diff --git a/thys/Word_Lib/Ancient_Numeral.thy b/thys/Word_Lib/Ancient_Numeral.thy
--- a/thys/Word_Lib/Ancient_Numeral.thy
+++ b/thys/Word_Lib/Ancient_Numeral.thy
@@ -1,229 +1,235 @@
+(*
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
+
 theory Ancient_Numeral
   imports Main Reversed_Bit_Lists
 begin
 
 definition Bit :: "int \<Rightarrow> bool \<Rightarrow> int"  (infixl "BIT" 90)
   where "k BIT b = (if b then 1 else 0) + k + k"
 
 lemma Bit_B0: "k BIT False = k + k"
    by (simp add: Bit_def)
 
 lemma Bit_B1: "k BIT True = k + k + 1"
    by (simp add: Bit_def)
 
 lemma Bit_B0_2t: "k BIT False = 2 * k"
   by (rule trans, rule Bit_B0) simp
 
 lemma Bit_B1_2t: "k BIT True = 2 * k + 1"
   by (rule trans, rule Bit_B1) simp
 
 lemma uminus_Bit_eq:
   "- k BIT b = (- k - of_bool b) BIT b"
   by (cases b) (simp_all add: Bit_def)
 
 lemma power_BIT: "2 ^ Suc n - 1 = (2 ^ n - 1) BIT True"
   by (simp add: Bit_B1)
 
 lemma bin_rl_simp [simp]: "bin_rest w BIT bin_last w = w"
   by (simp add: Bit_def)
 
 lemma bin_rest_BIT [simp]: "bin_rest (x BIT b) = x"
   by (simp add: Bit_def)
 
 lemma even_BIT [simp]: "even (x BIT b) \<longleftrightarrow> \<not> b"
   by (simp add: Bit_def)
 
 lemma bin_last_BIT [simp]: "bin_last (x BIT b) = b"
   by simp
 
 lemma BIT_eq_iff [iff]: "u BIT b = v BIT c \<longleftrightarrow> u = v \<and> b = c"
   by (auto simp: Bit_def) arith+
 
 lemma BIT_bin_simps [simp]:
   "numeral k BIT False = numeral (Num.Bit0 k)"
   "numeral k BIT True = numeral (Num.Bit1 k)"
   "(- numeral k) BIT False = - numeral (Num.Bit0 k)"
   "(- numeral k) BIT True = - numeral (Num.BitM k)"
   by (simp_all only: Bit_B0 Bit_B1 numeral.simps numeral_BitM)
 
 lemma BIT_special_simps [simp]:
   shows "0 BIT False = 0"
     and "0 BIT True = 1"
     and "1 BIT False = 2"
     and "1 BIT True = 3"
     and "(- 1) BIT False = - 2"
     and "(- 1) BIT True = - 1"
   by (simp_all add: Bit_def)
 
 lemma Bit_eq_0_iff: "w BIT b = 0 \<longleftrightarrow> w = 0 \<and> \<not> b"
   by (auto simp: Bit_def) arith
 
 lemma Bit_eq_m1_iff: "w BIT b = -1 \<longleftrightarrow> w = -1 \<and> b"
   by (auto simp: Bit_def) arith
 
 lemma expand_BIT:
   "numeral (Num.Bit0 w) = numeral w BIT False"
   "numeral (Num.Bit1 w) = numeral w BIT True"
   "- numeral (Num.Bit0 w) = (- numeral w) BIT False"
   "- numeral (Num.Bit1 w) = (- numeral (w + Num.One)) BIT True"
   by (simp_all add: BitM_inc_eq add_One)
 
 lemma less_Bits: "v BIT b < w BIT c \<longleftrightarrow> v < w \<or> v \<le> w \<and> \<not> b \<and> c"
   by (auto simp: Bit_def)
 
 lemma le_Bits: "v BIT b \<le> w BIT c \<longleftrightarrow> v < w \<or> v \<le> w \<and> (\<not> b \<or> c)"
   by (auto simp: Bit_def)
 
 lemma pred_BIT_simps [simp]:
   "x BIT False - 1 = (x - 1) BIT True"
   "x BIT True - 1 = x BIT False"
   by (simp_all add: Bit_B0_2t Bit_B1_2t)
 
 lemma succ_BIT_simps [simp]:
   "x BIT False + 1 = x BIT True"
   "x BIT True + 1 = (x + 1) BIT False"
   by (simp_all add: Bit_B0_2t Bit_B1_2t)
 
 lemma add_BIT_simps [simp]:
   "x BIT False + y BIT False = (x + y) BIT False"
   "x BIT False + y BIT True = (x + y) BIT True"
   "x BIT True + y BIT False = (x + y) BIT True"
   "x BIT True + y BIT True = (x + y + 1) BIT False"
   by (simp_all add: Bit_B0_2t Bit_B1_2t)
 
 lemma mult_BIT_simps [simp]:
   "x BIT False * y = (x * y) BIT False"
   "x * y BIT False = (x * y) BIT False"
   "x BIT True * y = (x * y) BIT False + y"
   by (simp_all add: Bit_B0_2t Bit_B1_2t algebra_simps)
 
 lemma B_mod_2': "X = 2 \<Longrightarrow> (w BIT True) mod X = 1 \<and> (w BIT False) mod X = 0"
   by (simp add: Bit_B0 Bit_B1)
 
 lemma bin_ex_rl: "\<exists>w b. w BIT b = bin"
   by (metis bin_rl_simp)
 
 lemma bin_exhaust: "(\<And>x b. bin = x BIT b \<Longrightarrow> Q) \<Longrightarrow> Q"
 by (metis bin_ex_rl)
 
 lemma bin_abs_lem: "bin = (w BIT b) \<Longrightarrow> bin \<noteq> -1 \<longrightarrow> bin \<noteq> 0 \<longrightarrow> nat \<bar>w\<bar> < nat \<bar>bin\<bar>"
   apply clarsimp
   apply (unfold Bit_def)
   apply (cases b)
    apply (clarsimp, arith)
   apply (clarsimp, arith)
   done
 
 lemma bin_induct:
   assumes PPls: "P 0"
     and PMin: "P (- 1)"
     and PBit: "\<And>bin bit. P bin \<Longrightarrow> P (bin BIT bit)"
   shows "P bin"
   apply (rule_tac P=P and a=bin and f1="nat \<circ> abs" in wf_measure [THEN wf_induct])
   apply (simp add: measure_def inv_image_def)
   apply (case_tac x rule: bin_exhaust)
   apply (frule bin_abs_lem)
   apply (auto simp add : PPls PMin PBit)
   done
 
 lemma Bit_div2: "(w BIT b) div 2 = w"
   by (fact bin_rest_BIT)
 
 lemma twice_conv_BIT: "2 * x = x BIT False"
   by (simp add: Bit_def)
 
 lemma BIT_lt0 [simp]: "x BIT b < 0 \<longleftrightarrow> x < 0"
 by(cases b)(auto simp add: Bit_def)
 
 lemma BIT_ge0 [simp]: "x BIT b \<ge> 0 \<longleftrightarrow> x \<ge> 0"
 by(cases b)(auto simp add: Bit_def)
 
 lemma bin_to_bl_aux_Bit_minus_simp [simp]:
   "0 < n \<Longrightarrow> bin_to_bl_aux n (w BIT b) bl = bin_to_bl_aux (n - 1) w (b # bl)"
   by (cases n) auto
 
 lemma bl_to_bin_BIT:
   "bl_to_bin bs BIT b = bl_to_bin (bs @ [b])"
   by (simp add: bl_to_bin_append Bit_def)
 
 lemma bin_nth_0_BIT: "bin_nth (w BIT b) 0 \<longleftrightarrow> b"
   by simp
 
 lemma bin_nth_Suc_BIT: "bin_nth (w BIT b) (Suc n) = bin_nth w n"
   by (simp add: bit_Suc)
 
 lemma bin_nth_minus [simp]: "0 < n \<Longrightarrow> bin_nth (w BIT b) n = bin_nth w (n - 1)"
   by (cases n) (simp_all add: bit_Suc)
 
 lemma bin_sign_simps [simp]:
   "bin_sign (w BIT b) = bin_sign w"
   by (simp add: bin_sign_def Bit_def)
 
 lemma bin_nth_Bit: "bin_nth (w BIT b) n \<longleftrightarrow> n = 0 \<and> b \<or> (\<exists>m. n = Suc m \<and> bin_nth w m)"
   by (cases n) auto
 
 lemmas sbintrunc_Suc_BIT [simp] =
   signed_take_bit_Suc [where a="w BIT b", simplified bin_last_BIT bin_rest_BIT] for w b
 
 lemmas sbintrunc_0_BIT_B0 [simp] =
   signed_take_bit_0 [where a="w BIT False", simplified bin_last_numeral_simps bin_rest_numeral_simps]
   for w
 
 lemmas sbintrunc_0_BIT_B1 [simp] =
   signed_take_bit_0 [where a="w BIT True", simplified bin_last_BIT bin_rest_numeral_simps]
   for w
 
 lemma sbintrunc_Suc_minus_Is:
   \<open>0 < n \<Longrightarrow>
   sbintrunc (n - 1) w = y \<Longrightarrow>
   sbintrunc n (w BIT b) = y BIT b\<close>
   by (cases n) (simp_all add: Bit_def signed_take_bit_Suc)
 
 lemma bin_cat_Suc_Bit: "bin_cat w (Suc n) (v BIT b) = bin_cat w n v BIT b"
   by (auto simp add: Bit_def concat_bit_Suc)
 
 lemma int_not_BIT [simp]: "NOT (w BIT b) = (NOT w) BIT (\<not> b)"
   by (simp add: not_int_def Bit_def)
 
 lemma int_and_Bits [simp]: "(x BIT b) AND (y BIT c) = (x AND y) BIT (b \<and> c)"
   using and_int_rec [of \<open>x BIT b\<close> \<open>y BIT c\<close>] by (auto simp add: Bit_B0_2t Bit_B1_2t)
 
 lemma int_or_Bits [simp]: "(x BIT b) OR (y BIT c) = (x OR y) BIT (b \<or> c)"
   using or_int_rec [of \<open>x BIT b\<close> \<open>y BIT c\<close>] by (auto simp add: Bit_B0_2t Bit_B1_2t)
 
 lemma int_xor_Bits [simp]: "(x BIT b) XOR (y BIT c) = (x XOR y) BIT ((b \<or> c) \<and> \<not> (b \<and> c))"
   using xor_int_rec [of \<open>x BIT b\<close> \<open>y BIT c\<close>] by (auto simp add: Bit_B0_2t Bit_B1_2t)
 
 lemma mod_BIT:
   "bin BIT bit mod 2 ^ Suc n = (bin mod 2 ^ n) BIT bit" for bit
 proof -
   have "2 * (bin mod 2 ^ n) + 1 = (2 * bin mod 2 ^ Suc n) + 1"
     by (simp add: mod_mult_mult1)
   also have "\<dots> = ((2 * bin mod 2 ^ Suc n) + 1) mod 2 ^ Suc n"
     by (simp add: ac_simps pos_zmod_mult_2)
   also have "\<dots> = (2 * bin + 1) mod 2 ^ Suc n"
     by (simp only: mod_simps)
   finally show ?thesis
     by (auto simp add: Bit_def)
 qed
 
 lemma minus_BIT_0: fixes x y :: int shows "x BIT b - y BIT False = (x - y) BIT b"
 by(simp add: Bit_def)
 
 lemma int_lsb_BIT [simp]: fixes x :: int shows
   "lsb (x BIT b) \<longleftrightarrow> b"
 by(simp add: lsb_int_def)
 
 lemma int_shiftr_BIT [simp]: fixes x :: int
   shows int_shiftr0: "x >> 0 = x"
   and int_shiftr_Suc: "x BIT b >> Suc n = x >> n"
 proof -
   show "x >> 0 = x" by (simp add: shiftr_int_def)
   show "x BIT b >> Suc n = x >> n" by (cases b)
    (simp_all add: shiftr_int_def Bit_def add.commute pos_zdiv_mult_2)
 qed
 
 lemma msb_BIT [simp]: "msb (x BIT b) = msb x"
 by(simp add: msb_int_def)
 
 end
\ No newline at end of file
diff --git a/thys/Word_Lib/Bit_Comprehension.thy b/thys/Word_Lib/Bit_Comprehension.thy
--- a/thys/Word_Lib/Bit_Comprehension.thy
+++ b/thys/Word_Lib/Bit_Comprehension.thy
@@ -1,247 +1,250 @@
-(*  Author:     Brian Huffman, PSU; Jeremy Dawson and Gerwin Klein, NICTA
-*)
+(*
+ * Copyright Brian Huffman, PSU; Jeremy Dawson and Gerwin Klein, NICTA
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
 
 section \<open>Comprehension syntax for bit expressions\<close>
 
 theory Bit_Comprehension
   imports "HOL-Library.Word"
 begin
 
 class bit_comprehension = ring_bit_operations +
   fixes set_bits :: \<open>(nat \<Rightarrow> bool) \<Rightarrow> 'a\<close>  (binder \<open>BITS \<close> 10)
   assumes set_bits_bit_eq: \<open>set_bits (bit a) = a\<close>
 begin
 
 lemma set_bits_False_eq [simp]:
   \<open>(BITS _. False) = 0\<close>
   using set_bits_bit_eq [of 0] by (simp add: bot_fun_def)
 
 end
 
 instantiation int :: bit_comprehension
 begin
 
 definition
   \<open>set_bits f = (
       if \<exists>n. \<forall>m\<ge>n. f m = f n then
       let n = LEAST n. \<forall>m\<ge>n. f m = f n
       in signed_take_bit n (horner_sum of_bool 2 (map f [0..<Suc n]))
      else 0 :: int)\<close>
 
 instance proof
   fix k :: int
-  from int_bit_bound [of k] 
+  from int_bit_bound [of k]
   obtain n where *: \<open>\<And>m. n \<le> m \<Longrightarrow> bit k m \<longleftrightarrow> bit k n\<close>
     and **: \<open>n > 0 \<Longrightarrow> bit k (n - 1) \<noteq> bit k n\<close>
     by blast
   then have ***: \<open>\<exists>n. \<forall>n'\<ge>n. bit k n' \<longleftrightarrow> bit k n\<close>
     by meson
   have l: \<open>(LEAST q. \<forall>m\<ge>q. bit k m \<longleftrightarrow> bit k q) = n\<close>
     apply (rule Least_equality)
     using * apply blast
     apply (metis "**" One_nat_def Suc_pred le_cases le0 neq0_conv not_less_eq_eq)
     done
   show \<open>set_bits (bit k) = k\<close>
     apply (simp only: *** set_bits_int_def horner_sum_bit_eq_take_bit l)
     apply simp
     apply (rule bit_eqI)
     apply (simp add: bit_signed_take_bit_iff min_def)
     apply (auto simp add: not_le bit_take_bit_iff dest: *)
     done
 qed
 
 end
 
 lemma int_set_bits_K_False [simp]: "(BITS _. False) = (0 :: int)"
   by (simp add: set_bits_int_def)
 
 lemma int_set_bits_K_True [simp]: "(BITS _. True) = (-1 :: int)"
   by (simp add: set_bits_int_def)
 
 instantiation word :: (len) bit_comprehension
 begin
 
 definition word_set_bits_def:
   \<open>(BITS n. P n) = (horner_sum of_bool 2 (map P [0..<LENGTH('a)]) :: 'a word)\<close>
 
 instance by standard
   (simp add: word_set_bits_def horner_sum_bit_eq_take_bit)
 
 end
 
 lemma bit_set_bits_word_iff:
   \<open>bit (set_bits P :: 'a::len word) n \<longleftrightarrow> n < LENGTH('a) \<and> P n\<close>
   by (auto simp add: word_set_bits_def bit_horner_sum_bit_word_iff)
 
 lemma set_bits_K_False [simp]:
   \<open>set_bits (\<lambda>_. False) = (0 :: 'a :: len word)\<close>
   by (rule bit_word_eqI) (simp add: bit_set_bits_word_iff)
 
 lemma set_bits_int_unfold':
   \<open>set_bits f =
     (if \<exists>n. \<forall>n'\<ge>n. \<not> f n' then
       let n = LEAST n. \<forall>n'\<ge>n. \<not> f n'
       in horner_sum of_bool 2 (map f [0..<n])
      else if \<exists>n. \<forall>n'\<ge>n. f n' then
       let n = LEAST n. \<forall>n'\<ge>n. f n'
       in signed_take_bit n (horner_sum of_bool 2 (map f [0..<n] @ [True]))
      else 0 :: int)\<close>
 proof (cases \<open>\<exists>n. \<forall>m\<ge>n. f m \<longleftrightarrow> f n\<close>)
   case True
   then obtain q where q: \<open>\<forall>m\<ge>q. f m \<longleftrightarrow> f q\<close>
     by blast
   define n where \<open>n = (LEAST n. \<forall>m\<ge>n. f m \<longleftrightarrow> f n)\<close>
-  have \<open>\<forall>m\<ge>n. f m \<longleftrightarrow> f n\<close> 
+  have \<open>\<forall>m\<ge>n. f m \<longleftrightarrow> f n\<close>
     unfolding n_def
     using q by (rule LeastI [of _ q])
   then have n: \<open>\<And>m. n \<le> m \<Longrightarrow> f m \<longleftrightarrow> f n\<close>
     by blast
   from n_def have n_eq: \<open>(LEAST q. \<forall>m\<ge>q. f m \<longleftrightarrow> f n) = n\<close>
     by (smt Least_equality Least_le \<open>\<forall>m\<ge>n. f m = f n\<close> dual_order.refl le_refl n order_refl)
   show ?thesis
   proof (cases \<open>f n\<close>)
     case False
-    with n have *: \<open>\<exists>n. \<forall>n'\<ge>n. \<not> f n'\<close> 
+    with n have *: \<open>\<exists>n. \<forall>n'\<ge>n. \<not> f n'\<close>
       by blast
     have **: \<open>(LEAST n. \<forall>n'\<ge>n. \<not> f n') = n\<close>
       using False n_eq by simp
     from * False show ?thesis
     apply (simp add: set_bits_int_def n_def [symmetric] ** del: upt.upt_Suc)
     apply (auto simp add: take_bit_horner_sum_bit_eq
       bit_horner_sum_bit_iff take_map
       signed_take_bit_def set_bits_int_def
       horner_sum_bit_eq_take_bit simp del: upt.upt_Suc)
     done
   next
     case True
     with n have *: \<open>\<exists>n. \<forall>n'\<ge>n. f n'\<close>
       by blast
     have ***: \<open>\<not> (\<exists>n. \<forall>n'\<ge>n. \<not> f n')\<close>
       apply (rule ccontr)
-      using * nat_le_linear by auto 
+      using * nat_le_linear by auto
     have **: \<open>(LEAST n. \<forall>n'\<ge>n. f n') = n\<close>
       using True n_eq by simp
     from * *** True show ?thesis
     apply (simp add: set_bits_int_def n_def [symmetric] ** del: upt.upt_Suc)
     apply (auto simp add: take_bit_horner_sum_bit_eq
       bit_horner_sum_bit_iff take_map
       signed_take_bit_def set_bits_int_def
       horner_sum_bit_eq_take_bit nth_append simp del: upt.upt_Suc)
     done
   qed
 next
   case False
   then show ?thesis
     by (auto simp add: set_bits_int_def)
 qed
 
-inductive wf_set_bits_int :: "(nat \<Rightarrow> bool) \<Rightarrow> bool" 
+inductive wf_set_bits_int :: "(nat \<Rightarrow> bool) \<Rightarrow> bool"
   for f :: "nat \<Rightarrow> bool"
 where
   zeros: "\<forall>n' \<ge> n. \<not> f n' \<Longrightarrow> wf_set_bits_int f"
 | ones: "\<forall>n' \<ge> n. f n' \<Longrightarrow> wf_set_bits_int f"
 
 lemma wf_set_bits_int_simps: "wf_set_bits_int f \<longleftrightarrow> (\<exists>n. (\<forall>n'\<ge>n. \<not> f n') \<or> (\<forall>n'\<ge>n. f n'))"
 by(auto simp add: wf_set_bits_int.simps)
 
 lemma wf_set_bits_int_const [simp]: "wf_set_bits_int (\<lambda>_. b)"
 by(cases b)(auto intro: wf_set_bits_int.intros)
 
-lemma wf_set_bits_int_fun_upd [simp]: 
+lemma wf_set_bits_int_fun_upd [simp]:
   "wf_set_bits_int (f(n := b)) \<longleftrightarrow> wf_set_bits_int f" (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   assume ?lhs
   then obtain n'
     where "(\<forall>n''\<ge>n'. \<not> (f(n := b)) n'') \<or> (\<forall>n''\<ge>n'. (f(n := b)) n'')"
     by(auto simp add: wf_set_bits_int_simps)
   hence "(\<forall>n''\<ge>max (Suc n) n'. \<not> f n'') \<or> (\<forall>n''\<ge>max (Suc n) n'. f n'')" by auto
   thus ?rhs by(auto simp only: wf_set_bits_int_simps)
 next
   assume ?rhs
   then obtain n' where "(\<forall>n''\<ge>n'. \<not> f n'') \<or> (\<forall>n''\<ge>n'. f n'')" (is "?wf f n'")
     by(auto simp add: wf_set_bits_int_simps)
   hence "?wf (f(n := b)) (max (Suc n) n')" by auto
   thus ?lhs by(auto simp only: wf_set_bits_int_simps)
 qed
 
 lemma wf_set_bits_int_Suc [simp]:
   "wf_set_bits_int (\<lambda>n. f (Suc n)) \<longleftrightarrow> wf_set_bits_int f" (is "?lhs \<longleftrightarrow> ?rhs")
 by(auto simp add: wf_set_bits_int_simps intro: le_SucI dest: Suc_le_D)
 
 context
   fixes f
   assumes wff: "wf_set_bits_int f"
 begin
 
 lemma int_set_bits_unfold_BIT:
   "set_bits f = of_bool (f 0) + (2 :: int) * set_bits (f \<circ> Suc)"
 using wff proof cases
   case (zeros n)
   show ?thesis
   proof(cases "\<forall>n. \<not> f n")
     case True
     hence "f = (\<lambda>_. False)" by auto
     thus ?thesis using True by(simp add: o_def)
   next
     case False
     then obtain n' where "f n'" by blast
     with zeros have "(LEAST n. \<forall>n'\<ge>n. \<not> f n') = Suc (LEAST n. \<forall>n'\<ge>Suc n. \<not> f n')"
       by(auto intro: Least_Suc)
     also have "(\<lambda>n. \<forall>n'\<ge>Suc n. \<not> f n') = (\<lambda>n. \<forall>n'\<ge>n. \<not> f (Suc n'))" by(auto dest: Suc_le_D)
     also from zeros have "\<forall>n'\<ge>n. \<not> f (Suc n')" by auto
     ultimately show ?thesis using zeros
       apply (simp (no_asm_simp) add: set_bits_int_unfold' exI
         del: upt.upt_Suc flip: map_map split del: if_split)
       apply (simp only: map_Suc_upt upt_conv_Cons)
       apply simp
       done
   qed
 next
   case (ones n)
   show ?thesis
   proof(cases "\<forall>n. f n")
     case True
     hence "f = (\<lambda>_. True)" by auto
     thus ?thesis using True by(simp add: o_def)
   next
     case False
     then obtain n' where "\<not> f n'" by blast
     with ones have "(LEAST n. \<forall>n'\<ge>n. f n') = Suc (LEAST n. \<forall>n'\<ge>Suc n. f n')"
       by(auto intro: Least_Suc)
     also have "(\<lambda>n. \<forall>n'\<ge>Suc n. f n') = (\<lambda>n. \<forall>n'\<ge>n. f (Suc n'))" by(auto dest: Suc_le_D)
     also from ones have "\<forall>n'\<ge>n. f (Suc n')" by auto
     moreover from ones have "(\<exists>n. \<forall>n'\<ge>n. \<not> f n') = False"
       by(auto intro!: exI[where x="max n m" for n m] simp add: max_def split: if_split_asm)
     moreover hence "(\<exists>n. \<forall>n'\<ge>n. \<not> f (Suc n')) = False"
       by(auto elim: allE[where x="Suc n" for n] dest: Suc_le_D)
     ultimately show ?thesis using ones
       apply (simp (no_asm_simp) add: set_bits_int_unfold' exI split del: if_split)
       apply (auto simp add: Let_def hd_map map_tl[symmetric] map_map[symmetric] map_Suc_upt upt_conv_Cons signed_take_bit_Suc
         not_le simp del: map_map)
       done
   qed
 qed
 
 lemma bin_last_set_bits [simp]:
   "odd (set_bits f :: int) = f 0"
   by (subst int_set_bits_unfold_BIT) simp_all
 
 lemma bin_rest_set_bits [simp]:
   "set_bits f div (2 :: int) = set_bits (f \<circ> Suc)"
   by (subst int_set_bits_unfold_BIT) simp_all
 
 lemma bin_nth_set_bits [simp]:
   "bit (set_bits f :: int) m \<longleftrightarrow> f m"
 using wff proof (induction m arbitrary: f)
-  case 0 
+  case 0
   then show ?case
     by (simp add: Bit_Comprehension.bin_last_set_bits)
 next
   case Suc
   from Suc.IH [of "f \<circ> Suc"] Suc.prems show ?case
     by (simp add: Bit_Comprehension.bin_rest_set_bits comp_def bit_Suc)
 qed
 
 end
 
 end
diff --git a/thys/Word_Lib/Bits_Int.thy b/thys/Word_Lib/Bits_Int.thy
--- a/thys/Word_Lib/Bits_Int.thy
+++ b/thys/Word_Lib/Bits_Int.thy
@@ -1,1490 +1,1493 @@
-(*  Author:     Jeremy Dawson and Gerwin Klein, NICTA
-*)
+(*
+ * Copyright Brian Huffman, PSU; Jeremy Dawson and Gerwin Klein, NICTA
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
 
 section \<open>Bitwise Operations on integers\<close>
 
 theory Bits_Int
   imports
     "HOL-Library.Word"
     Traditional_Infix_Syntax
 begin
 
 subsection \<open>Implicit bit representation of \<^typ>\<open>int\<close>\<close>
 
 abbreviation (input) bin_last :: "int \<Rightarrow> bool"
   where "bin_last \<equiv> odd"
 
 lemma bin_last_def:
   "bin_last w \<longleftrightarrow> w mod 2 = 1"
   by (fact odd_iff_mod_2_eq_one)
 
 abbreviation (input) bin_rest :: "int \<Rightarrow> int"
   where "bin_rest w \<equiv> w div 2"
 
 lemma bin_last_numeral_simps [simp]:
   "\<not> odd (0 :: int)"
   "odd (1 :: int)"
   "odd (- 1 :: int)"
   "odd (Numeral1 :: int)"
   "\<not> odd (numeral (Num.Bit0 w) :: int)"
   "odd (numeral (Num.Bit1 w) :: int)"
   "\<not> odd (- numeral (Num.Bit0 w) :: int)"
   "odd (- numeral (Num.Bit1 w) :: int)"
   by simp_all
 
 lemma bin_rest_numeral_simps [simp]:
   "bin_rest 0 = 0"
   "bin_rest 1 = 0"
   "bin_rest (- 1) = - 1"
   "bin_rest Numeral1 = 0"
   "bin_rest (numeral (Num.Bit0 w)) = numeral w"
   "bin_rest (numeral (Num.Bit1 w)) = numeral w"
   "bin_rest (- numeral (Num.Bit0 w)) = - numeral w"
   "bin_rest (- numeral (Num.Bit1 w)) = - numeral (w + Num.One)"
   by simp_all
 
 lemma bin_rl_eqI: "\<lbrakk>bin_rest x = bin_rest y; odd x = odd y\<rbrakk> \<Longrightarrow> x = y"
   by (auto elim: oddE)
 
-lemma [simp]: 
+lemma [simp]:
   shows bin_rest_lt0: "bin_rest i < 0 \<longleftrightarrow> i < 0"
   and  bin_rest_ge_0: "bin_rest i \<ge> 0 \<longleftrightarrow> i \<ge> 0"
   by auto
 
 lemma bin_rest_gt_0 [simp]: "bin_rest x > 0 \<longleftrightarrow> x > 1"
   by auto
 
 
 subsection \<open>Bit projection\<close>
 
 abbreviation (input) bin_nth :: \<open>int \<Rightarrow> nat \<Rightarrow> bool\<close>
   where \<open>bin_nth \<equiv> bit\<close>
 
 lemma bin_nth_eq_iff: "bin_nth x = bin_nth y \<longleftrightarrow> x = y"
   by (simp add: bit_eq_iff fun_eq_iff)
 
 lemma bin_eqI:
   "x = y" if "\<And>n. bin_nth x n \<longleftrightarrow> bin_nth y n"
   using that bin_nth_eq_iff [of x y] by (simp add: fun_eq_iff)
 
 lemma bin_eq_iff: "x = y \<longleftrightarrow> (\<forall>n. bin_nth x n = bin_nth y n)"
   by (fact bit_eq_iff)
 
 lemma bin_nth_zero [simp]: "\<not> bin_nth 0 n"
   by simp
 
 lemma bin_nth_1 [simp]: "bin_nth 1 n \<longleftrightarrow> n = 0"
   by (cases n) (simp_all add: bit_Suc)
 
 lemma bin_nth_minus1 [simp]: "bin_nth (- 1) n"
   by (induction n) (simp_all add: bit_Suc)
 
 lemma bin_nth_numeral: "bin_rest x = y \<Longrightarrow> bin_nth x (numeral n) = bin_nth y (pred_numeral n)"
   by (simp add: numeral_eq_Suc bit_Suc)
 
 lemmas bin_nth_numeral_simps [simp] =
   bin_nth_numeral [OF bin_rest_numeral_simps(2)]
   bin_nth_numeral [OF bin_rest_numeral_simps(5)]
   bin_nth_numeral [OF bin_rest_numeral_simps(6)]
   bin_nth_numeral [OF bin_rest_numeral_simps(7)]
   bin_nth_numeral [OF bin_rest_numeral_simps(8)]
 
 lemmas bin_nth_simps =
   bit_0 bit_Suc bin_nth_zero bin_nth_minus1
   bin_nth_numeral_simps
 
 lemma nth_2p_bin: "bin_nth (2 ^ n) m = (m = n)" \<comment> \<open>for use when simplifying with \<open>bin_nth_Bit\<close>\<close>
   by (auto simp add: bit_exp_iff)
-  
+
 lemma nth_rest_power_bin: "bin_nth ((bin_rest ^^ k) w) n = bin_nth w (n + k)"
   apply (induct k arbitrary: n)
    apply clarsimp
   apply clarsimp
   apply (simp only: bit_Suc [symmetric] add_Suc)
   done
 
 lemma bin_nth_numeral_unfold:
   "bin_nth (numeral (num.Bit0 x)) n \<longleftrightarrow> n > 0 \<and> bin_nth (numeral x) (n - 1)"
   "bin_nth (numeral (num.Bit1 x)) n \<longleftrightarrow> (n > 0 \<longrightarrow> bin_nth (numeral x) (n - 1))"
   by (cases n; simp)+
 
 
 subsection \<open>Truncating\<close>
 
 definition bin_sign :: "int \<Rightarrow> int"
   where "bin_sign k = (if k \<ge> 0 then 0 else - 1)"
 
 lemma bin_sign_simps [simp]:
   "bin_sign 0 = 0"
   "bin_sign 1 = 0"
   "bin_sign (- 1) = - 1"
   "bin_sign (numeral k) = 0"
   "bin_sign (- numeral k) = -1"
   by (simp_all add: bin_sign_def)
 
 lemma bin_sign_rest [simp]: "bin_sign (bin_rest w) = bin_sign w"
   by (simp add: bin_sign_def)
 
 abbreviation (input) bintrunc :: \<open>nat \<Rightarrow> int \<Rightarrow> int\<close>
   where \<open>bintrunc \<equiv> take_bit\<close>
 
 lemma bintrunc_mod2p: "bintrunc n w = w mod 2 ^ n"
   by (fact take_bit_eq_mod)
 
 abbreviation (input) sbintrunc :: \<open>nat \<Rightarrow> int \<Rightarrow> int\<close>
   where \<open>sbintrunc \<equiv> signed_take_bit\<close>
 
 abbreviation (input) norm_sint :: \<open>nat \<Rightarrow> int \<Rightarrow> int\<close>
   where \<open>norm_sint n \<equiv> signed_take_bit (n - 1)\<close>
 
 lemma sbintrunc_mod2p: "sbintrunc n w = (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n"
   by (simp add: bintrunc_mod2p signed_take_bit_eq_take_bit_shift)
-  
+
 lemma sbintrunc_eq_take_bit:
   \<open>sbintrunc n k = take_bit (Suc n) (k + 2 ^ n) - 2 ^ n\<close>
   by (fact signed_take_bit_eq_take_bit_shift)
 
 lemma sign_bintr: "bin_sign (bintrunc n w) = 0"
   by (simp add: bin_sign_def)
 
 lemma bintrunc_n_0: "bintrunc n 0 = 0"
   by (fact take_bit_of_0)
 
 lemma sbintrunc_n_0: "sbintrunc n 0 = 0"
   by (fact signed_take_bit_of_0)
 
 lemma sbintrunc_n_minus1: "sbintrunc n (- 1) = -1"
   by (fact signed_take_bit_of_minus_1)
 
