('a)\<rbrakk>
    \<Longrightarrow> x + y >> n = y >> n"
   by (subst word_plus_and_or_coroll; word_eqI, blast)
 
 lemma aligned_shift':
   "\<lbrakk>x < 2 ^ n; is_aligned (y :: 'a :: len word) n;n \<le> LENGTH('a)\<rbrakk>
    \<Longrightarrow> y + x >> n = y >> n"
   by (subst word_plus_and_or_coroll; word_eqI, blast)
 
 lemma neg_mask_add_mask:
   "((x:: 'a :: len word) && ~~ (mask n)) + (2 ^ n - 1) = x || mask n"
   unfolding mask_2pm1[symmetric]
   by (subst word_plus_and_or_coroll; word_eqI_solve)
 
 lemma subtract_mask:
   "p - (p && mask n) = (p && ~~ (mask n))"
   "p - (p && ~~ (mask n)) = (p && mask n)"
   by (simp add: field_simps word_plus_and_or_coroll2)+
 
 lemma and_neg_mask_plus_mask_mono: "(p && ~~ (mask n)) + mask n \<ge> p"
   apply (rule word_le_minus_cancel[where x = "p && ~~ (mask n)"])
    apply (clarsimp simp: subtract_mask)
    using word_and_le1[where a = "mask n" and y = p]
    apply (clarsimp simp: mask_def word_le_less_eq)
   apply (rule is_aligned_no_overflow'[folded mask_2pm1])
   apply (clarsimp simp: is_aligned_neg_mask)
   done
 
 lemma word_neg_and_le:
   "ptr \<le> (ptr && ~~ (mask n)) + (2 ^ n - 1)"
   by (simp add: and_neg_mask_plus_mask_mono mask_2pm1[symmetric])
 
 lemma aligned_less_plus_1:
   "\<lbrakk> is_aligned x n; n > 0 \<rbrakk> \<Longrightarrow> x < x + 1"
   apply (rule plus_one_helper2)
    apply (rule order_refl)
   apply (clarsimp simp: field_simps)
   apply (drule arg_cong[where f="\<lambda>x. x - 1"])
   apply (clarsimp simp: is_aligned_mask)
   apply (drule word_eqD[where x=0])
   apply simp
   done
 
 lemma aligned_add_offset_less:
   "\<lbrakk>is_aligned x n; is_aligned y n; x < y; z < 2 ^ n\<rbrakk> \<Longrightarrow> x + z < y"
   apply (cases "y = 0")
    apply simp
   apply (erule is_aligned_get_word_bits[where p=y], simp_all)
   apply (cases "z = 0", simp_all)
   apply (drule(2) aligned_at_least_t2n_diff[rotated -1])
   apply (drule plus_one_helper2)
    apply (rule less_is_non_zero_p1)
    apply (rule aligned_less_plus_1)
     apply (erule aligned_sub_aligned[OF _ _ order_refl],
            simp_all add: is_aligned_triv)[1]
    apply (cases n, simp_all)[1]
   apply (simp only: trans[OF diff_add_eq diff_diff_eq2[symmetric]])
   apply (drule word_less_add_right)
    apply (rule ccontr, simp add: linorder_not_le)
    apply (drule aligned_small_is_0, erule order_less_trans)
     apply (clarsimp simp: power_overflow)
    apply simp
   apply (erule order_le_less_trans[rotated],
          rule word_plus_mono_right)
    apply (erule word_le_minus_one_leq)
   apply (simp add: is_aligned_no_wrap' is_aligned_no_overflow field_simps)
   done
 
 lemma is_aligned_add_helper:
   "\<lbrakk> is_aligned p n; d < 2 ^ n \<rbrakk>
      \<Longrightarrow> (p + d && mask n = d) \<and> (p + d && (~~ (mask n)) = p)"
   apply (subst(asm) is_aligned_mask)
   apply (drule less_mask_eq)
   apply (rule context_conjI)
    apply (subst word_plus_and_or_coroll; word_eqI; blast)
   using word_plus_and_or_coroll2[where x="p + d" and w="mask n"] by simp
 
 lemma is_aligned_sub_helper:
   "\<lbrakk> is_aligned (p - d) n; d < 2 ^ n \<rbrakk>
      \<Longrightarrow> (p && mask n = d) \<and> (p && (~~ (mask n)) = p - d)"
   by (drule(1) is_aligned_add_helper, simp)
 
 lemma mask_twice:
   "(x && mask n) && mask m = x && mask (min m n)"
   by word_eqI_solve
 
 lemma is_aligned_after_mask:
   "\<lbrakk>is_aligned k m;m\<le> n\<rbrakk> \<Longrightarrow> is_aligned (k && mask n) m"
   by (rule is_aligned_andI1)
 
 lemma and_mask_plus:
   "\<lbrakk>is_aligned ptr m; m \<le> n; a < 2 ^ m\<rbrakk>
    \<Longrightarrow> ptr + a && mask n = (ptr && mask n) + a"
   apply (rule mask_eqI[where n = m])
    apply (simp add:mask_twice min_def)
     apply (simp add:is_aligned_add_helper)
     apply (subst is_aligned_add_helper[THEN conjunct1])
       apply (erule is_aligned_after_mask)
      apply simp
     apply simp
    apply simp
   apply (subgoal_tac "(ptr + a && mask n) && ~~ (mask m)
      = (ptr + a && ~~ (mask m) ) && mask n")
    apply (simp add:is_aligned_add_helper)
    apply (subst is_aligned_add_helper[THEN conjunct2])
      apply (simp add:is_aligned_after_mask)
     apply simp
    apply simp
   apply (simp add:word_bw_comms word_bw_lcs)
   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 && mask n) && ~~ (mask n) = 0"
   apply (clarsimp simp:mask_def)
   by (metis word_bool_alg.conj_cancel_right word_bool_alg.conj_zero_right word_bw_comms(1) word_bw_lcs(1))
 
 lemma and_and_not[simp]:"(a && b) && ~~ b = 0"
   apply (subst word_bw_assocs(1))
   apply clarsimp
   done
 
 lemma mask_shift_and_negate[simp]:"(w && mask n << m) && ~~ (mask n << m) = 0"
   apply (clarsimp simp:mask_def)
   by (metis (erased, hide_lams) mask_eq_x_eq_0 shiftl_over_and_dist word_bool_alg.conj_absorb word_bw_assocs(1))
 
 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 ex_mask_1[simp]: "(\<exists>x. mask x = 1)"
   apply (rule_tac x=1 in exI)
   apply (simp add:mask_def)
   done
 
 lemma not_switch:"~~ a = x \<Longrightarrow> a = ~~ x"
   by auto
 
 (* 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 && ~~ (mask n << m) || ((y && mask n) << m)) && ~~ (mask n << m) = x && ~~ (mask n << m)"
   by (induct n arbitrary: m) (auto simp: word_ao_dist)
 
 lemma bitfield_op_twice'':
   "\<lbrakk>~~ a = b << c; \<exists>x. b = mask x\<rbrakk> \<Longrightarrow> (x && a || (y && b << c)) && a = x && a"
   apply clarsimp
   apply (cut_tac n=xa and m=c and x=x and y=y in bitfield_op_twice)
   apply (clarsimp simp:mask_def)
   apply (drule not_switch)
   apply clarsimp
   done
 
 lemma bit_twiddle_min:
   "(y::'a::len word) xor (((x::'a::len word) xor y) && (if x < y then -1 else 0)) = min x y"
   by (metis (mono_tags) min_def word_bitwise_m1_simps(2) word_bool_alg.xor_left_commute
              word_bool_alg.xor_self word_bool_alg.xor_zero_right word_bw_comms(1)
              word_le_less_eq word_log_esimps(7))
 
 lemma bit_twiddle_max:
   "(x::'a::len word) xor (((x::'a::len word) xor y) && (if x < y then -1 else 0)) = max x y"
   by (metis (mono_tags) max_def word_bitwise_m1_simps(2) word_bool_alg.xor_left_self
             word_bool_alg.xor_zero_right word_bw_comms(1) word_le_less_eq word_log_esimps(7))
 
 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 (metis word_bool_alg.xor_assoc word_bool_alg.xor_commute word_bool_alg.xor_self
             word_bool_alg.xor_zero_right)
 
 lemma scast_nop1:
   "((scast ((of_int x)::('a::len) word))::'a sword) = of_int x"
   apply (clarsimp simp:scast_def word_of_int)
   by (metis len_signed sint_sbintrunc' word_sint.Rep_inverse)
 
 lemma scast_nop2:
   "((scast ((of_int x)::('a::len) sword))::'a word) = of_int x"
   apply (clarsimp simp:scast_def word_of_int)
   by (metis len_signed sint_sbintrunc' word_sint.Rep_inverse)
 
 lemmas scast_nop[simp] = scast_nop1 scast_nop2 scast_id
 
 lemma le_mask_imp_and_mask:
   "(x::'a::len word) \<le> mask n \<Longrightarrow> x && mask n = x"
   by (metis and_mask_eq_iff_le_mask)
 
 lemma or_not_mask_nop:
   "((x::'a::len word) || ~~ (mask n)) && mask n = x && 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) || y && mask n) && ~~ (mask m) = x && ~~ (mask m)"
   apply (subst word_ao_dist)
   apply (subgoal_tac "(y && mask n) && ~~ (mask m) = 0")
    apply simp
   by (metis (no_types, hide_lams) is_aligned_mask is_aligned_weaken word_and_not
             word_bool_alg.conj_zero_right word_bw_comms(1) word_bw_lcs(1))
 
 lemma and_mask_0_iff_le_mask:
   fixes w :: "'a::len word"
   shows "(w && ~~(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) && mask m) && mask n = x && mask n"
   by (metis mask_twice min_def)
 
 lemma uint_2_id:
   "LENGTH('a) \<ge> 2 \<Longrightarrow> uint (2::('a::len) word) = 2"
   apply clarsimp
   apply (subgoal_tac "2 \<in> uints (LENGTH('a))")
    apply (subst (asm) Word.word_ubin.set_iff_norm)
    apply simp
   apply (subst word_uint.set_iff_norm)
   apply clarsimp
   apply (rule int_mod_eq')
    apply simp
   apply (rule pow_2_gt)
   apply simp
   done
 
 lemma bintrunc_id:
   "\<lbrakk>m \<le> of_nat n; 0 < m\<rbrakk> \<Longrightarrow> bintrunc n m = m"
   by (simp add: bintrunc_mod2p le_less_trans)
 
 lemma shiftr1_unfold: "shiftr1 x = x >> 1"
   by (metis One_nat_def comp_apply funpow.simps(1) funpow.simps(2) id_apply shiftr_def)
 
 lemma shiftr1_is_div_2: "(x::('a::len) word) >> 1 = x div 2"
   apply (case_tac "LENGTH('a) = 1")
    apply simp
  apply (subgoal_tac "x = 0 \<or> x = 1")
     apply (erule disjE)
      apply (clarsimp simp:word_div_def)+
    apply (metis One_nat_def less_irrefl_nat sint_1_cases)
   apply clarsimp
   apply (subst word_div_def)
   apply clarsimp
   apply (subst bintrunc_id)
     apply (subgoal_tac "2 \<le> LENGTH('a)")
      apply simp
     apply (metis (no_types) le_0_eq le_SucE lens_not_0(2) not_less_eq_eq numeral_2_eq_2)
    apply simp
   apply (subst bin_rest_def[symmetric])
   apply (subst shiftr1_def[symmetric])
   apply (clarsimp simp:shiftr1_unfold)
   done
 
 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 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_less_dividend_word:"\<lbrakk>x \<noteq> 0; n \<noteq> 1\<rbrakk> \<Longrightarrow> (x::('a::len) word) div n < x"
   apply (case_tac "n = 0")
    apply clarsimp
    apply (subst div_by_0_word)
    apply (simp add:word_neq_0_conv)
   apply (subst word_arith_nat_div)
   apply (rule word_of_nat_less)
   apply (rule div_less_dividend)
    apply (metis (poly_guards_query) One_nat_def less_one nat_neq_iff unat_eq_1(2) unat_eq_zero)
   apply (simp add:unat_gt_0)
   done
 
 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 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 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 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 max_word_max plus_1_less)
 
 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> x !! 0"
   by (metis add.commute add_left_cancel max_word_max not_less word_and_1 word_bool_alg.conj_one_right word_bw_comms(1) word_overflow zero_neq_one)
 
 lemma word_lsb_nat:"lsb w = (unat w mod 2 = 1)"
   unfolding word_lsb_def bin_last_def
   by (metis (no_types, hide_lams) nat_mod_distrib nat_numeral not_mod_2_eq_1_eq_0 numeral_One uint_eq_0 uint_nonnegative unat_0 unat_def zero_le_numeral)
 
 lemma odd_iff_lsb:"odd (unat (x::('a::len) word)) = x !! 0"
   apply (simp add:even_iff_mod_2_eq_zero)
   apply (subst word_lsb_nat[unfolded One_nat_def, symmetric])
   apply (rule word_lsb_alt)
   done
 
 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 simp+
   apply (case_tac "((of_nat x)::('a::len) word) = of_nat y")
    apply (subst (asm) word_unat.norm_eq_iff[symmetric])
    apply simp+
   done
 
 lemma shiftr1_irrelevant_lsb:"(x::('a::len) word) !! 0 \<or> x >> 1 = (x + 1) >> 1"
   apply (case_tac "LENGTH('a) = 1")
    apply clarsimp
    apply (drule_tac x=x in degenerate_word[unfolded One_nat_def])
    apply (erule disjE)
     apply clarsimp+
   apply (subst (asm) shiftr1_is_div_2[unfolded One_nat_def])+
   apply (subst (asm) word_arith_nat_div)+
   apply clarsimp
   apply (subst (asm) bintrunc_id)
     apply (subgoal_tac "LENGTH('a) > 0")
      apply linarith
     apply clarsimp+
   apply (subst (asm) bintrunc_id)
     apply (subgoal_tac "LENGTH('a) > 0")
      apply linarith
     apply clarsimp+
   apply (case_tac "x + 1 = 0")
    apply (clarsimp simp:overflow_imp_lsb)
   apply (cut_tac x=x in word_overflow_unat)
   apply clarsimp
   apply (case_tac "even (unat x)")
    apply (subgoal_tac "unat x div 2 = Suc (unat x) div 2")
     apply metis
    apply (subst numeral_2_eq_2)+
    apply simp
   apply (simp add:odd_iff_lsb)
   done
 
 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>((x::('a::len) word) !! 0) \<Longrightarrow> x >> 1 = (x + 1) >> 1"
   by (metis shiftr1_irrelevant_lsb)
 
 lemma lsb_this_or_next:"\<not>(((x::('a::len) word) + 1) !! 0) \<Longrightarrow> x !! 0"
   by (metis (poly_guards_query) even_word_imp_odd_next odd_iff_lsb overflow_imp_lsb)
 
 (* 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) || (((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 (clarsimp simp:mask_def)
    apply (metis max_word_eq max_word_max mult_2_right)
   apply (metis add_diff_cancel_left' diff_less len_gt_0 mult_2_right)
   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) ||
    (((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 mask_or_not_mask:
   "x && mask n || x && ~~ (mask n) = x"
   apply (subst word_oa_dist, simp)
   apply (subst word_oa_dist2, simp)
   done
 
 lemma is_aligned_add_not_aligned:
   "\<lbrakk>is_aligned (p::'a::len word) n; \<not> is_aligned (q::'a::len word) n\<rbrakk> \<Longrightarrow> \<not> is_aligned (p + q) n"
   by (metis is_aligned_addD1)
 
 lemma word_gr0_conv_Suc: "(m::'a::len word) > 0 \<Longrightarrow> \<exists>n. m = n + 1"
   by (metis add.commute add_minus_cancel)
 
 lemma neg_mask_add_aligned:
   "\<lbrakk> is_aligned p n; q < 2 ^ n \<rbrakk> \<Longrightarrow> (p + q) && ~~ (mask n) = p && ~~ (mask n)"
   by (metis is_aligned_add_helper is_aligned_neg_mask_eq)
 
 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 (cut_tac n=x and 'a='a in max_word_max,
       clarsimp simp:max_word_def,
       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"
   from word_unat.Abs_inj_on
   have "inj_on ?of_nat {i. i < CARD('a word)}"
     by (simp add: unats_def card_word)
   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 le_max_word_ucast_id:
   assumes "(x::'a::len word) \<le> ucast (max_word::'b::len word)"
   shows "ucast ((ucast x)::'b word) = x"
 proof -
   {
     assume a1: "x \<le> word_of_int (uint (word_of_int (2 ^ LENGTH('b) - 1)::'b word))"
     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)
   }
   with assms show ?thesis
     unfolding ucast_def
     apply (subst word_ubin.eq_norm)
     apply (subst and_mask_bintr[symmetric])
     apply (subst and_mask_eq_iff_le_mask)
     apply (clarsimp simp: max_word_def mask_def)
     done
 qed
 
 lemma remdups_enum_upto:
   fixes s::"'a::len word"
   shows "remdups [s .e. e] = [s .e. e]"
   by simp
 
 lemma card_enum_upto:
   fixes s::"'a::len word"
   shows "card (set [s .e. e]) = Suc (unat e) - unat s"
   by (subst List.card_set) (simp add: remdups_enum_upto)
 
 lemma unat_mask:
   "unat (mask n :: 'a :: len word) = 2 ^ (min n (LENGTH('a))) - 1"
   apply (subst min.commute)
   apply (simp add: mask_def 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 word_shiftr_lt:
   fixes w :: "'a::len word"
   shows "unat (w >> n) < (2 ^ (LENGTH('a) - n))"
   apply (subst shiftr_div_2n')
   by (metis nat_mod_lem nat_zero_less_power_iff power_mod_div
       word_unat.Rep_inverse word_unat.eq_norm zero_less_numeral)
 
 lemma complement_nth_w2p:
   shows "n' < LENGTH('a) \<Longrightarrow> (~~ (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 && 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) && mask m) < 2 ^ m"
   by (rule word_unat_and_lt) (simp add: unat_mask word_size)
 