 lemma bintrunc_Suc_numeral:
   "bintrunc (Suc n) 1 = 1"
   "bintrunc (Suc n) (- 1) = 1 + 2 * bintrunc n (- 1)"
   "bintrunc (Suc n) (numeral (Num.Bit0 w)) = 2 * bintrunc n (numeral w)"
   "bintrunc (Suc n) (numeral (Num.Bit1 w)) = 1 + 2 * bintrunc n (numeral w)"
   "bintrunc (Suc n) (- numeral (Num.Bit0 w)) = 2 * bintrunc n (- numeral w)"
   "bintrunc (Suc n) (- numeral (Num.Bit1 w)) = 1 + 2 * bintrunc n (- numeral (w + Num.One))"
   by (simp_all add: take_bit_Suc)
 
 lemma sbintrunc_0_numeral [simp]:
   "sbintrunc 0 1 = -1"
   "sbintrunc 0 (numeral (Num.Bit0 w)) = 0"
   "sbintrunc 0 (numeral (Num.Bit1 w)) = -1"
   "sbintrunc 0 (- numeral (Num.Bit0 w)) = 0"
   "sbintrunc 0 (- numeral (Num.Bit1 w)) = -1"
   by simp_all
 
 lemma sbintrunc_Suc_numeral:
   "sbintrunc (Suc n) 1 = 1"
   "sbintrunc (Suc n) (numeral (Num.Bit0 w)) = 2 * sbintrunc n (numeral w)"
   "sbintrunc (Suc n) (numeral (Num.Bit1 w)) = 1 + 2 * sbintrunc n (numeral w)"
   "sbintrunc (Suc n) (- numeral (Num.Bit0 w)) = 2 * sbintrunc n (- numeral w)"
   "sbintrunc (Suc n) (- numeral (Num.Bit1 w)) = 1 + 2 * sbintrunc n (- numeral (w + Num.One))"
   by (simp_all add: signed_take_bit_Suc)
 
 lemma bin_sign_lem: "(bin_sign (sbintrunc n bin) = -1) = bit bin n"
   by (simp add: bin_sign_def)
 
 lemma nth_bintr: "bin_nth (bintrunc m w) n \<longleftrightarrow> n < m \<and> bin_nth w n"
   by (fact bit_take_bit_iff)
 
 lemma nth_sbintr: "bin_nth (sbintrunc m w) n = (if n < m then bin_nth w n else bin_nth w m)"
   by (simp add: bit_signed_take_bit_iff min_def)
 
 lemma bin_nth_Bit0:
   "bin_nth (numeral (Num.Bit0 w)) n \<longleftrightarrow>
     (\<exists>m. n = Suc m \<and> bin_nth (numeral w) m)"
   using bit_double_iff [of \<open>numeral w :: int\<close> n]
   by (auto intro: exI [of _ \<open>n - 1\<close>])
 
 lemma bin_nth_Bit1:
   "bin_nth (numeral (Num.Bit1 w)) n \<longleftrightarrow>
     n = 0 \<or> (\<exists>m. n = Suc m \<and> bin_nth (numeral w) m)"
   using even_bit_succ_iff [of \<open>2 * numeral w :: int\<close> n]
     bit_double_iff [of \<open>numeral w :: int\<close> n]
   by auto
 
 lemma bintrunc_bintrunc_l: "n \<le> m \<Longrightarrow> bintrunc m (bintrunc n w) = bintrunc n w"
   by (simp add: min.absorb2)
 
 lemma sbintrunc_sbintrunc_l: "n \<le> m \<Longrightarrow> sbintrunc m (sbintrunc n w) = sbintrunc n w"
   by (simp add: min_def)
 
 lemma bintrunc_bintrunc_ge: "n \<le> m \<Longrightarrow> bintrunc n (bintrunc m w) = bintrunc n w"
   by (rule bin_eqI) (auto simp: nth_bintr)
 
 lemma bintrunc_bintrunc_min [simp]: "bintrunc m (bintrunc n w) = bintrunc (min m n) w"
   by (rule bin_eqI) (auto simp: nth_bintr)
 
 lemma sbintrunc_sbintrunc_min [simp]: "sbintrunc m (sbintrunc n w) = sbintrunc (min m n) w"
   by (rule bin_eqI) (auto simp: nth_sbintr min.absorb1 min.absorb2)
 
 lemmas sbintrunc_Suc_Pls =
   signed_take_bit_Suc [where a="0::int", simplified bin_last_numeral_simps bin_rest_numeral_simps]
 
 lemmas sbintrunc_Suc_Min =
   signed_take_bit_Suc [where a="-1::int", simplified bin_last_numeral_simps bin_rest_numeral_simps]
 
 lemmas sbintrunc_Sucs = sbintrunc_Suc_Pls sbintrunc_Suc_Min
   sbintrunc_Suc_numeral
 
 lemmas sbintrunc_Pls =
   signed_take_bit_0 [where a="0::int", simplified bin_last_numeral_simps bin_rest_numeral_simps]
 
 lemmas sbintrunc_Min =
   signed_take_bit_0 [where a="-1::int", simplified bin_last_numeral_simps bin_rest_numeral_simps]
 
 lemmas sbintrunc_0_simps =
   sbintrunc_Pls sbintrunc_Min
 
 lemmas sbintrunc_simps = sbintrunc_0_simps sbintrunc_Sucs
 
 lemma bintrunc_minus: "0 < n \<Longrightarrow> bintrunc (Suc (n - 1)) w = bintrunc n w"
   by auto
 
 lemma sbintrunc_minus: "0 < n \<Longrightarrow> sbintrunc (Suc (n - 1)) w = sbintrunc n w"
   by auto
 
 lemmas sbintrunc_minus_simps =
   sbintrunc_Sucs [THEN [2] sbintrunc_minus [symmetric, THEN trans]]
 
 lemma sbintrunc_BIT_I:
   \<open>0 < n \<Longrightarrow>
   sbintrunc (n - 1) 0 = y \<Longrightarrow>
   sbintrunc n 0 = 2 * y\<close>
   by simp
 
 lemma sbintrunc_Suc_Is:
   \<open>sbintrunc n (- 1) = y \<Longrightarrow>
   sbintrunc (Suc n) (- 1) = 1 + 2 * y\<close>
   by auto
 
 lemma sbintrunc_Suc_lem: "sbintrunc (Suc n) x = y \<Longrightarrow> m = Suc n \<Longrightarrow> sbintrunc m x = y"
   by auto
 
 lemmas sbintrunc_Suc_Ialts =
   sbintrunc_Suc_Is [THEN sbintrunc_Suc_lem]
 
 lemma sbintrunc_bintrunc_lt: "m > n \<Longrightarrow> sbintrunc n (bintrunc m w) = sbintrunc n w"
   by (rule bin_eqI) (auto simp: nth_sbintr nth_bintr)
 
 lemma bintrunc_sbintrunc_le: "m \<le> Suc n \<Longrightarrow> bintrunc m (sbintrunc n w) = bintrunc m w"
   apply (rule bin_eqI)
   using le_Suc_eq less_Suc_eq_le apply (auto simp: nth_sbintr nth_bintr)
   done
 
 lemmas bintrunc_sbintrunc [simp] = order_refl [THEN bintrunc_sbintrunc_le]
 lemmas sbintrunc_bintrunc [simp] = lessI [THEN sbintrunc_bintrunc_lt]
 lemmas bintrunc_bintrunc [simp] = order_refl [THEN bintrunc_bintrunc_l]
 lemmas sbintrunc_sbintrunc [simp] = order_refl [THEN sbintrunc_sbintrunc_l]
 
 lemma bintrunc_sbintrunc' [simp]: "0 < n \<Longrightarrow> bintrunc n (sbintrunc (n - 1) w) = bintrunc n w"
   by (cases n) simp_all
 
 lemma sbintrunc_bintrunc' [simp]: "0 < n \<Longrightarrow> sbintrunc (n - 1) (bintrunc n w) = sbintrunc (n - 1) w"
   by (cases n) simp_all
 
 lemma bin_sbin_eq_iff: "bintrunc (Suc n) x = bintrunc (Suc n) y \<longleftrightarrow> sbintrunc n x = sbintrunc n y"
   apply (rule iffI)
    apply (rule box_equals [OF _ sbintrunc_bintrunc sbintrunc_bintrunc])
    apply simp
   apply (rule box_equals [OF _ bintrunc_sbintrunc bintrunc_sbintrunc])
   apply simp
   done
 
 lemma bin_sbin_eq_iff':
   "0 < n \<Longrightarrow> bintrunc n x = bintrunc n y \<longleftrightarrow> sbintrunc (n - 1) x = sbintrunc (n - 1) y"
   by (cases n) (simp_all add: bin_sbin_eq_iff)
 
 lemmas bintrunc_sbintruncS0 [simp] = bintrunc_sbintrunc' [unfolded One_nat_def]
 lemmas sbintrunc_bintruncS0 [simp] = sbintrunc_bintrunc' [unfolded One_nat_def]
 
 lemmas bintrunc_bintrunc_l' = le_add1 [THEN bintrunc_bintrunc_l]
 lemmas sbintrunc_sbintrunc_l' = le_add1 [THEN sbintrunc_sbintrunc_l]
 
 (* although bintrunc_minus_simps, if added to default simpset,
   tends to get applied where it's not wanted in developing the theories,
   we get a version for when the word length is given literally *)
 
 lemmas nat_non0_gr =
   trans [OF iszero_def [THEN Not_eq_iff [THEN iffD2]] refl]
 
 lemma bintrunc_numeral:
   "bintrunc (numeral k) x = of_bool (odd x) + 2 * bintrunc (pred_numeral k) (x div 2)"
   by (simp add: numeral_eq_Suc take_bit_Suc mod_2_eq_odd)
 
 lemma sbintrunc_numeral:
   "sbintrunc (numeral k) x = of_bool (odd x) + 2 * sbintrunc (pred_numeral k) (x div 2)"
   by (simp add: numeral_eq_Suc signed_take_bit_Suc mod2_eq_if)
 
 lemma bintrunc_numeral_simps [simp]:
   "bintrunc (numeral k) (numeral (Num.Bit0 w)) =
     2 * bintrunc (pred_numeral k) (numeral w)"
   "bintrunc (numeral k) (numeral (Num.Bit1 w)) =
     1 + 2 * bintrunc (pred_numeral k) (numeral w)"
   "bintrunc (numeral k) (- numeral (Num.Bit0 w)) =
     2 * bintrunc (pred_numeral k) (- numeral w)"
   "bintrunc (numeral k) (- numeral (Num.Bit1 w)) =
     1 + 2 * bintrunc (pred_numeral k) (- numeral (w + Num.One))"
   "bintrunc (numeral k) 1 = 1"
   by (simp_all add: bintrunc_numeral)
 
 lemma sbintrunc_numeral_simps [simp]:
   "sbintrunc (numeral k) (numeral (Num.Bit0 w)) =
     2 * sbintrunc (pred_numeral k) (numeral w)"
   "sbintrunc (numeral k) (numeral (Num.Bit1 w)) =
     1 + 2 * sbintrunc (pred_numeral k) (numeral w)"
   "sbintrunc (numeral k) (- numeral (Num.Bit0 w)) =
     2 * sbintrunc (pred_numeral k) (- numeral w)"
   "sbintrunc (numeral k) (- numeral (Num.Bit1 w)) =
     1 + 2 * sbintrunc (pred_numeral k) (- numeral (w + Num.One))"
   "sbintrunc (numeral k) 1 = 1"
   by (simp_all add: sbintrunc_numeral)
 
 lemma no_bintr_alt1: "bintrunc n = (\<lambda>w. w mod 2 ^ n :: int)"
   by (rule ext) (rule bintrunc_mod2p)
 
 lemma range_bintrunc: "range (bintrunc n) = {i. 0 \<le> i \<and> i < 2 ^ n}"
   by (auto simp add: take_bit_eq_mod image_iff) (metis mod_pos_pos_trivial)
 
 lemma no_sbintr_alt2: "sbintrunc n = (\<lambda>w. (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)"
   by (rule ext) (simp add : sbintrunc_mod2p)
 
 lemma range_sbintrunc: "range (sbintrunc n) = {i. - (2 ^ n) \<le> i \<and> i < 2 ^ n}"
 proof -
   have \<open>surj (\<lambda>k::int. k + 2 ^ n)\<close>
     by (rule surjI [of _ \<open>(\<lambda>k. k - 2 ^ n)\<close>]) simp
   moreover have \<open>sbintrunc n = ((\<lambda>k. k - 2 ^ n) \<circ> take_bit (Suc n) \<circ> (\<lambda>k. k + 2 ^ n))\<close>
     by (simp add: sbintrunc_eq_take_bit fun_eq_iff)
   ultimately show ?thesis
     apply (simp only: fun.set_map range_bintrunc)
     apply (auto simp add: image_iff)
     apply presburger
     done
 qed
-  
+
 lemma sbintrunc_inc:
   \<open>k + 2 ^ Suc n \<le> sbintrunc n k\<close> if \<open>k < - (2 ^ n)\<close>
   using that by (fact signed_take_bit_int_greater_eq)
-  
+
 lemma sbintrunc_dec:
   \<open>sbintrunc n k \<le> k - 2 ^ (Suc n)\<close> if \<open>k \<ge> 2 ^ n\<close>
   using that by (fact signed_take_bit_int_less_eq)
 
 lemma bintr_ge0: "0 \<le> bintrunc n w"
   by (simp add: bintrunc_mod2p)
 
 lemma bintr_lt2p: "bintrunc n w < 2 ^ n"
   by (simp add: bintrunc_mod2p)
 
 lemma bintr_Min: "bintrunc n (- 1) = 2 ^ n - 1"
   by (simp add: stable_imp_take_bit_eq)
-  
+
 lemma sbintr_ge: "- (2 ^ n) \<le> sbintrunc n w"
   by (simp add: sbintrunc_mod2p)
 
 lemma sbintr_lt: "sbintrunc n w < 2 ^ n"
   by (simp add: sbintrunc_mod2p)
 
 lemma sign_Pls_ge_0: "bin_sign bin = 0 \<longleftrightarrow> bin \<ge> 0"
   for bin :: int
   by (simp add: bin_sign_def)
 
 lemma sign_Min_lt_0: "bin_sign bin = -1 \<longleftrightarrow> bin < 0"
   for bin :: int
   by (simp add: bin_sign_def)
 
 lemma bin_rest_trunc: "bin_rest (bintrunc n bin) = bintrunc (n - 1) (bin_rest bin)"
   by (simp add: take_bit_rec [of n bin])
 
 lemma bin_rest_power_trunc:
   "(bin_rest ^^ k) (bintrunc n bin) = bintrunc (n - k) ((bin_rest ^^ k) bin)"
   by (induct k) (auto simp: bin_rest_trunc)
 
 lemma bin_rest_trunc_i: "bintrunc n (bin_rest bin) = bin_rest (bintrunc (Suc n) bin)"
   by (auto simp add: take_bit_Suc)
 
 lemma bin_rest_strunc: "bin_rest (sbintrunc (Suc n) bin) = sbintrunc n (bin_rest bin)"
   by (simp add: signed_take_bit_Suc)
 
 lemma bintrunc_rest [simp]: "bintrunc n (bin_rest (bintrunc n bin)) = bin_rest (bintrunc n bin)"
   by (induct n arbitrary: bin) (simp_all add: take_bit_Suc)
 
 lemma sbintrunc_rest [simp]: "sbintrunc n (bin_rest (sbintrunc n bin)) = bin_rest (sbintrunc n bin)"
   by (induct n arbitrary: bin) (simp_all add: signed_take_bit_Suc mod2_eq_if)
 
 lemma bintrunc_rest': "bintrunc n \<circ> bin_rest \<circ> bintrunc n = bin_rest \<circ> bintrunc n"
   by (rule ext) auto
 
 lemma sbintrunc_rest': "sbintrunc n \<circ> bin_rest \<circ> sbintrunc n = bin_rest \<circ> sbintrunc n"
   by (rule ext) auto
 
 lemma rco_lem: "f \<circ> g \<circ> f = g \<circ> f \<Longrightarrow> f \<circ> (g \<circ> f) ^^ n = g ^^ n \<circ> f"
   apply (rule ext)
   apply (induct_tac n)
    apply (simp_all (no_asm))
   apply (drule fun_cong)
   apply (unfold o_def)
   apply (erule trans)
   apply simp
   done
 
 lemmas rco_bintr = bintrunc_rest'
   [THEN rco_lem [THEN fun_cong], unfolded o_def]
 lemmas rco_sbintr = sbintrunc_rest'
   [THEN rco_lem [THEN fun_cong], unfolded o_def]
 
 
 subsection \<open>Splitting and concatenation\<close>
 
 definition bin_split :: \<open>nat \<Rightarrow> int \<Rightarrow> int \<times> int\<close>
   where [simp]: \<open>bin_split n k = (drop_bit n k, take_bit n k)\<close>
 
 lemma [code]:
   "bin_split (Suc n) w = (let (w1, w2) = bin_split n (w div 2) in (w1, of_bool (odd w) + 2 * w2))"
   "bin_split 0 w = (w, 0)"
   by (simp_all add: drop_bit_Suc take_bit_Suc mod_2_eq_odd)
 
 abbreviation (input) bin_cat :: \<open>int \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int\<close>
   where \<open>bin_cat k n l \<equiv> concat_bit n l k\<close>
 
 lemma bin_cat_eq_push_bit_add_take_bit:
   \<open>bin_cat k n l = push_bit n k + take_bit n l\<close>
   by (simp add: concat_bit_eq)
-  
+
 lemma bin_sign_cat: "bin_sign (bin_cat x n y) = bin_sign x"
 proof -
   have \<open>0 \<le> x\<close> if \<open>0 \<le> x * 2 ^ n + y mod 2 ^ n\<close>
   proof -
     have \<open>y mod 2 ^ n < 2 ^ n\<close>
       using pos_mod_bound [of \<open>2 ^ n\<close> y] by simp
     then have \<open>\<not> y mod 2 ^ n \<ge> 2 ^ n\<close>
       by (simp add: less_le)
     with that have \<open>x \<noteq> - 1\<close>
       by auto
     have *: \<open>- 1 \<le> (- (y mod 2 ^ n)) div 2 ^ n\<close>
       by (simp add: zdiv_zminus1_eq_if)
     from that have \<open>- (y mod 2 ^ n) \<le> x * 2 ^ n\<close>
       by simp
     then have \<open>(- (y mod 2 ^ n)) div 2 ^ n \<le> (x * 2 ^ n) div 2 ^ n\<close>
       using zdiv_mono1 zero_less_numeral zero_less_power by blast
     with * have \<open>- 1 \<le> x * 2 ^ n div 2 ^ n\<close> by simp
     with \<open>x \<noteq> - 1\<close> show ?thesis
       by simp
   qed
   then show ?thesis
     by (simp add: bin_sign_def not_le not_less bin_cat_eq_push_bit_add_take_bit push_bit_eq_mult take_bit_eq_mod)
 qed
 
 lemma bin_cat_assoc: "bin_cat (bin_cat x m y) n z = bin_cat x (m + n) (bin_cat y n z)"
   by (fact concat_bit_assoc)
 
 lemma bin_cat_assoc_sym: "bin_cat x m (bin_cat y n z) = bin_cat (bin_cat x (m - n) y) (min m n) z"
   by (fact concat_bit_assoc_sym)
 
 definition bin_rcat :: \<open>nat \<Rightarrow> int list \<Rightarrow> int\<close>
   where \<open>bin_rcat n = horner_sum (take_bit n) (2 ^ n) \<circ> rev\<close>
 
 lemma bin_rcat_eq_foldl:
   \<open>bin_rcat n = foldl (\<lambda>u v. bin_cat u n v) 0\<close>
 proof
   fix ks :: \<open>int list\<close>
   show \<open>bin_rcat n ks = foldl (\<lambda>u v. bin_cat u n v) 0 ks\<close>
     by (induction ks rule: rev_induct)
       (simp_all add: bin_rcat_def concat_bit_eq push_bit_eq_mult)
 qed
 
 fun bin_rsplit_aux :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> int list"
   where "bin_rsplit_aux n m c bs =
     (if m = 0 \<or> n = 0 then bs
      else
       let (a, b) = bin_split n c
       in bin_rsplit_aux n (m - n) a (b # bs))"
 
 definition bin_rsplit :: "nat \<Rightarrow> nat \<times> int \<Rightarrow> int list"
   where "bin_rsplit n w = bin_rsplit_aux n (fst w) (snd w) []"
 
 value \<open>bin_rsplit 1705 (3, 88)\<close>
 
 fun bin_rsplitl_aux :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> int list"
   where "bin_rsplitl_aux n m c bs =
     (if m = 0 \<or> n = 0 then bs
      else
       let (a, b) = bin_split (min m n) c
       in bin_rsplitl_aux n (m - n) a (b # bs))"
 
 definition bin_rsplitl :: "nat \<Rightarrow> nat \<times> int \<Rightarrow> int list"
   where "bin_rsplitl n w = bin_rsplitl_aux n (fst w) (snd w) []"
 
 declare bin_rsplit_aux.simps [simp del]
 declare bin_rsplitl_aux.simps [simp del]
 
 lemma bin_nth_cat:
   "bin_nth (bin_cat x k y) n =
     (if n < k then bin_nth y n else bin_nth x (n - k))"
   by (simp add: bit_concat_bit_iff)
 
 lemma bin_nth_drop_bit_iff:
   \<open>bin_nth (drop_bit n c) k \<longleftrightarrow> bin_nth c (n + k)\<close>
   by (simp add: bit_drop_bit_eq)
 
 lemma bin_nth_take_bit_iff:
   \<open>bin_nth (take_bit n c) k \<longleftrightarrow> k < n \<and> bin_nth c k\<close>
   by (fact bit_take_bit_iff)
 
 lemma bin_nth_split:
   "bin_split n c = (a, b) \<Longrightarrow>
     (\<forall>k. bin_nth a k = bin_nth c (n + k)) \<and>
     (\<forall>k. bin_nth b k = (k < n \<and> bin_nth c k))"
   by (auto simp add: bin_nth_drop_bit_iff bin_nth_take_bit_iff)
 
 lemma bin_cat_zero [simp]: "bin_cat 0 n w = bintrunc n w"
   by (simp add: bin_cat_eq_push_bit_add_take_bit)
 
 lemma bintr_cat1: "bintrunc (k + n) (bin_cat a n b) = bin_cat (bintrunc k a) n b"
   by (metis bin_cat_assoc bin_cat_zero)
 
 lemma bintr_cat: "bintrunc m (bin_cat a n b) =
     bin_cat (bintrunc (m - n) a) n (bintrunc (min m n) b)"
-  
+
   by (rule bin_eqI) (auto simp: bin_nth_cat nth_bintr)
 
 lemma bintr_cat_same [simp]: "bintrunc n (bin_cat a n b) = bintrunc n b"
   by (auto simp add : bintr_cat)
 
 lemma cat_bintr [simp]: "bin_cat a n (bintrunc n b) = bin_cat a n b"
   by (simp add: bin_cat_eq_push_bit_add_take_bit)
 
 lemma split_bintrunc: "bin_split n c = (a, b) \<Longrightarrow> b = bintrunc n c"
   by simp
 
 lemma bin_cat_split: "bin_split n w = (u, v) \<Longrightarrow> w = bin_cat u n v"
   by (auto simp add: bin_cat_eq_push_bit_add_take_bit bits_ident)
 
 lemma drop_bit_bin_cat_eq:
   \<open>drop_bit n (bin_cat v n w) = v\<close>
   by (rule bit_eqI) (simp add: bit_drop_bit_eq bit_concat_bit_iff)
 
 lemma take_bit_bin_cat_eq:
   \<open>take_bit n (bin_cat v n w) = take_bit n w\<close>
   by (rule bit_eqI) (simp add: bit_concat_bit_iff)
 
 lemma bin_split_cat: "bin_split n (bin_cat v n w) = (v, bintrunc n w)"
   by (simp add: drop_bit_bin_cat_eq take_bit_bin_cat_eq)
 
 lemma bin_split_zero [simp]: "bin_split n 0 = (0, 0)"
   by simp
 
 lemma bin_split_minus1 [simp]:
   "bin_split n (- 1) = (- 1, bintrunc n (- 1))"
   by simp
 
 lemma bin_split_trunc:
   "bin_split (min m n) c = (a, b) \<Longrightarrow>
     bin_split n (bintrunc m c) = (bintrunc (m - n) a, b)"
   apply (induct n arbitrary: m b c, clarsimp)
   apply (simp add: bin_rest_trunc Let_def split: prod.split_asm)
   apply (case_tac m)
    apply (auto simp: Let_def drop_bit_Suc take_bit_Suc mod_2_eq_odd split: prod.split_asm)
   done
 
 lemma bin_split_trunc1:
   "bin_split n c = (a, b) \<Longrightarrow>
     bin_split n (bintrunc m c) = (bintrunc (m - n) a, bintrunc m b)"
   apply (induct n arbitrary: m b c, clarsimp)
   apply (simp add: bin_rest_trunc Let_def split: prod.split_asm)
   apply (case_tac m)
    apply (auto simp: Let_def drop_bit_Suc take_bit_Suc mod_2_eq_odd split: prod.split_asm)
   done
 
 lemma bin_cat_num: "bin_cat a n b = a * 2 ^ n + bintrunc n b"
   by (simp add: bin_cat_eq_push_bit_add_take_bit push_bit_eq_mult)
 
 lemma bin_split_num: "bin_split n b = (b div 2 ^ n, b mod 2 ^ n)"
   by (simp add: drop_bit_eq_div take_bit_eq_mod)
 
 lemmas bin_rsplit_aux_simps = bin_rsplit_aux.simps bin_rsplitl_aux.simps
 lemmas rsplit_aux_simps = bin_rsplit_aux_simps
 
 lemmas th_if_simp1 = if_split [where P = "(=) l", THEN iffD1, THEN conjunct1, THEN mp] for l
 lemmas th_if_simp2 = if_split [where P = "(=) l", THEN iffD1, THEN conjunct2, THEN mp] for l
 
 lemmas rsplit_aux_simp1s = rsplit_aux_simps [THEN th_if_simp1]
 
 lemmas rsplit_aux_simp2ls = rsplit_aux_simps [THEN th_if_simp2]
 \<comment> \<open>these safe to \<open>[simp add]\<close> as require calculating \<open>m - n\<close>\<close>
 lemmas bin_rsplit_aux_simp2s [simp] = rsplit_aux_simp2ls [unfolded Let_def]
 lemmas rbscl = bin_rsplit_aux_simp2s (2)
 
 lemmas rsplit_aux_0_simps [simp] =
   rsplit_aux_simp1s [OF disjI1] rsplit_aux_simp1s [OF disjI2]
 
 lemma bin_rsplit_aux_append: "bin_rsplit_aux n m c (bs @ cs) = bin_rsplit_aux n m c bs @ cs"
   apply (induct n m c bs rule: bin_rsplit_aux.induct)
   apply (subst bin_rsplit_aux.simps)
   apply (subst bin_rsplit_aux.simps)
   apply (clarsimp split: prod.split)
   done
 
 lemma bin_rsplitl_aux_append: "bin_rsplitl_aux n m c (bs @ cs) = bin_rsplitl_aux n m c bs @ cs"
   apply (induct n m c bs rule: bin_rsplitl_aux.induct)
   apply (subst bin_rsplitl_aux.simps)
   apply (subst bin_rsplitl_aux.simps)
   apply (clarsimp split: prod.split)
   done
 
 lemmas rsplit_aux_apps [where bs = "[]"] =
   bin_rsplit_aux_append bin_rsplitl_aux_append
 
 lemmas rsplit_def_auxs = bin_rsplit_def bin_rsplitl_def
 
 lemmas rsplit_aux_alts = rsplit_aux_apps
   [unfolded append_Nil rsplit_def_auxs [symmetric]]
 
 lemma bin_split_minus: "0 < n \<Longrightarrow> bin_split (Suc (n - 1)) w = bin_split n w"
   by auto
 
 lemma bin_split_pred_simp [simp]:
   "(0::nat) < numeral bin \<Longrightarrow>
     bin_split (numeral bin) w =
       (let (w1, w2) = bin_split (numeral bin - 1) (bin_rest w)
        in (w1, of_bool (odd w) + 2 * w2))"
   by (simp add: take_bit_rec drop_bit_rec mod_2_eq_odd)
 
 lemma bin_rsplit_aux_simp_alt:
   "bin_rsplit_aux n m c bs =
     (if m = 0 \<or> n = 0 then bs
      else let (a, b) = bin_split n c in bin_rsplit n (m - n, a) @ b # bs)"
   apply (simp add: bin_rsplit_aux.simps [of n m c bs])
   apply (subst rsplit_aux_alts)
   apply (simp add: bin_rsplit_def)
   done
 
 lemmas bin_rsplit_simp_alt =
   trans [OF bin_rsplit_def bin_rsplit_aux_simp_alt]
 
 lemmas bthrs = bin_rsplit_simp_alt [THEN [2] trans]
 
 lemma bin_rsplit_size_sign' [rule_format]:
   "n > 0 \<Longrightarrow> rev sw = bin_rsplit n (nw, w) \<Longrightarrow> \<forall>v\<in>set sw. bintrunc n v = v"
   apply (induct sw arbitrary: nw w)
    apply clarsimp
   apply clarsimp
   apply (drule bthrs)
   apply (simp (no_asm_use) add: Let_def split: prod.split_asm if_split_asm)
   apply clarify
   apply simp
   done
 
 lemmas bin_rsplit_size_sign = bin_rsplit_size_sign' [OF asm_rl
   rev_rev_ident [THEN trans] set_rev [THEN equalityD2 [THEN subsetD]]]
 
 lemma bin_nth_rsplit [rule_format] :
   "n > 0 \<Longrightarrow> m < n \<Longrightarrow>
     \<forall>w k nw.
       rev sw = bin_rsplit n (nw, w) \<longrightarrow>
       k < size sw \<longrightarrow> bin_nth (sw ! k) m = bin_nth w (k * n + m)"
   apply (induct sw)
    apply clarsimp
   apply clarsimp
   apply (drule bthrs)
   apply (simp (no_asm_use) add: Let_def split: prod.split_asm if_split_asm)
   apply (erule allE, erule impE, erule exI)
   apply (case_tac k)
    apply clarsimp
    prefer 2
    apply clarsimp
    apply (erule allE)
    apply (erule (1) impE)
    apply (simp add: bit_drop_bit_eq ac_simps)
   apply (simp add: bit_take_bit_iff ac_simps)
   done
 
 lemma bin_rsplit_all: "0 < nw \<Longrightarrow> nw \<le> n \<Longrightarrow> bin_rsplit n (nw, w) = [bintrunc n w]"
   by (auto simp: bin_rsplit_def rsplit_aux_simp2ls split: prod.split dest!: split_bintrunc)
 
 lemma bin_rsplit_l [rule_format]:
   "\<forall>bin. bin_rsplitl n (m, bin) = bin_rsplit n (m, bintrunc m bin)"
   apply (rule_tac a = "m" in wf_less_than [THEN wf_induct])
   apply (simp (no_asm) add: bin_rsplitl_def bin_rsplit_def)
   apply (rule allI)
   apply (subst bin_rsplitl_aux.simps)
   apply (subst bin_rsplit_aux.simps)
   apply (clarsimp simp: Let_def split: prod.split)
   apply (simp add: ac_simps)
   apply (subst rsplit_aux_alts(1))
   apply (subst rsplit_aux_alts(2))
   apply clarsimp
   unfolding bin_rsplit_def bin_rsplitl_def
   apply (simp add: drop_bit_take_bit)
   apply (case_tac \<open>x < n\<close>)
   apply (simp_all add: not_less min_def)
   done
 
 lemma bin_rsplit_rcat [rule_format]:
   "n > 0 \<longrightarrow> bin_rsplit n (n * size ws, bin_rcat n ws) = map (bintrunc n) ws"
   apply (unfold bin_rsplit_def bin_rcat_eq_foldl)
   apply (rule_tac xs = ws in rev_induct)
    apply clarsimp
   apply clarsimp
   apply (subst rsplit_aux_alts)
   apply (simp add: drop_bit_bin_cat_eq take_bit_bin_cat_eq)
   done
 
 lemma bin_rsplit_aux_len_le [rule_format] :
   "\<forall>ws m. n \<noteq> 0 \<longrightarrow> ws = bin_rsplit_aux n nw w bs \<longrightarrow>
     length ws \<le> m \<longleftrightarrow> nw + length bs * n \<le> m * n"
 proof -
   have *: R
     if d: "i \<le> j \<or> m < j'"
     and R1: "i * k \<le> j * k \<Longrightarrow> R"
     and R2: "Suc m * k' \<le> j' * k' \<Longrightarrow> R"
     for i j j' k k' m :: nat and R
     using d
     apply safe
     apply (rule R1, erule mult_le_mono1)
     apply (rule R2, erule Suc_le_eq [THEN iffD2 [THEN mult_le_mono1]])
     done
   have **: "0 < sc \<Longrightarrow> sc - n + (n + lb * n) \<le> m * n \<longleftrightarrow> sc + lb * n \<le> m * n"
     for sc m n lb :: nat
     apply safe
      apply arith
     apply (case_tac "sc \<ge> n")
      apply arith
     apply (insert linorder_le_less_linear [of m lb])
     apply (erule_tac k=n and k'=n in *)
      apply arith
     apply simp
     done
   show ?thesis
     apply (induct n nw w bs rule: bin_rsplit_aux.induct)
     apply (subst bin_rsplit_aux.simps)
     apply (simp add: ** Let_def split: prod.split)
     done
 qed
 
 lemma bin_rsplit_len_le: "n \<noteq> 0 \<longrightarrow> ws = bin_rsplit n (nw, w) \<longrightarrow> length ws \<le> m \<longleftrightarrow> nw \<le> m * n"
   by (auto simp: bin_rsplit_def bin_rsplit_aux_len_le)
 
 lemma bin_rsplit_aux_len:
   "n \<noteq> 0 \<Longrightarrow> length (bin_rsplit_aux n nw w cs) = (nw + n - 1) div n + length cs"
   apply (induct n nw w cs rule: bin_rsplit_aux.induct)
   apply (subst bin_rsplit_aux.simps)
   apply (clarsimp simp: Let_def split: prod.split)
   apply (erule thin_rl)
   apply (case_tac m)
    apply simp
   apply (case_tac "m \<le> n")
    apply (auto simp add: div_add_self2)
   done
 
 lemma bin_rsplit_len: "n \<noteq> 0 \<Longrightarrow> length (bin_rsplit n (nw, w)) = (nw + n - 1) div n"
   by (auto simp: bin_rsplit_def bin_rsplit_aux_len)
 
 lemma bin_rsplit_aux_len_indep:
   "n \<noteq> 0 \<Longrightarrow> length bs = length cs \<Longrightarrow>
     length (bin_rsplit_aux n nw v bs) =
     length (bin_rsplit_aux n nw w cs)"
 proof (induct n nw w cs arbitrary: v bs rule: bin_rsplit_aux.induct)
   case (1 n m w cs v bs)
   show ?case
   proof (cases "m = 0")
     case True
     with \<open>length bs = length cs\<close> show ?thesis by simp
   next
     case False
     from "1.hyps" [of \<open>bin_split n w\<close> \<open>drop_bit n w\<close> \<open>take_bit n w\<close>] \<open>m \<noteq> 0\<close> \<open>n \<noteq> 0\<close>
     have hyp: "\<And>v bs. length bs = Suc (length cs) \<Longrightarrow>
       length (bin_rsplit_aux n (m - n) v bs) =
       length (bin_rsplit_aux n (m - n) (drop_bit n w) (take_bit n w # cs))"
-      using bin_rsplit_aux_len by fastforce 
+      using bin_rsplit_aux_len by fastforce
     from \<open>length bs = length cs\<close> \<open>n \<noteq> 0\<close> show ?thesis
       by (auto simp add: bin_rsplit_aux_simp_alt Let_def bin_rsplit_len split: prod.split)
   qed
 qed
 
 lemma bin_rsplit_len_indep:
   "n \<noteq> 0 \<Longrightarrow> length (bin_rsplit n (nw, v)) = length (bin_rsplit n (nw, w))"
   apply (unfold bin_rsplit_def)
   apply (simp (no_asm))
   apply (erule bin_rsplit_aux_len_indep)
   apply (rule refl)
   done
 
 
 subsection \<open>Logical operations\<close>
 
 primrec bin_sc :: "nat \<Rightarrow> bool \<Rightarrow> int \<Rightarrow> int"
   where
     Z: "bin_sc 0 b w = of_bool b + 2 * bin_rest w"
   | Suc: "bin_sc (Suc n) b w = of_bool (odd w) + 2 * bin_sc n b (w div 2)"
 
 lemma bin_nth_sc [simp]: "bit (bin_sc n b w) n \<longleftrightarrow> b"
   by (induction n arbitrary: w) (simp_all add: bit_Suc)
 
 lemma bin_sc_sc_same [simp]: "bin_sc n c (bin_sc n b w) = bin_sc n c w"
   by (induction n arbitrary: w) (simp_all add: bit_Suc)
 
 lemma bin_sc_sc_diff: "m \<noteq> n \<Longrightarrow> bin_sc m c (bin_sc n b w) = bin_sc n b (bin_sc m c w)"
   apply (induct n arbitrary: w m)
    apply (case_tac [!] m)
      apply auto
   done
 
 lemma bin_nth_sc_gen: "bin_nth (bin_sc n b w) m = (if m = n then b else bin_nth w m)"
   apply (induct n arbitrary: w m)
    apply (case_tac m; simp add: bit_Suc)
   apply (case_tac m; simp add: bit_Suc)
   done
 
 lemma bin_sc_eq:
   \<open>bin_sc n False = unset_bit n\<close>
   \<open>bin_sc n True = Bit_Operations.set_bit n\<close>
   by (simp_all add: fun_eq_iff bit_eq_iff)
     (simp_all add: bin_nth_sc_gen bit_set_bit_iff bit_unset_bit_iff)
 
 lemma bin_sc_nth [simp]: "bin_sc n (bin_nth w n) w = w"
   by (rule bit_eqI) (simp add: bin_nth_sc_gen)
 
 lemma bin_sign_sc [simp]: "bin_sign (bin_sc n b w) = bin_sign w"
 proof (induction n arbitrary: w)
   case 0
   then show ?case
     by (auto simp add: bin_sign_def) (use bin_rest_ge_0 in fastforce)
 next
   case (Suc n)
   from Suc [of \<open>w div 2\<close>]
   show ?case by (auto simp add: bin_sign_def split: if_splits)
 qed
 
 lemma bin_sc_bintr [simp]:
   "bintrunc m (bin_sc n x (bintrunc m w)) = bintrunc m (bin_sc n x w)"
   apply (cases x)
    apply (simp_all add: bin_sc_eq bit_eq_iff)
    apply (auto simp add: bit_take_bit_iff bit_set_bit_iff bit_unset_bit_iff)
   done
 
 lemma bin_clr_le: "bin_sc n False w \<le> w"
   by (simp add: bin_sc_eq unset_bit_less_eq)
 
 lemma bin_set_ge: "bin_sc n True w \<ge> w"
   by (simp add: bin_sc_eq set_bit_greater_eq)
 
 lemma bintr_bin_clr_le: "bintrunc n (bin_sc m False w) \<le> bintrunc n w"
   by (simp add: bin_sc_eq take_bit_unset_bit_eq unset_bit_less_eq)
 
 lemma bintr_bin_set_ge: "bintrunc n (bin_sc m True w) \<ge> bintrunc n w"
   by (simp add: bin_sc_eq take_bit_set_bit_eq set_bit_greater_eq)
 
 lemma bin_sc_FP [simp]: "bin_sc n False 0 = 0"
   by (induct n) auto
 
 lemma bin_sc_TM [simp]: "bin_sc n True (- 1) = - 1"
   by (induct n) auto
 
 lemmas bin_sc_simps = bin_sc.Z bin_sc.Suc bin_sc_TM bin_sc_FP
 
 lemma bin_sc_minus: "0 < n \<Longrightarrow> bin_sc (Suc (n - 1)) b w = bin_sc n b w"
   by auto
 
 lemmas bin_sc_Suc_minus =
   trans [OF bin_sc_minus [symmetric] bin_sc.Suc]
 
 lemma bin_sc_numeral [simp]:
   "bin_sc (numeral k) b w =
     of_bool (odd w) + 2 * bin_sc (pred_numeral k) b (w div 2)"
   by (simp add: numeral_eq_Suc)
 
 lemmas bin_sc_minus_simps =
   bin_sc_simps (2,3,4) [THEN [2] trans, OF bin_sc_minus [THEN sym]]
 
 instance int :: semiring_bit_syntax ..
 
 lemma test_bit_int_def [iff]:
   "i !! n \<longleftrightarrow> bin_nth i n"
   by (simp add: test_bit_eq_bit)
 
 lemma shiftl_int_def:
   "shiftl x n = x * 2 ^ n" for x :: int
   by (simp add: push_bit_int_def shiftl_eq_push_bit)
 
 lemma shiftr_int_def:
   "shiftr x n = x div 2 ^ n" for x :: int
   by (simp add: drop_bit_int_def shiftr_eq_drop_bit)
 
 
 subsubsection \<open>Basic simplification rules\<close>
 
 lemmas int_not_def = not_int_def
 
 lemma int_not_simps [simp]:
   "NOT (0::int) = -1"
   "NOT (1::int) = -2"
   "NOT (- 1::int) = 0"
   "NOT (numeral w::int) = - numeral (w + Num.One)"
   "NOT (- numeral (Num.Bit0 w)::int) = numeral (Num.BitM w)"
   "NOT (- numeral (Num.Bit1 w)::int) = numeral (Num.Bit0 w)"
   by (simp_all add: not_int_def)
 
 lemma int_not_not: "NOT (NOT x) = x"
   for x :: int
   by (fact bit.double_compl)
 
 lemma int_and_0 [simp]: "0 AND x = 0"
   for x :: int
   by (fact bit.conj_zero_left)
 
 lemma int_and_m1 [simp]: "-1 AND x = x"
   for x :: int
   by (fact bit.conj_one_left)
 
 lemma int_or_zero [simp]: "0 OR x = x"
   for x :: int
   by (fact bit.disj_zero_left)
 
 lemma int_or_minus1 [simp]: "-1 OR x = -1"
   for x :: int
   by (fact bit.disj_one_left)
 
 lemma int_xor_zero [simp]: "0 XOR x = x"
   for x :: int
   by (fact bit.xor_zero_left)
 
 
 subsubsection \<open>Binary destructors\<close>
 
 lemma bin_rest_NOT [simp]: "bin_rest (NOT x) = NOT (bin_rest x)"
   by (fact not_int_div_2)
 
 lemma bin_last_NOT [simp]: "bin_last (NOT x) \<longleftrightarrow> \<not> bin_last x"
   by simp
 
 lemma bin_rest_AND [simp]: "bin_rest (x AND y) = bin_rest x AND bin_rest y"
   by (subst and_int_rec) auto
 
 lemma bin_last_AND [simp]: "bin_last (x AND y) \<longleftrightarrow> bin_last x \<and> bin_last y"
   by (subst and_int_rec) auto
 
 lemma bin_rest_OR [simp]: "bin_rest (x OR y) = bin_rest x OR bin_rest y"
   by (subst or_int_rec) auto
 
 lemma bin_last_OR [simp]: "bin_last (x OR y) \<longleftrightarrow> bin_last x \<or> bin_last y"
   by (subst or_int_rec) auto
 
 lemma bin_rest_XOR [simp]: "bin_rest (x XOR y) = bin_rest x XOR bin_rest y"
   by (subst xor_int_rec) auto
 
 lemma bin_last_XOR [simp]: "bin_last (x XOR y) \<longleftrightarrow> (bin_last x \<or> bin_last y) \<and> \<not> (bin_last x \<and> bin_last y)"
   by (subst xor_int_rec) auto
 
 lemma bin_nth_ops:
   "\<And>x y. bin_nth (x AND y) n \<longleftrightarrow> bin_nth x n \<and> bin_nth y n"
   "\<And>x y. bin_nth (x OR y) n \<longleftrightarrow> bin_nth x n \<or> bin_nth y n"
   "\<And>x y. bin_nth (x XOR y) n \<longleftrightarrow> bin_nth x n \<noteq> bin_nth y n"
   "\<And>x. bin_nth (NOT x) n \<longleftrightarrow> \<not> bin_nth x n"
   by (simp_all add: bit_and_iff bit_or_iff bit_xor_iff bit_not_iff)
 
 
 subsubsection \<open>Derived properties\<close>
 
 lemma int_xor_minus1 [simp]: "-1 XOR x = NOT x"
   for x :: int
   by (fact bit.xor_one_left)
 
 lemma int_xor_extra_simps [simp]:
   "w XOR 0 = w"
   "w XOR -1 = NOT w"
   for w :: int
   by simp_all
 
 lemma int_or_extra_simps [simp]:
   "w OR 0 = w"
   "w OR -1 = -1"
   for w :: int
   by simp_all
 
 lemma int_and_extra_simps [simp]:
   "w AND 0 = 0"
   "w AND -1 = w"
   for w :: int
   by simp_all
 
 text \<open>Commutativity of the above.\<close>
 lemma bin_ops_comm:
   fixes x y :: int
   shows int_and_comm: "x AND y = y AND x"
     and int_or_comm:  "x OR y = y OR x"
     and int_xor_comm: "x XOR y = y XOR x"
   by (simp_all add: ac_simps)
 
 lemma bin_ops_same [simp]:
   "x AND x = x"
   "x OR x = x"
   "x XOR x = 0"
   for x :: int
   by simp_all
 
 lemmas bin_log_esimps =
   int_and_extra_simps  int_or_extra_simps  int_xor_extra_simps
   int_and_0 int_and_m1 int_or_zero int_or_minus1 int_xor_zero int_xor_minus1
 
 
 subsubsection \<open>Basic properties of logical (bit-wise) operations\<close>
 
 lemma bbw_ao_absorb: "x AND (y OR x) = x \<and> x OR (y AND x) = x"
   for x y :: int
   by (auto simp add: bin_eq_iff bin_nth_ops)
 
 lemma bbw_ao_absorbs_other:
   "x AND (x OR y) = x \<and> (y AND x) OR x = x"
   "(y OR x) AND x = x \<and> x OR (x AND y) = x"
   "(x OR y) AND x = x \<and> (x AND y) OR x = x"
   for x y :: int
   by (auto simp add: bin_eq_iff bin_nth_ops)
 
 lemmas bbw_ao_absorbs [simp] = bbw_ao_absorb bbw_ao_absorbs_other
 
 lemma int_xor_not: "(NOT x) XOR y = NOT (x XOR y) \<and> x XOR (NOT y) = NOT (x XOR y)"
   for x y :: int
   by (auto simp add: bin_eq_iff bin_nth_ops)
 
 lemma int_and_assoc: "(x AND y) AND z = x AND (y AND z)"
   for x y z :: int
   by (auto simp add: bin_eq_iff bin_nth_ops)
 
 lemma int_or_assoc: "(x OR y) OR z = x OR (y OR z)"
   for x y z :: int
   by (auto simp add: bin_eq_iff bin_nth_ops)
 
 lemma int_xor_assoc: "(x XOR y) XOR z = x XOR (y XOR z)"
   for x y z :: int
   by (auto simp add: bin_eq_iff bin_nth_ops)
 
 lemmas bbw_assocs = int_and_assoc int_or_assoc int_xor_assoc
 
 (* BH: Why are these declared as simp rules??? *)
 lemma bbw_lcs [simp]:
   "y AND (x AND z) = x AND (y AND z)"
   "y OR (x OR z) = x OR (y OR z)"
   "y XOR (x XOR z) = x XOR (y XOR z)"
   for x y :: int
   by (auto simp add: bin_eq_iff bin_nth_ops)
 
 lemma bbw_not_dist:
   "NOT (x OR y) = (NOT x) AND (NOT y)"
   "NOT (x AND y) = (NOT x) OR (NOT y)"
   for x y :: int
   by (auto simp add: bin_eq_iff bin_nth_ops)
 
 lemma bbw_oa_dist: "(x AND y) OR z = (x OR z) AND (y OR z)"
   for x y z :: int
   by (auto simp add: bin_eq_iff bin_nth_ops)
 
 lemma bbw_ao_dist: "(x OR y) AND z = (x AND z) OR (y AND z)"
   for x y z :: int
   by (auto simp add: bin_eq_iff bin_nth_ops)
 
 (*
 Why were these declared simp???
 declare bin_ops_comm [simp] bbw_assocs [simp]
 *)
 
 
 subsubsection \<open>Simplification with numerals\<close>
 
 text \<open>Cases for \<open>0\<close> and \<open>-1\<close> are already covered by other simp rules.\<close>
 
 lemma bin_rest_neg_numeral_BitM [simp]:
   "bin_rest (- numeral (Num.BitM w)) = - numeral w"
   by simp
 
 lemma bin_last_neg_numeral_BitM [simp]:
   "bin_last (- numeral (Num.BitM w))"
   by simp
 
 
 subsubsection \<open>Interactions with arithmetic\<close>
 
 lemma le_int_or: "bin_sign y = 0 \<Longrightarrow> x \<le> x OR y"
   for x y :: int
   by (simp add: bin_sign_def or_greater_eq split: if_splits)
 
 lemmas int_and_le =
   xtrans(3) [OF bbw_ao_absorbs (2) [THEN conjunct2, symmetric] le_int_or]
 
 text \<open>Interaction between bit-wise and arithmetic: good example of \<open>bin_induction\<close>.\<close>
 lemma bin_add_not: "x + NOT x = (-1::int)"
   by (simp add: not_int_def)
 
 lemma AND_mod: "x AND (2 ^ n - 1) = x mod 2 ^ n"
   for x :: int
   by (simp flip: take_bit_eq_mod add: take_bit_eq_mask mask_eq_exp_minus_1)
 
 
 subsubsection \<open>Truncating results of bit-wise operations\<close>
 
 lemma bin_trunc_ao:
   "bintrunc n x AND bintrunc n y = bintrunc n (x AND y)"
   "bintrunc n x OR bintrunc n y = bintrunc n (x OR y)"
   by simp_all
 
 lemma bin_trunc_xor: "bintrunc n (bintrunc n x XOR bintrunc n y) = bintrunc n (x XOR y)"
   by simp
 
 lemma bin_trunc_not: "bintrunc n (NOT (bintrunc n x)) = bintrunc n (NOT x)"
   by (fact take_bit_not_take_bit)
 
 text \<open>Want theorems of the form of \<open>bin_trunc_xor\<close>.\<close>
 lemma bintr_bintr_i: "x = bintrunc n y \<Longrightarrow> bintrunc n x = bintrunc n y"
   by auto
 
 lemmas bin_trunc_and = bin_trunc_ao(1) [THEN bintr_bintr_i]
 lemmas bin_trunc_or = bin_trunc_ao(2) [THEN bintr_bintr_i]
 
 
 subsubsection \<open>More lemmas\<close>
 
 lemma not_int_cmp_0 [simp]:
   fixes i :: int shows
   "0 < NOT i \<longleftrightarrow> i < -1"
   "0 \<le> NOT i \<longleftrightarrow> i < 0"
   "NOT i < 0 \<longleftrightarrow> i \<ge> 0"
   "NOT i \<le> 0 \<longleftrightarrow> i \<ge> -1"
 by(simp_all add: int_not_def) arith+
 
 lemma bbw_ao_dist2: "(x :: int) AND (y OR z) = x AND y OR x AND z"
   by (fact bit.conj_disj_distrib)
 
 lemmas int_and_ac = bbw_lcs(1) int_and_comm int_and_assoc
 
 lemma int_nand_same [simp]: fixes x :: int shows "x AND NOT x = 0"
   by simp
 
 lemma int_nand_same_middle: fixes x :: int shows "x AND y AND NOT x = 0"
   by (simp add: bit_eq_iff bit_and_iff bit_not_iff)
 
 lemma and_xor_dist: fixes x :: int shows
   "x AND (y XOR z) = (x AND y) XOR (x AND z)"
   by (fact bit.conj_xor_distrib)
 
 lemma int_and_lt0 [simp]:
   \<open>x AND y < 0 \<longleftrightarrow> x < 0 \<and> y < 0\<close> for x y :: int
   by (fact and_negative_int_iff)
 
-lemma int_and_ge0 [simp]: 
+lemma int_and_ge0 [simp]:
   \<open>x AND y \<ge> 0 \<longleftrightarrow> x \<ge> 0 \<or> y \<ge> 0\<close> for x y :: int
   by (fact and_nonnegative_int_iff)
-  
+
 lemma int_and_1: fixes x :: int shows "x AND 1 = x mod 2"
   by (fact and_one_eq)
 
 lemma int_1_and: fixes x :: int shows "1 AND x = x mod 2"
   by (fact one_and_eq)
 
-lemma int_or_lt0 [simp]: 
+lemma int_or_lt0 [simp]:
   \<open>x OR y < 0 \<longleftrightarrow> x < 0 \<or> y < 0\<close> for x y :: int
   by (fact or_negative_int_iff)
 
 lemma int_or_ge0 [simp]:
   \<open>x OR y \<ge> 0 \<longleftrightarrow> x \<ge> 0 \<and> y \<ge> 0\<close> for x y :: int
   by (fact or_nonnegative_int_iff)
-  
+
 lemma int_xor_lt0 [simp]:
   \<open>x XOR y < 0 \<longleftrightarrow> (x < 0) \<noteq> (y < 0)\<close> for x y :: int
   by (fact xor_negative_int_iff)
 
 lemma int_xor_ge0 [simp]:
   \<open>x XOR y \<ge> 0 \<longleftrightarrow> (x \<ge> 0 \<longleftrightarrow> y \<ge> 0)\<close> for x y :: int
   by (fact xor_nonnegative_int_iff)
-  
+
 lemma even_conv_AND:
   \<open>even i \<longleftrightarrow> i AND 1 = 0\<close> for i :: int
   by (simp add: and_one_eq mod2_eq_if)
 
 lemma bin_last_conv_AND:
   "bin_last i \<longleftrightarrow> i AND 1 \<noteq> 0"
   by (simp add: and_one_eq mod2_eq_if)
 
 lemma bitval_bin_last:
   "of_bool (bin_last i) = i AND 1"
   by (simp add: and_one_eq mod2_eq_if)
 
 lemma bin_sign_and:
   "bin_sign (i AND j) = - (bin_sign i * bin_sign j)"
 by(simp add: bin_sign_def)
 
 lemma int_not_neg_numeral: "NOT (- numeral n) = (Num.sub n num.One :: int)"
 by(simp add: int_not_def)
 
 lemma int_neg_numeral_pOne_conv_not: "- numeral (n + num.One) = (NOT (numeral n) :: int)"
 by(simp add: int_not_def)
 
 
 subsection \<open>Setting and clearing bits\<close>
 
 lemma int_shiftl_BIT: fixes x :: int
   shows int_shiftl0 [simp]: "x << 0 = x"
   and int_shiftl_Suc [simp]: "x << Suc n = 2 * (x << n)"
   by (auto simp add: shiftl_int_def)
 
 lemma int_0_shiftl [simp]: "0 << n = (0 :: int)"
 by(induct n) simp_all
 
 lemma bin_last_shiftl: "bin_last (x << n) \<longleftrightarrow> n = 0 \<and> bin_last x"
 by(cases n)(simp_all)
 
 lemma bin_rest_shiftl: "bin_rest (x << n) = (if n > 0 then x << (n - 1) else bin_rest x)"
 by(cases n)(simp_all)
 
 lemma bin_nth_shiftl [simp]: "bin_nth (x << n) m \<longleftrightarrow> n \<le> m \<and> bin_nth x (m - n)"
   by (simp add: bit_push_bit_iff_int shiftl_eq_push_bit)
 
 lemma bin_last_shiftr: "odd (x >> n) \<longleftrightarrow> x !! n" for x :: int
   by (simp add: shiftr_eq_drop_bit bit_iff_odd_drop_bit)
 
 lemma bin_rest_shiftr [simp]: "bin_rest (x >> n) = x >> Suc n"
   by (simp add: bit_eq_iff shiftr_eq_drop_bit drop_bit_Suc bit_drop_bit_eq drop_bit_half)
 
 lemma bin_nth_shiftr [simp]: "bin_nth (x >> n) m = bin_nth x (n + m)"
   by (simp add: shiftr_eq_drop_bit bit_drop_bit_eq)
 
 lemma bin_nth_conv_AND:
-  fixes x :: int shows 
+  fixes x :: int shows
   "bin_nth x n \<longleftrightarrow> x AND (1 << n) \<noteq> 0"
   by (simp add: bit_eq_iff)
     (auto simp add: shiftl_eq_push_bit bit_and_iff bit_push_bit_iff bit_exp_iff)
 
-lemma int_shiftl_numeral [simp]: 
+lemma int_shiftl_numeral [simp]:
   "(numeral w :: int) << numeral w' = numeral (num.Bit0 w) << pred_numeral w'"
   "(- numeral w :: int) << numeral w' = - numeral (num.Bit0 w) << pred_numeral w'"
 by(simp_all add: numeral_eq_Suc shiftl_int_def)
   (metis add_One mult_inc semiring_norm(11) semiring_norm(13) semiring_norm(2) semiring_norm(6) semiring_norm(87))+
 
 lemma int_shiftl_One_numeral [simp]:
   "(1 :: int) << numeral w = 2 << pred_numeral w"
   using int_shiftl_numeral [of Num.One w] by simp
 
 lemma shiftl_ge_0 [simp]: fixes i :: int shows "i << n \<ge> 0 \<longleftrightarrow> i \<ge> 0"
 by(induct n) simp_all
 
 lemma shiftl_lt_0 [simp]: fixes i :: int shows "i << n < 0 \<longleftrightarrow> i < 0"
 by (metis not_le shiftl_ge_0)
 
 lemma int_shiftl_test_bit: "(n << i :: int) !! m \<longleftrightarrow> m \<ge> i \<and> n !! (m - i)"
   by simp
 
 lemma int_0shiftr [simp]: "(0 :: int) >> x = 0"
 by(simp add: shiftr_int_def)
 
 lemma int_minus1_shiftr [simp]: "(-1 :: int) >> x = -1"
 by(simp add: shiftr_int_def div_eq_minus1)
 
 lemma int_shiftr_ge_0 [simp]: fixes i :: int shows "i >> n \<ge> 0 \<longleftrightarrow> i \<ge> 0"
   by (simp add: shiftr_eq_drop_bit)
 
 lemma int_shiftr_lt_0 [simp]: fixes i :: int shows "i >> n < 0 \<longleftrightarrow> i < 0"
 by (metis int_shiftr_ge_0 not_less)
 
 lemma int_shiftr_numeral [simp]:
   "(1 :: int) >> numeral w' = 0"
   "(numeral num.One :: int) >> numeral w' = 0"
   "(numeral (num.Bit0 w) :: int) >> numeral w' = numeral w >> pred_numeral w'"
   "(numeral (num.Bit1 w) :: int) >> numeral w' = numeral w >> pred_numeral w'"
   "(- numeral (num.Bit0 w) :: int) >> numeral w' = - numeral w >> pred_numeral w'"
   "(- numeral (num.Bit1 w) :: int) >> numeral w' = - numeral (Num.inc w) >> pred_numeral w'"
   by (simp_all add: shiftr_eq_drop_bit numeral_eq_Suc add_One drop_bit_Suc)
 
 lemma int_shiftr_numeral_Suc0 [simp]:
   "(1 :: int) >> Suc 0 = 0"
   "(numeral num.One :: int) >> Suc 0 = 0"
   "(numeral (num.Bit0 w) :: int) >> Suc 0 = numeral w"
   "(numeral (num.Bit1 w) :: int) >> Suc 0 = numeral w"
   "(- numeral (num.Bit0 w) :: int) >> Suc 0 = - numeral w"
   "(- numeral (num.Bit1 w) :: int) >> Suc 0 = - numeral (Num.inc w)"
   by (simp_all add: shiftr_eq_drop_bit drop_bit_Suc add_One)
 
 lemma bin_nth_minus_p2:
   assumes sign: "bin_sign x = 0"
   and y: "y = 1 << n"
   and m: "m < n"
   and x: "x < y"
   shows "bin_nth (x - y) m = bin_nth x m"
 proof -
   from sign y x have \<open>x \<ge> 0\<close> and \<open>y = 2 ^ n\<close> and \<open>x < 2 ^ n\<close>
     by (simp_all add: bin_sign_def shiftl_eq_push_bit push_bit_eq_mult split: if_splits)
   from \<open>0 \<le> x\<close> \<open>x < 2 ^ n\<close> \<open>m < n\<close> have \<open>bit x m \<longleftrightarrow> bit (x - 2 ^ n) m\<close>
   proof (induction m arbitrary: x n)
     case 0
     then show ?case
       by simp
   next
     case (Suc m)
     moreover define q where \<open>q = n - 1\<close>
     ultimately have n: \<open>n = Suc q\<close>
       by simp
     have \<open>(x - 2 ^ Suc q) div 2 = x div 2 - 2 ^ q\<close>
       by simp
     moreover from Suc.IH [of \<open>x div 2\<close> q] Suc.prems
     have \<open>bit (x div 2) m \<longleftrightarrow> bit (x div 2 - 2 ^ q) m\<close>
       by (simp add: n)
     ultimately show ?case
       by (simp add: bit_Suc n)
   qed
   with \<open>y = 2 ^ n\<close> show ?thesis
     by simp
 qed
 
 lemma bin_clr_conv_NAND:
   "bin_sc n False i = i AND NOT (1 << n)"
   by (induct n arbitrary: i) (rule bin_rl_eqI; simp)+
 
 lemma bin_set_conv_OR:
   "bin_sc n True i = i OR (1 << n)"
   by (induct n arbitrary: i) (rule bin_rl_eqI; simp)+
 
 
 subsection \<open>More lemmas on words\<close>
 
 lemma word_rcat_eq:
   \<open>word_rcat ws = word_of_int (bin_rcat (LENGTH('a::len)) (map uint ws))\<close>
   for ws :: \<open>'a::len word list\<close>
   apply (simp add: word_rcat_def bin_rcat_def rev_map)
   apply transfer
   apply (simp add: horner_sum_foldr foldr_map comp_def)
   done
 
 lemma sign_uint_Pls [simp]: "bin_sign (uint x) = 0"
   by (simp add: sign_Pls_ge_0)
 
 lemmas bin_log_bintrs = bin_trunc_not bin_trunc_xor bin_trunc_and bin_trunc_or
 
 \<comment> \<open>following definitions require both arithmetic and bit-wise word operations\<close>
 
 \<comment> \<open>to get \<open>word_no_log_defs\<close> from \<open>word_log_defs\<close>, using \<open>bin_log_bintrs\<close>\<close>
 lemmas wils1 = bin_log_bintrs [THEN word_of_int_eq_iff [THEN iffD2],
   folded uint_word_of_int_eq, THEN eq_reflection]
 
 \<comment> \<open>the binary operations only\<close>  (* BH: why is this needed? *)
 lemmas word_log_binary_defs =
   word_and_def word_or_def word_xor_def
 
 lemma setBit_no [simp]: "setBit (numeral bin) n = word_of_int (bin_sc n True (numeral bin))"
   by transfer (simp add: bin_sc_eq)
- 
+
 lemma clearBit_no [simp]:
   "clearBit (numeral bin) n = word_of_int (bin_sc n False (numeral bin))"
   by transfer (simp add: bin_sc_eq)
 
 lemma eq_mod_iff: "0 < n \<Longrightarrow> b = b mod n \<longleftrightarrow> 0 \<le> b \<and> b < n"
   for b n :: int
   by auto (metis pos_mod_conj)+
 
 lemma split_uint_lem: "bin_split n (uint w) = (a, b) \<Longrightarrow>
     a = take_bit (LENGTH('a) - n) a \<and> b = take_bit (LENGTH('a)) b"
   for w :: "'a::len word"
   by transfer (simp add: drop_bit_take_bit ac_simps)
 
 \<comment> \<open>limited hom result\<close>
 lemma word_cat_hom:
   "LENGTH('a::len) \<le> LENGTH('b::len) + LENGTH('c::len) \<Longrightarrow>
     (word_cat (word_of_int w :: 'b word) (b :: 'c word) :: 'a word) =
     word_of_int (bin_cat w (size b) (uint b))"
   by transfer (simp add: take_bit_concat_bit_eq)
 
 lemma bintrunc_shiftl:
   "take_bit n (m << i) = take_bit (n - i) m << i"
   for m :: int
   by (rule bit_eqI) (auto simp add: bit_take_bit_iff)
 
 lemma uint_shiftl:
   "uint (n << i) = take_bit (size n) (uint n << i)"
   by transfer (simp add: push_bit_take_bit shiftl_eq_push_bit)
 
 lemma bin_mask_conv_pow2:
   "mask n = 2 ^ n - (1 :: int)"
   by (fact mask_eq_exp_minus_1)
-  
+
 lemma bin_mask_ge0: "mask n \<ge> (0 :: int)"
   by (fact mask_nonnegative_int)
 
 lemma and_bin_mask_conv_mod: "x AND mask n = x mod 2 ^ n"
   for x :: int
   by (simp flip: take_bit_eq_mod add: take_bit_eq_mask)
 
-lemma bin_mask_numeral: 
+lemma bin_mask_numeral:
   "mask (numeral n) = (1 :: int) + 2 * mask (pred_numeral n)"
   by (fact mask_numeral)
 
 lemma bin_nth_mask [simp]: "bit (mask n :: int) i \<longleftrightarrow> i < n"
   by (simp add: bit_mask_iff)
 
 lemma bin_sign_mask [simp]: "bin_sign (mask n) = 0"
   by (simp add: bin_sign_def bin_mask_conv_pow2)
 
 lemma bin_mask_p1_conv_shift: "mask n + 1 = (1 :: int) << n"
   by (simp add: bin_mask_conv_pow2 shiftl_int_def)
 
 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 sint_range':
   \<open>- (2 ^ (LENGTH('a) - Suc 0)) \<le> sint x \<and> sint x < 2 ^ (LENGTH('a) - Suc 0)\<close>
   for x :: \<open>'a::len word\<close>
   apply transfer
   using sbintr_ge sbintr_lt apply auto
   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_eq word_sless_alt sbintrunc_If)
 