 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 td_gal_lt)
 
 lemma le_or_mask:
   "w \<le> w' \<Longrightarrow> w || mask x \<le> w' || mask x"
   by (metis neg_mask_add_mask add.commute le_word_or1 mask_2pm1 neg_mask_mono_le word_plus_mono_left)
 
 lemma le_shiftr1':
   "\<lbrakk> shiftr1 u \<le> shiftr1 v ; shiftr1 u \<noteq> shiftr1 v \<rbrakk> \<Longrightarrow> u \<le> v"
   apply (unfold word_le_def shiftr1_def word_ubin.eq_norm)
   apply (unfold bin_rest_trunc_i
                 trans [OF bintrunc_bintrunc_l word_ubin.norm_Rep,
                           unfolded word_ubin.norm_Rep,
                        OF order_refl [THEN le_SucI]])
   apply (case_tac "uint u" rule: bin_exhaust)
   apply (case_tac "uint v" rule: bin_exhaust)
   by (smt Bit_def bin_rest_BIT)
 
 lemma le_shiftr':
   "\<lbrakk> u >> n \<le> v >> n ; u >> n \<noteq> v >> n \<rbrakk> \<Longrightarrow> (u::'a::len0 word) \<le> v"
   apply (induct n; simp add: shiftr_def)
   apply (case_tac "(shiftr1 ^^ n) u = (shiftr1 ^^ n) v", simp)
   apply (fastforce dest: le_shiftr1')
   done
 
 lemma word_log2_nth_same:
   "w \<noteq> 0 \<Longrightarrow> w !! word_log2 w"
   unfolding word_log2_def
   using nth_length_takeWhile[where P=Not and xs="to_bl w"]
   apply (simp add: word_clz_def word_size to_bl_nth)
   apply (fastforce simp: linorder_not_less eq_zero_set_bl
                    dest: takeWhile_take_has_property)
   done
 
 lemma word_log2_nth_not_set:
   "\<lbrakk> word_log2 w < i ; i < size w \<rbrakk> \<Longrightarrow> \<not> w !! i"
   unfolding word_log2_def word_clz_def
   using takeWhile_take_has_property_nth[where P=Not and xs="to_bl w" and n="size w - Suc i"]
   by (fastforce simp add: to_bl_nth word_size)
 
 lemma word_log2_highest:
   assumes a: "w !! i"
   shows "i \<le> word_log2 w"
 proof -
   from a have "i < size w" by - (rule test_bit_size)
   with a show ?thesis
     by - (rule ccontr, simp add: word_log2_nth_not_set)
 qed
 
 lemma word_log2_max:
   "word_log2 w < size w"
   unfolding word_log2_def word_clz_def
   by simp
 
 lemma word_clz_0[simp]:
   "word_clz (0::'a::len word) = LENGTH('a)"
   unfolding word_clz_def
   by (simp add: takeWhile_replicate)
 
 lemma word_clz_minus_one[simp]:
   "word_clz (-1::'a::len word) = 0"
   unfolding word_clz_def
   by (simp add: max_word_bl[simplified max_word_minus] takeWhile_replicate)
 
 lemma word_add_no_overflow:"(x::'a::len word) < max_word \<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"
   unfolding ucast_def unat_def
   apply (subst int_word_uint)
   apply (subst mod_pos_pos_trivial; simp?)
   apply (rule lt2p_lem)
   apply (simp add: assms)
   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[THEN iffD1])
+    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 upward_cast)
+      apply (simp add: ucast_less_ucast[OF order.strict_implies_order] upward_cast)
       apply (simp add: ucast_nat_def[symmetric])
       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]
+
 (* casting a long word to a shorter word and casting back to the long word
    is equal to the original long word -- if the word is small enough.
   'l is the longer word.
   's is the shorter word.
 *)
 lemma bl_cast_long_short_long_ingoreLeadingZero_generic:
   "\<lbrakk> length (dropWhile Not (to_bl w)) \<le> LENGTH('s); LENGTH('s) \<le> LENGTH('l) \<rbrakk> \<Longrightarrow>
    (of_bl :: _ \<Rightarrow> 'l::len word) (to_bl ((of_bl::_ \<Rightarrow> 's::len word) (to_bl w))) = w"
   by (rule word_uint_eqI) (simp add: uint_of_bl_is_bl_to_bin uint_of_bl_is_bl_to_bin_drop)
 
 (*
  Casting between longer and shorter word.
   'l is the longer word.
   's is the shorter word.
  For example: 'l::len word is 128 word (full ipv6 address)
               's::len word is 16 word (address piece of ipv6 address in colon-text-representation)
 *)
 corollary ucast_short_ucast_long_ingoreLeadingZero:
   "\<lbrakk> length (dropWhile Not (to_bl w)) \<le> LENGTH('s); LENGTH('s) \<le> LENGTH('l) \<rbrakk> \<Longrightarrow>
    (ucast:: 's::len word \<Rightarrow> 'l::len word) ((ucast:: 'l::len word \<Rightarrow> 's::len word) w) = w"
   apply (subst Word.ucast_bl)+
   apply (rule bl_cast_long_short_long_ingoreLeadingZero_generic; simp)
   done
 
 lemma length_drop_mask:
   fixes w::"'a::len word"
   shows "length (dropWhile Not (to_bl (w AND mask n))) \<le> n"
 proof -
   have "length (takeWhile Not (replicate n False @ ls)) = n + length (takeWhile Not ls)"
     for ls n by(subst takeWhile_append2) simp+
   then show ?thesis
     unfolding bl_and_mask by (simp add: dropWhile_eq_drop)
 qed
 
 lemma minus_one_word:
   "(-1 :: 'a :: len word) = 2 ^ LENGTH('a) - 1"
   by simp
 
 lemma mask_exceed:
   "n \<ge> LENGTH('a) \<Longrightarrow> (x::'a::len word) && ~~ (mask n) = 0"
   by (simp add: and_not_mask shiftr_eq_0)
 
 lemma two_power_strict_part_mono:
   "strict_part_mono {..LENGTH('a) - 1} (\<lambda>x. (2 :: 'a :: len word) ^ x)"
 proof -
   { fix n
     have "n < LENGTH('a) \<Longrightarrow> strict_part_mono {..n} (\<lambda>x. (2 :: 'a :: len word) ^ x)"
     proof (induct n)
       case 0 then show ?case by simp
     next
       case (Suc n)
       from Suc.prems
       have "2 ^ n < (2 :: 'a :: len word) ^ Suc n"
         using power_strict_increasing unat_power_lower word_less_nat_alt by fastforce
       with Suc
       show ?case by (subst strict_part_mono_by_steps) simp
     qed
   }
   then show ?thesis by simp
 qed
 
 lemma word_shift_by_2:
   "x * 4 = (x::'a::len word) << 2"
   by (simp add: shiftl_t2n)
 
 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> 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> (p::'a::len word) !! n'"
   using upper_bits_unset_is_l2p [where p=p]
   by (cases "n < LENGTH('a)") auto
 
 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
 
 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 and_mask_eq_iff_shiftr_0 less_mask_eq p2_gt_0 semiring_normalization_rules(7)
             shiftl_shiftr_id shiftl_t2n)
 
 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 (rule word_uint.Rep_eqD)
   apply (simp only: uint_word_ariths uint_div uint_power_lower)
   apply (subst mod_pos_pos_trivial, fastforce, fastforce)+
   apply (subst mod_pos_pos_trivial)
     apply (simp add: le_diff_eq uint_2p_alt)
    apply (rule less_1_helper)
    apply (rule power_increasing; simp)
   apply (subst mod_pos_pos_trivial)
     apply (simp add: uint_2p_alt)
    apply (rule less_1_helper)
    apply (rule power_increasing; simp)
   apply (subst int_div_sub_1; simp add: uint_2p_alt)
   apply (subst power_0[symmetric])
   apply (simp add: uint_2p_alt le_imp_power_dvd power_sub_int)
   done
 
 
 lemma word_add_power_off:
   fixes a :: "'a :: len word"
   assumes ak: "a < k"
   and kw: "k < 2 ^ (LENGTH('a) - m)"
   and mw: "m < LENGTH('a)"
   and off: "off < 2 ^ m"
   shows "(a * 2 ^ m) + off < k * 2 ^ m"
 proof (cases "m = 0")
   case True
   then show ?thesis using off ak by simp
 next
   case False
 
   from ak have ak1: "a + 1 \<le> k" by (rule inc_le)
   then have "(a + 1) * 2 ^ m \<noteq> 0"
     apply -
     apply (rule word_power_nonzero)
     apply (erule order_le_less_trans  [OF _ kw])
     apply (rule mw)
     apply (rule less_is_non_zero_p1 [OF ak])
     done
   then have "(a * 2 ^ m) + off < ((a + 1) * 2 ^ m)" using kw mw
     apply -
     apply (simp add: distrib_right)
     apply (rule word_plus_strict_mono_right [OF off])
     apply (rule is_aligned_no_overflow'')
     apply (rule is_aligned_mult_triv2)
     apply assumption
     done
   also have "\<dots> \<le> k * 2 ^ m" using ak1 mw kw False
     apply -
     apply (erule word_mult_le_mono1)
     apply (simp add: p2_gt_0)
     apply (simp add: word_less_nat_alt)
     apply (rule nat_less_power_trans2[simplified])
     apply (simp add: word_less_nat_alt)
     apply simp
     done
   finally show ?thesis .
 qed
 
 lemma offset_not_aligned:
   "\<lbrakk> is_aligned (p::'a::len word) n; i > 0; i < 2 ^ n; n < LENGTH('a)\<rbrakk> \<Longrightarrow>
    \<not> is_aligned (p + of_nat i) n"
   apply (erule is_aligned_add_not_aligned)
   unfolding is_aligned_def
   by (metis le_unat_uoi nat_dvd_not_less order_less_imp_le unat_power_lower)
 
 lemma length_upto_enum_one:
   fixes x :: "'a :: len word"
   assumes lt1: "x < y" and lt2: "z < y" and lt3: "x \<le> z"
   shows "[x , y .e. z] = [x]"
   unfolding upto_enum_step_def
 proof (subst upto_enum_red, subst if_not_P [OF leD [OF lt3]], clarsimp, rule conjI)
   show "unat ((z - x) div (y - x)) = 0"
   proof (subst unat_div, rule div_less)
     have syx: "unat (y - x) = unat y - unat x"
       by (rule unat_sub [OF order_less_imp_le]) fact
     moreover have "unat (z - x) = unat z - unat x"
       by (rule unat_sub) fact
 
     ultimately show "unat (z - x) < unat (y - x)"
       using lt2 lt3 unat_mono word_less_minus_mono_left by blast
   qed
 
   then show "(z - x) div (y - x) * (y - x) = 0"
     by (metis mult_zero_left unat_0 word_unat.Rep_eqD)
 qed
 
 lemma max_word_mask:
   "(max_word :: 'a::len word) = mask LENGTH('a)"
   unfolding mask_def by (metis max_word_eq shiftl_1)
 
 lemmas mask_len_max = max_word_mask[symmetric]
 
 lemma is_aligned_alignUp[simp]:
   "is_aligned (alignUp p n) n"
   by (simp add: alignUp_def complement_def is_aligned_mask mask_def word_bw_assocs)
 
 lemma alignUp_le[simp]:
   "alignUp p n \<le> p + 2 ^ n - 1"
   unfolding alignUp_def by (rule word_and_le2)
 
 lemma complement_mask:
   "complement (2 ^ n - 1) = ~~ (mask n)"
   unfolding complement_def mask_def by simp
 
 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 complement_mask is_aligned_add_helper p_assoc_help power_2_ge_iff)
 
 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 -
   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" using sz
     by (meson Euclidean_Division.div_eq_0_iff le_m1_iff_lt measure_unat order_less_trans
               unat_less_power word_less_sub_le word_mod_less_divisor)
 
   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 complement_mask)
     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 shiftr_div_2n' word_shiftr_lt)
 
   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 min.absorb2)
     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 shiftr_div_2n' word_shiftr_lt)
 
     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 (metis unat_div unat_less_helper unat_power_lower)
 
     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, hide_lams) 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 unat_def 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 (clarsimp elim!: range_subset_lower [where x = x])
 
   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 (clarsimp elim!: range_subset_upper [where x = x])
 
     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 && ~~ (mask sz)"
   unfolding alignUp_def[unfolded complement_def]
   by (simp add:mask_def[symmetric,unfolded shiftl_t2n,simplified])
 
 lemma mask_out_first_mask_some:
   "\<lbrakk> x && ~~ (mask n) = y; n \<le> m \<rbrakk> \<Longrightarrow> x && ~~ (mask m) = y && ~~ (mask m)"
   by word_eqI_solve
 
 lemma gap_between_aligned:
   "\<lbrakk>a < (b :: 'a ::len word); is_aligned a n; is_aligned b n; n < LENGTH('a) \<rbrakk>
   \<Longrightarrow> a + (2^n - 1) < b"
   by (simp add: aligned_add_offset_less)
 
 lemma mask_out_add_aligned:
   assumes al: "is_aligned p n"
   shows "p + (q && ~~ (mask n)) = (p + q) && ~~ (mask n)"
   using mask_add_aligned [OF al]
   by (simp add: mask_out_sub_mask)
 
 lemma alignUp_def3:
   "alignUp a sz = 2^ sz + (a - 1 && ~~ (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 mask_lower_twice:
   "n \<le> m \<Longrightarrow> (x && ~~ (mask n)) && ~~ (mask m) = x && ~~ (mask m)"
   by word_eqI_solve
 
 lemma mask_lower_twice2:
   "(a && ~~ (mask n)) && ~~ (mask m) = a && ~~ (mask (max n m))"
   by word_eqI_solve
 
 lemma ucast_and_neg_mask:
   "ucast (x && ~~ (mask n)) = ucast x && ~~ (mask n)"
   by word_eqI_solve
 
 lemma ucast_and_mask:
   "ucast (x && mask n) = ucast x && mask n"
   by word_eqI_solve
 
 lemma ucast_mask_drop:
   "LENGTH('a :: len) \<le> n \<Longrightarrow> (ucast (x && mask n) :: 'a word) = ucast x"
   by word_eqI
 
 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 && ~~ (mask sz)) = (p - ((alignUp q sz) && ~~ (mask sz)))"
   apply (clarsimp simp only:word_and_le2 diff_conv_add_uminus)
   apply (subst mask_out_add_aligned[symmetric]; simp)
   apply (rule sum_to_zero)
   apply (subst add.commute)
   by (simp add: alignUp_distance and_mask_0_iff_le_mask is_aligned_neg_mask_eq mask_out_add_aligned)
 
 lemma map_bits_rev_to_bl:
   "map ((!!) x) [0..<size x] = rev (to_bl x)"
   by (auto simp: list_eq_iff_nth_eq test_bit_bl word_size)
 
 (* negating a mask which has been shifted to the very left *)
 lemma NOT_mask_shifted_lenword:
   "~~ (mask len << (LENGTH('a) - len) ::'a::len word) = mask (LENGTH('a) - len)"
   by (rule Word.word_bool_alg.compl_unique, simp) word_eqI_solve
 
 (* Comparisons between different word sizes. *)
 
 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 (simp add: is_down ucast_id ucast_ucast_a)
 
 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_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 word_le_not_less:
   "((b::'a::len word) \<le> a) = (\<not>(a < b))"
   by fastforce
 
 lemma ucast_or_distrib:
   fixes x :: "'a::len word"
   fixes y :: "'a::len word"
   shows "(ucast (x || y) :: ('b::len) word) = ucast x || ucast y"
   unfolding ucast_def Word.bitOR_word.abs_eq uint_or
   by blast
 
 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 && w' \<noteq> 0 \<Longrightarrow> w \<noteq> 0 \<and> w' \<noteq> 0"
   by auto
 
 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"
       by (fastforce intro!: list_of_false)
     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
 
 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 from_bool_eq_if':
   "((if P then 1 else 0) = from_bool Q) = (P = Q)"
   by (simp add: case_bool_If from_bool_def split: if_split)
 
 lemma word_exists_nth:
   "(w::'a::len word) \<noteq> 0 \<Longrightarrow> \<exists>i. w !! i"
   using word_log2_nth_same by blast
 
 lemma shiftr_le_0:
   "unat (w::'a::len word) < 2 ^ n \<Longrightarrow> w >> n = (0::'a::len word)"
   by (rule word_unat.Rep_eqD) (simp add: shiftr_div_2n')
 