 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)
   define r s where \<open>r = LENGTH('a) - 1\<close> \<open>s = LENGTH('b) - 1\<close>
   then have \<open>LENGTH('a) = Suc r\<close> \<open>LENGTH('b) = Suc s\<close>
     \<open>size a = Suc r\<close> \<open>size b = Suc r\<close>
     by (simp_all add: word_size)
   then show ?thesis
     using P[unfolded range_sbintrunc] abs_ab le
     apply clarsimp
     apply (transfer fixing: r s)
     apply (auto simp add: signed_take_bit_int_eq_self simp flip: signed_take_bit_eq_iff_take_bit_eq)
     done
 qed
 
 code_identifier
   code_module Bits_Int \<rightharpoonup>
   (SML) Bit_Operations and (OCaml) Bit_Operations and (Haskell) Bit_Operations and (Scala) Bit_Operations
 
 end
diff --git a/thys/Word_Lib/Bitwise.thy b/thys/Word_Lib/Bitwise.thy
--- a/thys/Word_Lib/Bitwise.thy
+++ b/thys/Word_Lib/Bitwise.thy
@@ -1,503 +1,506 @@
-(*  Authors:    Thomas Sewell, NICTA and Sascha Boehme, TU Muenchen
-*)
+(*
+ * Copyright Thomas Sewell, NICTA and Sascha Boehme, TU Muenchen
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
 
 theory Bitwise
   imports
     "HOL-Library.Word"
     More_Arithmetic
     Reversed_Bit_Lists
 begin
 
 text \<open>Helper constants used in defining addition\<close>
 
 definition xor3 :: "bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool"
   where "xor3 a b c = (a = (b = c))"
 
 definition carry :: "bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool"
   where "carry a b c = ((a \<and> (b \<or> c)) \<or> (b \<and> c))"
 
 lemma carry_simps:
   "carry True a b = (a \<or> b)"
   "carry a True b = (a \<or> b)"
   "carry a b True = (a \<or> b)"
   "carry False a b = (a \<and> b)"
   "carry a False b = (a \<and> b)"
   "carry a b False = (a \<and> b)"
   by (auto simp add: carry_def)
 
 lemma xor3_simps:
   "xor3 True a b = (a = b)"
   "xor3 a True b = (a = b)"
   "xor3 a b True = (a = b)"
   "xor3 False a b = (a \<noteq> b)"
   "xor3 a False b = (a \<noteq> b)"
   "xor3 a b False = (a \<noteq> b)"
   by (simp_all add: xor3_def)
 
 text \<open>Breaking up word equalities into equalities on their
   bit lists. Equalities are generated and manipulated in the
   reverse order to \<^const>\<open>to_bl\<close>.\<close>
 
 lemma bl_word_sub: "to_bl (x - y) = to_bl (x + (- y))"
   by simp
 
 lemma rbl_word_1: "rev (to_bl (1 :: 'a::len word)) = takefill False (LENGTH('a)) [True]"
   apply (rule_tac s="rev (to_bl (word_succ (0 :: 'a word)))" in trans)
    apply simp
   apply (simp only: rtb_rbl_ariths(1)[OF refl])
   apply simp
   apply (case_tac "LENGTH('a)")
    apply simp
   apply (simp add: takefill_alt)
   done
 
 lemma rbl_word_if: "rev (to_bl (if P then x else y)) = map2 (If P) (rev (to_bl x)) (rev (to_bl y))"
   by (simp add: split_def)
 
 lemma rbl_add_carry_Cons:
   "(if car then rbl_succ else id) (rbl_add (x # xs) (y # ys)) =
     xor3 x y car # (if carry x y car then rbl_succ else id) (rbl_add xs ys)"
   by (simp add: carry_def xor3_def)
 
 lemma rbl_add_suc_carry_fold:
   "length xs = length ys \<Longrightarrow>
     \<forall>car. (if car then rbl_succ else id) (rbl_add xs ys) =
       (foldr (\<lambda>(x, y) res car. xor3 x y car # res (carry x y car)) (zip xs ys) (\<lambda>_. [])) car"
   apply (erule list_induct2)
    apply simp
   apply (simp only: rbl_add_carry_Cons)
   apply simp
   done
 
 lemma to_bl_plus_carry:
   "to_bl (x + y) =
     rev (foldr (\<lambda>(x, y) res car. xor3 x y car # res (carry x y car))
       (rev (zip (to_bl x) (to_bl y))) (\<lambda>_. []) False)"
   using rbl_add_suc_carry_fold[where xs="rev (to_bl x)" and ys="rev (to_bl y)"]
   apply (simp add: word_add_rbl[OF refl refl])
   apply (drule_tac x=False in spec)
   apply (simp add: zip_rev)
   done
 
 definition "rbl_plus cin xs ys =
   foldr (\<lambda>(x, y) res car. xor3 x y car # res (carry x y car)) (zip xs ys) (\<lambda>_. []) cin"
 
 lemma rbl_plus_simps:
   "rbl_plus cin (x # xs) (y # ys) = xor3 x y cin # rbl_plus (carry x y cin) xs ys"
   "rbl_plus cin [] ys = []"
   "rbl_plus cin xs [] = []"
   by (simp_all add: rbl_plus_def)
 
 lemma rbl_word_plus: "rev (to_bl (x + y)) = rbl_plus False (rev (to_bl x)) (rev (to_bl y))"
   by (simp add: rbl_plus_def to_bl_plus_carry zip_rev)
 
 definition "rbl_succ2 b xs = (if b then rbl_succ xs else xs)"
 
 lemma rbl_succ2_simps:
   "rbl_succ2 b [] = []"
   "rbl_succ2 b (x # xs) = (b \<noteq> x) # rbl_succ2 (x \<and> b) xs"
   by (simp_all add: rbl_succ2_def)
 
 lemma twos_complement: "- x = word_succ (NOT x)"
   using arg_cong[OF word_add_not[where x=x], where f="\<lambda>a. a - x + 1"]
   by (simp add: word_succ_p1 word_sp_01[unfolded word_succ_p1] del: word_add_not)
 
 lemma rbl_word_neg: "rev (to_bl (- x)) = rbl_succ2 True (map Not (rev (to_bl x)))"
   for x :: \<open>'a::len word\<close>
   by (simp add: twos_complement word_succ_rbl[OF refl] bl_word_not rev_map rbl_succ2_def)
 
 lemma rbl_word_cat:
   "rev (to_bl (word_cat x y :: 'a::len word)) =
     takefill False (LENGTH('a)) (rev (to_bl y) @ rev (to_bl x))"
   by (simp add: word_cat_bl word_rev_tf)
 
 lemma rbl_word_slice:
   "rev (to_bl (slice n w :: 'a::len word)) =
     takefill False (LENGTH('a)) (drop n (rev (to_bl w)))"
   apply (simp add: slice_take word_rev_tf rev_take)
   apply (cases "n < LENGTH('b)", simp_all)
   done
 
 lemma rbl_word_ucast:
   "rev (to_bl (ucast x :: 'a::len word)) = takefill False (LENGTH('a)) (rev (to_bl x))"
   apply (simp add: to_bl_ucast takefill_alt)
   apply (simp add: rev_drop)
   apply (cases "LENGTH('a) < LENGTH('b)")
    apply simp_all
   done
 
 lemma rbl_shiftl:
   "rev (to_bl (w << n)) = takefill False (size w) (replicate n False @ rev (to_bl w))"
   by (simp add: bl_shiftl takefill_alt word_size rev_drop)
 
 lemma rbl_shiftr:
   "rev (to_bl (w >> n)) = takefill False (size w) (drop n (rev (to_bl w)))"
   by (simp add: shiftr_slice rbl_word_slice word_size)
 
 definition "drop_nonempty v n xs = (if n < length xs then drop n xs else [last (v # xs)])"
 
 lemma drop_nonempty_simps:
   "drop_nonempty v (Suc n) (x # xs) = drop_nonempty x n xs"
   "drop_nonempty v 0 (x # xs) = (x # xs)"
   "drop_nonempty v n [] = [v]"
   by (simp_all add: drop_nonempty_def)
 
 definition "takefill_last x n xs = takefill (last (x # xs)) n xs"
 
 lemma takefill_last_simps:
   "takefill_last z (Suc n) (x # xs) = x # takefill_last x n xs"
   "takefill_last z 0 xs = []"
   "takefill_last z n [] = replicate n z"
   by (simp_all add: takefill_last_def) (simp_all add: takefill_alt)
 
 lemma rbl_sshiftr:
   "rev (to_bl (w >>> n)) = takefill_last False (size w) (drop_nonempty False n (rev (to_bl w)))"
   apply (cases "n < size w")
    apply (simp add: bl_sshiftr takefill_last_def word_size
       takefill_alt rev_take last_rev
       drop_nonempty_def)
   apply (subgoal_tac "(w >>> n) = of_bl (replicate (size w) (msb w))")
    apply (simp add: word_size takefill_last_def takefill_alt
       last_rev word_msb_alt word_rev_tf
       drop_nonempty_def take_Cons')
    apply (case_tac "LENGTH('a)", simp_all)
   apply (rule word_eqI)
   apply (simp add: nth_sshiftr word_size test_bit_of_bl
       msb_nth)
   done
 
 lemma nth_word_of_int:
   "(word_of_int x :: 'a::len word) !! n = (n < LENGTH('a) \<and> bin_nth x n)"
   apply (simp add: test_bit_bl word_size to_bl_of_bin)
   apply (subst conj_cong[OF refl], erule bin_nth_bl)
   apply auto
   done
 
 lemma nth_scast:
   "(scast (x :: 'a::len word) :: 'b::len word) !! n =
     (n < LENGTH('b) \<and>
     (if n < LENGTH('a) - 1 then x !! n
      else x !! (LENGTH('a) - 1)))"
   apply transfer
   apply (auto simp add: bit_signed_take_bit_iff min_def)
   done
 
 lemma rbl_word_scast:
   "rev (to_bl (scast x :: 'a::len word)) = takefill_last False (LENGTH('a)) (rev (to_bl x))"
   apply (rule nth_equalityI)
    apply (simp add: word_size takefill_last_def)
   apply (clarsimp simp: nth_scast takefill_last_def
       nth_takefill word_size rev_nth to_bl_nth)
   apply (cases "LENGTH('b)")
    apply simp
   apply (clarsimp simp: less_Suc_eq_le linorder_not_less
       last_rev word_msb_alt[symmetric]
       msb_nth)
   done
 
 definition rbl_mul :: "bool list \<Rightarrow> bool list \<Rightarrow> bool list"
   where "rbl_mul xs ys = foldr (\<lambda>x sm. rbl_plus False (map ((\<and>) x) ys) (False # sm)) xs []"
 
 lemma rbl_mul_simps:
   "rbl_mul (x # xs) ys = rbl_plus False (map ((\<and>) x) ys) (False # rbl_mul xs ys)"
   "rbl_mul [] ys = []"
   by (simp_all add: rbl_mul_def)
 
 lemma takefill_le2: "length xs \<le> n \<Longrightarrow> takefill x m (takefill x n xs) = takefill x m xs"
   by (simp add: takefill_alt replicate_add[symmetric])
 
 lemma take_rbl_plus: "\<forall>n b. take n (rbl_plus b xs ys) = rbl_plus b (take n xs) (take n ys)"
   apply (simp add: rbl_plus_def take_zip[symmetric])
   apply (rule_tac list="zip xs ys" in list.induct)
    apply simp
   apply (clarsimp simp: split_def)
   apply (case_tac n, simp_all)
   done
 
 lemma word_rbl_mul_induct:
   "length xs \<le> size y \<Longrightarrow>
     rbl_mul xs (rev (to_bl y)) = take (length xs) (rev (to_bl (of_bl (rev xs) * y)))"
   for y :: "'a::len word"
 proof (induct xs)
   case Nil
   show ?case by (simp add: rbl_mul_simps)
 next
   case (Cons z zs)
 
   have rbl_word_plus': "to_bl (x + y) = rev (rbl_plus False (rev (to_bl x)) (rev (to_bl y)))"
     for x y :: "'a word"
     by (simp add: rbl_word_plus[symmetric])
 
   have mult_bit: "to_bl (of_bl [z] * y) = map ((\<and>) z) (to_bl y)"
     by (cases z) (simp cong: map_cong, simp add: map_replicate_const cong: map_cong)
 
   have shiftl: "of_bl xs * 2 * y = (of_bl xs * y) << 1" for xs
     by (simp add: shiftl_t2n)
 
   have zip_take_triv: "\<And>xs ys n. n = length ys \<Longrightarrow> zip (take n xs) ys = zip xs ys"
     by (rule nth_equalityI) simp_all
 
   from Cons show ?case
     apply (simp add: trans [OF of_bl_append add.commute]
         rbl_mul_simps rbl_word_plus' distrib_right mult_bit shiftl rbl_shiftl)
     apply (simp add: takefill_alt word_size rev_map take_rbl_plus min_def)
     apply (simp add: rbl_plus_def zip_take_triv)
     done
 qed
 
 lemma rbl_word_mul: "rev (to_bl (x * y)) = rbl_mul (rev (to_bl x)) (rev (to_bl y))"
   for x :: "'a::len word"
   using word_rbl_mul_induct[where xs="rev (to_bl x)" and y=y] by (simp add: word_size)
 
 text \<open>Breaking up inequalities into bitlist properties.\<close>
 
 definition
   "rev_bl_order F xs ys =
      (length xs = length ys \<and>
        ((xs = ys \<and> F)
           \<or> (\<exists>n < length xs. drop (Suc n) xs = drop (Suc n) ys
                    \<and> \<not> xs ! n \<and> ys ! n)))"
 
 lemma rev_bl_order_simps:
   "rev_bl_order F [] [] = F"
   "rev_bl_order F (x # xs) (y # ys) = rev_bl_order ((y \<and> \<not> x) \<or> ((y \<or> \<not> x) \<and> F)) xs ys"
    apply (simp_all add: rev_bl_order_def)
   apply (rule conj_cong[OF refl])
   apply (cases "xs = ys")
    apply (simp add: nth_Cons')
    apply blast
   apply (simp add: nth_Cons')
   apply safe
    apply (rule_tac x="n - 1" in exI)
    apply simp
   apply (rule_tac x="Suc n" in exI)
   apply simp
   done
 
 lemma rev_bl_order_rev_simp:
   "length xs = length ys \<Longrightarrow>
    rev_bl_order F (xs @ [x]) (ys @ [y]) = ((y \<and> \<not> x) \<or> ((y \<or> \<not> x) \<and> rev_bl_order F xs ys))"
   by (induct arbitrary: F rule: list_induct2) (auto simp: rev_bl_order_simps)
 
 lemma rev_bl_order_bl_to_bin:
   "length xs = length ys \<Longrightarrow>
     rev_bl_order True xs ys = (bl_to_bin (rev xs) \<le> bl_to_bin (rev ys)) \<and>
     rev_bl_order False xs ys = (bl_to_bin (rev xs) < bl_to_bin (rev ys))"
   apply (induct xs ys rule: list_induct2)
    apply (simp_all add: rev_bl_order_simps bl_to_bin_app_cat concat_bit_Suc)
   apply (auto simp add: bl_to_bin_def add1_zle_eq)
   done
 
 lemma word_le_rbl: "x \<le> y \<longleftrightarrow> rev_bl_order True (rev (to_bl x)) (rev (to_bl y))"
   for x y :: "'a::len word"
   by (simp add: rev_bl_order_bl_to_bin word_le_def)
 
 lemma word_less_rbl: "x < y \<longleftrightarrow> rev_bl_order False (rev (to_bl x)) (rev (to_bl y))"
   for x y :: "'a::len word"
   by (simp add: word_less_alt rev_bl_order_bl_to_bin)
 
 definition "map_last f xs = (if xs = [] then [] else butlast xs @ [f (last xs)])"
 
 lemma map_last_simps:
   "map_last f [] = []"
   "map_last f [x] = [f x]"
   "map_last f (x # y # zs) = x # map_last f (y # zs)"
   by (simp_all add: map_last_def)
 
 lemma word_sle_rbl:
   "x <=s y \<longleftrightarrow> rev_bl_order True (map_last Not (rev (to_bl x))) (map_last Not (rev (to_bl y)))"
   using word_msb_alt[where w=x] word_msb_alt[where w=y]
   apply (simp add: word_sle_msb_le word_le_rbl)
   apply (subgoal_tac "length (to_bl x) = length (to_bl y)")
    apply (cases "to_bl x", simp)
    apply (cases "to_bl y", simp)
    apply (clarsimp simp: map_last_def rev_bl_order_rev_simp)
    apply auto
   done
 
 lemma word_sless_rbl:
   "x <s y \<longleftrightarrow> rev_bl_order False (map_last Not (rev (to_bl x))) (map_last Not (rev (to_bl y)))"
   using word_msb_alt[where w=x] word_msb_alt[where w=y]
   apply (simp add: word_sless_msb_less word_less_rbl)
   apply (subgoal_tac "length (to_bl x) = length (to_bl y)")
    apply (cases "to_bl x", simp)
    apply (cases "to_bl y", simp)
    apply (clarsimp simp: map_last_def rev_bl_order_rev_simp)
    apply auto
   done
 
 text \<open>Lemmas for unpacking \<^term>\<open>rev (to_bl n)\<close> for numerals n and also
   for irreducible values and expressions.\<close>
 
 lemma rev_bin_to_bl_simps:
   "rev (bin_to_bl 0 x) = []"
   "rev (bin_to_bl (Suc n) (numeral (num.Bit0 nm))) = False # rev (bin_to_bl n (numeral nm))"
   "rev (bin_to_bl (Suc n) (numeral (num.Bit1 nm))) = True # rev (bin_to_bl n (numeral nm))"
   "rev (bin_to_bl (Suc n) (numeral (num.One))) = True # replicate n False"
   "rev (bin_to_bl (Suc n) (- numeral (num.Bit0 nm))) = False # rev (bin_to_bl n (- numeral nm))"
   "rev (bin_to_bl (Suc n) (- numeral (num.Bit1 nm))) =
     True # rev (bin_to_bl n (- numeral (nm + num.One)))"
   "rev (bin_to_bl (Suc n) (- numeral (num.One))) = True # replicate n True"
   "rev (bin_to_bl (Suc n) (- numeral (num.Bit0 nm + num.One))) =
     True # rev (bin_to_bl n (- numeral (nm + num.One)))"
   "rev (bin_to_bl (Suc n) (- numeral (num.Bit1 nm + num.One))) =
     False # rev (bin_to_bl n (- numeral (nm + num.One)))"
   "rev (bin_to_bl (Suc n) (- numeral (num.One + num.One))) =
     False # rev (bin_to_bl n (- numeral num.One))"
   by (simp_all add: bin_to_bl_aux_append bin_to_bl_zero_aux bin_to_bl_minus1_aux replicate_append_same)
 
 lemma to_bl_upt: "to_bl x = rev (map ((!!) x) [0 ..< size x])"
   apply (rule nth_equalityI)
    apply (simp add: word_size)
   apply (auto simp: to_bl_nth word_size rev_nth)
   done
 
 lemma rev_to_bl_upt: "rev (to_bl x) = map ((!!) x) [0 ..< size x]"
   by (simp add: to_bl_upt)
 
 lemma upt_eq_list_intros:
   "j \<le> i \<Longrightarrow> [i ..< j] = []"
   "i = x \<Longrightarrow> x < j \<Longrightarrow> [x + 1 ..< j] = xs \<Longrightarrow> [i ..< j] = (x # xs)"
   by (simp_all add: upt_eq_Cons_conv)
 
 
 subsection \<open>Tactic definition\<close>
 
 lemma if_bool_simps:
   "If p True y = (p \<or> y) \<and> If p False y = (\<not> p \<and> y) \<and>
     If p y True = (p \<longrightarrow> y) \<and> If p y False = (p \<and> y)"
   by auto
 
 ML \<open>
 structure Word_Bitwise_Tac =
 struct
 
 val word_ss = simpset_of \<^theory_context>\<open>Word\<close>;
 
 fun mk_nat_clist ns =
   fold_rev (Thm.mk_binop \<^cterm>\<open>Cons :: nat \<Rightarrow> _\<close>)
     ns \<^cterm>\<open>[] :: nat list\<close>;
 
 fun upt_conv ctxt ct =
   case Thm.term_of ct of
     (\<^const>\<open>upt\<close> $ n $ m) =>
       let
         val (i, j) = apply2 (snd o HOLogic.dest_number) (n, m);
         val ns = map (Numeral.mk_cnumber \<^ctyp>\<open>nat\<close>) (i upto (j - 1))
           |> mk_nat_clist;
         val prop =
           Thm.mk_binop \<^cterm>\<open>(=) :: nat list \<Rightarrow> _\<close> ct ns
           |> Thm.apply \<^cterm>\<open>Trueprop\<close>;
       in
         try (fn () =>
           Goal.prove_internal ctxt [] prop
             (K (REPEAT_DETERM (resolve_tac ctxt @{thms upt_eq_list_intros} 1
                 ORELSE simp_tac (put_simpset word_ss ctxt) 1))) |> mk_meta_eq) ()
       end
   | _ => NONE;
 
 val expand_upt_simproc =
   Simplifier.make_simproc \<^context> "expand_upt"
    {lhss = [\<^term>\<open>upt x y\<close>], proc = K upt_conv};
 
 fun word_len_simproc_fn ctxt ct =
   (case Thm.term_of ct of
     Const (\<^const_name>\<open>len_of\<close>, _) $ t =>
      (let
         val T = fastype_of t |> dest_Type |> snd |> the_single
         val n = Numeral.mk_cnumber \<^ctyp>\<open>nat\<close> (Word_Lib.dest_binT T);
         val prop =
           Thm.mk_binop \<^cterm>\<open>(=) :: nat \<Rightarrow> _\<close> ct n
           |> Thm.apply \<^cterm>\<open>Trueprop\<close>;
       in
         Goal.prove_internal ctxt [] prop (K (simp_tac (put_simpset word_ss ctxt) 1))
         |> mk_meta_eq |> SOME
       end handle TERM _ => NONE | TYPE _ => NONE)
   | _ => NONE);
 
 val word_len_simproc =
   Simplifier.make_simproc \<^context> "word_len"
    {lhss = [\<^term>\<open>len_of x\<close>], proc = K word_len_simproc_fn};
 
 (* convert 5 or nat 5 to Suc 4 when n_sucs = 1, Suc (Suc 4) when n_sucs = 2,
    or just 5 (discarding nat) when n_sucs = 0 *)
 
 fun nat_get_Suc_simproc_fn n_sucs ctxt ct =
   let
     val (f $ arg) = Thm.term_of ct;
     val n =
       (case arg of \<^term>\<open>nat\<close> $ n => n | n => n)
       |> HOLogic.dest_number |> snd;
     val (i, j) = if n > n_sucs then (n_sucs, n - n_sucs) else (n, 0);
     val arg' = funpow i HOLogic.mk_Suc (HOLogic.mk_number \<^typ>\<open>nat\<close> j);
     val _ = if arg = arg' then raise TERM ("", []) else ();
     fun propfn g =
       HOLogic.mk_eq (g arg, g arg')
       |> HOLogic.mk_Trueprop |> Thm.cterm_of ctxt;
     val eq1 =
       Goal.prove_internal ctxt [] (propfn I)
         (K (simp_tac (put_simpset word_ss ctxt) 1));
   in
     Goal.prove_internal ctxt [] (propfn (curry (op $) f))
       (K (simp_tac (put_simpset HOL_ss ctxt addsimps [eq1]) 1))
     |> mk_meta_eq |> SOME
   end handle TERM _ => NONE;
 
 fun nat_get_Suc_simproc n_sucs ts =
   Simplifier.make_simproc \<^context> "nat_get_Suc"
    {lhss = map (fn t => t $ \<^term>\<open>n :: nat\<close>) ts,
     proc = K (nat_get_Suc_simproc_fn n_sucs)};
 
 val no_split_ss =
   simpset_of (put_simpset HOL_ss \<^context>
     |> Splitter.del_split @{thm if_split});
 
 val expand_word_eq_sss =
   (simpset_of (put_simpset HOL_basic_ss \<^context> addsimps
        @{thms word_eq_rbl_eq word_le_rbl word_less_rbl word_sle_rbl word_sless_rbl}),
   map simpset_of [
    put_simpset no_split_ss \<^context> addsimps
     @{thms rbl_word_plus rbl_word_and rbl_word_or rbl_word_not
                                 rbl_word_neg bl_word_sub rbl_word_xor
                                 rbl_word_cat rbl_word_slice rbl_word_scast
                                 rbl_word_ucast rbl_shiftl rbl_shiftr rbl_sshiftr
                                 rbl_word_if},
    put_simpset no_split_ss \<^context> addsimps
     @{thms to_bl_numeral to_bl_neg_numeral to_bl_0 rbl_word_1},
    put_simpset no_split_ss \<^context> addsimps
     @{thms rev_rev_ident rev_replicate rev_map to_bl_upt word_size}
           addsimprocs [word_len_simproc],
    put_simpset no_split_ss \<^context> addsimps
     @{thms list.simps split_conv replicate.simps list.map
                                 zip_Cons_Cons zip_Nil drop_Suc_Cons drop_0 drop_Nil
                                 foldr.simps list.map zip.simps(1) zip_Nil zip_Cons_Cons takefill_Suc_Cons
                                 takefill_Suc_Nil takefill.Z rbl_succ2_simps
                                 rbl_plus_simps rev_bin_to_bl_simps append.simps
                                 takefill_last_simps drop_nonempty_simps
                                 rev_bl_order_simps}
           addsimprocs [expand_upt_simproc,
                        nat_get_Suc_simproc 4
                          [\<^term>\<open>replicate\<close>, \<^term>\<open>takefill x\<close>,
                           \<^term>\<open>drop\<close>, \<^term>\<open>bin_to_bl\<close>,
                           \<^term>\<open>takefill_last x\<close>,
                           \<^term>\<open>drop_nonempty x\<close>]],
     put_simpset no_split_ss \<^context> addsimps @{thms xor3_simps carry_simps if_bool_simps}
   ])
 
 fun tac ctxt =
   let
     val (ss, sss) = expand_word_eq_sss;
   in
     foldr1 (op THEN_ALL_NEW)
       ((CHANGED o safe_full_simp_tac (put_simpset ss ctxt)) ::
         map (fn ss => safe_full_simp_tac (put_simpset ss ctxt)) sss)
   end;
 
 end
 \<close>
 
 method_setup word_bitwise =
   \<open>Scan.succeed (fn ctxt => Method.SIMPLE_METHOD (Word_Bitwise_Tac.tac ctxt 1))\<close>
   "decomposer for word equalities and inequalities into bit propositions"
 
 end
diff --git a/thys/Word_Lib/Even_More_List.thy b/thys/Word_Lib/Even_More_List.thy
--- a/thys/Word_Lib/Even_More_List.thy
+++ b/thys/Word_Lib/Even_More_List.thy
@@ -1,108 +1,112 @@
+(*
+ * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
 
 section \<open>Lemmas on list operations\<close>
 
 theory Even_More_List
-  imports
-    Main
+  imports Main
 begin
 
 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 map_idem_upt_eq:
   \<open>map f [m..<n] = [m..<n]\<close> if \<open>\<And>q. m \<le> q \<Longrightarrow> q < n \<Longrightarrow> f q = q\<close>
 proof (cases \<open>n \<ge> m\<close>)
   case False
   then show ?thesis
     by simp
 next
   case True
   moreover define r where \<open>r = n - m\<close>
   ultimately have \<open>n = m + r\<close>
     by simp
   with that have \<open>\<And>q. m \<le> q \<Longrightarrow> q < m + r \<Longrightarrow> f q = q\<close>
     by simp
   then have \<open>map f [m..<m + r] = [m..<m + r]\<close>
     by (induction r) simp_all
   with \<open>n = m + r\<close> show ?thesis
     by simp
 qed
 
 lemma upt_zero_numeral_unfold:
   \<open>[0..<numeral n] = [0..<pred_numeral n] @ [pred_numeral n]\<close>
   by (simp add: numeral_eq_Suc)
 
 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 Min_eq_length_takeWhile:
   \<open>Min {m. P m} = length (takeWhile (Not \<circ> P) ([0..<n]))\<close>
   if *: \<open>\<And>m. P m \<Longrightarrow> m < n\<close> and \<open>\<exists>m. P m\<close>
 proof -
   from \<open>\<exists>m. P m\<close> obtain r where \<open>P r\<close> ..
   have \<open>Min {m. P m} = q + length (takeWhile (Not \<circ> P) ([q..<n]))\<close>
     if \<open>q \<le> n\<close> \<open>\<And>m. P m \<Longrightarrow> q \<le> m \<and> m < n\<close> for q
   using that proof (induction rule: inc_induct)
     case base
     from base [of r] \<open>P r\<close> show ?case
       by simp
   next
     case (step q)
     from \<open>q < n\<close> have *: \<open>[q..<n] = q # [Suc q..<n]\<close>
       by (simp add: upt_rec)
     show ?case
     proof (cases \<open>P q\<close>)
       case False
       then have \<open>Suc q \<le> m \<and> m < n\<close> if \<open>P m\<close> for m
         using that step.prems [of m] by (auto simp add: Suc_le_eq less_le)
       with \<open>\<not> P q\<close> show ?thesis
         by (simp add: * step.IH)
     next
       case True
       have \<open>{a. P a} \<subseteq> {0..n}\<close>
-        using step.prems by (auto simp add: less_imp_le_nat)  
+        using step.prems by (auto simp add: less_imp_le_nat)
       moreover have \<open>finite {0..n}\<close>
         by simp
       ultimately have \<open>finite {a. P a}\<close>
         by (rule finite_subset)
       with \<open>P q\<close> step.prems show ?thesis
         by (auto intro: Min_eqI simp add: *)
     qed
   qed
   from this [of 0] and that show ?thesis
     by simp
 qed
 
 lemma Max_eq_length_takeWhile:
   \<open>Max {m. P m} = n - Suc (length (takeWhile (Not \<circ> P) (rev [0..<n])))\<close>
   if *: \<open>\<And>m. P m \<Longrightarrow> m < n\<close> and \<open>\<exists>m. P m\<close>
 using * proof (induction n)
   case 0
   with \<open>\<exists>m. P m\<close> show ?case
     by auto
 next
   case (Suc n)
   show ?case
   proof (cases \<open>P n\<close>)
     case False
     then have \<open>m < n\<close> if \<open>P m\<close> for m
       using that Suc.prems [of m] by (auto simp add: less_le)
     with Suc.IH show ?thesis
       by auto
   next
     case True
     have \<open>{a. P a} \<subseteq> {0..n}\<close>
-      using Suc.prems by (auto simp add: less_Suc_eq_le) 
+      using Suc.prems by (auto simp add: less_Suc_eq_le)
     moreover have \<open>finite {0..n}\<close>
       by simp
     ultimately have \<open>finite {a. P a}\<close>
       by (rule finite_subset)
     with \<open>P n\<close> Suc.prems show ?thesis
       by (auto intro: Max_eqI simp add: less_Suc_eq_le)
   qed
 qed
 
 end
diff --git a/thys/Word_Lib/Examples.thy b/thys/Word_Lib/Examples.thy
--- a/thys/Word_Lib/Examples.thy
+++ b/thys/Word_Lib/Examples.thy
@@ -1,220 +1,226 @@
+(*
+ * Copyright Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
+
 theory Examples
   imports Bitwise Next_and_Prev Generic_set_bit Word_Syntax Signed_Division_Word
 begin
 
 text "modulus"
 
 lemma "(27 :: 4 word) = -5" by simp
 
 lemma "(27 :: 4 word) = 11" by simp
 
 lemma "27 \<noteq> (11 :: 6 word)" by simp
 
 text "signed"
 
 lemma "(127 :: 6 word) = -1" by simp
 
 text "number ring simps"
 
 lemma
   "27 + 11 = (38::'a::len word)"
   "27 + 11 = (6::5 word)"
   "7 * 3 = (21::'a::len word)"
   "11 - 27 = (-16::'a::len word)"
   "- (- 11) = (11::'a::len word)"
   "-40 + 1 = (-39::'a::len word)"
   by simp_all
 
 lemma "word_pred 2 = 1" by simp
 
 lemma "word_succ (- 3) = -2" by simp
 
 lemma "23 < (27::8 word)" by simp
 lemma "23 \<le> (27::8 word)" by simp
 lemma "\<not> 23 < (27::2 word)" by simp
 lemma "0 < (4::3 word)" by simp
 lemma "1 < (4::3 word)" by simp
 lemma "0 < (1::3 word)" by simp
 
 text "ring operations"
 
 lemma "a + 2 * b + c - b = (b + c) + (a :: 32 word)" by simp
 
 text "casting"
 
 lemma "uint (234567 :: 10 word) = 71" by simp
 lemma "uint (-234567 :: 10 word) = 953" by simp
 lemma "sint (234567 :: 10 word) = 71" by simp
 lemma "sint (-234567 :: 10 word) = -71" by simp
 lemma "uint (1 :: 10 word) = 1" by simp
 
 lemma "unat (-234567 :: 10 word) = 953" by simp
 lemma "unat (1 :: 10 word) = 1" by simp
 
 lemma "ucast (0b1010 :: 4 word) = (0b10 :: 2 word)" by simp
 lemma "ucast (0b1010 :: 4 word) = (0b1010 :: 10 word)" by simp
 lemma "scast (0b1010 :: 4 word) = (0b111010 :: 6 word)" by simp
 lemma "ucast (1 :: 4 word) = (1 :: 2 word)" by simp
 
 text "reducing goals to nat or int and arith:"
 lemma "i < x \<Longrightarrow> i < i + 1" for i x :: "'a::len word"
   by unat_arith
 lemma "i < x \<Longrightarrow> i < i + 1" for i x :: "'a::len word"
   by unat_arith
 
 text "bool lists"
 
 lemma "of_bl [True, False, True, True] = (0b1011::'a::len word)" by simp
 
 lemma "to_bl (0b110::4 word) = [False, True, True, False]" by simp
 
 lemma "of_bl (replicate 32 True) = (0xFFFFFFFF::32 word)"
   by (simp add: numeral_eq_Suc)
 
 text "bit operations"
 
 lemma "0b110 AND 0b101 = (0b100 :: 32 word)" by simp
 lemma "0b110 OR 0b011 = (0b111 :: 8 word)" by simp
 lemma "0xF0 XOR 0xFF = (0x0F :: 8 word)" by simp
 lemma "NOT (0xF0 :: 16 word) = 0xFF0F" by simp
 lemma "0 AND 5 = (0 :: 8 word)" by simp
 lemma "1 AND 1 = (1 :: 8 word)" by simp
 lemma "1 AND 0 = (0 :: 8 word)" by simp
 lemma "1 AND 5 = (1 :: 8 word)" by simp
 lemma "1 OR 6 = (7 :: 8 word)" by simp
 lemma "1 OR 1 = (1 :: 8 word)" by simp
 lemma "1 XOR 7 = (6 :: 8 word)" by simp
 lemma "1 XOR 1 = (0 :: 8 word)" by simp
 lemma "NOT 1 = (254 :: 8 word)" by simp
 lemma "NOT 0 = (255 :: 8 word)" by simp
 
 lemma "(-1 :: 32 word) = 0xFFFFFFFF" by simp
 
 lemma "(0b0010 :: 4 word) !! 1" by simp
 lemma "\<not> (0b0010 :: 4 word) !! 0" by simp
 lemma "\<not> (0b1000 :: 3 word) !! 4" by simp
 lemma "\<not> (1 :: 3 word) !! 2" by simp
 
 lemma "(0b11000 :: 10 word) !! n = (n = 4 \<or> n = 3)"
   by (auto simp add: bin_nth_Bit0 bin_nth_Bit1)
 
 lemma "set_bit 55 7 True = (183::'a::len word)" by simp
 lemma "set_bit 0b0010 7 True = (0b10000010::'a::len word)" by simp
 lemma "set_bit 0b0010 1 False = (0::'a::len word)" by simp
 lemma "set_bit 1 3 True = (0b1001::'a::len word)" by simp
 lemma "set_bit 1 0 False = (0::'a::len word)" by simp
 lemma "set_bit 0 3 True = (0b1000::'a::len word)" by simp
 lemma "set_bit 0 3 False = (0::'a::len word)" by simp
 
 lemma "odd (0b0101::'a::len word)" by simp
 lemma "even (0b1000::'a::len word)" by simp
 lemma "odd (1::'a::len word)" by simp
 lemma "even (0::'a::len word)" by simp
 
 lemma "\<not> msb (0b0101::4 word)" by simp
 lemma   "msb (0b1000::4 word)" by simp
 lemma "\<not> msb (1::4 word)" by simp
 lemma "\<not> msb (0::4 word)" by simp
 
 lemma "word_cat (27::4 word) (27::8 word) = (2843::'a::len word)" by simp
 lemma "word_cat (0b0011::4 word) (0b1111::6word) = (0b0011001111 :: 10 word)"
   by simp
 
 lemma "0b1011 << 2 = (0b101100::'a::len word)" by simp
 lemma "0b1011 >> 2 = (0b10::8 word)" by simp
 lemma "0b1011 >>> 2 = (0b10::8 word)" by simp
 lemma "1 << 2 = (0b100::'a::len word)" apply simp? oops
 
 lemma "slice 3 (0b101111::6 word) = (0b101::3 word)" by simp
 lemma "slice 3 (1::6 word) = (0::3 word)" apply simp? oops
 
 lemma "word_rotr 2 0b0110 = (0b1001::4 word)" by simp
 lemma "word_rotl 1 0b1110 = (0b1101::4 word)" by simp
 lemma "word_roti 2 0b1110 = (0b1011::4 word)" by simp
 lemma "word_roti (- 2) 0b0110 = (0b1001::4 word)" by simp
 lemma "word_rotr 2 0 = (0::4 word)" by simp
 lemma "word_rotr 2 1 = (0b0100::4 word)" apply simp? oops
 lemma "word_rotl 2 1 = (0b0100::4 word)" apply simp? oops
 lemma "word_roti (- 2) 1 = (0b0100::4 word)" apply simp? oops
 
 lemma "(x AND 0xff00) OR (x AND 0x00ff) = (x::16 word)"
 proof -
   have "(x AND 0xff00) OR (x AND 0x00ff) = x AND (0xff00 OR 0x00ff)"
     by (simp only: word_ao_dist2)
   also have "0xff00 OR 0x00ff = (-1::16 word)"
     by simp
   also have "x AND -1 = x"
     by simp
   finally show ?thesis .
 qed
 