 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)
 
 lemma max_word_not_0[simp]:
   "max_word \<noteq> 0"
   by (simp add: max_word_minus)
 
 lemma ucast_zero_is_aligned:
   "UCAST('a::len \<rightarrow> 'b::len) w = 0 \<Longrightarrow> n \<le> LENGTH('b) \<Longrightarrow> is_aligned w n"
   by (clarsimp simp: is_aligned_mask word_eq_iff word_size nth_ucast)
 
 lemma unat_ucast_eq_unat_and_mask:
   "unat (UCAST('b::len \<rightarrow> 'a::len) w) = unat (w && mask LENGTH('a))"
   proof -
     have "unat (UCAST('b \<rightarrow> 'a) w) = unat (UCAST('a \<rightarrow> 'b) (UCAST('b \<rightarrow> 'a) w))"
       by (cases "LENGTH('a) < LENGTH('b)"; simp add: is_down ucast_ucast_a unat_ucast_up_simp)
     thus ?thesis using ucast_ucast_mask by simp
   qed
 
 lemma unat_max_word_pos[simp]: "0 < unat max_word"
   by (auto simp: unat_gt_0)
 
 
 (* 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 (cases n)
   case 0 thus ?thesis using eq by simp
 next
   case (Suc n')
   have m_lt: "m < LENGTH('a)" using Suc ws by simp
   have xylt: "x < 2^m" "y < 2^m" using le m_lt unfolding mask_2pm1 by auto
   have lenm: "n \<le> LENGTH('a) - m" using ws by simp
   show ?thesis
     using eq xylt
     apply (fold shiftl_t2n[where n=n, simplified mult.commute])
     apply (simp only: word_bl.Rep_inject[symmetric] bl_shiftl)
     apply (erule ssubst[OF less_is_drop_replicate])+
     apply (clarsimp elim!: drop_eq_mono[OF lenm])
     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])
 
 (* Sign extension from bit n. *)
 
 lemma sign_extend_bitwise_if:
   "i < size w \<Longrightarrow> sign_extend e w !! i \<longleftrightarrow> (if i < e then w !! i else w !! e)"
   by (simp add: sign_extend_def neg_mask_test_bit word_size)
 
 lemma sign_extend_bitwise_disj:
   "i < size w \<Longrightarrow> sign_extend e w !! i \<longleftrightarrow> i \<le> e \<and> w !! i \<or> e \<le> i \<and> w !! e"
   by (auto simp: sign_extend_bitwise_if)
 
 lemma sign_extend_bitwise_cases:
   "i < size w \<Longrightarrow> sign_extend e w !! i \<longleftrightarrow> (i \<le> e \<longrightarrow> w !! i) \<and> (e \<le> i \<longrightarrow> w !! e)"
   by (auto simp: sign_extend_bitwise_if)
 
 lemmas sign_extend_bitwise_if'[word_eqI_simps] = sign_extend_bitwise_if[simplified word_size]
 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 w !! n then w || ~~ (mask (Suc n)) else w && mask (Suc n))"
   by word_eqI (auto dest: less_antisym)
 
 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 (rule iffI)
    apply (word_eqI, rename_tac i)
    apply (case_tac "n < i"; simp add: sign_extended_def word_size)
   apply (erule subst, rule sign_extended_sign_extend)
   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 word_eqI
 
 lemma sign_extended_high_bits:
   "\<lbrakk> sign_extended e p; j < size p; e \<le> i; i < j \<rbrakk> \<Longrightarrow> p !! i = p !! j"
   by (drule (1) sign_extended_weaken; simp add: sign_extended_def)
 
 lemma sign_extend_eq:
   "w && mask (Suc n) = v && mask (Suc n) \<Longrightarrow> sign_extend n w = sign_extend n v"
   by word_eqI_solve
 
 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> f !! e"
     using True e less_2p_is_upper_bits_unset[THEN iffD1, OF f]
     by (fastforce simp: word_size)
   hence i: "(p + f) !! e = p !! e"
     by (simp add: and_or)
   have fm: "f && mask e = f"
     by (fastforce intro: subst[where P="\<lambda>f. f && 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 && ~~ (mask m))"
   by (fastforce simp: sign_extended_def word_size neg_mask_test_bit)
 
 (* Uints *)
 
 lemma uints_mono_iff: "uints l \<subseteq> uints m \<longleftrightarrow> l \<le> m"
   using power_increasing_iff[of "2::int" l m]
   apply (auto simp: uints_num subset_iff simp del: power_increasing_iff)
   by (meson less_irrefl not_less zle2p)
 
 lemmas uints_monoI = uints_mono_iff[THEN iffD2]
 
 lemma Bit_in_uints_Suc: "w BIT c \<in> uints (Suc m)" if "w \<in> uints m"
   using that
   by (auto simp: uints_num Bit_def)
 
 lemma Bit_in_uintsI: "w BIT c \<in> uints m" if "w \<in> uints (m - 1)" "m > 0"
   using Bit_in_uints_Suc[OF that(1)] that(2)
   by auto
 
 lemma bin_cat_in_uintsI: "bin_cat a n b \<in> uints m" if "a \<in> uints l" "m \<ge> l + n"
   using that
 proof (induction n arbitrary: b m)
   case 0
   then have "uints l \<subseteq> uints m"
     by (intro uints_monoI) auto
   then show ?case using 0 by auto
 next
   case (Suc n)
   then show ?case
     by (auto intro!: Bit_in_uintsI)
 qed
 
 lemma bin_cat_cong: "bin_cat a n b = bin_cat c m d"
   if "n = m" "a = c" "bintrunc m b = bintrunc m d"
   using that(3) unfolding that(1,2)
 proof (induction m arbitrary: b d)
   case (Suc m)
   show ?case
     using Suc.prems by (auto intro: Suc.IH)
 qed simp
 
 lemma bin_cat_eqD1: "bin_cat a n b = bin_cat c n d \<Longrightarrow> a = c"
   by (induction n arbitrary: b d) auto
 
 lemma bin_cat_eqD2: "bin_cat a n b = bin_cat c n d \<Longrightarrow> bintrunc n b = bintrunc n d"
   by (induction n arbitrary: b d) auto
 
 lemma bin_cat_inj: "(bin_cat a n b) = bin_cat c n d \<longleftrightarrow> a = c \<and> bintrunc n b = bintrunc n d"
   by (auto intro: bin_cat_cong bin_cat_eqD1 bin_cat_eqD2)
 
 lemma word_of_int_bin_cat_eq_iff:
   "(word_of_int (bin_cat (uint a) LENGTH('b) (uint b))::'c::len0 word) =
   word_of_int (bin_cat (uint c) LENGTH('b) (uint d)) \<longleftrightarrow> b = d \<and> a = c"
   if "LENGTH('a) + LENGTH('b) \<le> LENGTH('c)"
   for a::"'a::len0 word" and b::"'b::len0 word"
   by (subst word_uint.Abs_inject)
      (auto simp: bin_cat_inj intro!: that bin_cat_in_uintsI)
 
 lemma word_cat_inj: "(word_cat a b::'c::len0 word) = word_cat c d \<longleftrightarrow> a = c \<and> b = d"
   if "LENGTH('a) + LENGTH('b) \<le> LENGTH('c)"
   for a::"'a::len0 word" and b::"'b::len0 word"
   by (auto simp: word_cat_def word_uint.Abs_inject word_of_int_bin_cat_eq_iff that)
 
 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
 
 (* 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 unat_minus_one_word:
   "unat (-1 :: 'a :: len word) = 2 ^ LENGTH('a) - 1"
   by (subst minus_one_word)
      (subst unat_sub_if', clarsimp)
 
 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_ctz_le:
   "word_ctz (w :: ('a::len word)) \<le> LENGTH('a)"
   apply (clarsimp simp: word_ctz_def)
   apply (rule nat_le_Suc_less_imp[where y="LENGTH('a) + 1" , simplified])
   apply (rule order_le_less_trans[OF List.length_takeWhile_le])
   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)
   apply (rule order_less_le_trans[OF length_takeWhile_less])
    apply fastforce+
   done
 
 lemma word_ctz_not_minus_1:
   "1 < LENGTH('a) \<Longrightarrow> of_nat (word_ctz (w :: 'a :: len word)) \<noteq> (- 1 :: 'a::len word)"
   by (metis (mono_tags) One_nat_def add.right_neutral add_Suc_right le_diff_conv le_less_trans
                         n_less_equal_power_2 not_le suc_le_pow_2 unat_minus_one_word unat_of_nat_len
                         word_ctz_le)
 
 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^n + mask n"
   by (simp add: mask_def)
 
 lemma is_aligned_no_overflow_mask:
   "is_aligned x n \<Longrightarrow> x \<le> x + mask n"
   by (simp add: mask_def) (erule is_aligned_no_overflow')
 
 lemma is_aligned_mask_offset_unat:
   fixes off :: "('a::len) word"
   and     x :: "'a word"
   assumes al: "is_aligned x sz"
   and   offv: "off \<le> mask sz"
   shows  "unat x + unat off < 2 ^ LENGTH('a)"
 proof cases
   assume szv: "sz < LENGTH('a)"
   from al obtain k where xv: "x = 2 ^ sz * (of_nat k)"
     and kl: "k < 2 ^ (LENGTH('a) - sz)"
     by (auto elim: is_alignedE)
 
   from offv szv have offv': "unat off < 2 ^ sz"
     by (simp add: mask_2pm1 unat_less_power)
 
   show ?thesis using szv
     using al is_aligned_no_wrap''' offv' by blast
 next
   assume "\<not> sz < LENGTH('a)"
   with al have "x = 0" by - word_eqI
   thus ?thesis by simp
 qed
 
 lemma of_bl_max:
   "(of_bl xs :: 'a::len word) \<le> mask (length xs)"
   apply (induct xs)
    apply simp
   apply (simp add: of_bl_Cons mask_Suc)
   apply (rule conjI; clarsimp)
    apply (erule word_plus_mono_right)
    apply (rule is_aligned_no_overflow_mask)
    apply (rule is_aligned_triv)
   apply (simp add: word_le_nat_alt)
   apply (subst unat_add_lem')
    apply (rule is_aligned_mask_offset_unat)
     apply (rule is_aligned_triv)
    apply (simp add: mask_def)
   apply simp
   done
 
 lemma mask_over_length:
   "LENGTH('a) \<le> n \<Longrightarrow> mask n = (-1::'a::len word)"
   by (simp add: mask_def)
 
 lemma is_aligned_over_length:
   "\<lbrakk> is_aligned p n; LENGTH('a) \<le> n \<rbrakk> \<Longrightarrow> (p::'a::len word) = 0"
   by (simp add: is_aligned_mask mask_over_length)
 
 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 is_aligned_add_step_le:
   "\<lbrakk> is_aligned (a::'a::len word) n; is_aligned b n; a < b; b \<le> a + mask n \<rbrakk> \<Longrightarrow> False"
   apply (simp flip: not_le)
   apply (erule notE)
   apply (cases "LENGTH('a) \<le> n")
    apply (drule (1) is_aligned_over_length)+
    apply (drule mask_over_length)
    apply clarsimp
   apply (clarsimp simp: word_le_nat_alt not_less)
   apply (subst (asm) unat_plus_simple[THEN iffD1], erule is_aligned_no_overflow_mask)
   apply (clarsimp simp: is_aligned_def dvd_def word_le_nat_alt)
   apply (drule le_imp_less_Suc)
   apply (simp add: Suc_2p_unat_mask)
   by (metis Groups.mult_ac(2) Suc_leI linorder_not_less mult_le_mono order_refl times_nat.simps(2))
 
 lemma power_2_mult_step_le:
   "\<lbrakk>n' \<le> n; 2 ^ n' * k' < 2 ^ n * k\<rbrakk> \<Longrightarrow> 2 ^ n' * (k' + 1) \<le> 2 ^ n * (k::nat)"
   apply (cases "n'=n", simp)
    apply (metis Suc_leI le_refl mult_Suc_right mult_le_mono semiring_normalization_rules(7))
   apply (drule (1) le_neq_trans)
   apply clarsimp
   apply (subgoal_tac "\<exists>m. n = n' + m")
    prefer 2
    apply (simp add: le_Suc_ex)
   apply (clarsimp simp: power_add)
   by (metis Suc_leI mult.assoc mult_Suc_right nat_mult_le_cancel_disj)
 
 lemma aligned_mask_step:
   "\<lbrakk> n' \<le> n; p' \<le> p + mask n; is_aligned p n; is_aligned p' n' \<rbrakk> \<Longrightarrow>
    (p'::'a::len word) + mask n' \<le> p + mask n"
   apply (cases "LENGTH('a) \<le> n")
    apply (frule (1) is_aligned_over_length)
    apply (drule mask_over_length)
    apply clarsimp
   apply (simp add: not_le)
   apply (simp add: word_le_nat_alt unat_plus_simple)
   apply (subst unat_plus_simple[THEN iffD1], erule is_aligned_no_overflow_mask)+
   apply (subst (asm) unat_plus_simple[THEN iffD1], erule is_aligned_no_overflow_mask)
   apply (clarsimp simp: is_aligned_def dvd_def)
   apply (rename_tac k k')
   apply (thin_tac "unat p = x" for p x)+
   apply (subst Suc_le_mono[symmetric])
   apply (simp only: Suc_2p_unat_mask)
   apply (drule le_imp_less_Suc, subst (asm) Suc_2p_unat_mask, assumption)
   apply (erule (1) power_2_mult_step_le)
   done
 
 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 && b = 0"
   by word_eqI_solve
 
 lemma word_and_or_mask_aligned:
   "\<lbrakk> is_aligned a n; b \<le> mask n \<rbrakk> \<Longrightarrow> a + b = a || b"
   by (simp add: aligned_mask_disjoint word_plus_and_or_coroll)
 
 lemmas word_and_or_mask_aligned2 =
   word_and_or_mask_aligned[where a=b and b=a for a b,
                            simplified add.commute word_bool_alg.disj.commute]
 
 lemma is_aligned_ucastI:
   "is_aligned w n \<Longrightarrow> is_aligned (ucast w) n"
   by (clarsimp simp: word_eqI_simps)
 
 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 is_aligned_ucastI ucast_le_maskI ucast_or_distrib word_and_or_mask_aligned2)
 
 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::len0 word) \<le> mask n"
   by (clarsimp simp: le_mask_high_bits word_size nth_ucast)
 
 lemma shiftl_inj:
   "\<lbrakk> x << n = y << n; x \<le> mask (LENGTH('a)-n); y \<le> mask (LENGTH('a)-n) \<rbrakk> \<Longrightarrow>
    x = (y :: 'a :: len word)"
   apply word_eqI
   apply (rename_tac n')
   apply (case_tac "LENGTH('a) - n \<le> n'", simp)
   by (metis add.commute add.right_neutral diff_add_inverse le_diff_conv linorder_not_less zero_order(1))
 
 lemma distinct_word_add_ucast_shift_inj:
   "\<lbrakk> p + (UCAST('a::len \<rightarrow> 'b::len) off << n) = p' + (ucast off' << n);
      is_aligned p n'; is_aligned p' n'; n' = n + LENGTH('a); n' < LENGTH('b) \<rbrakk>
    \<Longrightarrow> p' = p \<and> off' = off"
   apply (simp add: word_and_or_mask_aligned le_mask_shiftl_le_mask[where n="LENGTH('a)"]
                    ucast_leq_mask)
   apply (simp add: is_aligned_nth)
   apply (rule conjI; word_eqI)
    apply (metis add.commute test_bit_conj_lt diff_add_inverse le_diff_conv nat_less_le)
   apply (rename_tac i)
   apply (erule_tac x="i+n" in allE)
   apply simp
   done
 
 lemma aligned_add_mask_lessD:
   "\<lbrakk> x + mask n < y; is_aligned x n \<rbrakk> \<Longrightarrow> x < y" for y::"'a::len word"
   by (metis is_aligned_no_overflow' mask_2pm1 order_le_less_trans)
 
 lemma aligned_add_mask_less_eq:
   "\<lbrakk> is_aligned x n; is_aligned y n;  n < LENGTH('a) \<rbrakk> \<Longrightarrow> (x + mask n < y) = (x < y)"
   for y::"'a::len word"
   using aligned_add_mask_lessD is_aligned_add_step_le word_le_not_less by blast
 
 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 max_word_not_less[simp]:
    "\<not> max_word < x"
   by (simp add: not_less)
 
 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 fold_eq_0_to_bool:
   "(v = 0) = (\<not> to_bool v)"
   by (simp add: to_bool_def)
 
 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:
   "m && (2 ^ n - 1) = 0 \<Longrightarrow> (x + y * m) && ~~(mask n) = (x && ~~(mask n)) + y * m"
   by (metis Groups.add_ac(2) is_aligned_mask mask_def mask_eqs(5) mask_out_add_aligned
             mult_zero_right shiftl_1 word_bw_comms(1) word_log_esimps(1))
 
 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: 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 && mask n = 0 \<Longrightarrow> x + y && ~~(mask n) = (x && ~~(mask n)) + y"
   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
 
 lemma add_right_shift:
   "\<lbrakk> x && mask n = 0; y && 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_def)
   done
 
 lemma sub_right_shift:
   "\<lbrakk> x && mask n = 0; y && 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 && mask n = mask n \<Longrightarrow> (x && y) && mask n = x && mask n"
   by (simp add: word_bool_alg.conj.assoc)
 
 lemma and_and_mask_simple_not:
   "y && mask n = 0 \<Longrightarrow> (x && y) && mask n = 0"
   by (simp add: word_bool_alg.conj.assoc)
 
 lemma word_and_le':
   "b \<le> c \<Longrightarrow> (a :: 'a :: len word) && b \<le> c"
   by (metis word_and_le1 order_trans)
 
 lemma word_and_less':
   "b < c \<Longrightarrow> (a :: 'a :: len word) && b < c"
   by (metis word_and_le1 xtr7)
 
 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 ucast_ucast_mask_eq:
   "\<lbrakk> UCAST('a::len \<rightarrow> 'b::len) x = y; x && mask LENGTH('b) = x \<rbrakk> \<Longrightarrow> x = ucast y"
   by word_eqI_solve
 
 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 word_eqI_solve
 
 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 && mask n = 0; m \<le> n \<rbrakk> \<Longrightarrow> x && mask m = 0"
   by (metis mask_twice2 word_and_notzeroD)
 
 lemma mask_len_id [simp]:
   "(x :: 'a :: len word) && 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) && 1 = x \<rbrakk> \<Longrightarrow> x = 0 \<or> x = 1"
   by (metis word_and_1)
 
 lemma pow_mono_leq_imp_lt:
   "x \<le> y \<Longrightarrow> x < 2 ^ y"
   by (simp add: le_less_trans)
 
 lemma unat_of_nat_ctz_mw:
   "unat (of_nat (word_ctz (w :: 'a :: len word)) :: 'a :: len word) = word_ctz w"
   using word_ctz_le[where w=w, simplified] unat_of_nat_eq[where x="word_ctz w" and 'a="'a"]
         pow_mono_leq_imp_lt
   by simp
 
 lemma unat_of_nat_ctz_smw:
   "unat (of_nat (word_ctz (w :: 'a :: len word)) :: 'a :: len sword) = word_ctz w"
   using word_ctz_le[where w=w, simplified] unat_of_nat_eq[where x="word_ctz w" and 'a="'a"]
         pow_mono_leq_imp_lt
   by (metis le_unat_uoi le_unat_uoi linorder_neqE_nat nat_less_le scast_of_nat
             word_unat.Rep_inverse)
 
 lemma shiftr_and_eq_shiftl:
   "(w >> n) && x = y \<Longrightarrow> w && (x << n) = (y << n)" for y :: "'a:: len word"
   by (smt and_not_mask is_aligned_neg_mask_eq is_aligned_shift shift_over_ao_dists(4)
           word_bool_alg.conj_assoc word_bw_comms(1))
 
 lemma neg_mask_combine:
   "~~(mask a) && ~~(mask b) = ~~(mask (max a b))"
   by (auto simp: word_ops_nth_size word_size intro!: word_eqI)
 
 lemma neg_mask_twice:
   "x && ~~(mask n) && ~~(mask m) = x && ~~(mask (max n m))"
   by (metis neg_mask_combine)
 
 lemma multiple_mask_trivia:
   "n \<ge> m \<Longrightarrow> (x && ~~(mask n)) + (x && mask n && ~~(mask m)) = x && ~~(mask m)"
   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 add_mask_lower_bits':
   "\<lbrakk> len = LENGTH('a); is_aligned (x :: 'a :: len word) n;
      \<forall>n' \<ge> n. n' < len \<longrightarrow> \<not> p !! n' \<rbrakk>
    \<Longrightarrow> x + p && ~~(mask n) = x"
   using add_mask_lower_bits by auto
 
 lemma neg_mask_in_mask_range:
   "is_aligned ptr bits \<Longrightarrow> (ptr' && ~~(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 le_word_or2 neg_mask_add_mask word_bool_alg.conj.right_idem)
    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_def)
   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 (simp only: is_aligned_add_or flip: neg_mask_in_mask_range)
   by (metis less_mask_eq mask_subsume)
 
 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_def)
    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_def)
    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_def)
   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 (erule nonemptyE)
   apply simp
   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_def)
 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 (intro range_subsetI)
      apply (rule is_aligned_no_wrap' [OF al xsz])
     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_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)
   by (simp add: aligned_add_mask_less_eq is_aligned_weaken algebra_split_simps)
 
 lemma aligned_mask_ranges_disjoint:
   "\<lbrakk> is_aligned (p :: 'a :: len word) n; is_aligned (p' :: 'a :: len word) n';
      p && ~~(mask n') \<noteq> p'; p' && ~~(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 && mask n >> m" in spec)
   apply (clarsimp simp: shiftr_less_t2n and_mask_less_size wsst_TYs multiple_mask_trivia
                         word_bw_assocs neg_mask_twice max_absorb2 shiftr_shiftl1)
   done
 
 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)
 
 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 (clarsimp simp: word_le_def uint_nat of_nat_inverse)
 
 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 && ~~(mask sz)) + 2 ^ sz - 1"
   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))"
   apply (case_tac "LENGTH('a) = 1")
    apply (clarsimp simp: ucast_def)
    apply (metis (hide_lams, mono_tags) One_nat_def len_signed plus_word.abs_eq
                                        uint_word_arith_bintrs(1) word_ubin.Abs_norm)
   apply (clarsimp simp: ucast_def)
   apply (metis le_refl len_signed plus_word.abs_eq uint_word_arith_bintrs(1) wi_bintr)
   done
 
 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, hide_lams) 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) && (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) && mask m) = unat (x && mask m)"
   by (metis ucast_and_mask unat_ucast_up_simp)
 
 lemma small_powers_of_2:
   "x \<ge> 3 \<Longrightarrow> x < 2 ^ (x - 1)"
   by (induct x; simp add: suc_le_pow_2)
 
 lemma word_clz_sint_upper[simp]:
   "LENGTH('a) \<ge> 3 \<Longrightarrow> sint (of_nat (word_clz (w :: 'a :: len word)) :: 'a sword) \<le> LENGTH('a)"
   using small_powers_of_2
   by (smt One_nat_def diff_less le_less_trans len_gt_0 len_signed lessI n_less_equal_power_2
            not_msb_from_less of_nat_mono sint_eq_uint uint_nat unat_of_nat_eq unat_power_lower
            word_clz_max word_of_nat_less wsst_TYs(3))
 
 lemma word_clz_sint_lower[simp]:
   "LENGTH('a) \<ge> 3
    \<Longrightarrow> - sint (of_nat (word_clz (w :: 'a :: len word)) :: 'a signed word) \<le> LENGTH('a)"
   apply (subst sint_eq_uint)
    using small_powers_of_2 uint_nat
    apply (simp add: order_le_less_trans[OF word_clz_max] not_msb_from_less word_of_nat_less
                     word_size)
   by (simp add: uint_nat)
 
 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_def 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"
   using less_not_refl3 le_step_down_nat le_trans less_or_eq_imp_le word_shiftr_lt
   by (metis (no_types, lifting))
 
 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"
   by (simp add: mask_shift multi_shift_simps(5) shiftr_shiftr)
 
 lemma of_int_uint_ucast:
    "of_int (uint (x :: 'a::len word)) = (ucast x :: 'b::len word)"
   by (simp add: ucast_def word_of_int)
 
 lemma mod_mask_drop:
   "\<lbrakk> m = 2 ^ n; 0 < m; mask n && msk = mask n \<rbrakk>
    \<Longrightarrow> (x mod m) && msk = x mod m"
   by (simp add: word_mod_2p_is_mask word_bw_assocs)
 
 lemma mask_eq_ucast_eq:
   "\<lbrakk> x && 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 && ~~(mask n)"
   by (metis (no_types) AND_NOT_mask_plus_AND_mask_eq add_right_imp_eq diff_add_cancel
                        and_mask_eq_iff_shiftr_0 shiftr_mask_le word_bool_alg.conj.commute)
 
 lemma neg_mask_diff_bound:
   "sz'\<le> sz \<Longrightarrow> (ptr && ~~(mask sz')) - (ptr && ~~(mask sz)) \<le> 2 ^ sz - 2 ^ sz'"
   (is "_ \<Longrightarrow> ?lhs \<le> ?rhs")
 proof -
   assume lt: "sz' \<le> sz"
   hence "?lhs = ptr && (mask sz && ~~(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_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 of_bl_length2:
   "length xs + c < LENGTH('a) \<Longrightarrow> of_bl xs * 2^c < (2::'a::len word) ^ (length xs + c)"
   by (simp add: of_bl_length word_less_power_trans2)
 
 lemma mask_out_eq_0:
   "\<lbrakk> idx < 2 ^ sz; sz < LENGTH('a) \<rbrakk> \<Longrightarrow> (of_nat idx :: 'a :: len word) && ~~(mask sz) = 0"
   by (simp add: Word_Lemmas.of_nat_power less_mask_eq mask_eq_0_eq_x)
 
 lemma is_aligned_neg_mask_eq':
   "is_aligned ptr sz = (ptr && ~~(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) && ~~(mask sz)) + unat (ptr && 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:
   "\<lbrakk> unat (x :: 'a :: len word) < 2 ^ (m - n); m < LENGTH('a) \<rbrakk> \<Longrightarrow> unat (x << n) < 2 ^ m"
   by (metis (no_types) Word_Lemmas.of_nat_power diff_le_self le_less_trans shiftl_less_t2n
                        unat_less_power word_unat.Rep_inverse)
 
 lemma unat_is_aligned_add:
   "\<lbrakk> is_aligned p n; unat d < 2 ^ n \<rbrakk>
    \<Longrightarrow> unat (p + d && mask n) = unat d \<and> unat (p + d && ~~(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) && ~~(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) && mask high_bits = x >> low_bits"
   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 && mask low_bits = 0) = (x = 0)"
   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) && mask n = q && 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 from_to_bool_last_bit:
   "from_bool (to_bool (x && 1)) = x && 1"
   by (metis from_bool_to_bool_iff word_and_1)
 
 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) && mask sz) + unat z < 2 ^ sz; sz < LENGTH('a) \<rbrakk>
    \<Longrightarrow> x \<le> x + z"
   apply (subgoal_tac "(x && ~~(mask sz)) + (x && mask sz) \<le> (x && ~~(mask sz)) + ((x && 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 Word_Lemmas.of_nat_power is_aligned_no_wrap' by force
 
 lemma mul_not_mask_eq_neg_shiftl:
   "~~(mask n) = -1 << n"
   by (simp add: NOT_mask shiftl_t2n)
 
 lemma shiftr_mul_not_mask_eq_and_not_mask:
   "(x >> n) * ~~(mask n) = - (x && ~~(mask n))"
   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) && ~~(mask n) = p + (x && ~~(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) && ~~(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) && mask (m::nat) = mask (min m n)"
   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)
 
 lemmas unat_ucast_mask = unat_ucast_eq_unat_and_mask[where w=a for a]
 
 lemma t2n_mask_eq_if:
   "2 ^ n && mask m = (if n < m then 2 ^ n else 0)"
   by (rule word_eqI, auto simp add: word_size nth_w2p split: if_split)
 
 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 (max_word :: '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 && mask a >> b) :: 'a :: len word) = p && 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 && mask a >> b) :: 'a :: len word) = unat (p && mask a >> b)"
   by (metis linear ucast_ucast_mask_shift unat_ucast_up_simp)
 
 lemma mask_overlap_zero:
   "a \<le> b \<Longrightarrow> (p && mask a) && ~~(mask b) = 0"
   by (metis NOT_mask_AND_mask mask_lower_twice2 max_def)
 
 lemma mask_shifl_overlap_zero:
   "a + c \<le> b \<Longrightarrow> (p && mask a << c) && ~~(mask b) = 0"
   by (metis and_not_mask le_mask_iff mask_shiftl_decompose shiftl_0 shiftl_over_and_dist
             shiftr_mask_le word_and_le' word_bool_alg.conj_commute)
 
 lemma mask_overlap_zero':
   "a \<ge> b \<Longrightarrow> (p && ~~(mask a)) && mask b = 0"
   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 && ~~(mask a)) + ((p && mask a) && ~~(mask b)) + (p && mask b) = p"
   by (simp add: add.commute multiple_mask_trivia word_bool_alg.conj_commute
                 word_bool_alg.conj_left_commute word_plus_and_or_coroll2)
 
 lemma mask_shift_eq_mask_mask:
   "(p && mask a >> b << b) = (p && mask a) && ~~(mask b)"
   by (simp add: and_not_mask)
 
 lemma mask_shift_sum:
   "\<lbrakk> a \<ge> b; unat n = unat (p && mask b) \<rbrakk>
    \<Longrightarrow> (p && ~~(mask a)) + (p && mask a >> b) * (1 << b) + n = (p :: 'a :: len word)"
   by (metis and_not_mask mask_rshift_mult_eq_rshift_lshift mask_split_sum_twice word_unat.Rep_eqD)
 
 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 (metis scast_def word_of_int)
 
 lemma scast_of_nat_to_signed [simp]:
   "scast (of_nat x :: 'a :: len word) = (of_nat x :: 'a signed word)"
   by (metis cast_simps(23) scast_scast_id(2))
 
 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 && mask n \<in> of_nat ` set [0 ..< 2 ^ n]"
 proof -
   have "x && mask n \<in> {0 .. 2 ^ n - 1}"
     by (simp add: mask_def word_and_le1)
   also
   have "... = of_nat ` {0 .. 2 ^ n - 1}"
     apply (rule set_eqI, rule iffI)
      apply (clarsimp simp: image_iff)
      apply (rule_tac x="unat x" in bexI; simp)
      using len
      apply (simp add: word_le_nat_alt unat_2tp_if unat_minus_one)
     using len
     apply (clarsimp simp: word_le_nat_alt unat_2tp_if unat_minus_one)
     apply (subst unat_of_nat_eq; simp add: nat_le_Suc_less)
     apply (erule less_le_trans)
     apply simp
     done
   also have "{0::nat .. 2^n - 1} = set [0 ..< 2^n]" by (auto simp: nat_le_Suc_less)
   finally show ?thesis .
 qed
 
 lemma sint_of_nat_ge_zero:
   "x < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> sint (of_nat x :: 'a :: len word) \<ge> 0"
   by (simp add: Word_Lemmas.of_nat_power not_msb_from_less sint_eq_uint)
 
 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 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)"
   by (smt Word_Lemmas.of_nat_power diff_less le_less_trans len_gt_0 len_of_numeral_defs(2)
           nat_power_minus_less of_nat_le_iff sint_eq_uint_2pl uint_nat unat_of_nat_len)
 
 lemma int_eq_sint:
   "x < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> sint (of_nat x :: 'a :: len word) = int x"
   by (smt Word_Lemmas.of_nat_power diff_less le_less_trans len_gt_0 len_of_numeral_defs(2)
           nat_less_le sint_eq_uint_2pl uint_nat unat_lt2p unat_of_nat_len unat_power_lower)
 
 lemma sint_ctz:
   "LENGTH('a) > 2
    \<Longrightarrow> 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> LENGTH('a)"
   apply (subgoal_tac "LENGTH('a) < 2 ^ (LENGTH('a) - 1)")
    apply (rule conjI)
     apply (metis len_signed order_le_less_trans sint_of_nat_ge_zero word_ctz_le)
    apply (metis int_eq_sint len_signed sint_of_nat_le word_ctz_le)
   by (rule small_powers_of_2, simp)
 
 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 (metis (mono_tags) eq_or_less_helperD not_less of_nat_numeral power_add
                         semiring_1_class.of_nat_power unat_pow_le_intro word_unat.Rep_inverse)
 
 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 && mask (size w - n)"
   by (cases "n \<le> size w"; clarsimp simp: word_and_le2 and_mask shiftl_zero_size)
 