 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
 
 text "proofs using bitwise expansion"
 
 lemma "(x AND 0xff00) OR (x AND 0x00ff) = (x::16 word)"
   by word_bitwise
 
 lemma "(x AND NOT 3) >> 4 << 2 = ((x >> 2) AND NOT 3)"
   for x :: "10 word"
   by word_bitwise
 
 lemma "((x AND -8) >> 3) AND 7 = (x AND 56) >> 3"
   for x :: "12 word"
   by word_bitwise
 
 text "some problems require further reasoning after bit expansion"
 
 lemma "x \<le> 42 \<Longrightarrow> x \<le> 89"
   for x :: "8 word"
   apply word_bitwise
   apply blast
   done
 
 lemma "(x AND 1023) = 0 \<Longrightarrow> x \<le> -1024"
   for x :: \<open>32 word\<close>
   apply word_bitwise
   apply clarsimp
   done
 
 text "operations like shifts by non-numerals will expose some internal list
  representations but may still be easy to solve"
 
 lemma shiftr_overflow: "32 \<le> a \<Longrightarrow> b >> a = 0"
   for b :: \<open>32 word\<close>
   apply word_bitwise
   apply simp
   done
 
 (* testing for presence of word_bitwise *)
 lemma "((x :: 32 word) >> 3) AND 7 = (x AND 56) >> 3"
   by word_bitwise
 
 (* Tests *)
 lemma
   "( 4 :: 32 word) sdiv  4 =  1"
   "(-4 :: 32 word) sdiv  4 = -1"
   "(-3 :: 32 word) sdiv  4 =  0"
   "( 3 :: 32 word) sdiv -4 =  0"
   "(-3 :: 32 word) sdiv -4 =  0"
   "(-5 :: 32 word) sdiv -4 =  1"
   "( 5 :: 32 word) sdiv -4 = -1"
   by (simp_all add: sdiv_word_def signed_divide_int_def)
 
 lemma
   "( 4 :: 32 word) smod  4 =   0"
   "( 3 :: 32 word) smod  4 =   3"
   "(-3 :: 32 word) smod  4 =  -3"
   "( 3 :: 32 word) smod -4 =   3"
   "(-3 :: 32 word) smod -4 =  -3"
   "(-5 :: 32 word) smod -4 =  -1"
   "( 5 :: 32 word) smod -4 =   1"
   by (simp_all add: smod_word_def signed_modulo_int_def signed_divide_int_def)
 
 lemma "1 < (1024::32 word) \<and> 1 \<le> (1024::32 word)"
   by simp
 
 end
diff --git a/thys/Word_Lib/Generic_set_bit.thy b/thys/Word_Lib/Generic_set_bit.thy
--- a/thys/Word_Lib/Generic_set_bit.thy
+++ b/thys/Word_Lib/Generic_set_bit.thy
@@ -1,180 +1,185 @@
-(*  Author:     Jeremy Dawson, NICTA
-*)
+(*
+ * Copyright Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
+
+(* Author: Jeremy Dawson, NICTA *)
 
 section \<open>Operation variant for setting and unsetting bits\<close>
 
 theory Generic_set_bit
   imports
     "HOL-Library.Word"
     Bits_Int
     Most_significant_bit
 begin
 
 class set_bit = ring_bit_operations +
   fixes set_bit :: \<open>'a \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> 'a\<close>
   assumes set_bit_eq: \<open>set_bit a n b = (if b then Bit_Operations.set_bit else unset_bit) n a\<close>
 
 instantiation int :: set_bit
 begin
 
 definition set_bit_int :: \<open>int \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> int\<close>
   where \<open>set_bit i n b = bin_sc n b i\<close>
 
 instance
   by standard (simp add: set_bit_int_def bin_sc_eq)
 
 end
 
 lemma int_set_bit_0 [simp]: fixes x :: int shows
   "set_bit x 0 b = of_bool b + 2 * (x div 2)"
   by (auto simp add: set_bit_int_def intro: bin_rl_eqI)
 
 lemma int_set_bit_Suc: fixes x :: int shows
   "set_bit x (Suc n) b = of_bool (odd x) + 2 * set_bit (x div 2) n b"
   by (auto simp add: set_bit_int_def intro: bin_rl_eqI)
 
 lemma bin_last_set_bit:
   "bin_last (set_bit x n b) = (if n > 0 then bin_last x else b)"
   by (cases n) (simp_all add: int_set_bit_Suc)
 
-lemma bin_rest_set_bit: 
+lemma bin_rest_set_bit:
   "bin_rest (set_bit x n b) = (if n > 0 then set_bit (x div 2) (n - 1) b else x div 2)"
   by (cases n) (simp_all add: int_set_bit_Suc)
 
 lemma int_set_bit_numeral: fixes x :: int shows
   "set_bit x (numeral w) b = of_bool (odd x) + 2 * set_bit (x div 2) (pred_numeral w) b"
   by (simp add: set_bit_int_def)
 
 lemmas int_set_bit_numerals [simp] =
-  int_set_bit_numeral[where x="numeral w'"] 
+  int_set_bit_numeral[where x="numeral w'"]
   int_set_bit_numeral[where x="- numeral w'"]
   int_set_bit_numeral[where x="Numeral1"]
   int_set_bit_numeral[where x="1"]
   int_set_bit_numeral[where x="0"]
   int_set_bit_Suc[where x="numeral w'"]
   int_set_bit_Suc[where x="- numeral w'"]
   int_set_bit_Suc[where x="Numeral1"]
   int_set_bit_Suc[where x="1"]
   int_set_bit_Suc[where x="0"]
   for w'
 
 lemma msb_set_bit [simp]: "msb (set_bit (x :: int) n b) \<longleftrightarrow> msb x"
 by(simp add: msb_conv_bin_sign set_bit_int_def)
 
 instantiation word :: (len) set_bit
 begin
 
 definition set_bit_word :: \<open>'a word \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> 'a word\<close>
   where word_set_bit_def: \<open>set_bit a n x = word_of_int (bin_sc n x (uint a))\<close>
 
 instance
   apply standard
   apply (simp add: word_set_bit_def bin_sc_eq Bit_Operations.set_bit_def unset_bit_def)
   apply transfer
   apply simp
   done
 
 end
 
 lemma set_bit_unfold:
   \<open>set_bit w n b = (if b then Bit_Operations.set_bit n w else unset_bit n w)\<close>
   for w :: \<open>'a::len word\<close>
   by (simp add: set_bit_eq)
 
 lemma bit_set_bit_word_iff [bit_simps]:
   \<open>bit (set_bit w m b) n \<longleftrightarrow> (if m = n then n < LENGTH('a) \<and> b else bit w n)\<close>
   for w :: \<open>'a::len word\<close>
   by (auto simp add: set_bit_unfold bit_unset_bit_iff bit_set_bit_iff exp_eq_zero_iff not_le bit_imp_le_length)
 
 lemma word_set_nth [simp]: "set_bit w n (test_bit w n) = w"
   for w :: "'a::len word"
   by (auto simp: word_test_bit_def word_set_bit_def)
 
 lemma test_bit_set: "(set_bit w n x) !! n \<longleftrightarrow> n < size w \<and> x"
   for w :: "'a::len word"
   by (auto simp: word_size word_test_bit_def word_set_bit_def nth_bintr)
 
 lemma test_bit_set_gen:
   "test_bit (set_bit w n x) m = (if m = n then n < size w \<and> x else test_bit w m)"
   for w :: "'a::len word"
   apply (unfold word_size word_test_bit_def word_set_bit_def)
   apply (clarsimp simp add: nth_bintr bin_nth_sc_gen)
   apply (auto elim!: test_bit_size [unfolded word_size]
       simp add: word_test_bit_def [symmetric])
   done
 
 lemma word_set_set_same [simp]: "set_bit (set_bit w n x) n y = set_bit w n y"
   for w :: "'a::len word"
   by (rule word_eqI) (simp add : test_bit_set_gen word_size)
 
 lemma word_set_set_diff:
   fixes w :: "'a::len word"
   assumes "m \<noteq> n"
   shows "set_bit (set_bit w m x) n y = set_bit (set_bit w n y) m x"
   by (rule word_eqI) (auto simp: test_bit_set_gen word_size assms)
 
 lemma set_bit_word_of_int: "set_bit (word_of_int x) n b = word_of_int (bin_sc n b x)"
   unfolding word_set_bit_def
   by (rule word_eqI)(simp add: word_size bin_nth_sc_gen nth_bintr)
 
 lemma word_set_numeral [simp]:
   "set_bit (numeral bin::'a::len word) n b =
     word_of_int (bin_sc n b (numeral bin))"
   unfolding word_numeral_alt by (rule set_bit_word_of_int)
 
 lemma word_set_neg_numeral [simp]:
   "set_bit (- numeral bin::'a::len word) n b =
     word_of_int (bin_sc n b (- numeral bin))"
   unfolding word_neg_numeral_alt by (rule set_bit_word_of_int)
 
 lemma word_set_bit_0 [simp]: "set_bit 0 n b = word_of_int (bin_sc n b 0)"
   unfolding word_0_wi by (rule set_bit_word_of_int)
 
 lemma word_set_bit_1 [simp]: "set_bit 1 n b = word_of_int (bin_sc n b 1)"
   unfolding word_1_wi by (rule set_bit_word_of_int)
 
 lemma word_set_nth_iff: "set_bit w n b = w \<longleftrightarrow> w !! n = b \<or> n \<ge> size w"
   for w :: "'a::len word"
   apply (rule iffI)
    apply (rule disjCI)
    apply (drule word_eqD)
    apply (erule sym [THEN trans])
    apply (simp add: test_bit_set)
   apply (erule disjE)
    apply clarsimp
   apply (rule word_eqI)
   apply (clarsimp simp add : test_bit_set_gen)
   apply (drule test_bit_size)
   apply force
   done
 
 lemma word_clr_le: "w \<ge> set_bit w n False"
   for w :: "'a::len word"
   apply (simp add: word_set_bit_def word_le_def)
   apply transfer
   apply (rule order_trans)
    apply (rule bintr_bin_clr_le)
   apply simp
   done
 
 lemma word_set_ge: "w \<le> set_bit w n True"
   for w :: "'a::len word"
   apply (simp add: word_set_bit_def word_le_def)
   apply transfer
   apply (rule order_trans [OF _ bintr_bin_set_ge])
   apply simp
   done
 
 lemma set_bit_beyond:
   "size x \<le> n \<Longrightarrow> set_bit x n b = x" for x :: "'a :: len word"
   by (auto intro: word_eqI simp add: test_bit_set_gen 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]
 
 end
diff --git a/thys/Word_Lib/Guide.thy b/thys/Word_Lib/Guide.thy
--- a/thys/Word_Lib/Guide.thy
+++ b/thys/Word_Lib/Guide.thy
@@ -1,348 +1,354 @@
+(*
+ * Copyright Florian Haftmann
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
+
 (*<*)
 theory Guide
   imports Word_Lib_Sumo
 begin
 
 hide_const (open) Generic_set_bit.set_bit
 
 (*>*)
 section \<open>A short overview over bit operations and word types\<close>
 
 subsection \<open>Basic theories and key ideas\<close>
 
 text \<open>
   When formalizing bit operations, it is tempting to represent
   bit values as explicit lists over a binary type. This however
   is a bad idea, mainly due to the inherent ambiguities in
   representation concerning repeating leading bits.
 
   Hence this approach avoids such explicit lists altogether
   following an algebraic path:
 
     \<^item> Bit values are represented by numeric types: idealized
       unbounded bit values can be represented by type \<^typ>\<open>int\<close>,
       bounded bit values by quotient types over \<^typ>\<open>int\<close>, aka \<^typ>\<open>'a word\<close>.
 
     \<^item> (A special case are idealized unbounded bit values ending
       in @{term [source] 0} which can be represented by type \<^typ>\<open>nat\<close> but
       only support a restricted set of operations).
 
   The most fundamental ideas are developed in theory \<^theory>\<open>HOL.Parity\<close>
   (which is part of \<^theory>\<open>Main\<close>):
 
     \<^item> Multiplication by \<^term>\<open>2 :: int\<close> is a bit shift to the left and
 
     \<^item> Division by \<^term>\<open>2 :: int\<close> is a bit shift to the right.
 
     \<^item> Concerning bounded bit values, iterated shifts to the left
     may result in eliminating all bits by shifting them all
     beyond the boundary.  The property \<^prop>\<open>(2 :: int) ^ n \<noteq> 0\<close>
     represents that \<^term>\<open>n\<close> is \<^emph>\<open>not\<close> beyond that boundary.
 
     \<^item> The projection on a single bit is then @{thm [mode=iff] bit_iff_odd [where ?'a = int, no_vars]}.
 
     \<^item> This leads to the most fundamental properties of bit values:
 
       \<^item> Equality rule: @{thm [display, mode=iff] bit_eq_iff [where ?'a = int, no_vars]}
 
       \<^item> Induction rule: @{thm [display, mode=iff] bits_induct [where ?'a = int, no_vars]}
 
     \<^item> Characteristic properties \<^prop>\<open>bit (f x) n \<longleftrightarrow> P x n\<close>
       are available in fact collection \<^text>\<open>bit_simps\<close>.
 
   On top of this, the following generic operations are provided
   after import of theory \<^theory>\<open>HOL-Library.Bit_Operations\<close>:
 
     \<^item> Singleton \<^term>\<open>n\<close>th bit: \<^term>\<open>(2 :: int) ^ n\<close>
 
     \<^item> Bit mask upto bit \<^term>\<open>n\<close>: @{thm mask_eq_exp_minus_1 [where ?'a = int, no_vars]}
 
     \<^item> Left shift: @{thm push_bit_eq_mult [where ?'a = int, no_vars]}
 
     \<^item> Right shift: @{thm drop_bit_eq_div [where ?'a = int, no_vars]}
 
     \<^item> Truncation: @{thm take_bit_eq_mod [where ?'a = int, no_vars]}
 
     \<^item> Negation: @{thm [mode=iff] bit_not_iff [where ?'a = int, no_vars]}
 
     \<^item> And: @{thm [mode=iff] bit_and_iff [where ?'a = int, no_vars]}
 
     \<^item> Or: @{thm [mode=iff] bit_or_iff [where ?'a = int, no_vars]}
 
     \<^item> Xor: @{thm [mode=iff] bit_xor_iff [where ?'a = int, no_vars]}
 
     \<^item> Set a single bit: @{thm set_bit_def [where ?'a = int, no_vars]}
 
     \<^item> Unset a single bit: @{thm unset_bit_def [where ?'a = int, no_vars]}
 
     \<^item> Flip a single bit: @{thm flip_bit_def [where ?'a = int, no_vars]}
 
     \<^item> Signed truncation, or modulus centered around \<^term>\<open>0::int\<close>:
         @{thm [display] signed_take_bit_def [where ?'a = int, no_vars]}
 
     \<^item> (Bounded) conversion from and to a list of bits:
         @{thm [display] horner_sum_bit_eq_take_bit [where ?'a = int, no_vars]}
 
   Proper word types are introduced in theory \<^theory>\<open>HOL-Library.Word\<close>, with
   the following specific operations:
 
     \<^item> Standard arithmetic:
         @{term \<open>(+) :: 'a::len word \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>},
         @{term \<open>uminus :: 'a::len word \<Rightarrow> 'a word\<close>},
         @{term \<open>(-) :: 'a::len word \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>},
         @{term \<open>(*) :: 'a::len word \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>},
         @{term \<open>0 :: 'a::len word\<close>}, @{term \<open>1 :: 'a::len word\<close>}, numerals etc.
 
     \<^item> Standard bit operations: see above.
 
     \<^item> Conversion with unsigned interpretation of words:
 
         \<^item> @{term [source] \<open>unsigned :: 'a::len word \<Rightarrow> 'b::semiring_1\<close>}
 
         \<^item> Important special cases as abbreviations:
 
             \<^item> @{term [source] \<open>unat :: 'a::len word \<Rightarrow> nat\<close>}
 
             \<^item> @{term [source] \<open>uint :: 'a::len word \<Rightarrow> int\<close>}
 
             \<^item> @{term [source] \<open>ucast :: 'a::len word \<Rightarrow> 'b::len word\<close>}
 
     \<^item> Conversion with signed interpretation of words:
 
         \<^item> @{term [source] \<open>signed :: 'a::len word \<Rightarrow> 'b::ring_1\<close>}
 
         \<^item> Important special cases as abbreviations:
 
             \<^item> @{term [source] \<open>sint :: 'a::len word \<Rightarrow> int\<close>}
 
             \<^item> @{term [source] \<open>scast :: 'a::len word \<Rightarrow> 'b::len word\<close>}
 
     \<^item> Operations with unsigned interpretation of words:
 
         \<^item> @{thm [mode=iff] word_le_nat_alt [no_vars]}
 
         \<^item> @{thm [mode=iff] word_less_nat_alt [no_vars]}
 
         \<^item> @{thm unat_div_distrib [no_vars]}
 
         \<^item> @{thm unat_drop_bit_eq [no_vars]}
 
         \<^item> @{thm unat_mod_distrib [no_vars]}
 
         \<^item> @{thm [mode=iff] udvd_iff_dvd [no_vars]}
 
     \<^item> Operations with signed interpretation of words:
 
         \<^item> @{thm [mode=iff] word_sle_eq [no_vars]}
 
         \<^item> @{thm [mode=iff] word_sless_alt [no_vars]}
 
         \<^item> @{thm sint_signed_drop_bit_eq [no_vars]}
 
     \<^item> Rotation and reversal:
 
         \<^item> @{term [source] \<open>word_rotl :: nat \<Rightarrow> 'a::len word \<Rightarrow> 'a word\<close>}
 
         \<^item> @{term [source] \<open>word_rotr :: nat \<Rightarrow> 'a::len word \<Rightarrow> 'a word\<close>}
 
         \<^item> @{term [source] \<open>word_roti :: int \<Rightarrow> 'a::len word \<Rightarrow> 'a word\<close>}
 
         \<^item> @{term [source] \<open>word_reverse :: 'a::len word \<Rightarrow> 'a word\<close>}
 
     \<^item> Concatenation: @{term [source, display] \<open>word_cat :: 'a::len word \<Rightarrow> 'b::len word \<Rightarrow> 'c::len word\<close>}
 
   For proofs about words the following default strategies are applicable:
 
     \<^item> Using bit extensionality (facts \<^text>\<open>bit_eq_iff\<close>, \<^text>\<open>bit_eqI\<close>; fact
       collection \<^text>\<open>bit_simps\<close>).
 
     \<^item> Using the @{method transfer} method.
 \<close>
 
 subsection \<open>More library theories\<close>
 
 text \<open>
   Note: currently, the theories listed here are hardly separate
   entities since they import each other in various ways.
   Always inspect them to understand what you pull in if you
   want to import one.
 
     \<^descr>[Syntax]
 
       \<^descr>[\<^theory>\<open>Word_Lib.Hex_Words\<close>]
         Printing word numerals as hexadecimal numerals.
 
       \<^descr>[\<^theory>\<open>Word_Lib.Type_Syntax\<close>]
         Pretty type-sensitive syntax for cast operations.
 
       \<^descr>[\<^theory>\<open>Word_Lib.Word_Syntax\<close>]
         Specific ASCII syntax for prominent bit operations on word.
 
     \<^descr>[Proof tools]
 
       \<^descr>[\<^theory>\<open>Word_Lib.Norm_Words\<close>]
         Rewriting word numerals to normal forms.
 
       \<^descr>[\<^theory>\<open>Word_Lib.Bitwise\<close>]
         Method @{method word_bitwise} decomposes word equalities and inequalities into bit propositions.
 
       \<^descr>[\<^theory>\<open>Word_Lib.Word_EqI\<close>]
         Method @{method word_eqI_solve} decomposes word equalities and inequalities into bit propositions.
 
     \<^descr>[Operations]
 
       \<^descr>[\<^theory>\<open>Word_Lib.Signed_Division_Word\<close>]
 
         Signed division on word:
 
           \<^item> @{term [source] \<open>(sdiv) :: 'a::len word \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>}
 
           \<^item> @{term [source] \<open>(smod) :: 'a::len word \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>}
 
       \<^descr>[\<^theory>\<open>Word_Lib.Aligned\<close>] \
 
           \<^item> @{thm [mode=iff] is_aligned_iff_udvd [no_vars]}
 
       \<^descr>[\<^theory>\<open>Word_Lib.Least_significant_bit\<close>]
-    
+
         The least significant bit as an alias:
         @{thm [mode=iff] lsb_odd [where ?'a = int, no_vars]}
-    
+
       \<^descr>[\<^theory>\<open>Word_Lib.Most_significant_bit\<close>]
-    
+
         The most significant bit:
-    
+
           \<^item> @{thm [mode=iff] msb_int_def [of k]}
-    
+
           \<^item> @{thm [mode=iff] word_msb_sint [no_vars]}
-    
+
           \<^item> @{thm [mode=iff] msb_word_iff_sless_0 [no_vars]}
-    
+
           \<^item> @{thm [mode=iff] msb_word_iff_bit [no_vars]}
 
       \<^descr>[\<^theory>\<open>Word_Lib.Traditional_Infix_Syntax\<close>]
-    
+
         Clones of existing operations decorated with
         traditional syntax:
-    
+
           \<^item> @{thm test_bit_eq_bit [no_vars]}
-    
+
           \<^item> @{thm shiftl_eq_push_bit [no_vars]}
-    
+
           \<^item> @{thm shiftr_eq_drop_bit [no_vars]}
 
           \<^item> @{thm sshiftr_eq [no_vars]}
 
       \<^descr>[\<^theory>\<open>Word_Lib.Next_and_Prev\<close>] \
 
           \<^item> @{thm word_next_unfold [no_vars]}
 
           \<^item> @{thm word_prev_unfold [no_vars]}
 
       \<^descr>[\<^theory>\<open>Word_Lib.Enumeration_Word\<close>]
 
         More on explicit enumeration of word types.
 
       \<^descr>[\<^theory>\<open>Word_Lib.More_Word_Operations\<close>]
 
         Even more operations on word.
 
     \<^descr>[Types]
 
       \<^descr>[\<^theory>\<open>Word_Lib.Signed_Words\<close>]
 
           Formal tagging of word types with a \<^text>\<open>signed\<close> marker.
 
     \<^descr>[Lemmas]
 
       \<^descr>[\<^theory>\<open>Word_Lib.More_Word\<close>]
 
           More lemmas on words.
 
       \<^descr>[\<^theory>\<open>Word_Lib.Word_Lemmas\<close>]
 
           More lemmas on words, covering many other theories mentioned here.
 
     \<^descr>[Words of fixed length]
 
       \<^descr>[\<^theory>\<open>Word_Lib.Word_8\<close>]
 
           for 8-bit words.
 
       \<^descr>[\<^theory>\<open>Word_Lib.Word_16\<close>]
 
           for 16-bit words.
 
       \<^descr>[\<^theory>\<open>Word_Lib.Word_32\<close>]
 
           for 32-bit words.
 
       \<^descr>[\<^theory>\<open>Word_Lib.Word_64\<close>]
 
           for 64-bit words.
 \<close>
 
 
 subsection \<open>More library sessions\<close>
 
 text \<open>
   \<^descr>[\<^text>\<open>Native_Word\<close>] Makes machine words and machine arithmetic available for code generation.
   It provides a common abstraction that hides the differences between the different target languages.
   The code generator maps these operations to the APIs of the target languages.
 \<close>
 
 subsection \<open>Legacy theories\<close>
 
 text \<open>
   The following theories contain material which has been
   factored out since it is not recommended to use it in
   new applications, mostly because matters can be expressed
   succinctly using already existing operations.
 
   This section gives some indication how to migrate away
   from those theories.  However theorem coverage may still
   be terse in some cases.
 
   \<^descr>[\<^theory>\<open>Word_Lib.Word_Lib_Sumo\<close>]
 
     An entry point importing any relevant theory in that session.  Intended
     for backward compatibility: start importing this theory when
     migrating applications to Isabelle2021, and later sort out
     what you really need.
 
   \<^descr>[\<^theory>\<open>Word_Lib.Generic_set_bit\<close>]
 
     Kind of an alias: @{thm set_bit_eq [no_vars]}
 
   \<^descr>[\<^theory>\<open>Word_Lib.Typedef_Morphisms\<close>]
 
     A low-level extension to HOL typedef providing
     conversions along type morphisms.  The @{method transfer} method
     seems to be sufficient for most applications though.
 
   \<^descr>[\<^theory>\<open>Word_Lib.Bit_Comprehension\<close>]
 
     Comprehension syntax for bit values over predicates
     \<^typ>\<open>nat \<Rightarrow> bool\<close>.  For \<^typ>\<open>'a::len word\<close>, straightforward
     alternatives exist; difficult to handle for \<^typ>\<open>int\<close>.
 
   \<^descr>[\<^theory>\<open>Word_Lib.Reversed_Bit_Lists\<close>]
 
     Representation of bit values as explicit list in
     \<^emph>\<open>reversed\<close> order.
 