 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_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: mask_def NOT_eq flip: mul_not_mask_eq_neg_shiftl)
 
 lemma ucast_eq_mask:
   "(UCAST('a::len \<rightarrow> 'b::len) x = UCAST('a \<rightarrow> 'b) y) =
    (x && mask LENGTH('b) = y && mask LENGTH('b))"
   by (rule iffI; word_eqI_solve)
 
 context
   fixes w :: "'a::len word"
 begin
 
 private lemma sbintrunc_uint_ucast:
   assumes "Suc n = LENGTH('b::len)"
   shows "sbintrunc n (uint (ucast w :: 'b word)) = sbintrunc n (uint w)"
   by (metis assms sbintrunc_bintrunc ucast_def word_ubin.eq_norm)
 
 private lemma test_bit_sbintrunc:
   assumes "i < LENGTH('a)"
   shows "(word_of_int (sbintrunc n (uint w)) :: 'a word) !! i
            = (if n < i then w !! n else w !! i)"
   using assms by (simp add: nth_sbintr)
                  (simp add: test_bit_bin)
 
 private lemma test_bit_sbintrunc_ucast:
   assumes len_a: "i < LENGTH('a)"
   shows "(word_of_int (sbintrunc (LENGTH('b) - 1) (uint (ucast w :: 'b word))) :: 'a word) !! i
           = (if LENGTH('b::len) \<le> i then w !! (LENGTH('b) - 1) else w !! i)"
   apply (subst sbintrunc_uint_ucast)
    apply simp
   apply (subst test_bit_sbintrunc)
    apply (rule len_a)
   apply (rule if_cong[OF _ refl refl])
   using leD less_linear by fastforce
 
 lemma scast_ucast_high_bits:
   "scast (ucast w :: 'b::len word) = w
      \<longleftrightarrow> (\<forall> i \<in> {LENGTH('b) ..< size w}. w !! i = w !! (LENGTH('b) - 1))"
   unfolding scast_def sint_uint word_size
   apply (subst word_eq_iff)
   apply (rule iffI)
    apply (rule ballI)
    apply (drule_tac x=i in spec)
    apply (subst (asm) test_bit_sbintrunc_ucast; simp)
   apply (rule allI)
   apply (case_tac "n < LENGTH('a)")
    apply (subst test_bit_sbintrunc_ucast)
     apply simp
    apply (case_tac "n \<ge> LENGTH('b)")
     apply (drule_tac x=n in bspec)
   by auto
 
 lemma scast_ucast_mask_compare:
   "scast (ucast w :: 'b::len word) = w
    \<longleftrightarrow> (w \<le> mask (LENGTH('b) - 1) \<or> ~~(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 && mask LENGTH('b))"
   by word_eqI_solve
 
 lemma ucast_NOT:
   "ucast (~~x) = ~~(ucast x) && mask (LENGTH('a))" for x::"'a::len word"
   by word_eqI
 
 lemma ucast_NOT_down:
   "is_down UCAST('a::len \<rightarrow> 'b::len) \<Longrightarrow> UCAST('a \<rightarrow> 'b) (~~x) = ~~(UCAST('a \<rightarrow> 'b) x)"
   by word_eqI
 
 lemma of_bl_mult_and_not_mask_eq:
   "\<lbrakk>is_aligned (a :: 'a::len word) n; length b + m \<le> n\<rbrakk>
    \<Longrightarrow> a + of_bl b * (2^m) && ~~(mask n) = a"
   by (smt add.left_neutral add_diff_cancel_right' add_mask_lower_bits and_mask_plus is_aligned_mask
           is_aligned_weaken le_less_trans of_bl_length2 subtract_mask(1))
 
 lemma bin_to_bl_of_bl_eq:
   "\<lbrakk>is_aligned (a::'a::len word) n; length b + c \<le> n; length b + c < LENGTH('a)\<rbrakk>
   \<Longrightarrow> bin_to_bl (length b) (uint ((a + of_bl b * 2^c) >> c)) = b"
   apply (subst word_plus_and_or_coroll)
    apply (erule is_aligned_get_word_bits)
     apply (rule is_aligned_AND_less_0)
      apply (simp add: is_aligned_mask)
     apply (rule order_less_le_trans)
      apply (rule of_bl_length2)
      apply simp
     apply (simp add: two_power_increasing)
    apply simp
   apply (rule nth_equalityI)
    apply (simp only: len_bin_to_bl)
   apply (clarsimp simp only: len_bin_to_bl nth_bin_to_bl word_test_bit_def[symmetric])
   apply (simp add: nth_shiftr nth_shiftl
                    shiftl_t2n[where n=c, simplified mult.commute, simplified, symmetric])
   apply (simp add: is_aligned_nth[THEN iffD1, rule_format] test_bit_of_bl nth_rev)
   apply arith
   done
 
 end
diff --git a/thys/Word_Lib/Word_Lemmas_32.thy b/thys/Word_Lib/Word_Lemmas_32.thy
--- a/thys/Word_Lib/Word_Lemmas_32.thy
+++ b/thys/Word_Lib/Word_Lemmas_32.thy
@@ -1,312 +1,308 @@
 (*
- * Copyright 2014, NICTA
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
- * This software may be distributed and modified according to the terms of
- * the BSD 2-Clause license. Note that NO WARRANTY is provided.
- * See "LICENSE_BSD2.txt" for details.
- *
- * @TAG(NICTA_BSD)
+ * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "Lemmas for Word Length 32"
 
 theory Word_Lemmas_32
 imports
   Word_Lemmas
   Word_Setup_32
 begin
 
 lemma ucast_8_32_inj:
   "inj (ucast ::  8 word \<Rightarrow> 32 word)"
   by (rule down_ucast_inj) (clarsimp simp: is_down_def target_size source_size)
 
 lemma upto_2_helper:
   "{0..<2 :: 32 word} = {0, 1}"
   by (safe; simp) unat_arith
 
 lemmas upper_bits_unset_is_l2p_32 = upper_bits_unset_is_l2p [where 'a=32, folded word_bits_def]
 lemmas le_2p_upper_bits_32 = le_2p_upper_bits [where 'a=32, folded word_bits_def]
 lemmas le2p_bits_unset_32 = le2p_bits_unset[where 'a=32, folded word_bits_def]
 
 lemma word_bits_len_of:
   "len_of TYPE (32) = word_bits"
   by (simp add: word_bits_conv)
 
 lemmas unat_power_lower32' = unat_power_lower[where 'a=32]
 lemmas unat_power_lower32 [simp] = unat_power_lower32'[unfolded word_bits_len_of]
 
 lemmas word32_less_sub_le' = word_less_sub_le[where 'a = 32]
 lemmas word32_less_sub_le[simp] = word32_less_sub_le' [folded word_bits_def]
 
 lemma word_bits_size:
   "size (w::word32) = word_bits"
   by (simp add: word_bits_def word_size)
 
 lemmas word32_power_less_1' = word_power_less_1[where 'a = 32]
 lemmas word32_power_less_1[simp] = word32_power_less_1'[folded word_bits_def]
 
 lemma of_nat32_0:
   "\<lbrakk>of_nat n = (0::word32); n < 2 ^ word_bits\<rbrakk> \<Longrightarrow> n = 0"
   by (erule of_nat_0, simp add: word_bits_def)
 
 lemma unat_mask_2_less_4:
   "unat (p && mask 2 :: word32) < 4"
   apply (rule unat_less_helper)
   apply (rule order_le_less_trans, rule word_and_le1)
   apply (simp add: mask_def)
   done
 
 lemmas unat_of_nat32' = unat_of_nat_eq[where 'a=32]
 lemmas unat_of_nat32 = unat_of_nat32'[unfolded word_bits_len_of]
 
 lemmas word_power_nonzero_32 = word_power_nonzero [where 'a=32, folded word_bits_def]
 
 lemmas unat_mult_simple = iffD1 [OF unat_mult_lem [where 'a = 32, unfolded word_bits_len_of]]
 
 lemmas div_power_helper_32 = div_power_helper [where 'a=32, folded word_bits_def]
 
 lemma n_less_word_bits:
   "(n < word_bits) = (n < 32)"
   by (simp add: word_bits_def)
 
 lemmas of_nat_less_pow_32 = of_nat_power [where 'a=32, folded word_bits_def]
 
 lemma lt_word_bits_lt_pow:
   "sz < word_bits \<Longrightarrow> sz < 2 ^ word_bits"
   by (simp add: word_bits_conv)
 
 lemma unat_less_word_bits:
   fixes y :: word32
   shows "x < unat y \<Longrightarrow> x < 2 ^ word_bits"
   unfolding word_bits_def
   by (rule order_less_trans [OF _ unat_lt2p])
 
 lemmas unat_mask_word32' = unat_mask[where 'a=32]
 lemmas unat_mask_word32 = unat_mask_word32'[folded word_bits_def]
 
 lemma unat_less_2p_word_bits:
   "unat (x :: 32 word) < 2 ^ word_bits"
   apply (simp only: word_bits_def)
   apply (rule unat_lt2p)
   done
 
 lemma Suc_unat_mask_div:
   "Suc (unat (mask sz div word_size::word32)) = 2 ^ (min sz word_bits - 2)"
   apply (case_tac "sz < word_bits")
    apply (case_tac "2\<le>sz")
     apply (clarsimp simp: word_size_def word_bits_def min_def mask_def)
     apply (drule (2) Suc_div_unat_helper
            [where 'a=32 and sz=sz and us=2, simplified, symmetric])
    apply (simp add: not_le word_size_def word_bits_def)
    apply (case_tac sz, simp add: unat_word_ariths)
    apply (case_tac nat, simp add: unat_word_ariths
                                   unat_mask_word32 min_def word_bits_def)
    apply simp
   apply (simp add: unat_word_ariths
                    unat_mask_word32 min_def word_bits_def word_size_def)
   done
 
 lemmas word32_minus_one_le' = word_minus_one_le[where 'a=32]
 lemmas word32_minus_one_le = word32_minus_one_le'[simplified]
 
 lemma ucast_not_helper:
   fixes a::word8
   assumes a: "a \<noteq> 0xFF"
   shows "ucast a \<noteq> (0xFF::word32)"
 proof
   assume "ucast a = (0xFF::word32)"
   also
   have "(0xFF::word32) = ucast (0xFF::word8)" by simp
   finally
   show False using a
     apply -
     apply (drule up_ucast_inj, simp)
     apply simp
     done
 qed
 
 lemma less_4_cases:
   "(x::word32) < 4 \<Longrightarrow> x=0 \<or> x=1 \<or> x=2 \<or> x=3"
   apply clarsimp
   apply (drule word_less_cases, erule disjE, simp, simp)+
   done
 
 lemma unat_ucast_8_32:
   fixes x :: "word8"
   shows "unat (ucast x :: word32) = unat x"
   unfolding ucast_def unat_def
   apply (subst int_word_uint)
   apply (subst mod_pos_pos_trivial)
     apply simp
    apply (rule lt2p_lem)
    apply simp
   apply simp
   done
 
 lemma if_then_1_else_0:
   "((if P then 1 else 0) = (0 :: word32)) = (\<not> P)"
   by simp
 
 lemma if_then_0_else_1:
   "((if P then 0 else 1) = (0 :: word32)) = (P)"
   by simp
 
 lemmas if_then_simps = if_then_0_else_1 if_then_1_else_0
 
 lemma ucast_le_ucast_8_32:
   "(ucast x \<le> (ucast y :: word32)) = (x \<le> (y :: word8))"
   by (simp add: ucast_le_ucast)
 
 lemma in_16_range:
   "0 \<in> S \<Longrightarrow> r \<in> (\<lambda>x. r + x * (16 :: word32)) ` S"
   "n - 1 \<in> S \<Longrightarrow> (r + (16 * n - 16)) \<in> (\<lambda>x :: word32. r + x * 16) ` S"
   by (clarsimp simp: image_def elim!: bexI[rotated])+
 
 lemma eq_2_32_0:
   "(2 ^ 32 :: word32) = 0"
   by simp
 
 lemma x_less_2_0_1:
   fixes x :: word32 shows
   "x < 2 \<Longrightarrow> x = 0 \<or> x = 1"
   by (rule x_less_2_0_1') auto
 
 lemmas mask_32_max_word  = max_word_mask [symmetric, where 'a=32, simplified]
 
 lemma of_nat32_n_less_equal_power_2:
  "n < 32 \<Longrightarrow> ((of_nat n)::32 word) < 2 ^ n"
   by (rule of_nat_n_less_equal_power_2, clarsimp simp: word_size)
 
 lemma word_rsplit_0:
   "word_rsplit (0 :: word32) = [0, 0, 0, 0 :: word8]"
   apply (simp add: word_rsplit_def bin_rsplit_def Let_def)
   done
 
 lemma unat_ucast_10_32 :
   fixes x :: "10 word"
   shows "unat (ucast x :: word32) = unat x"
   unfolding ucast_def unat_def
   apply (subst int_word_uint)
   apply (subst mod_pos_pos_trivial)
     apply simp
    apply (rule lt2p_lem)
    apply simp
   apply simp
   done
 
 lemma bool_mask [simp]:
   fixes x :: word32
   shows "(0 < x && 1) = (x && 1 = 1)"
   by (rule bool_mask') auto
 
 lemma word32_bounds:
   "- (2 ^ (size (x :: word32) - 1)) = (-2147483648 :: int)"
   "((2 ^ (size (x :: word32) - 1)) - 1) = (2147483647 :: int)"
   "- (2 ^ (size (y :: 32 signed word) - 1)) = (-2147483648 :: int)"
   "((2 ^ (size (y :: 32 signed word) - 1)) - 1) = (2147483647 :: int)"
   by (simp_all add: word_size)
 
 lemma word_ge_min:"sint (x::32 word) \<ge> -2147483648"
   by (metis sint_ge word32_bounds(1) word_size)
 
 lemmas signed_arith_ineq_checks_to_eq_word32'
     = signed_arith_ineq_checks_to_eq[where 'a=32]
       signed_arith_ineq_checks_to_eq[where 'a="32 signed"]
 
 lemmas signed_arith_ineq_checks_to_eq_word32
     = signed_arith_ineq_checks_to_eq_word32' [unfolded word32_bounds]
 
 lemmas signed_mult_eq_checks32_to_64'
     = signed_mult_eq_checks_double_size[where 'a=32 and 'b=64]
       signed_mult_eq_checks_double_size[where 'a="32 signed" and 'b=64]
 
 lemmas signed_mult_eq_checks32_to_64 = signed_mult_eq_checks32_to_64'[simplified]
 
 lemmas sdiv_word32_max' = sdiv_word_max [where 'a=32] sdiv_word_max [where 'a="32 signed"]
 lemmas sdiv_word32_max = sdiv_word32_max'[simplified word_size, simplified]
 
 lemmas sdiv_word32_min' = sdiv_word_min [where 'a=32] sdiv_word_min [where 'a="32 signed"]
 lemmas sdiv_word32_min = sdiv_word32_min' [simplified word_size, simplified]
 
 lemmas sint32_of_int_eq' = sint_of_int_eq [where 'a=32]
 lemmas sint32_of_int_eq = sint32_of_int_eq' [simplified]
 
 lemma ucast_of_nats [simp]:
   "(ucast (of_nat x :: word32) :: sword32) = (of_nat x)"
   "(ucast (of_nat x :: word32) :: sword16) = (of_nat x)"
   "(ucast (of_nat x :: word32) :: sword8) = (of_nat x)"
   "(ucast (of_nat x :: word16) :: sword16) = (of_nat x)"
   "(ucast (of_nat x :: word16) :: sword8) = (of_nat x)"
   "(ucast (of_nat x :: word8) :: sword8) = (of_nat x)"
   by (auto simp: ucast_of_nat is_down)
 
 lemmas signed_shift_guard_simpler_32'
     = power_strict_increasing_iff[where b="2 :: nat" and y=31]
 lemmas signed_shift_guard_simpler_32 = signed_shift_guard_simpler_32'[simplified]
 
 lemma word32_31_less:
   "31 < len_of TYPE (32 signed)" "31 > (0 :: nat)"
   "31 < len_of TYPE (32)" "31 > (0 :: nat)"
   by auto
 
 lemmas signed_shift_guard_to_word_32
     = signed_shift_guard_to_word[OF word32_31_less(1-2)]
     signed_shift_guard_to_word[OF word32_31_less(3-4)]
 
 lemma le_step_down_word_3:
   fixes x :: "32 word"
   shows "\<lbrakk>x \<le>  y; x \<noteq> y; y < 2 ^ 32 - 1\<rbrakk> \<Longrightarrow> x \<le> y - 1"
   by (rule le_step_down_word_2, assumption+)
 
 lemma shiftr_1:
   "(x::word32) >> 1 = 0 \<Longrightarrow> x < 2"
   by word_bitwise clarsimp
 
 lemma has_zero_byte:
   "~~ (((((v::word32) && 0x7f7f7f7f) + 0x7f7f7f7f) || v) || 0x7f7f7f7f) \<noteq> 0
     \<Longrightarrow> v && 0xff000000 = 0 \<or> v && 0xff0000 = 0 \<or> v && 0xff00 = 0 \<or> v && 0xff = 0"
   apply clarsimp
   apply word_bitwise
   by metis
 
 lemma mask_step_down_32:
   "(b::32word) && 0x1 = (1::32word) \<Longrightarrow> (\<exists>x. x < 32 \<and> mask x = b >> 1) \<Longrightarrow> (\<exists>x. mask x = b)"
   apply clarsimp
   apply (rule_tac x="x + 1" in exI)
   apply (subgoal_tac "x \<le> 31")
    apply (erule le_step_down_nat, clarsimp simp:mask_def, word_bitwise, clarsimp+)+
    apply (clarsimp simp:mask_def, word_bitwise, clarsimp)
   apply clarsimp
   done
 
 lemma unat_of_int_32:
   "\<lbrakk>i \<ge> 0; i \<le>2 ^ 31\<rbrakk> \<Longrightarrow> (unat ((of_int i)::sword32)) = nat i"
   unfolding unat_def
   apply (subst eq_nat_nat_iff, clarsimp+)
   apply (simp add: word_of_int uint_word_of_int)
   done
 
 lemmas word_ctz_not_minus_1_32 = word_ctz_not_minus_1[where 'a=32, simplified]
 
 (* Helper for packing then unpacking a 64-bit variable. *)
 lemma cast_chunk_assemble_id_64[simp]:
   "(((ucast ((ucast (x::64 word))::32 word))::64 word) || (((ucast ((ucast (x >> 32))::32 word))::64 word) << 32)) = x"
   by (simp add:cast_chunk_assemble_id)
 
 (* Another variant of packing and unpacking a 64-bit variable. *)
 lemma cast_chunk_assemble_id_64'[simp]:
   "(((ucast ((scast (x::64 word))::32 word))::64 word) || (((ucast ((scast (x >> 32))::32 word))::64 word) << 32)) = x"
   by (simp add:cast_chunk_scast_assemble_id)
 
 (* Specialisations of down_cast_same for adding to local simpsets. *)
 lemma cast_down_u64: "(scast::64 word \<Rightarrow> 32 word) = (ucast::64 word \<Rightarrow> 32 word)"
   apply (subst down_cast_same[symmetric])
    apply (simp add:is_down)+
   done
 
 lemma cast_down_s64: "(scast::64 sword \<Rightarrow> 32 word) = (ucast::64 sword \<Rightarrow> 32 word)"
   apply (subst down_cast_same[symmetric])
    apply (simp add:is_down)+
   done
 
 end
diff --git a/thys/Word_Lib/Word_Lemmas_64.thy b/thys/Word_Lib/Word_Lemmas_64.thy
--- a/thys/Word_Lib/Word_Lemmas_64.thy
+++ b/thys/Word_Lib/Word_Lemmas_64.thy
@@ -1,294 +1,290 @@
 (*
- * Copyright 2014, NICTA
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
- * This software may be distributed and modified according to the terms of
- * the BSD 2-Clause license. Note that NO WARRANTY is provided.
- * See "LICENSE_BSD2.txt" for details.
- *
- * @TAG(NICTA_BSD)
+ * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "Lemmas for Word Length 64"
 
 theory Word_Lemmas_64
 imports
   Word_Lemmas
   Word_Setup_64
 begin
 
 lemma ucast_8_64_inj:
   "inj (ucast ::  8 word \<Rightarrow> 64 word)"
   by (rule down_ucast_inj) (clarsimp simp: is_down_def target_size source_size)
 
 lemma upto_2_helper:
   "{0..<2 :: 64 word} = {0, 1}"
   by (safe; simp) unat_arith
 
 lemmas upper_bits_unset_is_l2p_64 = upper_bits_unset_is_l2p [where 'a=64, folded word_bits_def]
 lemmas le_2p_upper_bits_64 = le_2p_upper_bits [where 'a=64, folded word_bits_def]
 lemmas le2p_bits_unset_64 = le2p_bits_unset[where 'a=64, folded word_bits_def]
 
 lemma word_bits_len_of:
   "len_of TYPE (64) = word_bits"
   by (simp add: word_bits_conv)
 
 lemmas unat_power_lower64' = unat_power_lower[where 'a=64]
 lemmas unat_power_lower64 [simp] = unat_power_lower64'[unfolded word_bits_len_of]
 
 lemmas word64_less_sub_le' = word_less_sub_le[where 'a = 64]
 lemmas word64_less_sub_le[simp] = word64_less_sub_le' [folded word_bits_def]
 
 lemma word_bits_size:
   "size (w::word64) = word_bits"
   by (simp add: word_bits_def word_size)
 
 lemmas word64_power_less_1' = word_power_less_1[where 'a = 64]
 lemmas word64_power_less_1[simp] = word64_power_less_1'[folded word_bits_def]
 
 lemma of_nat64_0:
   "\<lbrakk>of_nat n = (0::word64); n < 2 ^ word_bits\<rbrakk> \<Longrightarrow> n = 0"
   by (erule of_nat_0, simp add: word_bits_def)
 
 lemma unat_mask_2_less_4:
   "unat (p && mask 2 :: word64) < 4"
   apply (rule unat_less_helper)
   apply (rule order_le_less_trans, rule word_and_le1)
   apply (simp add: mask_def)
   done
 
 lemmas unat_of_nat64' = unat_of_nat_eq[where 'a=64]
 lemmas unat_of_nat64 = unat_of_nat64'[unfolded word_bits_len_of]
 
 lemmas word_power_nonzero_64 = word_power_nonzero [where 'a=64, folded word_bits_def]
 
 lemmas unat_mult_simple = iffD1 [OF unat_mult_lem [where 'a = 64, unfolded word_bits_len_of]]
 
 lemmas div_power_helper_64 = div_power_helper [where 'a=64, folded word_bits_def]
 
 lemma n_less_word_bits:
   "(n < word_bits) = (n < 64)"
   by (simp add: word_bits_def)
 
 lemmas of_nat_less_pow_64 = of_nat_power [where 'a=64, folded word_bits_def]
 
 lemma lt_word_bits_lt_pow:
   "sz < word_bits \<Longrightarrow> sz < 2 ^ word_bits"
   by (simp add: word_bits_conv)
 
 lemma unat_less_word_bits:
   fixes y :: word64
   shows "x < unat y \<Longrightarrow> x < 2 ^ word_bits"
   unfolding word_bits_def
   by (rule order_less_trans [OF _ unat_lt2p])
 
 lemmas unat_mask_word64' = unat_mask[where 'a=64]
 lemmas unat_mask_word64 = unat_mask_word64'[folded word_bits_def]
 
 lemma unat_less_2p_word_bits:
   "unat (x :: 64 word) < 2 ^ word_bits"
   apply (simp only: word_bits_def)
   apply (rule unat_lt2p)
   done
 
 lemma Suc_unat_mask_div:
   "Suc (unat (mask sz div word_size::word64)) = 2 ^ (min sz word_bits - 3)"
   apply (case_tac "sz < word_bits")
    apply (case_tac "3\<le>sz")
     apply (clarsimp simp: word_size_def word_bits_def min_def mask_def)
     apply (drule (2) Suc_div_unat_helper
            [where 'a=64 and sz=sz and us=3, simplified, symmetric])
    apply (simp add: not_le word_size_def word_bits_def)
    apply (case_tac sz, simp add: unat_word_ariths)
    apply (case_tac nat, simp add: unat_word_ariths
                                   unat_mask_word64 min_def word_bits_def)
    apply (case_tac nata, simp add: unat_word_ariths unat_mask_word64 word_bits_def)
    apply simp
   apply (simp add: unat_word_ariths
                    unat_mask_word64 min_def word_bits_def word_size_def)
   done
 
 lemmas word64_minus_one_le' = word_minus_one_le[where 'a=64]
 lemmas word64_minus_one_le = word64_minus_one_le'[simplified]
 
 lemma ucast_not_helper:
   fixes a::word8
   assumes a: "a \<noteq> 0xFF"
   shows "ucast a \<noteq> (0xFF::word64)"
 proof
   assume "ucast a = (0xFF::word64)"
   also
   have "(0xFF::word64) = ucast (0xFF::word8)" by simp
   finally
   show False using a
     apply -
     apply (drule up_ucast_inj, simp)
     apply simp
     done
 qed
 
 lemma less_4_cases:
   "(x::word64) < 4 \<Longrightarrow> x=0 \<or> x=1 \<or> x=2 \<or> x=3"
   apply clarsimp
   apply (drule word_less_cases, erule disjE, simp, simp)+
   done
 
 lemma if_then_1_else_0:
   "((if P then 1 else 0) = (0 :: word64)) = (\<not> P)"
   by simp
 
 lemma if_then_0_else_1:
   "((if P then 0 else 1) = (0 :: word64)) = (P)"
   by simp
 
 lemmas if_then_simps = if_then_0_else_1 if_then_1_else_0
 
 lemma ucast_le_ucast_8_64:
   "(ucast x \<le> (ucast y :: word64)) = (x \<le> (y :: word8))"
   by (simp add: ucast_le_ucast)
 
 lemma in_16_range:
   "0 \<in> S \<Longrightarrow> r \<in> (\<lambda>x. r + x * (16 :: word64)) ` S"
   "n - 1 \<in> S \<Longrightarrow> (r + (16 * n - 16)) \<in> (\<lambda>x :: word64. r + x * 16) ` S"
   by (clarsimp simp: image_def elim!: bexI[rotated])+
 
 lemma eq_2_64_0:
   "(2 ^ 64 :: word64) = 0"
   by simp
 
 lemma x_less_2_0_1:
   fixes x :: word64 shows
   "x < 2 \<Longrightarrow> x = 0 \<or> x = 1"
   by (rule x_less_2_0_1') auto
 
 lemmas mask_64_max_word  = max_word_mask [symmetric, where 'a=64, simplified]
 
 lemma of_nat64_n_less_equal_power_2:
  "n < 64 \<Longrightarrow> ((of_nat n)::64 word) < 2 ^ n"
   by (rule of_nat_n_less_equal_power_2, clarsimp simp: word_size)
 
 lemma word_rsplit_0:
   "word_rsplit (0 :: word64) = [0, 0, 0, 0, 0, 0, 0, 0 :: word8]"
   apply (simp add: word_rsplit_def bin_rsplit_def Let_def)
   done
 
 lemma unat_ucast_10_64 :
   fixes x :: "10 word"
   shows "unat (ucast x :: word64) = unat x"
   unfolding ucast_def unat_def
   apply (subst int_word_uint)
   apply (subst mod_pos_pos_trivial)
     apply simp
    apply (rule lt2p_lem)
    apply simp
   apply simp
   done
 
 lemma bool_mask [simp]:
   fixes x :: word64
   shows "(0 < x && 1) = (x && 1 = 1)"
   by (rule bool_mask') auto
 
 lemma word64_bounds:
   "- (2 ^ (size (x :: word64) - 1)) = (-9223372036854775808 :: int)"
   "((2 ^ (size (x :: word64) - 1)) - 1) = (9223372036854775807 :: int)"
   "- (2 ^ (size (y :: 64 signed word) - 1)) = (-9223372036854775808 :: int)"
   "((2 ^ (size (y :: 64 signed word) - 1)) - 1) = (9223372036854775807 :: int)"
   by (simp_all add: word_size)
 
 lemma word_ge_min:"sint (x::64 word) \<ge> -9223372036854775808"
   by (metis sint_ge word64_bounds(1) word_size)
 
 lemmas signed_arith_ineq_checks_to_eq_word64'
     = signed_arith_ineq_checks_to_eq[where 'a=64]
       signed_arith_ineq_checks_to_eq[where 'a="64 signed"]
 
 lemmas signed_arith_ineq_checks_to_eq_word64
     = signed_arith_ineq_checks_to_eq_word64' [unfolded word64_bounds]
 
 lemmas signed_mult_eq_checks64_to_64'
     = signed_mult_eq_checks_double_size[where 'a=64 and 'b=64]
       signed_mult_eq_checks_double_size[where 'a="64 signed" and 'b=64]
 
 lemmas signed_mult_eq_checks64_to_64 = signed_mult_eq_checks64_to_64'[simplified]
 
 lemmas sdiv_word64_max' = sdiv_word_max [where 'a=64] sdiv_word_max [where 'a="64 signed"]
 lemmas sdiv_word64_max = sdiv_word64_max'[simplified word_size, simplified]
 
 lemmas sdiv_word64_min' = sdiv_word_min [where 'a=64] sdiv_word_min [where 'a="64 signed"]
 lemmas sdiv_word64_min = sdiv_word64_min' [simplified word_size, simplified]
 
 lemmas sint64_of_int_eq' = sint_of_int_eq [where 'a=64]
 lemmas sint64_of_int_eq = sint64_of_int_eq' [simplified]
 
 lemma ucast_of_nats [simp]:
   "(ucast (of_nat x :: word64) :: sword64) = (of_nat x)"
   "(ucast (of_nat x :: word64) :: sword16) = (of_nat x)"
   "(ucast (of_nat x :: word64) :: sword8) = (of_nat x)"
   "(ucast (of_nat x :: word16) :: sword16) = (of_nat x)"
   "(ucast (of_nat x :: word16) :: sword8) = (of_nat x)"
   "(ucast (of_nat x :: word8) :: sword8) = (of_nat x)"
   by (auto simp: ucast_of_nat is_down)
 
 lemmas signed_shift_guard_simpler_64'
     = power_strict_increasing_iff[where b="2 :: nat" and y=31]
 lemmas signed_shift_guard_simpler_64 = signed_shift_guard_simpler_64'[simplified]
 
 lemma word64_31_less:
   "31 < len_of TYPE (64 signed)" "31 > (0 :: nat)"
   "31 < len_of TYPE (64)" "31 > (0 :: nat)"
   by auto
 
 lemmas signed_shift_guard_to_word_64
     = signed_shift_guard_to_word[OF word64_31_less(1-2)]
     signed_shift_guard_to_word[OF word64_31_less(3-4)]
 
 lemma le_step_down_word_3:
   fixes x :: "64 word"
   shows "\<lbrakk>x \<le>  y; x \<noteq> y; y < 2 ^ 64 - 1\<rbrakk> \<Longrightarrow> x \<le> y - 1"
   by (rule le_step_down_word_2, assumption+)
 
 lemma shiftr_1:
   "(x::word64) >> 1 = 0 \<Longrightarrow> x < 2"
   by word_bitwise clarsimp
 
 lemma mask_step_down_64:
   "(b::64word) && 0x1 = (1::64word) \<Longrightarrow> (\<exists>x. x < 64 \<and> mask x = b >> 1) \<Longrightarrow> (\<exists>x. mask x = b)"
   apply clarsimp
   apply (rule_tac x="x + 1" in exI)
   apply (subgoal_tac "x \<le> 63")
    apply (erule le_step_down_nat, clarsimp simp:mask_def, word_bitwise, clarsimp+)+
    apply (clarsimp simp:mask_def, word_bitwise, clarsimp)
   apply clarsimp
   done
 
 lemma unat_of_int_64:
   "\<lbrakk>i \<ge> 0; i \<le> 2 ^ 63\<rbrakk> \<Longrightarrow> (unat ((of_int i)::sword64)) = nat i"
   unfolding unat_def
   apply (subst eq_nat_nat_iff, clarsimp+)
   apply (simp add: word_of_int uint_word_of_int)
   done
 
 lemmas word_ctz_not_minus_1_64 = word_ctz_not_minus_1[where 'a=64, simplified]
 
 (* Helper for packing then unpacking a 64-bit variable. *)
 lemma cast_chunk_assemble_id_64[simp]:
   "(((ucast ((ucast (x::64 word))::32 word))::64 word) || (((ucast ((ucast (x >> 32))::32 word))::64 word) << 32)) = x"
   by (simp add:cast_chunk_assemble_id)
 
 (* Another variant of packing and unpacking a 64-bit variable. *)
 lemma cast_chunk_assemble_id_64'[simp]:
   "(((ucast ((scast (x::64 word))::32 word))::64 word) || (((ucast ((scast (x >> 32))::32 word))::64 word) << 32)) = x"
   by (simp add:cast_chunk_scast_assemble_id)
 
 (* Specialisations of down_cast_same for adding to local simpsets. *)
 lemma cast_down_u64: "(scast::64 word \<Rightarrow> 32 word) = (ucast::64 word \<Rightarrow> 32 word)"
   apply (subst down_cast_same[symmetric])
    apply (simp add:is_down)+
   done
 
 lemma cast_down_s64: "(scast::64 sword \<Rightarrow> 32 word) = (ucast::64 sword \<Rightarrow> 32 word)"
   apply (subst down_cast_same[symmetric])
    apply (simp add:is_down)+
   done
 
 end
diff --git a/thys/Word_Lib/Word_Lib.thy b/thys/Word_Lib/Word_Lib.thy
--- a/thys/Word_Lib/Word_Lib.thy
+++ b/thys/Word_Lib/Word_Lib.thy
@@ -1,705 +1,701 @@
 (*
- * Copyright 2014, NICTA
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
- * This software may be distributed and modified according to the terms of
- * the BSD 2-Clause license. Note that NO WARRANTY is provided.
- * See "LICENSE_BSD2.txt" for details.
- *
- * @TAG(NICTA_BSD)
+ * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "Additional Word Operations"
 
 theory Word_Lib
 imports
   Word_Syntax
 begin
 
 definition
   ptr_add :: "'a :: len word \<Rightarrow> nat \<Rightarrow> 'a word" where
   "ptr_add ptr n \<equiv> ptr + of_nat n"
 
 definition
   complement :: "'a :: len word \<Rightarrow> 'a word"  where
  "complement x \<equiv> ~~ x"
 
 definition
   alignUp :: "'a::len word \<Rightarrow> nat \<Rightarrow> 'a word" where
  "alignUp x n \<equiv> x + 2 ^ n - 1 && complement (2 ^ n - 1)"
 
 (* 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}"
 
 (* Haskellish names/syntax *)
 notation (input)
   test_bit ("testBit")
 
 definition
   w2byte :: "'a :: len word \<Rightarrow> 8 word" where
   "w2byte \<equiv> ucast"
 
 
 (*
  * Signed division: when the result of a division is negative,
  * we will round towards zero instead of towards minus infinity.
  *)
 class signed_div =
   fixes sdiv :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "sdiv" 70)
   fixes smod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "smod" 70)
 
 instantiation int :: signed_div
 begin
 definition "(a :: int) sdiv b \<equiv> sgn (a * b) * (abs a div abs b)"
 definition "(a :: int) smod b \<equiv> a - (a sdiv b) * b"
 instance ..
 end
 
 instantiation word :: (len) signed_div
 begin
 definition "(a :: ('a::len) word) sdiv b = word_of_int (sint a sdiv sint b)"
 definition "(a :: ('a::len) word) smod b = word_of_int (sint a smod sint b)"
 instance ..
 end
 