     This should rarely be necessary: the \<^const>\<open>bit\<close> projection
     should be sufficient in most cases.  In case explicit lists
     are needed, existing operations can be used:
 
     @{thm [display] horner_sum_bit_eq_take_bit [where ?'a = int, no_vars]}
 
   \<^descr>[\<^theory>\<open>Word_Lib.Many_More\<close>]
 
     Collection of operations and theorems which are kept for backward
     compatibility and not used in other theories in session \<^text>\<open>Word_Lib\<close>.
 \<close>
 
 (*<*)
 end
 (*>*)
diff --git a/thys/Word_Lib/Least_significant_bit.thy b/thys/Word_Lib/Least_significant_bit.thy
--- a/thys/Word_Lib/Least_significant_bit.thy
+++ b/thys/Word_Lib/Least_significant_bit.thy
@@ -1,89 +1,94 @@
-(*  Author:     Jeremy Dawson, NICTA
-*)
+(*
+ * Copyright Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
+
+(* Author: Jeremy Dawson, NICTA *)
 
 section \<open>Operation variant for the least significant bit\<close>
 
 theory Least_significant_bit
   imports
     "HOL-Library.Word"
     Bits_Int
 begin
 
 class lsb = semiring_bits +
   fixes lsb :: \<open>'a \<Rightarrow> bool\<close>
   assumes lsb_odd: \<open>lsb = odd\<close>
 
 instantiation int :: lsb
 begin
 
 definition lsb_int :: \<open>int \<Rightarrow> bool\<close>
   where \<open>lsb i = i !! 0\<close> for i :: int
 
 instance
   by standard (simp add: fun_eq_iff lsb_int_def)
 
 end
 
 lemma bin_last_conv_lsb: "bin_last = lsb"
   by (simp add: lsb_odd)
 
 lemma int_lsb_numeral [simp]:
   "lsb (0 :: int) = False"
   "lsb (1 :: int) = True"
   "lsb (Numeral1 :: int) = True"
   "lsb (- 1 :: int) = True"
   "lsb (- Numeral1 :: int) = True"
   "lsb (numeral (num.Bit0 w) :: int) = False"
   "lsb (numeral (num.Bit1 w) :: int) = True"
   "lsb (- numeral (num.Bit0 w) :: int) = False"
   "lsb (- numeral (num.Bit1 w) :: int) = True"
   by (simp_all add: lsb_int_def)
 
 instantiation word :: (len) lsb
 begin
 
 definition lsb_word :: \<open>'a word \<Rightarrow> bool\<close>
   where word_lsb_def: \<open>lsb a \<longleftrightarrow> odd (uint a)\<close> for a :: \<open>'a word\<close>
 
 instance
   apply standard
   apply (simp add: fun_eq_iff word_lsb_def)
   apply transfer apply simp
   done
 
 end
-  
+
 lemma lsb_word_eq:
   \<open>lsb = (odd :: 'a word \<Rightarrow> bool)\<close> for w :: \<open>'a::len word\<close>
   by (fact lsb_odd)
 
 lemma word_lsb_alt: "lsb w = test_bit w 0"
   for w :: "'a::len word"
   by (auto simp: word_test_bit_def word_lsb_def)
 
 lemma word_lsb_1_0 [simp]: "lsb (1::'a::len word) \<and> \<not> lsb (0::'b::len word)"
   unfolding word_lsb_def by simp
 
 lemma word_lsb_int: "lsb w \<longleftrightarrow> uint w mod 2 = 1"
   apply (simp add: lsb_odd flip: odd_iff_mod_2_eq_one)
   apply transfer
   apply simp
   done
 
 lemmas word_ops_lsb = lsb0 [unfolded word_lsb_alt]
 
 lemma word_lsb_numeral [simp]:
   "lsb (numeral bin :: 'a::len word) \<longleftrightarrow> bin_last (numeral bin)"
   unfolding word_lsb_alt test_bit_numeral by simp
 
 lemma word_lsb_neg_numeral [simp]:
   "lsb (- numeral bin :: 'a::len word) \<longleftrightarrow> bin_last (- numeral bin)"
   by (simp add: word_lsb_alt)
 
 lemma word_lsb_nat:"lsb w = (unat w mod 2 = 1)"
   apply (simp add: word_lsb_def Groebner_Basis.algebra(31))
   apply transfer
   apply (simp add: even_nat_iff)
   done
 
 end
diff --git a/thys/Word_Lib/Legacy_Aliases.thy b/thys/Word_Lib/Legacy_Aliases.thy
--- a/thys/Word_Lib/Legacy_Aliases.thy
+++ b/thys/Word_Lib/Legacy_Aliases.thy
@@ -1,20 +1,24 @@
-
+(*
+ * Copyright Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
 
 section \<open>Legacy aliases\<close>
 
 theory Legacy_Aliases
   imports "HOL-Library.Word"
 begin
 
 definition
   complement :: "'a :: len word \<Rightarrow> 'a word"  where
  "complement x \<equiv> NOT x"
 
 lemma complement_mask:
   "complement (2 ^ n - 1) = NOT (mask n)"
   unfolding complement_def mask_eq_decr_exp by simp
 
 lemmas less_def = less_eq [symmetric]
 lemmas le_def = not_less [symmetric, where ?'a = nat]
 
 end
diff --git a/thys/Word_Lib/Many_More.thy b/thys/Word_Lib/Many_More.thy
--- a/thys/Word_Lib/Many_More.thy
+++ b/thys/Word_Lib/Many_More.thy
@@ -1,679 +1,684 @@
+(*
+ * Copyright Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
 
 theory Many_More
   imports
-    Main 
+    Main
     "HOL-Library.Word"
     More_Word
     Even_More_List
 begin
 
 lemma nat_less_mult_monoish: "\<lbrakk> a < b; c < (d :: nat) \<rbrakk> \<Longrightarrow> (a + 1) * (c + 1) <= b * d"
   apply (drule Suc_leI)+
   apply (drule(1) mult_le_mono)
   apply simp
   done
 
 lemma if_and_helper:
   "(If x v v') AND v'' = If x (v AND v'') (v' AND v'')"
   by (rule if_distrib)
 
 lemma eq_eqI:
   "a = b \<Longrightarrow> (a = x) = (b = x)"
   by simp
 
 lemma map2_Cons_2_3:
   "(map2 f xs (y # ys) = (z # zs)) = (\<exists>x xs'. xs = x # xs' \<and> f x y = z \<and> map2 f xs' ys = zs)"
   by (case_tac xs, simp_all)
 
 lemma map2_xor_replicate_False:
   "map2 (\<lambda>x y. x \<longleftrightarrow> \<not> y) xs (replicate n False) = take n xs"
   apply (induct xs arbitrary: n, simp)
   apply (case_tac n; simp)
   done
 
 lemma plus_Collect_helper:
   "(+) x ` {xa. P (xa :: 'a :: len word)} = {xa. P (xa - x)}"
   by (fastforce simp add: image_def)
 
 lemma plus_Collect_helper2:
   "(+) (- x) ` {xa. P (xa :: 'a :: len word)} = {xa. P (x + xa)}"
   using plus_Collect_helper [of "- x" P] by (simp add: ac_simps)
 
 lemma range_subset_eq2:
   "{a :: 'a :: len word .. b} \<noteq> {} \<Longrightarrow> ({a .. b} \<subseteq> {c .. d}) = (c \<le> a \<and> b \<le> d)"
   by simp
 
 lemma nat_mod_power_lem:
   fixes a :: nat
   shows "1 < a \<Longrightarrow> a ^ n mod a ^ m = (if m \<le> n then 0 else a ^ n)"
   apply (clarsimp)
   apply (clarsimp simp add: le_iff_add power_add)
   done
 
 lemma i_hate_words_helper:
   "i \<le> (j - k :: nat) \<Longrightarrow> i \<le> j"
   by simp
 
 lemma i_hate_words:
   "unat (a :: 'a word) \<le> unat (b :: 'a :: len word) - Suc 0
     \<Longrightarrow> a \<noteq> -1"
   apply (frule i_hate_words_helper)
   apply (subst(asm) word_le_nat_alt[symmetric])
   apply (clarsimp simp only: word_minus_one_le)
   apply (simp only: linorder_not_less[symmetric])
   apply (erule notE)
   apply (rule diff_Suc_less)
   apply (subst neq0_conv[symmetric])
   apply (subst unat_eq_0)
   apply (rule notI, drule arg_cong[where f="(+) 1"])
   apply simp
   done
 
 lemma If_eq_obvious:
   "x \<noteq> z \<Longrightarrow> ((if P then x else y) = z) = (\<not> P \<and> y = z)"
   by simp
 
 lemma Some_to_the:
   "v = Some x \<Longrightarrow> x = the v"
   by simp
 
 lemma dom_if_Some:
   "dom (\<lambda>x. if P x then Some (f x) else g x) = {x. P x} \<union> dom g"
   by fastforce
 
 lemma dom_insert_absorb:
   "x \<in> dom f \<Longrightarrow> insert x (dom f) = dom f"
   by (fact insert_absorb)
 
 lemma emptyE2:
   "\<lbrakk> S = {}; x \<in> S \<rbrakk> \<Longrightarrow> P"
   by simp
 
 lemma ptr_add_image_multI:
   "\<lbrakk> \<And>x y. (x * val = y * val') = (x * val'' = y); x * val'' \<in> S \<rbrakk> \<Longrightarrow>
      ptr_add ptr (x * val) \<in> (\<lambda>p. ptr_add ptr (p * val')) ` S"
   by (auto simp add: image_iff) metis
 
 lemmas map_prod_split_imageI'
   = map_prod_imageI[where f="case_prod f" and g="case_prod g"
                     and a="(a, b)" and b="(c, d)" for a b c d f g]
 lemmas map_prod_split_imageI = map_prod_split_imageI'[simplified]
 
 lemma dom_if:
   "dom (\<lambda>a. if a \<in> addrs then Some (f a) else g a)  = addrs \<union> dom g"
   by (auto simp: dom_def split: if_split)
 
 lemmas arg_cong_Not = arg_cong [where f=Not]
 
 lemma drop_append_miracle:
   "n = length xs \<Longrightarrow> drop n (xs @ ys) = ys"
   by simp
 
 lemma foldr_does_nothing_to_xf:
   "\<lbrakk> \<And>x s. x \<in> set xs \<Longrightarrow> xf (f x s) = xf s \<rbrakk> \<Longrightarrow> xf (foldr f xs s) = xf s"
   by (induct xs, simp_all)
 
 lemma mod_mod_power_int:
   fixes k :: int
   shows "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ (min m n)"
   by (fact mod_exp_eq)
 
 lemma le_step_down_nat:"\<lbrakk>(i::nat) \<le> n; i = n \<longrightarrow> P; i \<le> n - 1 \<longrightarrow> P\<rbrakk> \<Longrightarrow> P"
   by arith
 
 lemma le_step_down_int:"\<lbrakk>(i::int) \<le> n; i = n \<longrightarrow> P; i \<le> n - 1 \<longrightarrow> P\<rbrakk> \<Longrightarrow> P"
   by arith
 
 lemma replicate_numeral [simp]: "replicate (numeral k) x = x # replicate (pred_numeral k) x"
   by (simp add: numeral_eq_Suc)
 
 lemma list_exhaust_size_gt0:
   assumes "\<And>a list. y = a # list \<Longrightarrow> P"
   shows "0 < length y \<Longrightarrow> P"
   apply (cases y)
    apply simp
   apply (rule assms)
   apply fastforce
   done
 
 lemma list_exhaust_size_eq0:
   assumes "y = [] \<Longrightarrow> P"
   shows "length y = 0 \<Longrightarrow> P"
   apply (cases y)
    apply (rule assms)
    apply simp
   apply simp
   done
 
 lemma size_Cons_lem_eq: "y = xa # list \<Longrightarrow> size y = Suc k \<Longrightarrow> size list = k"
   by auto
 
 lemma takeWhile_take_has_property:
   "n \<le> length (takeWhile P xs) \<Longrightarrow> \<forall>x \<in> set (take n xs). P x"
   by (induct xs arbitrary: n; simp split: if_split_asm) (case_tac n, simp_all)
 
 lemma takeWhile_take_has_property_nth:
   "\<lbrakk> n < length (takeWhile P xs) \<rbrakk> \<Longrightarrow> P (xs ! n)"
   by (induct xs arbitrary: n; simp split: if_split_asm) (case_tac n, simp_all)
 
 lemma takeWhile_replicate:
   "takeWhile f (replicate len x) = (if f x then replicate len x else [])"
   by (induct_tac len) auto
 
 lemma takeWhile_replicate_empty:
   "\<not> f x \<Longrightarrow> takeWhile f (replicate len x) = []"
   by (simp add: takeWhile_replicate)
 
 lemma takeWhile_replicate_id:
   "f x \<Longrightarrow> takeWhile f (replicate len x) = replicate len x"
   by (simp add: takeWhile_replicate)
 
 lemma nth_rev: "n < length xs \<Longrightarrow> rev xs ! n = xs ! (length xs - 1 - n)"
   using rev_nth by simp
 
 lemma nth_rev_alt: "n < length ys \<Longrightarrow> ys ! n = rev ys ! (length ys - Suc n)"
   by (simp add: nth_rev)
 
 lemma hd_butlast: "length xs > 1 \<Longrightarrow> hd (butlast xs) = hd xs"
   by (cases xs) auto
 
 lemma split_upt_on_n:
   "n < m \<Longrightarrow> [0 ..< m] = [0 ..< n] @ [n] @ [Suc n ..< m]"
   by (metis append_Cons append_Nil less_Suc_eq_le less_imp_le_nat upt_add_eq_append'
             upt_rec zero_less_Suc)
 
 lemma drop_eq_mono:
   assumes le: "m \<le> n"
   assumes drop: "drop m xs = drop m ys"
   shows "drop n xs = drop n ys"
 proof -
   have ex: "\<exists>p. n = p + m" by (rule exI[of _ "n - m"]) (simp add: le)
   then obtain p where p: "n = p + m" by blast
   show ?thesis unfolding p drop_drop[symmetric] drop by simp
 qed
 
 lemma drop_Suc_nth:
   "n < length xs \<Longrightarrow> drop n xs = xs!n # drop (Suc n) xs"
   by (simp add: Cons_nth_drop_Suc)
 
 lemma and_len: "xs = ys \<Longrightarrow> xs = ys \<and> length xs = length ys"
   by auto
 
 lemma tl_if: "tl (if p then xs else ys) = (if p then tl xs else tl ys)"
   by auto
 
 lemma hd_if: "hd (if p then xs else ys) = (if p then hd xs else hd ys)"
   by auto
 
 lemma if_single: "(if xc then [xab] else [an]) = [if xc then xab else an]"
   by auto
 
 \<comment> \<open>note -- \<open>if_Cons\<close> can cause blowup in the size, if \<open>p\<close> is complex, so make a simproc\<close>
 lemma if_Cons: "(if p then x # xs else y # ys) = If p x y # If p xs ys"
   by auto
 
 lemma list_of_false:
   "True \<notin> set xs \<Longrightarrow> xs = replicate (length xs) False"
   by (induct xs, simp_all)
 
 lemma list_all2_induct [consumes 1, case_names Nil Cons]:
   assumes lall: "list_all2 Q xs ys"
   and     nilr: "P [] []"
   and    consr: "\<And>x xs y ys. \<lbrakk>list_all2 Q xs ys; Q x y; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)"
   shows  "P xs ys"
   using lall
 proof (induct rule: list_induct2 [OF list_all2_lengthD [OF lall]])
   case 1 then show ?case by auto fact+
 next
   case (2 x xs y ys)
 
   show ?case
   proof (rule consr)
     from "2.prems" show "list_all2 Q xs ys" and "Q x y" by simp_all
     then show "P xs ys" by (intro "2.hyps")
   qed
 qed
 
 lemma replicate_minus:
   "k < n \<Longrightarrow> replicate n False = replicate (n - k) False @ replicate k False"
   by (subst replicate_add [symmetric]) simp
 
 lemma cart_singleton_empty:
   "(S \<times> {e} = {}) = (S = {})"
   by blast
 
 lemma MinI:
   assumes fa: "finite A"
   and     ne: "A \<noteq> {}"
   and     xv: "m \<in> A"
   and    min: "\<forall>y \<in> A. m \<le> y"
   shows "Min A = m" using fa ne xv min
 proof (induct A arbitrary: m rule: finite_ne_induct)
   case singleton then show ?case by simp
 next
   case (insert y F)
 
   from insert.prems have yx: "m \<le> y" and fx: "\<forall>y \<in> F. m \<le> y" by auto
   have "m \<in> insert y F" by fact
   then show ?case
   proof
     assume mv: "m = y"
 
     have mlt: "m \<le> Min F"
       by (rule iffD2 [OF Min_ge_iff [OF insert.hyps(1) insert.hyps(2)] fx])
 
     show ?case
       apply (subst Min_insert [OF insert.hyps(1) insert.hyps(2)])
       apply (subst mv [symmetric])
       apply (auto simp: min_def mlt)
       done
   next
     assume "m \<in> F"
     then have mf: "Min F = m"
       by (rule insert.hyps(4) [OF _ fx])
 
     show ?case
       apply (subst Min_insert [OF insert.hyps(1) insert.hyps(2)])
       apply (subst mf)
       apply (rule iffD2 [OF _ yx])
       apply (auto simp: min_def)
       done
   qed
 qed
 
 lemma power_numeral: "a ^ numeral k = a * a ^ (pred_numeral k)"
   by (simp add: numeral_eq_Suc)
 
 lemma funpow_numeral [simp]: "f ^^ numeral k = f \<circ> f ^^ (pred_numeral k)"
   by (simp add: numeral_eq_Suc)
 
 lemma funpow_minus_simp: "0 < n \<Longrightarrow> f ^^ n = f \<circ> f ^^ (n - 1)"
   by (auto dest: gr0_implies_Suc)
 
 lemma rco_alt: "(f \<circ> g) ^^ n \<circ> f = f \<circ> (g \<circ> f) ^^ n"
   apply (rule ext)
   apply (induct n)
    apply (simp_all add: o_def)
   done
 
 lemma union_sub:
   "\<lbrakk>B \<subseteq> A; C \<subseteq> B\<rbrakk> \<Longrightarrow> (A - B) \<union> (B - C) = (A - C)"
   by fastforce
 
 lemma insert_sub:
   "x \<in> xs \<Longrightarrow> (insert x (xs - ys)) = (xs - (ys - {x}))"
   by blast
 
 lemma ran_upd:
   "\<lbrakk> inj_on f (dom f); f y = Some z \<rbrakk> \<Longrightarrow> ran (\<lambda>x. if x = y then None else f x) = ran f - {z}"
   unfolding ran_def
   apply (rule set_eqI)
   apply simp
   by (metis domI inj_on_eq_iff option.sel)
 
 lemma if_apply_def2:
   "(if P then F else G) = (\<lambda>x. (P \<longrightarrow> F x) \<and> (\<not> P \<longrightarrow> G x))"
   by simp
 
 lemma case_bool_If:
   "case_bool P Q b = (if b then P else Q)"
   by simp
 
 lemma if_f:
   "(if a then f b else f c) = f (if a then b else c)"
   by simp
 
 lemma size_if: "size (if p then xs else ys) = (if p then size xs else size ys)"
   by (fact if_distrib)
 
 lemma if_Not_x: "(if p then \<not> x else x) = (p = (\<not> x))"
   by auto
 
 lemma if_x_Not: "(if p then x else \<not> x) = (p = x)"
   by auto
 
 lemma if_same_and: "(If p x y \<and> If p u v) = (if p then x \<and> u else y \<and> v)"
   by auto
 
 lemma if_same_eq: "(If p x y  = (If p u v)) = (if p then x = u else y = v)"
   by auto
 
 lemma if_same_eq_not: "(If p x y = (\<not> If p u v)) = (if p then x = (\<not> u) else y = (\<not> v))"
   by auto
 
 lemma the_elemI: "y = {x} \<Longrightarrow> the_elem y = x"
   by simp
 
 lemma nonemptyE: "S \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> R) \<Longrightarrow> R"
   by auto
 
 lemmas xtr1 = xtrans(1)
 lemmas xtr2 = xtrans(2)
 lemmas xtr3 = xtrans(3)
 lemmas xtr4 = xtrans(4)
 lemmas xtr5 = xtrans(5)
 lemmas xtr6 = xtrans(6)
 lemmas xtr7 = xtrans(7)
 lemmas xtr8 = xtrans(8)
 
 lemmas if_fun_split = if_apply_def2
 
 lemma not_empty_eq:
   "(S \<noteq> {}) = (\<exists>x. x \<in> S)"
   by auto
 
 lemma range_subset_lower:
   fixes c :: "'a ::linorder"
   shows "\<lbrakk> {a..b} \<subseteq> {c..d}; x \<in> {a..b} \<rbrakk> \<Longrightarrow> c \<le> a"
   apply (frule (1) subsetD)
   apply (rule classical)
   apply clarsimp
   done
 
 lemma range_subset_upper:
   fixes c :: "'a ::linorder"
   shows "\<lbrakk> {a..b} \<subseteq> {c..d}; x \<in> {a..b} \<rbrakk> \<Longrightarrow> b \<le> d"
   apply (frule (1) subsetD)
   apply (rule classical)
   apply clarsimp
   done
 
 lemma range_subset_eq:
   fixes a::"'a::linorder"
   assumes non_empty: "a \<le> b"
   shows "({a..b} \<subseteq> {c..d}) = (c \<le> a \<and> b \<le> d)"
   apply (insert non_empty)
   apply (rule iffI)
    apply (frule range_subset_lower [where x=a], simp)
    apply (drule range_subset_upper [where x=a], simp)
    apply simp
   apply auto
   done
 
 lemma range_eq:
   fixes a::"'a::linorder"
   assumes non_empty: "a \<le> b"
   shows "({a..b} = {c..d}) = (a = c \<and> b = d)"
   by (metis atLeastatMost_subset_iff eq_iff non_empty)
 
 lemma range_strict_subset_eq:
   fixes a::"'a::linorder"
   assumes non_empty: "a \<le> b"
   shows "({a..b} \<subset> {c..d}) = (c \<le> a \<and> b \<le> d \<and> (a = c \<longrightarrow> b \<noteq> d))"
   apply (insert non_empty)
   apply (subst psubset_eq)
   apply (subst range_subset_eq, assumption+)
   apply (subst range_eq, assumption+)
   apply simp
   done
 
 lemma range_subsetI:
   fixes x :: "'a :: order"
   assumes xX: "X \<le> x"
   and     yY: "y \<le> Y"
   shows   "{x .. y} \<subseteq> {X .. Y}"
   using xX yY by auto
 
 lemma set_False [simp]:
   "(set bs \<subseteq> {False}) = (True \<notin> set bs)" by auto
 
 lemma int_not_emptyD:
   "A \<inter> B \<noteq> {} \<Longrightarrow> \<exists>x. x \<in> A \<and> x \<in> B"
   by (erule contrapos_np, clarsimp simp: disjoint_iff_not_equal)
 
 definition
   sum_map :: "('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> 'a + 'c \<Rightarrow> 'b + 'd" where
  "sum_map f g x \<equiv> case x of Inl v \<Rightarrow> Inl (f v) | Inr v' \<Rightarrow> Inr (g v')"
 
 lemma sum_map_simps[simp]:
   "sum_map f g (Inl v) = Inl (f v)"
   "sum_map f g (Inr w) = Inr (g w)"
   by (simp add: sum_map_def)+
 
 lemma if_Some_None_eq_None:
   "((if P then Some v else None) = None) = (\<not> P)"
   by simp
 
 lemma CollectPairFalse [iff]:
   "{(a,b). False} = {}"
   by (simp add: split_def)
 
 lemma if_conj_dist:
   "((if b then w else x) \<and> (if b then y else z) \<and> X) =
   ((if b then w \<and> y else x \<and> z) \<and> X)"
   by simp
 
 lemma if_P_True1:
   "Q \<Longrightarrow> (if P then True else Q)"
   by simp
 
 lemma if_P_True2:
   "Q \<Longrightarrow> (if P then Q else True)"
   by simp
 
 lemmas nat_simps = diff_add_inverse2 diff_add_inverse
 
 lemmas nat_iffs = le_add1 le_add2
 
 lemma nat_min_simps:
   "(a::nat) \<le> b \<Longrightarrow> min b a = a"
   "a \<le> b \<Longrightarrow> min a b = a"
   by simp_all
 
 lemmas zadd_diff_inverse =
   trans [OF diff_add_cancel [symmetric] add.commute]
 
 lemmas add_diff_cancel2 =
   add.commute [THEN diff_eq_eq [THEN iffD2]]
 
 lemmas mcl = mult_cancel_left [THEN iffD1, THEN make_pos_rule]
 
 lemma pl_pl_rels: "a + b = c + d \<Longrightarrow> a \<ge> c \<and> b \<le> d \<or> a \<le> c \<and> b \<ge> d"
   for a b c d :: nat
   by arith
 
 lemmas pl_pl_rels' = add.commute [THEN [2] trans, THEN pl_pl_rels]
 
 lemma iszero_minus:
   \<open>iszero (- z) \<longleftrightarrow> iszero z\<close>
   by (simp add: iszero_def)
 
 lemma diff_le_eq': "a - b \<le> c \<longleftrightarrow> a \<le> b + c"
   for a b c :: int
   by arith
 
 lemma zless2: "0 < (2 :: int)"
   by (fact zero_less_numeral)
 
 lemma zless2p: "0 < (2 ^ n :: int)"
   by arith
 
 lemma zle2p: "0 \<le> (2 ^ n :: int)"
   by arith
 
 lemma ex_eq_or: "(\<exists>m. n = Suc m \<and> (m = k \<or> P m)) \<longleftrightarrow> n = Suc k \<or> (\<exists>m. n = Suc m \<and> P m)"
   by auto
 
 lemma power_minus_simp: "0 < n \<Longrightarrow> a ^ n = a * a ^ (n - 1)"
   by (auto dest: gr0_implies_Suc)
 
 lemma n2s_ths:
   \<open>2 + n = Suc (Suc n)\<close>
   \<open>n + 2 = Suc (Suc n)\<close>
   by (fact add_2_eq_Suc add_2_eq_Suc')+
 
 lemma s2n_ths:
   \<open>Suc (Suc n) = 2 + n\<close>
   \<open>Suc (Suc n) = n + 2\<close>
   by simp_all
 
 lemma gt_or_eq_0: "0 < y \<or> 0 = y"
   for y :: nat
   by arith
 
 lemma sum_imp_diff: "j = k + i \<Longrightarrow> j - i = k"
   for k :: nat
   by simp
 
 lemma le_diff_eq': "a \<le> c - b \<longleftrightarrow> b + a \<le> c"
   for a b c :: int
   by arith
 
 lemma less_diff_eq': "a < c - b \<longleftrightarrow> b + a < c"
   for a b c :: int
   by arith
 
 lemma diff_less_eq': "a - b < c \<longleftrightarrow> a < b + c"
   for a b c :: int
   by arith
 
 lemma axxbyy: "a + m + m = b + n + n \<Longrightarrow> a = 0 \<or> a = 1 \<Longrightarrow> b = 0 \<or> b = 1 \<Longrightarrow> a = b \<and> m = n"
   for a b m n :: int
   by arith
 
 lemma minus_eq: "m - k = m \<longleftrightarrow> k = 0 \<or> m = 0"
   for k m :: nat
   by arith
 
 lemma pl_pl_mm: "a + b = c + d \<Longrightarrow> a - c = d - b"
   for a b c d :: nat
   by arith
 
 lemmas pl_pl_mm' = add.commute [THEN [2] trans, THEN pl_pl_mm]
 
 lemma less_le_mult': "w * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> (w + 1) * c \<le> b * c"
   for b c w :: int
   apply (rule mult_right_mono)
    apply (rule zless_imp_add1_zle)
    apply (erule (1) mult_right_less_imp_less)
   apply assumption
   done
 
 lemma less_le_mult: "w * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> w * c + c \<le> b * c"
   for b c w :: int
   using less_le_mult' [of w c b] by (simp add: algebra_simps)
 
 lemmas less_le_mult_minus = iffD2 [OF le_diff_eq less_le_mult,
   simplified left_diff_distrib]
 
 lemma gen_minus: "0 < n \<Longrightarrow> f n = f (Suc (n - 1))"
   by auto
 
 lemma mpl_lem: "j \<le> i \<Longrightarrow> k < j \<Longrightarrow> i - j + k < i"
   for i j k :: nat
   by arith
 
 lemmas dme = div_mult_mod_eq
 lemmas dtle = div_times_less_eq_dividend
 lemmas th2 = order_trans [OF order_refl [THEN [2] mult_le_mono] div_times_less_eq_dividend]
 
 lemmas sdl = div_nat_eqI
 
 lemma given_quot: "f > 0 \<Longrightarrow> (f * l + (f - 1)) div f = l"
   for f l :: nat
   by (rule div_nat_eqI) (simp_all)
 
 lemma given_quot_alt: "f > 0 \<Longrightarrow> (l * f + f - Suc 0) div f = l"
   for f l :: nat
   apply (frule given_quot)
   apply (rule trans)
    prefer 2
    apply (erule asm_rl)
   apply (rule_tac f="\<lambda>n. n div f" in arg_cong)
   apply (simp add : ac_simps)
   done
 
 lemma x_power_minus_1:
   fixes x :: "'a :: {ab_group_add, power, numeral, one}"
   shows "x + (2::'a) ^ n - (1::'a) = x + (2 ^ n - 1)" by simp
 
 lemma nat_diff_add:
   fixes i :: nat
   shows "\<lbrakk> i + j = k \<rbrakk> \<Longrightarrow> i = k - j"
   by arith
 
 lemma pow_2_gt: "n \<ge> 2 \<Longrightarrow> (2::int) < 2 ^ n"
   by (induct n) auto
 
 lemma sum_to_zero:
   "(a :: 'a :: ring) + b = 0 \<Longrightarrow> a = (- b)"
   by (drule arg_cong[where f="\<lambda> x. x - a"], simp)
 
 lemma arith_is_1:
   "\<lbrakk> x \<le> Suc 0; x > 0 \<rbrakk> \<Longrightarrow> x = 1"
   by arith
 
 lemma suc_le_pow_2:
   "1 < (n::nat) \<Longrightarrow> Suc n < 2 ^ n"
   by (induct n; clarsimp)
      (case_tac "n = 1"; clarsimp)
 
 lemma nat_le_Suc_less_imp:
   "x < y \<Longrightarrow> x \<le> y - Suc 0"
   by arith
 
 lemma power_sub_int:
   "\<lbrakk> m \<le> n; 0 < b \<rbrakk> \<Longrightarrow> b ^ n div b ^ m = (b ^ (n - m) :: int)"
   apply (subgoal_tac "\<exists>n'. n = m + n'")
    apply (clarsimp simp: power_add)
   apply (rule exI[where x="n - m"])
   apply simp
   done
 
 lemma nat_Suc_less_le_imp:
   "(k::nat) < Suc n \<Longrightarrow> k \<le> n"
   by auto
 
 lemma nat_add_less_by_max:
   "\<lbrakk> (x::nat) \<le> xmax ; y < k - xmax \<rbrakk> \<Longrightarrow> x + y < k"
   by simp
 
 lemma nat_le_Suc_less:
   "0 < y \<Longrightarrow> (x \<le> y - Suc 0) = (x < y)"
   by arith
 
 lemma nat_power_minus_less:
   "a < 2 ^ (x - n) \<Longrightarrow> (a :: nat) < 2 ^ x"
   by (erule order_less_le_trans) simp
 
 lemma less_le_mult_nat':
   "w * c < b * c ==> 0 \<le> c ==> Suc w * c \<le> b * (c::nat)"
   apply (rule mult_right_mono)
    apply (rule Suc_leI)
    apply (erule (1) mult_right_less_imp_less)
   apply assumption
   done
 
 lemma less_le_mult_nat:
   \<open>0 < c \<and> w < b \<Longrightarrow> c + w * c \<le> b * c\<close> for b c w :: nat
   using less_le_mult_nat' [of w c b] by simp
 
 lemma p_assoc_help:
   fixes p :: "'a::{ring,power,numeral,one}"
   shows "p + 2^sz - 1 = p + (2^sz - 1)"
   by simp
 
 lemma pow_mono_leq_imp_lt:
   "x \<le> y \<Longrightarrow> x < 2 ^ y"
   by (simp add: le_less_trans)
 
 lemma small_powers_of_2:
   "x \<ge> 3 \<Longrightarrow> x < 2 ^ (x - 1)"
   by (induct x; simp add: suc_le_pow_2)
 
 lemma nat_less_power_trans2:
   fixes n :: nat
   shows "\<lbrakk>n < 2 ^ (m - k); k \<le> m\<rbrakk> \<Longrightarrow> n * 2 ^ k  < 2 ^ m"
   by (subst mult.commute, erule (1) nat_less_power_trans)
 
 lemma nat_move_sub_le: "(a::nat) + b \<le> c \<Longrightarrow> a \<le> c - b"
   by arith
 
 lemma plus_minus_one_rewrite:
   "v + (- 1 :: ('a :: {ring, one, uminus})) \<equiv> v - 1"
   by (simp add: field_simps)
 
 lemma Suc_0_lt_2p_len_of: "Suc 0 < 2 ^ LENGTH('a :: len)"
   by (metis One_nat_def len_gt_0 lessI numeral_2_eq_2 one_less_power)
 
 end
diff --git a/thys/Word_Lib/More_Arithmetic.thy b/thys/Word_Lib/More_Arithmetic.thy
--- a/thys/Word_Lib/More_Arithmetic.thy
+++ b/thys/Word_Lib/More_Arithmetic.thy
@@ -1,136 +1,141 @@
+(*
+ * Copyright Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
 
 section \<open>Arithmetic lemmas\<close>
 
 theory More_Arithmetic
   imports Main "HOL-Library.Type_Length" "HOL-Library.Bit_Operations"
 begin
 
 declare iszero_0 [intro]
 
 declare min.absorb1 [simp] min.absorb2 [simp]
 
 lemma n_less_equal_power_2 [simp]:
   "n < 2 ^ n"
   by (fact less_exp)
 
 lemma min_pm [simp]: "min a b + (a - b) = a"
   for a b :: nat
   by arith
 
 lemma min_pm1 [simp]: "a - b + min a b = a"
   for a b :: nat
   by arith
 
 lemma rev_min_pm [simp]: "min b a + (a - b) = a"
   for a b :: nat
   by arith
 
 lemma rev_min_pm1 [simp]: "a - b + min b a = a"
   for a b :: nat
   by arith
 
 lemma min_minus [simp]: "min m (m - k) = m - k"
   for m k :: nat
   by arith
 
 lemma min_minus' [simp]: "min (m - k) m = m - k"
   for m k :: nat
   by arith
 
 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 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 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 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)
   apply (metis Suc_leI mult.assoc mult_Suc_right nat_mult_le_cancel_disj)
   done
 
 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 diff_diff_less:
   "(i < m - (m - (n :: nat))) = (i < m \<and> i < n)"
   by auto
 
 lemma small_powers_of_2:
   \<open>x < 2 ^ (x - 1)\<close> if \<open>x \<ge> 3\<close> for x :: nat
 proof -
   define m where \<open>m = x - 3\<close>
   with that have \<open>x = m + 3\<close>
     by simp
   moreover have \<open>m + 3 < 4 * 2 ^ m\<close>
     by (induction m) simp_all
   ultimately show ?thesis
     by simp
 qed
 