 (* Tests *)
 lemma
   "( 4 :: word32) sdiv  4 =  1"
   "(-4 :: word32) sdiv  4 = -1"
   "(-3 :: word32) sdiv  4 =  0"
   "( 3 :: word32) sdiv -4 =  0"
   "(-3 :: word32) sdiv -4 =  0"
   "(-5 :: word32) sdiv -4 =  1"
   "( 5 :: word32) sdiv -4 = -1"
   by (simp_all add: sdiv_word_def sdiv_int_def)
 
 lemma
   "( 4 :: word32) smod  4 =   0"
   "( 3 :: word32) smod  4 =   3"
   "(-3 :: word32) smod  4 =  -3"
   "( 3 :: word32) smod -4 =   3"
   "(-3 :: word32) smod -4 =  -3"
   "(-5 :: word32) smod -4 =  -1"
   "( 5 :: word32) smod -4 =   1"
   by (simp_all add: smod_word_def smod_int_def sdiv_int_def)
 
 
 (* 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)))"
 
 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 w !! n then w || ~~ (mask n) else w && mask n"
 
 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> w !! i = w !! n"
 
 
 lemma ptr_add_0 [simp]:
   "ptr_add ref 0 = ref "
   unfolding ptr_add_def by simp
 
 lemma shiftl_power:
   "(shiftl1 ^^ x) (y::'a::len word) = 2 ^ x * y"
   apply (induct x)
    apply simp
   apply (simp add: shiftl1_2t)
   done
 
 lemmas of_bl_reasoning = to_bl_use_of_bl of_bl_append
 
 lemma uint_of_bl_is_bl_to_bin_drop:
   "length (dropWhile Not l) \<le> LENGTH('a) \<Longrightarrow> uint (of_bl l :: 'a::len word) = bl_to_bin l"
   apply (simp add: of_bl_def)
   apply (rule word_uint.Abs_inverse)
   apply (simp add: uints_num bl_to_bin_ge0)
   apply (rule order_less_le_trans)
   apply (rule bl_to_bin_lt2p_drop)
   apply (simp)
   done
 
 corollary uint_of_bl_is_bl_to_bin:
   "length l\<le>LENGTH('a) \<Longrightarrow> uint ((of_bl::bool list\<Rightarrow> ('a :: len) word) l) = bl_to_bin l"
   apply(rule uint_of_bl_is_bl_to_bin_drop)
   using le_trans length_dropWhile_le by blast
 
 
 lemma bin_to_bl_or:
   "bin_to_bl n (a OR b) = map2 (\<or>) (bin_to_bl n a) (bin_to_bl n b)"
   using bl_or_aux_bin[where n=n and v=a and w=b and bs="[]" and cs="[]"]
   by simp
 
 lemma word_ops_nth [simp]:
   shows
   word_or_nth:  "(x || y) !! n = (x !! n \<or> y !! n)" and
   word_and_nth: "(x && y) !! n = (x !! n \<and> y !! n)" and
   word_xor_nth: "(x xor y) !! n = (x !! n \<noteq> y !! n)"
   by ((cases "n < size x",
       auto dest: test_bit_size simp: word_ops_nth_size word_size)[1])+
 
 (* simp del to avoid warning on the simp add in iff *)
 declare test_bit_1 [simp del, iff]
 
 (* test: *)
 lemma "1 < (1024::32 word) \<and> 1 \<le> (1024::32 word)" by simp
 
 lemma and_not_mask:
   "w AND NOT (mask n) = (w >> n) << n"
   apply (rule word_eqI)
   apply (simp add : word_ops_nth_size word_size)
   apply (simp add : nth_shiftr nth_shiftl)
   by auto
 
 lemma and_mask:
   "w AND mask n = (w << (size w - n)) >> (size w - n)"
   apply (rule word_eqI)
   apply (simp add : word_ops_nth_size word_size)
   apply (simp add : nth_shiftr nth_shiftl)
   by auto
 
 lemma AND_twice [simp]:
   "(w && m) && m = w && m"
   by (simp add: word_eqI)
 
 lemma word_combine_masks:
   "w && m = z \<Longrightarrow> w && m' = z' \<Longrightarrow> w && (m || m') = (z || z')"
   by (auto simp: word_eq_iff)
 
 lemma nth_w2p_same:
   "(2^n :: 'a :: len word) !! n = (n < LENGTH('a))"
   by (simp add : nth_w2p)
 
 lemma p2_gt_0:
   "(0 < (2 ^ n :: 'a :: len word)) = (n < LENGTH('a))"
   apply (simp add : word_gt_0)
   apply safe
    apply (erule swap)
    apply (rule word_eqI)
    apply (simp add : nth_w2p)
   apply (drule word_eqD)
   apply simp
   apply (erule notE)
   apply (erule nth_w2p_same [THEN iffD2])
   done
 
 lemmas uint_2p_alt = uint_2p [unfolded p2_gt_0]
 
 lemma shiftr_div_2n_w: "n < size w \<Longrightarrow> w >> n = w div (2^n :: 'a :: len word)"
   apply (unfold word_div_def)
   apply (simp add : uint_2p_alt word_size)
   apply (rule word_uint.Rep_inverse' [THEN sym])
   apply (rule shiftr_div_2n)
   done
 
 lemmas less_def = less_eq [symmetric]
 lemmas le_def = not_less [symmetric, where ?'a = nat]
 
 lemmas p2_eq_0 [simp] = trans [OF eq_commute
   iffD2 [OF Not_eq_iff p2_gt_0, folded le_def, unfolded word_gt_0 not_not]]
 
 lemma neg_mask_is_div':
   "n < size w \<Longrightarrow> w AND NOT (mask n) = ((w div (2 ^ n)) * (2 ^ n))"
   by (simp add : and_not_mask shiftr_div_2n_w shiftl_t2n word_size)
 
 lemma neg_mask_is_div:
   "w AND NOT (mask n) = (w div 2^n) * 2^n"
   apply (cases "n < size w")
    apply (erule neg_mask_is_div')
   apply (simp add: word_size)
   apply (frule p2_gt_0 [THEN Not_eq_iff [THEN iffD2], THEN iffD2])
   apply (simp add: word_gt_0  del: p2_eq_0)
   apply (rule word_eqI)
   apply (simp add: word_ops_nth_size word_size)
   done
 
 lemma and_mask_arith':
   "0 < n \<Longrightarrow> w AND mask n = (w * 2^(size w - n)) div 2^(size w - n)"
   by (simp add: and_mask shiftr_div_2n_w shiftl_t2n word_size mult.commute)
 
 lemmas p2len = iffD2 [OF p2_eq_0 order_refl]
 
 lemma and_mask_arith:
   "w AND mask n = (w * 2^(size w - n)) div 2^(size w - n)"
   apply (cases "0 < n")
    apply (erule and_mask_arith')
   apply (simp add: word_size)
   apply (simp add: word_of_int_hom_syms word_div_def)
   done
 
 lemma mask_2pm1: "mask n = 2 ^ n - 1"
   by (simp add : mask_def)
 
 lemma add_mask_fold:
   "x + 2 ^ n - 1 = x + mask n"
   by (simp add: mask_def)
 
 lemma word_and_mask_le_2pm1: "w && mask n \<le> 2 ^ n - 1"
   by (simp add: mask_2pm1[symmetric] word_and_le1)
 
 lemma is_aligned_AND_less_0:
   "u && mask n = 0 \<Longrightarrow> v < 2^n \<Longrightarrow> u && v = 0"
   apply (drule less_mask_eq)
   apply (simp add: mask_2pm1)
   apply (rule word_eqI)
   apply (clarsimp simp add: word_size)
   apply (drule_tac x=na in word_eqD)
   apply (drule_tac x=na in word_eqD)
   apply simp
   done
 
 lemma le_shiftr1:
   "u <= v \<Longrightarrow> shiftr1 u <= shiftr1 v"
   apply (unfold word_le_def shiftr1_def word_ubin.eq_norm)
   apply (unfold bin_rest_trunc_i
                 trans [OF bintrunc_bintrunc_l word_ubin.norm_Rep,
                           unfolded word_ubin.norm_Rep,
                        OF order_refl [THEN le_SucI]])
   apply (case_tac "uint u" rule: bin_exhaust)
   apply (rename_tac bs bit)
   apply (case_tac "uint v" rule: bin_exhaust)
   apply (rename_tac bs' bit')
   apply (case_tac "bit")
    apply (case_tac "bit'", auto simp: less_eq_int_code le_Bits)[1]
   apply (case_tac bit')
    apply (simp add: le_Bits less_eq_int_code)
   apply (auto simp: le_Bits less_eq_int_code)
   done
 
 lemma le_shiftr:
   "u \<le> v \<Longrightarrow> u >> (n :: nat) \<le> (v :: 'a :: len0 word) >> n"
   apply (unfold shiftr_def)
   apply (induct_tac "n")
    apply auto
   apply (erule le_shiftr1)
   done
 
 lemma shiftr_mask_le:
   "n <= m \<Longrightarrow> mask n >> m = 0"
   apply (rule word_eqI)
   apply (simp add: word_size nth_shiftr)
   done
 
 lemmas shiftr_mask = order_refl [THEN shiftr_mask_le, simp]
 
 lemma word_leI:
   "(\<And>n.  \<lbrakk>n < size (u::'a::len0 word); u !! n \<rbrakk> \<Longrightarrow> (v::'a::len0 word) !! n) \<Longrightarrow> u <= v"
   apply (rule xtr4)
    apply (rule word_and_le2)
   apply (rule word_eqI)
   apply (simp add: word_ao_nth)
   apply safe
     apply assumption
    apply (erule_tac [2] asm_rl)
   apply (unfold word_size)
   by auto
 
 lemma le_mask_iff:
   "(w \<le> mask n) = (w >> n = 0)"
   apply safe
    apply (rule word_le_0_iff [THEN iffD1])
    apply (rule xtr3)
     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)
   apply (case_tac "n <= n'")
   by auto
 
 lemma and_mask_eq_iff_shiftr_0:
   "(w AND mask n = w) = (w >> n = 0)"
   apply (unfold test_bit_eq_iff [THEN sym])
   apply (rule iffI)
    apply (rule ext)
    apply (rule_tac [2] ext)
    apply (auto simp add : word_ao_nth nth_shiftr)
     apply (drule arg_cong)
     apply (drule iffD2)
      apply assumption
     apply (simp add : word_ao_nth)
    prefer 2
    apply (simp add : word_size test_bit_bin)
   apply (drule_tac f = "%u. u !! (x - n)" in arg_cong)
   apply (simp add : nth_shiftr)
   apply (case_tac "n <= x")
    apply auto
   done
 
 lemmas and_mask_eq_iff_le_mask = trans
   [OF and_mask_eq_iff_shiftr_0 le_mask_iff [THEN sym]]
 
 lemma mask_shiftl_decompose:
   "mask m << n = mask (m + n) && ~~ (mask n)"
   by (auto intro!: word_eqI simp: and_not_mask nth_shiftl nth_shiftr word_size)
 
 lemma one_bit_shiftl: "set_bit 0 n True = (1 :: 'a :: len word) << n"
   apply (rule word_eqI)
   apply (auto simp add: test_bit_set_gen nth_shiftl word_size
               simp del: word_set_bit_0 shiftl_1)
   done
 
 lemmas one_bit_pow = trans [OF one_bit_shiftl shiftl_1]
 
 lemmas bin_sc_minus_simps =
   bin_sc_simps (2,3,4) [THEN [2] trans, OF bin_sc_minus [THEN sym]]
 
 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
 
 lemma NOT_mask: "NOT (mask n) = -(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
 
 lemmas gt0_iff_gem1 = iffD1 [OF iffD1 [OF iff_left_commute le_m1_iff_lt] order_refl]
 
 lemmas power_2_ge_iff = trans [OF gt0_iff_gem1 [THEN sym] p2_gt_0]
 
 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])
 
 lemmas mask_lt_2pn =
   le_mask_iff_lt_2n [THEN iffD1, THEN iffD1, OF _ order_refl]
 
 lemma bang_eq:
   fixes x :: "'a::len0 word"
   shows "(x = y) = (\<forall>n. x !! n = y !! n)"
   by (subst test_bit_eq_iff[symmetric]) fastforce
 
 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 shiftl_over_and_dist:
   fixes a::"'a::len word"
   shows "(a AND b) << c = (a << c) AND (b << c)"
   apply(rule word_eqI)
   apply(simp add: word_ao_nth nth_shiftl, safe)
   done
 
 lemma shiftr_over_and_dist:
   fixes a::"'a::len word"
   shows "a AND b >> c = (a >> c) AND (b >> c)"
   apply(rule word_eqI)
   apply(simp add:nth_shiftr word_ao_nth)
   done
 
 lemma sshiftr_over_and_dist:
   fixes a::"'a::len word"
   shows "a AND b >>> c = (a >>> c) AND (b >>> c)"
   apply(rule word_eqI)
   apply(simp add:nth_sshiftr word_ao_nth word_size)
   done
 
 lemma shiftl_over_or_dist:
   fixes a::"'a::len word"
   shows "a OR b << c = (a << c) OR (b << c)"
   apply(rule word_eqI)
   apply(simp add:nth_shiftl word_ao_nth, safe)
   done
 
 lemma shiftr_over_or_dist:
   fixes a::"'a::len word"
   shows "a OR b >> c = (a >> c) OR (b >> c)"
   apply(rule word_eqI)
   apply(simp add:nth_shiftr word_ao_nth)
   done
 
 lemma sshiftr_over_or_dist:
   fixes a::"'a::len word"
   shows "a OR b >>> c = (a >>> c) OR (b >>> c)"
   apply(rule word_eqI)
   apply(simp add:nth_sshiftr word_ao_nth word_size)
   done
 
 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)"
   apply(rule word_eqI)
   apply(auto simp:word_size nth_shiftl add.commute add.left_commute)
   done
 
 lemma shiftr_shiftr:
   fixes a::"'a::len word"
   shows "a >> b >> c = a >> (b + c)"
   apply(rule word_eqI)
   apply(simp add:word_size nth_shiftr add.left_commute add.commute)
   done
 
 lemma shiftl_shiftr1:
   fixes a::"'a::len word"
   shows "c \<le> b \<Longrightarrow> a << b >> c = a AND (mask (size a - b)) << (b - c)"
   apply(rule word_eqI)
   apply(auto simp:nth_shiftr nth_shiftl word_size word_ao_nth)
   done
 
 lemma shiftl_shiftr2:
   fixes a::"'a::len word"
   shows "b < c \<Longrightarrow> a << b >> c = (a >> (c - b)) AND (mask (size a - c))"
   apply(rule word_eqI)
   apply(auto simp:nth_shiftr nth_shiftl word_size word_ao_nth)
   done
 
 lemma shiftr_shiftl1:
   fixes a::"'a::len word"
   shows "c \<le> b \<Longrightarrow> a >> b << c = (a >> (b - c)) AND (NOT (mask c))"
   apply(rule word_eqI)
   apply(auto simp:nth_shiftr nth_shiftl word_size word_ops_nth_size)
   done
 
 lemma shiftr_shiftl2:
   fixes a::"'a::len word"
   shows "b < c \<Longrightarrow> a >> b << c = (a << (c - b)) AND (NOT (mask c))"
   apply(rule word_eqI)
   apply(auto simp:nth_shiftr nth_shiftl word_size word_ops_nth_size)
   done
 
 lemmas multi_shift_simps =
   shiftl_shiftl shiftr_shiftr
   shiftl_shiftr1 shiftl_shiftr2
   shiftr_shiftl1 shiftr_shiftl2
 
 lemma word_and_max_word:
   fixes a::"'a::len word"
   shows "x = max_word \<Longrightarrow> a AND x = a"
   by simp
 
 (* Simplifying with word_and_max_word and max_word_def works for arbitrary word sizes,
    but the conditional rewrite can be slow when combined with other common rewrites on
    word expressions. If we are willing to limit our attention to common word sizes,
    the following will usually be much faster. *)
 lemmas word_and_max_simps[simplified max_word_def, simplified] =
   word_and_max[where 'a=8] word_and_max[where 'a=16] word_and_max[where 'a=32] word_and_max[where 'a=64]
 
 lemma word_and_1_bl:
   fixes x::"'a::len word"
   shows "(x AND 1) = of_bl [x !! 0]"
   by (simp add: word_and_1)
 
 lemma word_1_and_bl:
   fixes x::"'a::len word"
   shows "(1 AND x) = of_bl [x !! 0]"
   by (subst word_bw_comms) (simp add: word_and_1)
 
 lemma scast_scast_id [simp]:
   "scast (scast x :: ('a::len) signed word) = (x :: 'a word)"
   "scast (scast y :: ('a::len) word) = (y :: 'a signed word)"
   by (auto simp: is_up scast_up_scast_id)
 
 lemma scast_ucast_id [simp]:
     "scast (ucast (x :: 'a::len word) :: 'a signed word) = x"
   by (metis down_cast_same is_down len_signed order_refl scast_scast_id(1))
 
 lemma ucast_scast_id [simp]:
     "ucast (scast (x :: 'a::len signed word) :: 'a word) = x"
   by (metis scast_scast_id(2) scast_ucast_id)
 
 lemma scast_of_nat [simp]:
     "scast (of_nat x :: 'a::len signed word) = (of_nat x :: 'a word)"
   by (metis (hide_lams, no_types) len_signed scast_def uint_sint
          word_of_nat word_ubin.Abs_norm word_ubin.eq_norm)
 
 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)"
   apply (rule sym)
   apply (subst word_unat.inverse_norm)
   apply (simp add: ucast_def word_of_int[symmetric]
                    of_nat_nat[symmetric] unat_def[symmetric])
   apply (simp add: unat_of_nat)
   apply (rule nat_int.Rep_eqD)
   apply (simp only: zmod_int)
   apply (rule mod_mod_cancel)
   by (simp add: is_down le_imp_power_dvd)
 
 (* 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)
 
 lemma word_sint_1 [simp]:
   "sint (1::'a::len word) = (if LENGTH('a) = 1 then -1 else 1)"
   apply (case_tac "LENGTH('a) \<le> 1")
    apply (metis diff_is_0_eq sbintrunc_0_numeral(1) sint_n1 word_1_wi
            word_m1_wi word_msb_1 word_msb_n1 word_sbin.Abs_norm)
   apply (metis bin_nth_1 diff_is_0_eq neq0_conv sbintrunc_minus_simps(4)
             sint_word_ariths(8) uint_1 word_msb_1 word_msb_nth)
   done
 
 lemma scast_1':
   "(scast (1::'a::len word) :: 'b::len word) =
    (word_of_int (sbintrunc (LENGTH('a::len) - Suc 0) (1::int)))"
   by (metis One_nat_def scast_def sint_word_ariths(8))
 
 lemma scast_1 [simp]:
   "(scast (1::'a::len word) :: 'b::len word) = (if LENGTH('a) = 1 then -1 else 1)"
   by (clarsimp simp: scast_1')
      (metis Suc_pred len_gt_0 nat.exhaust sbintrunc_Suc_numeral(1) uint_1 word_uint.Rep_inverse')
 
 lemma scast_eq_scast_id [simp]:
   "((scast (a :: 'a::len signed word) :: 'a word) = scast b) = (a = b)"
   by (metis ucast_scast_id)
 
 lemma ucast_eq_ucast_id [simp]:
   "((ucast (a :: 'a::len word) :: 'a signed word) = ucast b) = (a = b)"
   by (metis scast_ucast_id)
 
 lemma scast_ucast_norm [simp]:
   "(ucast (a :: 'a::len word) = (b :: 'a signed word)) = (a = scast b)"
   "((b :: 'a signed word) = ucast (a :: 'a::len word)) = (a = scast b)"
   by (metis scast_ucast_id ucast_scast_id)+
 
 lemma of_bl_drop:
   "of_bl (drop n xs) = (of_bl xs && mask (length xs - n))"
   apply (clarsimp simp: bang_eq test_bit_of_bl rev_nth cong: rev_conj_cong)
   apply (safe; simp add: word_size to_bl_nth)
   done
 
 lemma of_int_uint [simp]:
   "of_int (uint x) = x"
   by (metis word_of_int word_uint.Rep_inverse')
 
 lemma shiftr_mask2:
   "n \<le> LENGTH('a) \<Longrightarrow> (mask n >> m :: ('a :: len) word) = mask (n - m)"
   apply (rule word_eqI)
   apply (simp add: nth_shiftr word_size)
   apply arith
   done
 
 corollary word_plus_and_or_coroll:
   "x && y = 0 \<Longrightarrow> x + y = x || y"
   using word_plus_and_or[where x=x and y=y]
   by simp
 
 corollary word_plus_and_or_coroll2:
   "(x && w) + (x && ~~ w) = x"
   apply (subst word_plus_and_or_coroll)
    apply (rule word_eqI, simp add: word_size word_ops_nth_size)
   apply (rule word_eqI, simp add: word_size word_ops_nth_size)
   apply blast
   done
 
 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
 
 lemmas less_le_mult_nat = less_le_mult_nat'[simplified distrib_right, simplified]
 
 (* FIXME: these should eventually be moved to HOL/Word. *)
 lemmas extra_sle_sless_unfolds [simp] =
     word_sle_def[where a=0 and b=1]
     word_sle_def[where a=0 and b="numeral n"]
     word_sle_def[where a=1 and b=0]
     word_sle_def[where a=1 and b="numeral n"]
     word_sle_def[where a="numeral n" and b=0]
     word_sle_def[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 to_bl_1:
   "to_bl (1::'a::len word) = replicate (LENGTH('a) - 1) False @ [True]"
 proof -
   have "to_bl (1 :: 'a::len word) = to_bl (mask 1 :: 'a::len word)"
     by (simp add: mask_def)
 
   also have "\<dots> = replicate (LENGTH('a) - 1) False @ [True]"
     by (cases "LENGTH('a)"; clarsimp simp: to_bl_mask)
 
   finally show ?thesis .
 qed
 
 lemma list_of_false:
   "True \<notin> set xs \<Longrightarrow> xs = replicate (length xs) False"
   by (induct xs, simp_all)
 
 lemma eq_zero_set_bl:
   "(w = 0) = (True \<notin> set (to_bl w))"
   using list_of_false word_bl.Rep_inject by fastforce
 
 lemma diff_diff_less:
   "(i < m - (m - (n :: nat))) = (i < m \<and> i < n)"
   by auto
 
 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
 
 end
diff --git a/thys/Word_Lib/Word_Next.thy b/thys/Word_Lib/Word_Next.thy
--- a/thys/Word_Lib/Word_Next.thy
+++ b/thys/Word_Lib/Word_Next.thy
@@ -1,84 +1,86 @@
+(* SPDX-License-Identifier: BSD-3-Clause *)
+
 section\<open>Increment and Decrement Machine Words Without Wrap-Around\<close>
 
 theory Word_Next
 imports
   Aligned
 begin
 
 text\<open>Previous and next words addresses, without wrap around.\<close>
 
 definition word_next :: "'a::len word \<Rightarrow> 'a::len word" where
   "word_next a \<equiv> if a = max_word then max_word else a + 1"
 
 definition word_prev :: "'a::len word \<Rightarrow> 'a::len word" where
   "word_prev a \<equiv> if a = 0 then 0 else a - 1"
 
 text\<open>Examples:\<close>
 
 lemma "word_next (2:: 8 word) = 3" by eval
 lemma "word_next (255:: 8 word) = 255" by eval
 lemma "word_prev (2:: 8 word) = 1" by eval
 lemma "word_prev (0:: 8 word) = 0" by eval
 
 
 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
   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> max_word \<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
 
 
 lemma word_Suc_leq: fixes k::"'a::len word" shows "k \<noteq> max_word \<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> max_word \<Longrightarrow> x + 1 \<le> k \<longleftrightarrow> x < k"
   by (meson not_less word_Suc_leq)
 
 lemma word_lessThan_Suc_atMost: fixes k::"'a::len word" shows "k \<noteq> max_word \<Longrightarrow> {..< k + 1} = {..k}"
   by(simp add: lessThan_def atMost_def word_Suc_leq)
 
 lemma word_atLeastLessThan_Suc_atLeastAtMost:
   fixes l::"'a::len word" shows "u \<noteq> max_word \<Longrightarrow> {l..< u + 1} = {l..u}"
   by (simp add: atLeastAtMost_def atLeastLessThan_def word_lessThan_Suc_atMost)
 
 lemma word_atLeastAtMost_Suc_greaterThanAtMost: fixes l::"'a::len word"
   shows "m \<noteq> max_word \<Longrightarrow> {m<..u} = {m + 1..u}"
   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> max_word" and "l \<le> m" and "m \<le> u"
   shows "{l..m} \<union> {m+1..u} = {l..u}"
   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
 
 lemma word_adjacent_union:
   "word_next e = s' \<Longrightarrow> s \<le> e \<Longrightarrow> s' \<le> e' \<Longrightarrow> {s..e} \<union> {s'..e'} = {s .. e'}"
   by (metis Un_absorb2 atLeastatMost_subset_iff ivl_disj_un_two(7) max_word_max
             word_atLeastLessThan_Suc_atLeastAtMost word_le_less_eq word_next_def linorder_not_le)
 
 end
diff --git a/thys/Word_Lib/Word_Setup_32.thy b/thys/Word_Lib/Word_Setup_32.thy
--- a/thys/Word_Lib/Word_Setup_32.thy
+++ b/thys/Word_Lib/Word_Setup_32.thy
@@ -1,47 +1,43 @@
 (*
- * Copyright 2014, NICTA
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
- * This software may be distributed and modified according to the terms of
- * the BSD 2-Clause license. Note that NO WARRANTY is provided.
- * See "LICENSE_BSD2.txt" for details.
- *
- * @TAG(NICTA_BSD)
+ * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "32-Bit Machine Word Setup"
 
 theory Word_Setup_32
 imports Word_Enum
 begin
 
 text \<open>This theory defines standard platform-specific word size and alignment.\<close>
 
 type_synonym machine_word_len = 32
 type_synonym machine_word = "machine_word_len word"
 
 definition word_bits :: nat
 where
   "word_bits = LENGTH(machine_word_len)"
 
 text \<open>The following two are numerals so they can be used as nats and words.\<close>
 definition word_size_bits :: "'a :: numeral"
 where
   "word_size_bits = 2"
 
 definition word_size :: "'a :: numeral"
 where
   "word_size = 4"
 
 lemma word_bits_conv[code]:
   "word_bits = 32"
   unfolding word_bits_def by simp
 
 lemma word_size_word_size_bits:
   "(word_size::nat) = 2 ^ word_size_bits"
   unfolding word_size_def word_size_bits_def by simp
 
 lemma word_bits_word_size_conv:
   "word_bits = word_size * 8"
   unfolding word_bits_def word_size_def by simp
 
 end
diff --git a/thys/Word_Lib/Word_Setup_64.thy b/thys/Word_Lib/Word_Setup_64.thy
--- a/thys/Word_Lib/Word_Setup_64.thy
+++ b/thys/Word_Lib/Word_Setup_64.thy
@@ -1,47 +1,43 @@
 (*
- * Copyright 2014, NICTA
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
- * This software may be distributed and modified according to the terms of
- * the BSD 2-Clause license. Note that NO WARRANTY is provided.
- * See "LICENSE_BSD2.txt" for details.
- *
- * @TAG(NICTA_BSD)
+ * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "64-Bit Machine Word Setup"
 
 theory Word_Setup_64
 imports Word_Enum
 begin
 
 text \<open>This theory defines standard platform-specific word size and alignment.\<close>
 
 type_synonym machine_word_len = 64
 type_synonym machine_word = "machine_word_len word"
 
 definition word_bits :: nat
 where
   "word_bits = LENGTH(machine_word_len)"
 
 text \<open>The following two are numerals so they can be used as nats and words.\<close>
 definition word_size_bits :: "'a :: numeral"
 where
   "word_size_bits = 3"
 
 definition word_size :: "'a :: numeral"
 where
   "word_size = 8"
 
 lemma word_bits_conv[code]:
   "word_bits = 64"
   unfolding word_bits_def by simp
 
 lemma word_size_word_size_bits:
   "(word_size::nat) = 2 ^ word_size_bits"
   unfolding word_size_def word_size_bits_def by simp
 
 lemma word_bits_word_size_conv:
   "word_bits = word_size * 8"
   unfolding word_bits_def word_size_def by simp
 
 end
diff --git a/thys/Word_Lib/Word_Syntax.thy b/thys/Word_Lib/Word_Syntax.thy
--- a/thys/Word_Lib/Word_Syntax.thy
+++ b/thys/Word_Lib/Word_Syntax.thy
@@ -1,68 +1,64 @@
 (*
- * Copyright 2014, NICTA
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
- * This software may be distributed and modified according to the terms of
- * the BSD 2-Clause license. Note that NO WARRANTY is provided.
- * See "LICENSE_BSD2.txt" for details.
- *
- * @TAG(NICTA_BSD)
+ * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "Additional Syntax for Word Bit Operations"
 
 theory Word_Syntax
 imports
   "HOL-Word.More_Word"
   WordBitwise_Signed
   Hex_Words
   Norm_Words
   Word_Type_Syntax
 begin
 
 text \<open>Additional bit and type syntax that forces word types.\<close>
 
 type_synonym word8 = "8 word"
 type_synonym word16 = "16 word"
 type_synonym word32 = "32 word"
 type_synonym word64 = "64 word"
 
 lemma len8: "len_of (x :: 8 itself) = 8" by simp
 lemma len16: "len_of (x :: 16 itself) = 16" by simp
 lemma len32: "len_of (x :: 32 itself) = 32" by simp
 lemma len64: "len_of (x :: 64 itself) = 64" by simp
 
 
 abbreviation
   wordNOT  :: "'a::len0 word \<Rightarrow> 'a word"      ("~~ _" [70] 71)
 where
   "~~ x == NOT x"
 
 abbreviation
   wordAND  :: "'a::len0 word \<Rightarrow> 'a word \<Rightarrow> 'a word" (infixr "&&" 64)
 where
   "a && b == a AND b"
 
 abbreviation
   wordOR   :: "'a::len0 word \<Rightarrow> 'a word \<Rightarrow> 'a word" (infixr "||"  59)
 where
   "a || b == a OR b"
 
 abbreviation
   wordXOR  :: "'a::len0 word \<Rightarrow> 'a word \<Rightarrow> 'a word" (infixr "xor" 59)
 where
   "a xor b == a XOR b"
 
 (* testing for presence of word_bitwise *)
 lemma "((x :: word32) >> 3) AND 7 = (x AND 56) >> 3"
   by word_bitwise
 
 (* FIXME: move to Word distribution *)
 lemma bin_nth_minus_Bit0[simp]:
   "0 < n \<Longrightarrow> bin_nth (numeral (num.Bit0 w)) n = bin_nth (numeral w) (n - 1)"
   by (cases n; simp)
 
 lemma bin_nth_minus_Bit1[simp]:
   "0 < n \<Longrightarrow> bin_nth (numeral (num.Bit1 w)) n = bin_nth (numeral w) (n - 1)"
   by (cases n; simp)
 
 end
diff --git a/thys/Word_Lib/Word_Type_Syntax.thy b/thys/Word_Lib/Word_Type_Syntax.thy
--- a/thys/Word_Lib/Word_Type_Syntax.thy
+++ b/thys/Word_Lib/Word_Type_Syntax.thy
@@ -1,62 +1,58 @@
 (*
- * Copyright 2017, Data61/CSIRO
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
- * This software may be distributed and modified according to the terms of
- * the BSD 2-Clause license. Note that NO WARRANTY is provided.
- * See "LICENSE_BSD2.txt" for details.
- *
- * @TAG(DATA61_BSD)
+ * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "Displaying Phantom Types for Word Operations"
 
 theory Word_Type_Syntax
 imports "HOL-Word.Word"
 begin
 
 ML \<open>
 structure Word_Syntax =
 struct
 
   val show_word_types = Attrib.setup_config_bool @{binding show_word_types} (K true)
 
   fun tr' cnst ctxt typ ts = if Config.get ctxt show_word_types then
       case (Term.binder_types typ, Term.body_type typ) of
         ([Type (@{type_name "word"}, [S])], Type (@{type_name "word"}, [T])) =>
           list_comb
             (Syntax.const cnst $ Syntax_Phases.term_of_typ ctxt S $ Syntax_Phases.term_of_typ ctxt T
             , ts)
         | _ => raise Match
   else raise Match
 
 end
 \<close>
 
 
 syntax
   "_Ucast" :: "type \<Rightarrow> type \<Rightarrow> logic" ("(1UCAST/(1'(_ \<rightarrow> _')))")
 translations
   "UCAST('s \<rightarrow> 't)" => "CONST ucast :: ('s word \<Rightarrow> 't word)"
 typed_print_translation
   \<open> [(@{const_syntax ucast}, Word_Syntax.tr' @{syntax_const "_Ucast"})] \<close>
 
 
 syntax
   "_Scast" :: "type \<Rightarrow> type \<Rightarrow> logic" ("(1SCAST/(1'(_ \<rightarrow> _')))")
 translations
   "SCAST('s \<rightarrow> 't)" => "CONST scast :: ('s word \<Rightarrow> 't word)"
 typed_print_translation
   \<open> [(@{const_syntax scast}, Word_Syntax.tr' @{syntax_const "_Scast"})] \<close>
 
 
 syntax
   "_Revcast" :: "type \<Rightarrow> type \<Rightarrow> logic" ("(1REVCAST/(1'(_ \<rightarrow> _')))")
 translations
   "REVCAST('s \<rightarrow> 't)" => "CONST revcast :: ('s word \<Rightarrow> 't word)"
 typed_print_translation
   \<open> [(@{const_syntax revcast}, Word_Syntax.tr' @{syntax_const "_Revcast"})] \<close>
 
 (* Further candidates for showing word sizes, but with different arities:
    slice, word_cat, word_split, word_rcat, word_rsplit *)
 
 end