 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
@@ -1,410 +1,415 @@
+(*
+ * Copyright Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
 
 section \<open>Lemmas on division\<close>
 
 theory More_Divides
   imports
     "HOL-Library.Word"
 begin
 
 declare div_eq_dividend_iff [simp]
 
 lemma int_div_same_is_1 [simp]:
   \<open>a div b = a \<longleftrightarrow> b = 1\<close> if \<open>0 < a\<close> for a b :: int
   using that 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)
 
 lemma int_div_minus_is_minus1 [simp]:
   \<open>a div b = - a \<longleftrightarrow> b = - 1\<close> if \<open>0 > a\<close> for a b :: int
   using that by (metis div_minus_right equation_minus_iff int_div_same_is_1 neg_0_less_iff_less)
 
 lemma nat_div_eq_Suc_0_iff: "n div m = Suc 0 \<longleftrightarrow> m \<le> n \<and> n < 2 * m"
   apply auto
   using div_greater_zero_iff apply fastforce
    apply (metis One_nat_def div_greater_zero_iff dividend_less_div_times mult.right_neutral mult_Suc mult_numeral_1 numeral_2_eq_2 zero_less_numeral)
   apply (simp add: div_nat_eqI)
   done
 
 lemma diff_mod_le:
   \<open>a - a mod b \<le> d - b\<close> if \<open>a < d\<close> \<open>b dvd d\<close> for a b d :: nat
   using that
   apply(subst minus_mod_eq_mult_div)
   apply(clarsimp simp: dvd_def)
   apply(cases \<open>b = 0\<close>)
    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_times_less_eq_dividend)
    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 one_mod_exp_eq_one [simp]:
   "1 mod (2 * 2 ^ n) = (1::int)"
   using power_gt1 [of 2 n] by (auto intro: mod_pos_pos_trivial)
 
 lemma int_mod_lem: "0 < n \<Longrightarrow> 0 \<le> b \<and> b < n \<longleftrightarrow> b mod n = b"
   for b n :: int
   apply safe
     apply (erule (1) mod_pos_pos_trivial)
    apply (erule_tac [!] subst)
    apply auto
   done
 
 lemma int_mod_ge': "b < 0 \<Longrightarrow> 0 < n \<Longrightarrow> b + n \<le> b mod n"
   for b n :: int
   by (metis add_less_same_cancel2 int_mod_ge mod_add_self2)
 
 lemma int_mod_le': "0 \<le> b - n \<Longrightarrow> b mod n \<le> b - n"
   for b n :: int
   by (metis minus_mod_self2 zmod_le_nonneg_dividend)
 
 lemma emep1: "even n \<Longrightarrow> even d \<Longrightarrow> 0 \<le> d \<Longrightarrow> (n + 1) mod d = (n mod d) + 1"
   for n d :: int
   by (auto simp add: pos_zmod_mult_2 add.commute dvd_def)
 
 lemma m1mod2k: "- 1 mod 2 ^ n = (2 ^ n - 1 :: int)"
   by (rule zmod_minus1) simp
 
 lemma sb_inc_lem: "a + 2^k < 0 \<Longrightarrow> a + 2^k + 2^(Suc k) \<le> (a + 2^k) mod 2^(Suc k)"
   for a :: int
   using int_mod_ge' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k"]
   by simp
 
 lemma sb_inc_lem': "a < - (2^k) \<Longrightarrow> a + 2^k + 2^(Suc k) \<le> (a + 2^k) mod 2^(Suc k)"
   for a :: int
   by (rule sb_inc_lem) simp
 
 lemma sb_dec_lem: "0 \<le> - (2 ^ k) + a \<Longrightarrow> (a + 2 ^ k) mod (2 * 2 ^ k) \<le> - (2 ^ k) + a"
   for a :: int
   using int_mod_le'[where n = "2 ^ (Suc k)" and b = "a + 2 ^ k"] by simp
 
 lemma sb_dec_lem': "2 ^ k \<le> a \<Longrightarrow> (a + 2 ^ k) mod (2 * 2 ^ k) \<le> - (2 ^ k) + a"
   for a :: int
   by (rule sb_dec_lem) simp
 
 lemma mod_2_neq_1_eq_eq_0: "k mod 2 \<noteq> 1 \<longleftrightarrow> k mod 2 = 0"
   for k :: int
   by (fact not_mod_2_eq_1_eq_0)
 
 lemma z1pmod2: "(2 * b + 1) mod 2 = (1::int)"
   for b :: int
   by arith
 
 lemma p1mod22k': "(1 + 2 * b) mod (2 * 2 ^ n) = 1 + 2 * (b mod 2 ^ n)"
   for b :: int
   by (rule pos_zmod_mult_2) simp
 
 lemma p1mod22k: "(2 * b + 1) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n) + 1"
   for b :: int
   by (simp add: p1mod22k' add.commute)
 
 lemma pos_mod_sign2:
   \<open>0 \<le> a mod 2\<close> for a :: int
   by simp
 
 lemma pos_mod_bound2:
   \<open>a mod 2 < 2\<close> for a :: int
   by simp
 
 lemma nmod2: "n mod 2 = 0 \<or> n mod 2 = 1"
   for n :: int
   by arith
 
 lemma eme1p:
   "even n \<Longrightarrow> even d \<Longrightarrow> 0 \<le> d \<Longrightarrow> (1 + n) mod d = 1 + n mod d" for n d :: int
   using emep1 [of n d] by (simp add: ac_simps)
 
 lemma m1mod22k:
   \<open>- 1 mod (2 * 2 ^ n) = 2 * 2 ^ n - (1::int)\<close>
   by (simp add: zmod_minus1)
 
 lemma z1pdiv2: "(2 * b + 1) div 2 = b"
   for b :: int
   by arith
 
 lemma zdiv_le_dividend:
   \<open>0 \<le> a \<Longrightarrow> 0 < b \<Longrightarrow> a div b \<le> a\<close> for a b :: int
   by (metis div_by_1 int_one_le_iff_zero_less zdiv_mono2 zero_less_one)
 
 lemma axxmod2: "(1 + x + x) mod 2 = 1 \<and> (0 + x + x) mod 2 = 0"
   for x :: int
   by arith
 
 lemma axxdiv2: "(1 + x + x) div 2 = x \<and> (0 + x + x) div 2 = x"
   for x :: int
   by arith
 
 lemmas rdmods =
   mod_minus_eq [symmetric]
   mod_diff_left_eq [symmetric]
   mod_diff_right_eq [symmetric]
   mod_add_left_eq [symmetric]
   mod_add_right_eq [symmetric]
   mod_mult_right_eq [symmetric]
   mod_mult_left_eq [symmetric]
 
 lemma mod_plus_right: "(a + x) mod m = (b + x) mod m \<longleftrightarrow> a mod m = b mod m"
   for a b m x :: nat
   by (induct x) (simp_all add: mod_Suc, arith)
 
 lemma nat_minus_mod: "(n - n mod m) mod m = 0"
   for m n :: nat
   by (induct n) (simp_all add: mod_Suc)
 
 lemmas nat_minus_mod_plus_right =
   trans [OF nat_minus_mod mod_0 [symmetric],
     THEN mod_plus_right [THEN iffD2], simplified]
 
 lemmas push_mods' = mod_add_eq
   mod_mult_eq mod_diff_eq
   mod_minus_eq
 
 lemmas push_mods = push_mods' [THEN eq_reflection]
 lemmas pull_mods = push_mods [symmetric] rdmods [THEN eq_reflection]
 
 lemma nat_mod_eq: "b < n \<Longrightarrow> a mod n = b mod n \<Longrightarrow> a mod n = b"
   for a b n :: nat
   by (induct a) auto
 
 lemmas nat_mod_eq' = refl [THEN [2] nat_mod_eq]
 
 lemma nat_mod_lem: "0 < n \<Longrightarrow> b < n \<longleftrightarrow> b mod n = b"
   for b n :: nat
   apply safe
    apply (erule nat_mod_eq')
   apply (erule subst)
   apply (erule mod_less_divisor)
   done
 
 lemma mod_nat_add: "x < z \<Longrightarrow> y < z \<Longrightarrow> (x + y) mod z = (if x + y < z then x + y else x + y - z)"
   for x y z :: nat
   apply (rule nat_mod_eq)
    apply auto
   apply (rule trans)
    apply (rule le_mod_geq)
    apply simp
   apply (rule nat_mod_eq')
   apply arith
   done
 
 lemma mod_nat_sub: "x < z \<Longrightarrow> (x - y) mod z = x - y"
   for x y :: nat
   by (rule nat_mod_eq') arith
 
 lemma int_mod_eq: "0 \<le> b \<Longrightarrow> b < n \<Longrightarrow> a mod n = b mod n \<Longrightarrow> a mod n = b"
   for a b n :: int
   by (metis mod_pos_pos_trivial)
 
 lemma zmde:
   \<open>b * (a div b) = a - a mod b\<close> for a b :: \<open>'a::{group_add,semiring_modulo}\<close>
-  using mult_div_mod_eq [of b a] by (simp add: eq_diff_eq) 
+  using mult_div_mod_eq [of b a] by (simp add: eq_diff_eq)
 
 (* already have this for naturals, div_mult_self1/2, but not for ints *)
 lemma zdiv_mult_self: "m \<noteq> 0 \<Longrightarrow> (a + m * n) div m = a div m + n"
   for a m n :: int
   by simp
 
 lemma mod_power_lem: "a > 1 \<Longrightarrow> a ^ n mod a ^ m = (if m \<le> n then 0 else a ^ n)"
   for a :: int
   by (simp add: mod_eq_0_iff_dvd le_imp_power_dvd)
 
 lemma nonneg_mod_div: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> (a mod b) \<and> 0 \<le> a div b"
   for a b :: int
   by (cases "b = 0") (auto intro: pos_imp_zdiv_nonneg_iff [THEN iffD2])
 
 lemma mod_exp_less_eq_exp:
   \<open>a mod 2 ^ n < 2 ^ n\<close> for a :: int
   by (rule pos_mod_bound) simp
 
 lemma div_mult_le:
   \<open>a div b * b \<le> a\<close> for a b :: nat
   by (fact div_times_less_eq_dividend)
 
 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 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 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)
 
 lemmas m2pths = pos_mod_sign mod_exp_less_eq_exp
 
 lemmas int_mod_eq' = mod_pos_pos_trivial (* FIXME delete *)
 
 lemmas int_mod_le = zmod_le_nonneg_dividend (* FIXME: delete *)
 
 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 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 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 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 td_gal_lt:
   \<open>0 < c \<Longrightarrow> a < b * c \<longleftrightarrow> a div c < b\<close>
   for a b c :: nat
   apply (auto dest: less_mult_imp_div_less)
   apply (metis div_le_mono div_mult_self_is_m leD leI)
   done
 
 lemma td_gal:
   \<open>0 < c \<Longrightarrow> b * c \<le> a  \<longleftrightarrow> b \<le> a div c\<close>
   for a b c :: nat
   by (meson not_le td_gal_lt)
 
 end
diff --git a/thys/Word_Lib/More_Misc.thy b/thys/Word_Lib/More_Misc.thy
--- a/thys/Word_Lib/More_Misc.thy
+++ b/thys/Word_Lib/More_Misc.thy
@@ -1,10 +1,15 @@
+(*
+ * Copyright Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
 
 section \<open>Miscellaneous lemmas\<close>
 
 theory More_Misc
 imports Main
 begin
 
 lemmas ls_splits = prod.split prod.split_asm if_split_asm
 
 end
diff --git a/thys/Word_Lib/More_Sublist.thy b/thys/Word_Lib/More_Sublist.thy
--- a/thys/Word_Lib/More_Sublist.thy
+++ b/thys/Word_Lib/More_Sublist.thy
@@ -1,82 +1,87 @@
+(*
+ * Copyright Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
 
 section \<open>Lemmas on sublists\<close>
 
 theory More_Sublist
   imports
     "HOL-Library.Sublist"
 begin
 
 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
 
 lemma sublist_equal_part:
   "prefix xs ys \<Longrightarrow> take (length xs) ys = xs"
   by (clarsimp simp: prefix_def)
 
 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 map_prefixI:
   "prefix xs ys \<Longrightarrow> prefix (map f xs) (map f ys)"
   by (clarsimp simp: prefix_def)
 
 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 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
 
 end
diff --git a/thys/Word_Lib/More_Word.thy b/thys/Word_Lib/More_Word.thy
--- a/thys/Word_Lib/More_Word.thy
+++ b/thys/Word_Lib/More_Word.thy
@@ -1,1802 +1,1807 @@
+(*
+ * Copyright Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
 
 section \<open>Lemmas on words\<close>
 
 theory More_Word
   imports
     "HOL-Library.Word"
     More_Arithmetic
     More_Divides
 begin
 
 lemma unat_power_lower [simp]:
   "unat ((2::'a::len word) ^ n) = 2 ^ n" if "n < LENGTH('a::len)"
   using that by transfer simp
 
 lemma unat_p2: "n < LENGTH('a :: len) \<Longrightarrow> unat (2 ^ n :: 'a word) = 2 ^ n"
   by (fact unat_power_lower)
 
 lemma word_div_lt_eq_0:
   "x < y \<Longrightarrow> x div y = 0" for x :: "'a :: len word"
   by transfer simp
 
 lemma word_div_eq_1_iff: "n div m = 1 \<longleftrightarrow> n \<ge> m \<and> unat n < 2 * unat (m :: 'a :: len word)"
   apply (simp only: word_arith_nat_defs word_le_nat_alt word_of_nat_eq_iff flip: nat_div_eq_Suc_0_iff)
   apply (simp flip: unat_div unsigned_take_bit_eq)
   done
 
 lemma shiftl_power:
   "(shiftl1 ^^ x) (y::'a::len word) = 2 ^ x * y"
   apply (induct x)
    apply simp
   apply (simp add: shiftl1_2t)
   done
 
 lemma AND_twice [simp]:
   "(w AND m) AND m = w AND m"
   by (fact and.right_idem)
 
 lemma word_combine_masks:
   "w AND m = z \<Longrightarrow> w AND m' = z' \<Longrightarrow> w AND (m OR m') = (z OR z')"
   for w m m' z z' :: \<open>'a::len word\<close>
   by (simp add: bit.conj_disj_distrib)
 
 lemma p2_gt_0:
   "(0 < (2 ^ n :: 'a :: len word)) = (n < LENGTH('a))"
   by (simp add : word_gt_0 not_le)
 
 lemma uint_2p_alt:
   \<open>n < LENGTH('a::len) \<Longrightarrow> uint ((2::'a::len word) ^ n) = 2 ^ n\<close>
-  using p2_gt_0 [of n, where ?'a = 'a] by (simp add: uint_2p) 
+  using p2_gt_0 [of n, where ?'a = 'a] by (simp add: uint_2p)
 
 lemma p2_eq_0:
   \<open>(2::'a::len word) ^ n = 0 \<longleftrightarrow> LENGTH('a::len) \<le> n\<close>
   by (fact exp_eq_zero_iff)
 
 lemma p2len:
   \<open>(2 :: 'a word) ^ LENGTH('a::len) = 0\<close>
   by simp
 
 lemma neg_mask_is_div:
   "w AND NOT (mask n) = (w div 2^n) * 2^n"
   for w :: \<open>'a::len word\<close>
   by (rule bit_word_eqI)
     (auto simp add: bit_simps simp flip: push_bit_eq_mult drop_bit_eq_div)
 
 lemma neg_mask_is_div':
   "n < size w \<Longrightarrow> w AND NOT (mask n) = ((w div (2 ^ n)) * (2 ^ n))"
   for w :: \<open>'a::len word\<close>
   by (rule neg_mask_is_div)
 
 lemma and_mask_arith:
   "w AND mask n = (w * 2^(size w - n)) div 2^(size w - n)"
   for w :: \<open>'a::len word\<close>
   by (rule bit_word_eqI)
     (auto simp add: bit_simps word_size simp flip: push_bit_eq_mult drop_bit_eq_div)
 
 lemma and_mask_arith':
   "0 < n \<Longrightarrow> w AND mask n = (w * 2^(size w - n)) div 2^(size w - n)"
   for w :: \<open>'a::len word\<close>
   by (rule and_mask_arith)
-  
+
 lemma mask_2pm1: "mask n = 2 ^ n - (1 :: 'a::len word)"
   by (fact mask_eq_decr_exp)
 
 lemma add_mask_fold:
   "x + 2 ^ n - 1 = x + mask n"
   for x :: \<open>'a::len word\<close>
   by (simp add: mask_eq_decr_exp)
 
 lemma word_and_mask_le_2pm1: "w AND mask n \<le> 2 ^ n - 1"
   for w :: \<open>'a::len word\<close>
   by (simp add: mask_2pm1[symmetric] word_and_le1)
 
 lemma is_aligned_AND_less_0:
   "u AND mask n = 0 \<Longrightarrow> v < 2^n \<Longrightarrow> u AND v = 0"
   for u v :: \<open>'a::len word\<close>
   apply (drule less_mask_eq)
   apply (simp flip: take_bit_eq_mask)
   apply (simp add: bit_eq_iff)
   apply (auto simp add: bit_simps)
   done
 
 lemma le_shiftr1:
   \<open>shiftr1 u \<le> shiftr1 v\<close> if \<open>u \<le> v\<close>
 using that proof transfer
   fix k l :: int
   assume \<open>take_bit LENGTH('a) k \<le> take_bit LENGTH('a) l\<close>
   then have \<open>take_bit LENGTH('a) (drop_bit 1 (take_bit LENGTH('a) k))
     \<le> take_bit LENGTH('a) (drop_bit 1 (take_bit LENGTH('a) l))\<close>
     apply (simp add: take_bit_drop_bit min_def)
     apply (simp add: drop_bit_eq_div)
     done
   then show \<open>take_bit LENGTH('a) (take_bit LENGTH('a) k div 2)
     \<le> take_bit LENGTH('a) (take_bit LENGTH('a) l div 2)\<close>
     by (simp add: drop_bit_eq_div)
 qed
 
 lemma and_mask_eq_iff_le_mask:
   \<open>w AND mask n = w \<longleftrightarrow> w \<le> mask n\<close>
   for w :: \<open>'a::len word\<close>
   apply (simp flip: take_bit_eq_mask)
   apply (cases \<open>n \<ge> LENGTH('a)\<close>; transfer)
    apply (simp_all add: not_le min_def)
    apply (simp_all add: mask_eq_exp_minus_1)
   apply auto
    apply (metis take_bit_int_less_exp)
   apply (metis min_def nat_less_le take_bit_int_eq_self_iff take_bit_take_bit)
   done
 
 lemma less_eq_mask_iff_take_bit_eq_self:
   \<open>w \<le> mask n \<longleftrightarrow> take_bit n w = w\<close>
   for w :: \<open>'a::len word\<close>
   by (simp add: and_mask_eq_iff_le_mask take_bit_eq_mask)
 
 lemma NOT_eq:
   "NOT (x :: 'a :: len word) = - x - 1"
   apply (cut_tac x = "x" in word_add_not)
   apply (drule add.commute [THEN trans])
   apply (drule eq_diff_eq [THEN iffD2])
   by simp
 
 lemma NOT_mask: "NOT (mask n :: 'a::len word) = - (2 ^ n)"
   by (simp add : NOT_eq mask_2pm1)
 
 lemma le_m1_iff_lt: "(x > (0 :: 'a :: len word)) = ((y \<le> x - 1) = (y < x))"
   by uint_arith
 
 lemma gt0_iff_gem1:
   \<open>0 < x \<longleftrightarrow> x - 1 < x\<close>
   for x :: \<open>'a::len word\<close>
   by (metis add.right_neutral diff_add_cancel less_irrefl measure_unat unat_arith_simps(2) word_neq_0_conv word_sub_less_iff)
 
 lemma power_2_ge_iff:
   \<open>2 ^ n - (1 :: 'a::len word) < 2 ^ n \<longleftrightarrow> n < LENGTH('a)\<close>
   using gt0_iff_gem1 p2_gt_0 by blast
 
 lemma le_mask_iff_lt_2n:
   "n < len_of TYPE ('a) = (((w :: 'a :: len word) \<le> mask n) = (w < 2 ^ n))"
   unfolding mask_2pm1 by (rule trans [OF p2_gt_0 [THEN sym] le_m1_iff_lt])
 
 lemma mask_lt_2pn:
   \<open>n < LENGTH('a) \<Longrightarrow> mask n < (2 :: 'a::len word) ^ n\<close>
   by (simp add: mask_eq_exp_minus_1 power_2_ge_iff)
 
 lemma word_unat_power:
   "(2 :: 'a :: len word) ^ n = of_nat (2 ^ n)"
   by simp
 
 lemma of_nat_mono_maybe:
   assumes xlt: "x < 2 ^ len_of TYPE ('a)"
   shows   "y < x \<Longrightarrow> of_nat y < (of_nat x :: 'a :: len word)"
   apply (subst word_less_nat_alt)
   apply (subst unat_of_nat)+
   apply (subst mod_less)
    apply (erule order_less_trans [OF _ xlt])
   apply (subst mod_less [OF xlt])
   apply assumption
   done
 
 lemma word_and_max_word:
   fixes a::"'a::len word"
   shows "x = max_word \<Longrightarrow> a AND x = a"
   by simp
 
 lemma word_and_full_mask_simp:
   \<open>x AND mask LENGTH('a) = x\<close> for x :: \<open>'a::len word\<close>
 proof (rule bit_eqI)
   fix n
   assume \<open>2 ^ n \<noteq> (0 :: 'a word)\<close>
   then have \<open>n < LENGTH('a)\<close>
     by simp
   then show \<open>bit (x AND Bit_Operations.mask LENGTH('a)) n \<longleftrightarrow> bit x n\<close>
     by (simp add: bit_and_iff bit_mask_iff)
 qed
 
 lemma of_int_uint:
   "of_int (uint x) = x"
   by (fact word_of_int_uint)
 
 corollary word_plus_and_or_coroll:
   "x AND y = 0 \<Longrightarrow> x + y = x OR y"
   for x y :: \<open>'a::len word\<close>
   using word_plus_and_or[where x=x and y=y]
   by simp
 
 corollary word_plus_and_or_coroll2:
   "(x AND w) + (x AND NOT w) = x"
   for x w :: \<open>'a::len word\<close>
   apply (subst disjunctive_add)
    apply (simp add: bit_simps)
   apply (simp flip: bit.conj_disj_distrib)
   done
 
 lemma nat_mask_eq:
   \<open>nat (mask n) = mask n\<close>
   by (simp add: nat_eq_iff of_nat_mask_eq)
 
 lemma unat_mask_eq:
   \<open>unat (mask n :: 'a::len word) = mask (min LENGTH('a) n)\<close>
   by transfer (simp add: nat_mask_eq)
 
 lemma word_plus_mono_left:
   fixes x :: "'a :: len word"
   shows "\<lbrakk>y \<le> z; x \<le> x + z\<rbrakk> \<Longrightarrow> y + x \<le> z + x"
   by unat_arith
 
 lemma less_Suc_unat_less_bound:
   "n < Suc (unat (x :: 'a :: len word)) \<Longrightarrow> n < 2 ^ LENGTH('a)"
   by (auto elim!: order_less_le_trans intro: Suc_leI)
 
 lemma up_ucast_inj:
   "\<lbrakk> ucast x = (ucast y::'b::len word); LENGTH('a) \<le> len_of TYPE ('b) \<rbrakk> \<Longrightarrow> x = (y::'a::len word)"
   by transfer (simp add: min_def split: if_splits)
 
 lemmas ucast_up_inj = up_ucast_inj
 
 lemma up_ucast_inj_eq:
   "LENGTH('a) \<le> len_of TYPE ('b) \<Longrightarrow> (ucast x = (ucast y::'b::len word)) = (x = (y::'a::len word))"
   by (fastforce dest: up_ucast_inj)
 
 lemma no_plus_overflow_neg:
   "(x :: 'a :: len word) < -y \<Longrightarrow> x \<le> x + y"
   by (metis diff_minus_eq_add less_imp_le sub_wrap_lt)
 
 lemma ucast_ucast_eq:
   "\<lbrakk> ucast x = (ucast (ucast y::'a word)::'c::len word); LENGTH('a) \<le> LENGTH('b);
      LENGTH('b) \<le> LENGTH('c) \<rbrakk> \<Longrightarrow>
    x = ucast y" for x :: "'a::len word" and y :: "'b::len word"
   apply transfer
   apply (cases \<open>LENGTH('c) = LENGTH('a)\<close>)
    apply (auto simp add: min_def)
   done
 
 lemma ucast_0_I:
   "x = 0 \<Longrightarrow> ucast x = 0"
   by simp
 
 lemma word_add_offset_less:
   fixes x :: "'a :: len word"
   assumes yv: "y < 2 ^ n"
   and     xv: "x < 2 ^ m"
   and     mnv: "sz < LENGTH('a :: len)"
   and    xv': "x < 2 ^ (LENGTH('a :: len) - n)"
   and     mn: "sz = m + n"
   shows   "x * 2 ^ n + y < 2 ^ sz"
 proof (subst mn)
   from mnv mn have nv: "n < LENGTH('a)" and mv: "m < LENGTH('a)"  by auto
 
   have uy: "unat y < 2 ^ n"
     by (rule order_less_le_trans [OF unat_mono [OF yv] order_eq_refl],
         rule unat_power_lower[OF nv])
 
   have ux: "unat x < 2 ^ m"
     by (rule order_less_le_trans [OF unat_mono [OF xv] order_eq_refl],
         rule unat_power_lower[OF mv])
 
   then show "x * 2 ^ n + y < 2 ^ (m + n)" using ux uy nv mnv xv'
     apply (subst word_less_nat_alt)
     apply (subst unat_word_ariths)+
     apply (subst mod_less)
      apply simp
      apply (subst mult.commute)
      apply (rule nat_less_power_trans [OF _ order_less_imp_le [OF nv]])
      apply (rule order_less_le_trans [OF unat_mono [OF xv']])
      apply (cases "n = 0"; simp)
     apply (subst unat_power_lower[OF nv])
     apply (subst mod_less)
      apply (erule order_less_le_trans [OF nat_add_offset_less], assumption)
       apply (rule mn)
      apply simp
     apply (simp add: mn mnv)
     apply (erule nat_add_offset_less; simp)
     done
 qed
 
 lemma word_less_power_trans:
   fixes n :: "'a :: len word"
   assumes nv: "n < 2 ^ (m - k)"
   and     kv: "k \<le> m"
   and     mv: "m < len_of TYPE ('a)"
   shows "2 ^ k * n < 2 ^ m"
   using nv kv mv
   apply -
   apply (subst word_less_nat_alt)
   apply (subst unat_word_ariths)
   apply (subst mod_less)
    apply simp
    apply (rule nat_less_power_trans)
     apply (erule order_less_trans [OF unat_mono])
     apply simp
    apply simp
   apply simp
   apply (rule nat_less_power_trans)
    apply (subst unat_power_lower[where 'a = 'a, symmetric])
     apply simp
    apply (erule unat_mono)
   apply simp
   done
 
 lemma  word_less_power_trans2:
   fixes n :: "'a::len word"
   shows "\<lbrakk>n < 2 ^ (m - k); k \<le> m; m < LENGTH('a)\<rbrakk> \<Longrightarrow> n * 2 ^ k < 2 ^ m"
   by (subst field_simps, rule word_less_power_trans)
 
 lemma Suc_unat_diff_1:
   fixes x :: "'a :: len word"
   assumes lt: "1 \<le> x"
   shows "Suc (unat (x - 1)) = unat x"
 proof -
   have "0 < unat x"
     by (rule order_less_le_trans [where y = 1], simp, subst unat_1 [symmetric],
         rule iffD1 [OF word_le_nat_alt lt])
 
   then show ?thesis
     by ((subst unat_sub [OF lt])+, simp only:  unat_1)
 qed
 
 lemma word_eq_unatI:
   \<open>v = w\<close> if \<open>unat v = unat w\<close>
   using that by transfer (simp add: nat_eq_iff)
 
 lemma word_div_sub:
   fixes x :: "'a :: len word"
   assumes yx: "y \<le> x"
   and     y0: "0 < y"
   shows "(x - y) div y = x div y - 1"
   apply (rule word_eq_unatI)
   apply (subst unat_div)
   apply (subst unat_sub [OF yx])
   apply (subst unat_sub)
    apply (subst word_le_nat_alt)
    apply (subst unat_div)
    apply (subst le_div_geq)
      apply (rule order_le_less_trans [OF _ unat_mono [OF y0]])
      apply simp
     apply (subst word_le_nat_alt [symmetric], rule yx)
    apply simp
   apply (subst unat_div)
   apply (subst le_div_geq [OF _ iffD1 [OF word_le_nat_alt yx]])
    apply (rule order_le_less_trans [OF _ unat_mono [OF y0]])
    apply simp
   apply simp
   done
 
 lemma word_mult_less_mono1:
   fixes i :: "'a :: len word"
   assumes ij: "i < j"
   and    knz: "0 < k"
   and    ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
   shows  "i * k < j * k"
 proof -
   from ij ujk knz have jk: "unat i * unat k < 2 ^ len_of TYPE ('a)"
     by (auto intro: order_less_subst2 simp: word_less_nat_alt elim: mult_less_mono1)
 
   then show ?thesis using ujk knz ij
     by (auto simp: word_less_nat_alt iffD1 [OF unat_mult_lem])
 qed
 
 lemma word_mult_less_dest:
   fixes i :: "'a :: len word"
   assumes ij: "i * k < j * k"
   and    uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
   and    ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
   shows  "i < j"
   using uik ujk ij
   by (auto simp: word_less_nat_alt iffD1 [OF unat_mult_lem] elim: mult_less_mono1)
 
 lemma word_mult_less_cancel:
   fixes k :: "'a :: len word"
   assumes knz: "0 < k"
   and    uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
   and    ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
   shows "(i * k < j * k) = (i < j)"
   by (rule iffI [OF word_mult_less_dest [OF _ uik ujk] word_mult_less_mono1 [OF _ knz ujk]])
 
 lemma Suc_div_unat_helper:
   assumes szv: "sz < LENGTH('a :: len)"
   and   usszv: "us \<le> sz"
   shows "2 ^ (sz - us) = Suc (unat (((2::'a :: len word) ^ sz - 1) div 2 ^ us))"
 proof -
   note usv = order_le_less_trans [OF usszv szv]
 
   from usszv obtain q where qv: "sz = us + q" by (auto simp: le_iff_add)
 
   have "Suc (unat (((2:: 'a word) ^ sz - 1) div 2 ^ us)) =
     (2 ^ us + unat ((2:: 'a word) ^ sz - 1)) div 2 ^ us"
     apply (subst unat_div unat_power_lower[OF usv])+
     apply (subst div_add_self1, simp+)
     done
 
   also have "\<dots> = ((2 ^ us - 1) + 2 ^ sz) div 2 ^ us" using szv
     by (simp add: unat_minus_one)
 
   also have "\<dots> = 2 ^ q + ((2 ^ us - 1) div 2 ^ us)"
     apply (subst qv)
     apply (subst power_add)
     apply (subst div_mult_self2; simp)
     done
 
   also have "\<dots> = 2 ^ (sz - us)" using qv by simp
 
   finally show ?thesis ..
 qed
 
 lemma enum_word_nth_eq:
   \<open>(Enum.enum :: 'a::len word list) ! n = word_of_nat n\<close>
     if \<open>n < 2 ^ LENGTH('a)\<close>
     for n
   using that by (simp add: enum_word_def)
 
 lemma length_enum_word_eq:
   \<open>length (Enum.enum :: 'a::len word list) = 2 ^ LENGTH('a)\<close>
   by (simp add: enum_word_def)
 
 lemma unat_lt2p [iff]:
   \<open>unat x < 2 ^ LENGTH('a)\<close> for x :: \<open>'a::len word\<close>
   by transfer simp
 
 lemma of_nat_unat [simp]:
   "of_nat \<circ> unat = id"
   by (rule ext, simp)
 
 lemma Suc_unat_minus_one [simp]:
   "x \<noteq> 0 \<Longrightarrow> Suc (unat (x - 1)) = unat x"
   by (metis Suc_diff_1 unat_gt_0 unat_minus_one)
 
 lemma word_add_le_dest:
   fixes i :: "'a :: len word"
   assumes le: "i + k \<le> j + k"
   and    uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
   and    ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
   shows  "i \<le> j"
   using uik ujk le
   by (auto simp: word_le_nat_alt iffD1 [OF unat_add_lem] elim: add_le_mono1)
 
 lemma word_add_le_mono1:
   fixes i :: "'a :: len word"
   assumes ij: "i \<le> j"
   and    ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
   shows  "i + k \<le> j + k"
 proof -
   from ij ujk have jk: "unat i + unat k < 2 ^ len_of TYPE ('a)"
     by (auto elim: order_le_less_subst2 simp: word_le_nat_alt elim: add_le_mono1)
 
   then show ?thesis using ujk ij
     by (auto simp: word_le_nat_alt iffD1 [OF unat_add_lem])
 qed
 
 lemma word_add_le_mono2:
   fixes i :: "'a :: len word"
   shows "\<lbrakk>i \<le> j; unat j + unat k < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow> k + i \<le> k + j"
   by (subst field_simps, subst field_simps, erule (1) word_add_le_mono1)
 
 lemma word_add_le_iff:
   fixes i :: "'a :: len word"
   assumes uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
   and     ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
   shows  "(i + k \<le> j + k) = (i \<le> j)"
 proof
   assume "i \<le> j"
   show "i + k \<le> j + k" by (rule word_add_le_mono1) fact+
 next
   assume "i + k \<le> j + k"
   show "i \<le> j" by (rule word_add_le_dest) fact+
 qed
 
 lemma word_add_less_mono1:
   fixes i :: "'a :: len word"
   assumes ij: "i < j"
   and    ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
   shows  "i + k < j + k"
 proof -
   from ij ujk have jk: "unat i + unat k < 2 ^ len_of TYPE ('a)"
     by (auto elim: order_le_less_subst2 simp: word_less_nat_alt elim: add_less_mono1)
 
   then show ?thesis using ujk ij
     by (auto simp: word_less_nat_alt iffD1 [OF unat_add_lem])
 qed
 
 lemma word_add_less_dest:
   fixes i :: "'a :: len word"
   assumes le: "i + k < j + k"
   and    uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
   and    ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
   shows  "i < j"
   using uik ujk le
   by (auto simp: word_less_nat_alt iffD1 [OF unat_add_lem] elim: add_less_mono1)
 
 lemma word_add_less_iff:
   fixes i :: "'a :: len word"
   assumes uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
   and     ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
   shows  "(i + k < j + k) = (i < j)"
 proof
   assume "i < j"
   show "i + k < j + k" by (rule word_add_less_mono1) fact+
 next
   assume "i + k < j + k"
   show "i < j" by (rule word_add_less_dest) fact+
 qed
 
 lemma word_mult_less_iff:
   fixes i :: "'a :: len word"
   assumes knz: "0 < k"
   and     uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
   and     ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
   shows  "(i * k < j * k) = (i < j)"
   using assms by (rule word_mult_less_cancel)
 
 lemma word_le_imp_diff_le:
   fixes n :: "'a::len word"
   shows "\<lbrakk>k \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> n - k \<le> m"
   by (auto simp: unat_sub word_le_nat_alt)
 
 lemma word_less_imp_diff_less:
   fixes n :: "'a::len word"
   shows "\<lbrakk>k \<le> n; n < m\<rbrakk> \<Longrightarrow> n - k < m"
   by (clarsimp simp: unat_sub word_less_nat_alt
              intro!: less_imp_diff_less)
 
 lemma word_mult_le_mono1:
   fixes i :: "'a :: len word"
   assumes ij: "i \<le> j"
   and    knz: "0 < k"
   and    ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
   shows  "i * k \<le> j * k"
 proof -
   from ij ujk knz have jk: "unat i * unat k < 2 ^ len_of TYPE ('a)"
     by (auto elim: order_le_less_subst2 simp: word_le_nat_alt elim: mult_le_mono1)
 
   then show ?thesis using ujk knz ij
     by (auto simp: word_le_nat_alt iffD1 [OF unat_mult_lem])
 qed
 
 lemma word_mult_le_iff:
   fixes i :: "'a :: len word"
   assumes knz: "0 < k"
   and     uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
   and     ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
   shows  "(i * k \<le> j * k) = (i \<le> j)"
 proof
   assume "i \<le> j"
   show "i * k \<le> j * k" by (rule word_mult_le_mono1) fact+
 next
   assume p: "i * k \<le> j * k"
 
   have "0 < unat k" using knz by (simp add: word_less_nat_alt)
   then show "i \<le> j" using p
     by (clarsimp simp: word_le_nat_alt iffD1 [OF unat_mult_lem uik]
       iffD1 [OF unat_mult_lem ujk])
 qed
 
 lemma word_diff_less:
   fixes n :: "'a :: len word"
   shows "\<lbrakk>0 < n; 0 < m; n \<le> m\<rbrakk> \<Longrightarrow> m - n < m"
   apply (subst word_less_nat_alt)
   apply (subst unat_sub)
    apply assumption
   apply (rule diff_less)
    apply (simp_all add: word_less_nat_alt)
   done
 
 lemma word_add_increasing:
   fixes x :: "'a :: len word"
   shows "\<lbrakk> p + w \<le> x; p \<le> p + w \<rbrakk> \<Longrightarrow> p \<le> x"
   by unat_arith
 
 lemma word_random:
   fixes x :: "'a :: len word"
   shows "\<lbrakk> p \<le> p + x'; x \<le> x' \<rbrakk> \<Longrightarrow> p \<le> p + x"
   by unat_arith
 
 lemma word_sub_mono:
   "\<lbrakk> a \<le> c; d \<le> b; a - b \<le> a; c - d \<le> c \<rbrakk>
     \<Longrightarrow> (a - b) \<le> (c - d :: 'a :: len word)"
   by unat_arith
 
 lemma power_not_zero:
   "n < LENGTH('a::len) \<Longrightarrow> (2 :: 'a word) ^ n \<noteq> 0"
   by (metis p2_gt_0 word_neq_0_conv)
 
 lemma word_gt_a_gt_0:
   "a < n \<Longrightarrow> (0 :: 'a::len word) < n"
   apply (case_tac "n = 0")
    apply clarsimp
   apply (clarsimp simp: word_neq_0_conv)
   done
 
 lemma word_power_less_1 [simp]:
   "sz < LENGTH('a::len) \<Longrightarrow> (2::'a word) ^ sz - 1 < 2 ^ sz"
   apply (simp add: word_less_nat_alt)
   apply (subst unat_minus_one)
   apply simp_all
   done
 
 lemma word_sub_1_le:
   "x \<noteq> 0 \<Longrightarrow> x - 1 \<le> (x :: ('a :: len) word)"
   apply (subst no_ulen_sub)
   apply simp
   apply (cases "uint x = 0")
    apply (simp add: uint_0_iff)
   apply (insert uint_ge_0[where x=x])
   apply arith
   done
 
 lemma push_bit_word_eq_nonzero:
   \<open>push_bit n w \<noteq> 0\<close> if \<open>w < 2 ^ m\<close> \<open>m + n < LENGTH('a)\<close> \<open>w \<noteq> 0\<close>
     for w :: \<open>'a::len word\<close>
   using that
   apply (simp only: word_neq_0_conv word_less_nat_alt
                     mod_0 unat_word_ariths
                     unat_power_lower word_le_nat_alt)
   apply (metis add_diff_cancel_right' gr0I gr_implies_not0 less_or_eq_imp_le min_def push_bit_eq_0_iff take_bit_nat_eq_self_iff take_bit_push_bit take_bit_take_bit unsigned_push_bit_eq)
   done
 
 lemma unat_less_power:
   fixes k :: "'a::len word"
   assumes szv: "sz < LENGTH('a)"
   and     kv:  "k < 2 ^ sz"
   shows   "unat k < 2 ^ sz"
   using szv unat_mono [OF kv] by simp
 
 lemma unat_mult_power_lem:
   assumes kv: "k < 2 ^ (LENGTH('a::len) - sz)"
   shows "unat (2 ^ sz * of_nat k :: (('a::len) word)) = 2 ^ sz * k"
 proof (cases \<open>sz < LENGTH('a)\<close>)
   case True
   with assms show ?thesis
     by (simp add: unat_word_ariths take_bit_eq_mod mod_simps)
       (simp add: take_bit_nat_eq_self_iff nat_less_power_trans flip: take_bit_eq_mod)
 next
   case False
   with assms show ?thesis
     by simp
 qed
 
 lemma word_plus_mcs_4:
   "\<lbrakk>v + x \<le> w + x; x \<le> v + x\<rbrakk> \<Longrightarrow> v \<le> (w::'a::len word)"
   by uint_arith
 
 lemma word_plus_mcs_3:
   "\<lbrakk>v \<le> w; x \<le> w + x\<rbrakk> \<Longrightarrow> v + x \<le> w + (x::'a::len word)"
   by unat_arith
 
 lemma word_le_minus_one_leq:
   "x < y \<Longrightarrow> x \<le> y - 1" for x :: "'a :: len word"
-  by transfer (metis le_less_trans less_irrefl take_bit_decr_eq take_bit_nonnegative zle_diff1_eq) 
+  by transfer (metis le_less_trans less_irrefl take_bit_decr_eq take_bit_nonnegative zle_diff1_eq)
 
 lemma word_less_sub_le[simp]:
   fixes x :: "'a :: len word"
   assumes nv: "n < LENGTH('a)"
   shows "(x \<le> 2 ^ n - 1) = (x < 2 ^ n)"
   using le_less_trans word_le_minus_one_leq nv power_2_ge_iff by blast
 
 lemma unat_of_nat_len:
   "x < 2 ^ LENGTH('a) \<Longrightarrow> unat (of_nat x :: 'a::len word) = x"
   by (simp add: take_bit_nat_eq_self_iff)
 
 lemma unat_of_nat_eq:
   "x < 2 ^ LENGTH('a) \<Longrightarrow> unat (of_nat x ::'a::len word) = x"
   by (rule unat_of_nat_len)
 
 lemma unat_eq_of_nat:
   "n < 2 ^ LENGTH('a) \<Longrightarrow> (unat (x :: 'a::len word) = n) = (x = of_nat n)"
   by transfer
     (auto simp add: take_bit_of_nat nat_eq_iff take_bit_nat_eq_self_iff intro: sym)
 
 lemma alignUp_div_helper:
   fixes a :: "'a::len word"
   assumes kv: "k < 2 ^ (LENGTH('a) - n)"
   and     xk: "x = 2 ^ n * of_nat k"
   and    le: "a \<le> x"
   and    sz: "n < LENGTH('a)"
   and   anz: "a mod 2 ^ n \<noteq> 0"
   shows "a div 2 ^ n < of_nat k"
 proof -
   have kn: "unat (of_nat k :: 'a word) * unat ((2::'a word) ^ n) < 2 ^ LENGTH('a)"
     using xk kv sz
     apply (subst unat_of_nat_eq)
      apply (erule order_less_le_trans)
      apply simp
     apply (subst unat_power_lower, simp)
     apply (subst mult.commute)
     apply (rule nat_less_power_trans)
      apply simp
     apply simp
     done
 
   have "unat a div 2 ^ n * 2 ^ n \<noteq> unat a"
   proof -
     have "unat a = unat a div 2 ^ n * 2 ^ n + unat a mod 2 ^ n"
       by (simp add: div_mult_mod_eq)
     also have "\<dots> \<noteq> unat a div 2 ^ n * 2 ^ n" using sz anz
       by (simp add: unat_arith_simps)
     finally show ?thesis ..
   qed
 
   then have "a div 2 ^ n * 2 ^ n < a" using sz anz
     apply (subst word_less_nat_alt)
     apply (subst unat_word_ariths)
     apply (subst unat_div)
     apply simp
     apply (rule order_le_less_trans [OF mod_less_eq_dividend])
     apply (erule order_le_neq_trans [OF div_mult_le])
     done
 
   also from xk le have "\<dots> \<le> of_nat k * 2 ^ n" by (simp add: field_simps)
   finally show ?thesis using sz kv
     apply -
     apply (erule word_mult_less_dest [OF _ _ kn])
     apply (simp add: unat_div)
     apply (rule order_le_less_trans [OF div_mult_le])
     apply (rule unat_lt2p)
     done
 qed
 
 lemma mask_out_sub_mask:
   "(x AND NOT (mask n)) = x - (x AND (mask n))"
   for x :: \<open>'a::len word\<close>
   by (simp add: field_simps word_plus_and_or_coroll2)
 
 lemma subtract_mask:
   "p - (p AND mask n) = (p AND NOT (mask n))"
   "p - (p AND NOT (mask n)) = (p AND mask n)"
   for p :: \<open>'a::len word\<close>
   by (simp add: field_simps word_plus_and_or_coroll2)+
 
 lemma take_bit_word_eq_self_iff:
   \<open>take_bit n w = w \<longleftrightarrow> n \<ge> LENGTH('a) \<or> w < 2 ^ n\<close>
   for w :: \<open>'a::len word\<close>
   using take_bit_int_eq_self_iff [of n \<open>take_bit LENGTH('a) (uint w)\<close>]
   by (transfer fixing: n) auto
 
 lemma word_power_increasing:
   assumes x: "2 ^ x < (2 ^ y::'a::len word)" "x < LENGTH('a::len)" "y < LENGTH('a::len)"
   shows "x < y" using x
   using assms by transfer simp
 
 lemma mask_twice:
   "(x AND mask n) AND mask m = x AND mask (min m n)"
   for x :: \<open>'a::len word\<close>
   by (simp flip: take_bit_eq_mask)
 
 lemma plus_one_helper[elim!]:
   "x < n + (1 :: 'a :: len word) \<Longrightarrow> x \<le> n"
   apply (simp add: word_less_nat_alt word_le_nat_alt field_simps)
   apply (case_tac "1 + n = 0")
    apply simp_all
   apply (subst(asm) unatSuc, assumption)
   apply arith
   done
 
 lemma plus_one_helper2:
   "\<lbrakk> x \<le> n; n + 1 \<noteq> 0 \<rbrakk> \<Longrightarrow> x < n + (1 :: 'a :: len word)"
   by (simp add: word_less_nat_alt word_le_nat_alt field_simps
                 unatSuc)
 
 lemma less_x_plus_1:
   fixes x :: "'a :: len word" shows
   "x \<noteq> 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:
   \<open>{..< k + 1} = {..k}\<close> if \<open>k \<noteq> - 1\<close> for k :: \<open>'a::len word\<close>
   using that by (simp add: lessThan_def atMost_def word_Suc_leq)
 
 lemma word_atLeastLessThan_Suc_atLeastAtMost:
   \<open>{l ..< u + 1} = {l..u}\<close> if \<open>u \<noteq> - 1\<close> for l :: \<open>'a::len word\<close>
   using that by (simp add: atLeastAtMost_def atLeastLessThan_def word_lessThan_Suc_atMost)
 
 lemma word_atLeastAtMost_Suc_greaterThanAtMost:
   \<open>{m<..u} = {m + 1..u}\<close> if \<open>m \<noteq> - 1\<close> for m :: \<open>'a::len word\<close>
   using that by (simp add: greaterThanAtMost_def greaterThan_def atLeastAtMost_def atLeast_def word_Suc_le)
 
 lemma word_atLeastLessThan_Suc_atLeastAtMost_union:
   fixes l::"'a::len word"
   assumes "m \<noteq> 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 max_word_less_eq_iff [simp]:
   \<open>- 1 \<le> w \<longleftrightarrow> w = - 1\<close> for w :: \<open>'a::len word\<close>
   by (fact word_order.extremum_unique)
 
 lemma word_or_zero:
   "(a OR b = 0) = (a = 0 \<and> b = 0)"
   for a b :: \<open>'a::len word\<close>
   by (fact or_eq_0_iff)
 
 lemma word_2p_mult_inc:
   assumes x: "2 * 2 ^ n < (2::'a::len word) * 2 ^ m"
   assumes suc_n: "Suc n < LENGTH('a::len)"
   shows "2^n < (2::'a::len word)^m"
   by (smt suc_n le_less_trans lessI nat_less_le nat_mult_less_cancel_disj p2_gt_0
           power_Suc power_Suc unat_power_lower word_less_nat_alt x)
 
 lemma power_overflow:
   "n \<ge> LENGTH('a) \<Longrightarrow> 2 ^ n = (0 :: 'a::len word)"
   by simp
 
 lemmas extra_sle_sless_unfolds [simp] =
     word_sle_eq[where a=0 and b=1]
     word_sle_eq[where a=0 and b="numeral n"]
     word_sle_eq[where a=1 and b=0]
     word_sle_eq[where a=1 and b="numeral n"]
     word_sle_eq[where a="numeral n" and b=0]
     word_sle_eq[where a="numeral n" and b=1]
     word_sless_alt[where a=0 and b=1]
     word_sless_alt[where a=0 and b="numeral n"]
     word_sless_alt[where a=1 and b=0]
     word_sless_alt[where a=1 and b="numeral n"]
     word_sless_alt[where a="numeral n" and b=0]
     word_sless_alt[where a="numeral n" and b=1]
   for n
 
 lemma word_sint_1:
   "sint (1::'a::len word) = (if LENGTH('a) = 1 then -1 else 1)"
   by (fact signed_1)
 
 lemma ucast_of_nat:
   "is_down (ucast :: 'a :: len word \<Rightarrow> 'b :: len word)
     \<Longrightarrow> ucast (of_nat n :: 'a word) = (of_nat n :: 'b word)"
   by transfer simp
 
 lemma scast_1':
   "(scast (1::'a::len word) :: 'b::len word) =
    (word_of_int (signed_take_bit (LENGTH('a::len) - Suc 0) (1::int)))"
   by transfer simp
 
 lemma scast_1:
   "(scast (1::'a::len word) :: 'b::len word) = (if LENGTH('a) = 1 then -1 else 1)"
   by (fact signed_1)
 
 lemma unat_minus_one_word:
   "unat (-1 :: 'a :: len word) = 2 ^ LENGTH('a) - 1"
   apply (simp only: flip: mask_eq_exp_minus_1)
   apply transfer
   apply (simp add: take_bit_minus_one_eq_mask nat_mask_eq)
   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 two_power_increasing:
   "\<lbrakk> n \<le> m; m < LENGTH('a) \<rbrakk> \<Longrightarrow> (2 :: 'a :: len word) ^ n \<le> 2 ^ m"
   by (simp add: word_le_nat_alt)
 
 lemma word_leq_le_minus_one:
   "\<lbrakk> x \<le> y; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x - 1 < (y :: 'a :: len word)"
   apply (simp add: word_less_nat_alt word_le_nat_alt)
   apply (subst unat_minus_one)
    apply assumption
   apply (cases "unat x")
    apply (simp add: unat_eq_zero)
   apply arith
   done
 
 lemma neg_mask_combine:
   "NOT(mask a) AND NOT(mask b) = NOT(mask (max a b) :: 'a::len word)"
   by (rule bit_word_eqI) (auto simp add: bit_simps)
 
 lemma neg_mask_twice:
   "x AND NOT(mask n) AND NOT(mask m) = x AND NOT(mask (max n m))"
   for x :: \<open>'a::len word\<close>
   by (rule bit_word_eqI) (auto simp add: bit_simps)
 
 lemma multiple_mask_trivia:
   "n \<ge> m \<Longrightarrow> (x AND NOT(mask n)) + (x AND mask n AND NOT(mask m)) = x AND NOT(mask m)"
   for x :: \<open>'a::len word\<close>
   apply (rule trans[rotated], rule_tac w="mask n" in word_plus_and_or_coroll2)
   apply (simp add: word_bw_assocs word_bw_comms word_bw_lcs neg_mask_twice
                    max_absorb2)
   done
 
 lemma word_of_nat_less:
   "\<lbrakk> n < unat x \<rbrakk> \<Longrightarrow> of_nat n < x"
   apply (simp add: word_less_nat_alt)
   apply (erule order_le_less_trans[rotated])
   apply (simp add: take_bit_eq_mod)
   done
 
 lemma unat_mask:
   "unat (mask n :: 'a :: len word) = 2 ^ (min n (LENGTH('a))) - 1"
   apply (subst min.commute)
   apply (simp add: mask_eq_decr_exp not_less min_def  split: if_split_asm)
   apply (intro conjI impI)
    apply (simp add: unat_sub_if_size)
    apply (simp add: power_overflow word_size)
   apply (simp add: unat_sub_if_size)
   done
 
 lemma mask_over_length:
   "LENGTH('a) \<le> n \<Longrightarrow> mask n = (-1::'a::len word)"
   by (simp add: mask_eq_decr_exp)
 
 lemma Suc_2p_unat_mask:
   "n < LENGTH('a) \<Longrightarrow> Suc (2 ^ n * k + unat (mask n :: 'a::len word)) = 2 ^ n * (k+1)"
   by (simp add: unat_mask)
 
 lemma sint_of_nat_ge_zero:
   "x < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> sint (of_nat x :: 'a :: len word) \<ge> 0"
   by (simp add: bit_iff_odd)
 
 lemma int_eq_sint:
   "x < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> sint (of_nat x :: 'a :: len word) = int x"
   apply transfer
   apply (rule signed_take_bit_int_eq_self)
    apply simp_all
   apply (metis negative_zle numeral_power_eq_of_nat_cancel_iff)
   done
 
 lemma sint_of_nat_le:
   "\<lbrakk> b < 2 ^ (LENGTH('a) - 1); a \<le> b \<rbrakk>
    \<Longrightarrow> sint (of_nat a :: 'a :: len word) \<le> sint (of_nat b :: 'a :: len word)"
   apply (cases \<open>LENGTH('a)\<close>)
   apply simp_all
   apply transfer
   apply (subst signed_take_bit_eq_if_positive)
    apply (simp add: bit_simps)
   apply (metis bit_take_bit_iff nat_less_le order_less_le_trans take_bit_nat_eq_self_iff)
   apply (subst signed_take_bit_eq_if_positive)
     apply (simp add: bit_simps)
   apply (metis bit_take_bit_iff nat_less_le take_bit_nat_eq_self_iff)
     apply (simp flip: of_nat_take_bit add: take_bit_nat_eq_self)
   done
 
 lemma word_le_not_less:
   "((b::'a::len word) \<le> a) = (\<not>(a < b))"
   by fastforce
 
 lemma less_is_non_zero_p1:
   fixes a :: "'a :: len word"
   shows "a < k \<Longrightarrow> a + 1 \<noteq> 0"
   apply (erule contrapos_pn)
   apply (drule max_word_wrap)
   apply (simp add: not_less)
   done
 
 lemma unat_add_lem':
   "(unat x + unat y < 2 ^ LENGTH('a)) \<Longrightarrow>
     (unat (x + y :: 'a :: len word) = unat x + unat y)"
   by (subst unat_add_lem[symmetric], assumption)
 
 lemma word_less_two_pow_divI:
   "\<lbrakk> (x :: 'a::len word) < 2 ^ (n - m); m \<le> n; n < LENGTH('a) \<rbrakk> \<Longrightarrow> x < 2 ^ n div 2 ^ m"
   apply (simp add: word_less_nat_alt)
   apply (subst unat_word_ariths)
   apply (subst mod_less)
    apply (rule order_le_less_trans [OF div_le_dividend])
    apply (rule unat_lt2p)
   apply (simp add: power_sub)
   done
 
 lemma word_less_two_pow_divD:
   "\<lbrakk> (x :: 'a::len word) < 2 ^ n div 2 ^ m \<rbrakk>
      \<Longrightarrow> n \<ge> m \<and> (x < 2 ^ (n - m))"
   apply (cases "n < LENGTH('a)")
    apply (cases "m < LENGTH('a)")
     apply (simp add: word_less_nat_alt)
     apply (subst(asm) unat_word_ariths)
     apply (subst(asm) mod_less)
      apply (rule order_le_less_trans [OF div_le_dividend])
      apply (rule unat_lt2p)
     apply (clarsimp dest!: less_two_pow_divD)
    apply (simp add: power_overflow)
    apply (simp add: word_div_def)
   apply (simp add: power_overflow word_div_def)
   done
 
 lemma of_nat_less_two_pow_div_set:
   "\<lbrakk> n < LENGTH('a) \<rbrakk> \<Longrightarrow>
    {x. x < (2 ^ n div 2 ^ m :: 'a::len word)}
       = of_nat ` {k. k < 2 ^ n div 2 ^ m}"
   apply (simp add: image_def)
   apply (safe dest!: word_less_two_pow_divD less_two_pow_divD
              intro!: word_less_two_pow_divI)
    apply (rule_tac x="unat x" in exI)
    apply (simp add: power_sub[symmetric])
    apply (subst unat_power_lower[symmetric, where 'a='a])
     apply simp
    apply (erule unat_mono)
   apply (subst word_unat_power)
   apply (rule of_nat_mono_maybe)
    apply (rule power_strict_increasing)
     apply simp
    apply simp
   apply assumption
   done
 
 lemma ucast_less:
   "LENGTH('b) < LENGTH('a) \<Longrightarrow>
    (ucast (x :: 'b :: len word) :: ('a :: len word)) < 2 ^ LENGTH('b)"
   by transfer simp
 
 lemma ucast_range_less:
   "LENGTH('a :: len) < LENGTH('b :: len) \<Longrightarrow>
    range (ucast :: 'a word \<Rightarrow> 'b word) = {x. x < 2 ^ len_of TYPE ('a)}"
   apply safe
    apply (erule ucast_less)
   apply (simp add: image_def)
   apply (rule_tac x="ucast x" in exI)
   apply (rule bit_word_eqI)
   apply (auto simp add: bit_simps)
   apply (metis bit_take_bit_iff less_mask_eq not_less take_bit_eq_mask)
   done
 
 lemma word_power_less_diff:
   "\<lbrakk>2 ^ n * q < (2::'a::len word) ^ m; q < 2 ^ (LENGTH('a) - n)\<rbrakk> \<Longrightarrow> q < 2 ^ (m - n)"
   apply (case_tac "m \<ge> LENGTH('a)")
    apply (simp add: power_overflow)
   apply (case_tac "n \<ge> LENGTH('a)")
    apply (simp add: power_overflow)
   apply (cases "n = 0")
    apply simp
   apply (subst word_less_nat_alt)
   apply (subst unat_power_lower)
    apply simp
   apply (rule nat_power_less_diff)
   apply (simp add: word_less_nat_alt)
   apply (subst (asm) iffD1 [OF unat_mult_lem])
    apply (simp add:nat_less_power_trans)
   apply simp
   done
 
 lemma word_less_sub_1:
   "x < (y :: 'a :: len word) \<Longrightarrow> x \<le> y - 1"
   by (fact word_le_minus_one_leq)
 
 lemma word_sub_mono2:
   "\<lbrakk> a + b \<le> c + d; c \<le> a; b \<le> a + b; d \<le> c + d \<rbrakk>
     \<Longrightarrow> b \<le> (d :: 'a :: len word)"
   apply (drule(1) word_sub_mono)
     apply simp
    apply simp
   apply simp
   done
 
 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 two_power_eq:
   "\<lbrakk>n < LENGTH('a); m < LENGTH('a)\<rbrakk>
    \<Longrightarrow> ((2::'a::len word) ^ n = 2 ^ m) = (n = m)"
   apply safe
   apply (rule order_antisym)
    apply (simp add: power_le_mono[where 'a='a])+
   done
 
 lemma unat_less_helper:
   "x < of_nat n \<Longrightarrow> unat x < n"
   apply (simp add: word_less_nat_alt)
   apply (erule order_less_le_trans)
   apply (simp add: take_bit_eq_mod)
   done
 
 lemma nat_uint_less_helper:
   "nat (uint y) = z \<Longrightarrow> x < y \<Longrightarrow> nat (uint x) < z"
   apply (erule subst)
   apply (subst unat_eq_nat_uint [symmetric])
   apply (subst unat_eq_nat_uint [symmetric])
   by (simp add: unat_mono)
 
 lemma of_nat_0:
   "\<lbrakk>of_nat n = (0::'a::len word); n < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow> n = 0"
   by transfer (simp add: take_bit_eq_mod)
 
 lemma of_nat_inj:
   "\<lbrakk>x < 2 ^ LENGTH('a); y < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow>
    (of_nat x = (of_nat y :: 'a :: len word)) = (x = y)"
   by (metis unat_of_nat_len)
 
 lemma div_to_mult_word_lt:
   "\<lbrakk> (x :: 'a :: len word) \<le> y div z \<rbrakk> \<Longrightarrow> x * z \<le> y"
   apply (cases "z = 0")
    apply simp
   apply (simp add: word_neq_0_conv)
   apply (rule order_trans)
    apply (erule(1) word_mult_le_mono1)
    apply (simp add: unat_div)
    apply (rule order_le_less_trans [OF div_mult_le])
    apply simp
   apply (rule word_div_mult_le)
   done
 
 lemma ucast_ucast_mask:
   "(ucast :: 'a :: len word \<Rightarrow> 'b :: len word) (ucast x) = x AND mask (len_of TYPE ('a))"
   apply (simp flip: take_bit_eq_mask)
   apply transfer
   apply (simp add: ac_simps)
   done
 
 lemma ucast_ucast_len:
   "\<lbrakk> x < 2 ^ LENGTH('b) \<rbrakk> \<Longrightarrow> ucast (ucast x::'b::len word) = (x::'a::len word)"
   apply (subst ucast_ucast_mask)
   apply (erule less_mask_eq)
   done
 
 lemma ucast_ucast_id:
   "LENGTH('a) < LENGTH('b) \<Longrightarrow> ucast (ucast (x::'a::len word)::'b::len word) = x"
   by (auto intro: ucast_up_ucast_id simp: is_up_def source_size_def target_size_def word_size)
 
 lemma unat_ucast:
   "unat (ucast x :: ('a :: len) word) = unat x mod 2 ^ (LENGTH('a))"
 proof -
   have \<open>2 ^ LENGTH('a) = nat (2 ^ LENGTH('a))\<close>
     by simp
   moreover have \<open>unat (ucast x :: 'a word) = unat x mod nat (2 ^ LENGTH('a))\<close>
     by transfer (simp flip: nat_mod_distrib take_bit_eq_mod)
   ultimately show ?thesis
     by (simp only:)
 qed
 
 lemma ucast_less_ucast:
   "LENGTH('a) \<le> LENGTH('b) \<Longrightarrow>
    (ucast x < ((ucast (y :: 'a::len word)) :: 'b::len word)) = (x < y)"
   apply (simp add: word_less_nat_alt unat_ucast)
   apply (subst mod_less)
    apply(rule less_le_trans[OF unat_lt2p], simp)
   apply (subst mod_less)
    apply(rule less_le_trans[OF unat_lt2p], simp)
   apply simp
   done
 
 \<comment> \<open>This weaker version was previously called @{text ucast_less_ucast}. We retain it to
     support existing proofs.\<close>
 lemmas ucast_less_ucast_weak = ucast_less_ucast[OF order.strict_implies_order]
 
 lemma unat_Suc2:
   fixes n :: "'a :: len word"
   shows
   "n \<noteq> -1 \<Longrightarrow> unat (n + 1) = Suc (unat n)"
   apply (subst add.commute, rule unatSuc)
   apply (subst eq_diff_eq[symmetric], simp add: minus_equation_iff)
   done
 
 lemma word_div_1:
   "(n :: 'a :: len word) div 1 = n"
   by (fact bits_div_by_1)
 
 lemma word_minus_one_le:
   "-1 \<le> (x :: 'a :: len word) = (x = -1)"
   by (fact word_order.extremum_unique)
 
 lemma up_scast_inj:
       "\<lbrakk> scast x = (scast y :: 'b :: len word); size x \<le> LENGTH('b) \<rbrakk>
          \<Longrightarrow> x = y"
   apply transfer
   apply (cases \<open>LENGTH('a)\<close>)
   apply simp_all
   apply (metis order_refl take_bit_signed_take_bit take_bit_tightened)
   done
 
 lemma up_scast_inj_eq:
   "LENGTH('a) \<le> len_of TYPE ('b) \<Longrightarrow>
   (scast x = (scast y::'b::len word)) = (x = (y::'a::len word))"
   by (fastforce dest: up_scast_inj simp: word_size)
 
 lemma word_le_add:
   fixes x :: "'a :: len word"
   shows "x \<le> y \<Longrightarrow> \<exists>n. y = x + of_nat n"
   by (rule exI [where x = "unat (y - x)"]) simp
 
 lemma word_plus_mcs_4':
   fixes x :: "'a :: len word"
   shows "\<lbrakk>x + v \<le> x + w; x \<le> x + v\<rbrakk> \<Longrightarrow> v \<le> w"
   apply (rule word_plus_mcs_4)
    apply (simp add: add.commute)
   apply (simp add: add.commute)
   done
 
 lemma unat_eq_1:
   \<open>unat x = Suc 0 \<longleftrightarrow> x = 1\<close>
   by (auto intro!: unsigned_word_eqI [where ?'a = nat])
 
 lemma word_unat_Rep_inject1:
   \<open>unat x = unat 1 \<longleftrightarrow> x = 1\<close>
   by (simp add: unat_eq_1)
 
 lemma and_not_mask_twice:
   "(w AND NOT (mask n)) AND NOT (mask m) = w AND NOT (mask (max m n))"
   for w :: \<open>'a::len word\<close>
   by (rule bit_word_eqI) (auto simp add: bit_simps)
 
 lemma word_less_cases:
   "x < y \<Longrightarrow> x = y - 1 \<or> x < y - (1 ::'a::len word)"
   apply (drule word_less_sub_1)
   apply (drule order_le_imp_less_or_eq)
   apply auto
   done
 
 lemma mask_and_mask:
   "mask a AND mask b = (mask (min a b) :: 'a::len word)"
   by (simp flip: take_bit_eq_mask ac_simps)
 
 lemma mask_eq_0_eq_x:
   "(x AND w = 0) = (x AND NOT w = x)"
   for x w :: \<open>'a::len word\<close>
   using word_plus_and_or_coroll2[where x=x and w=w]
   by auto
 
 lemma mask_eq_x_eq_0:
   "(x AND w = x) = (x AND NOT w = 0)"
   for x w :: \<open>'a::len word\<close>
   using word_plus_and_or_coroll2[where x=x and w=w]
   by auto
 
 lemma compl_of_1: "NOT 1 = (-2 :: 'a :: len word)"
   by (fact not_one)
 
 lemma split_word_eq_on_mask:
   "(x = y) = (x AND m = y AND m \<and> x AND NOT m = y AND NOT m)"
   for x y m :: \<open>'a::len word\<close>
   apply transfer
   apply (simp add: bit_eq_iff)
   apply (auto simp add: bit_simps ac_simps)
   done
 
 lemma word_FF_is_mask:
   "0xFF = (mask 8 :: 'a::len word)"
   by (simp add: mask_eq_decr_exp)
 
 lemma word_1FF_is_mask:
   "0x1FF = (mask 9 :: 'a::len word)"
   by (simp add: mask_eq_decr_exp)
 
 lemma ucast_of_nat_small:
   "x < 2 ^ LENGTH('a) \<Longrightarrow> ucast (of_nat x :: 'a :: len word) = (of_nat x :: 'b :: len word)"
   apply transfer
   apply (auto simp add: take_bit_of_nat min_def not_le)
   apply (metis linorder_not_less min_def take_bit_nat_eq_self take_bit_take_bit)
   done
 
 lemma word_le_make_less:
   fixes x :: "'a :: len word"
   shows "y \<noteq> -1 \<Longrightarrow> (x \<le> y) = (x < (y + 1))"
   apply safe
   apply (erule plus_one_helper2)
   apply (simp add: eq_diff_eq[symmetric])
   done
 
 lemmas finite_word = finite [where 'a="'a::len word"]
 
 lemma word_to_1_set:
   "{0 ..< (1 :: 'a :: len word)} = {0}"
   by fastforce
 
 lemma word_leq_minus_one_le:
   fixes x :: "'a::len word"
   shows "\<lbrakk>y \<noteq> 0; x \<le> y - 1 \<rbrakk> \<Longrightarrow> x < y"
   using le_m1_iff_lt word_neq_0_conv by blast
 
 lemma word_count_from_top:
   "n \<noteq> 0 \<Longrightarrow> {0 ..< n :: 'a :: len word} = {0 ..< n - 1} \<union> {n - 1}"
   apply (rule set_eqI, rule iffI)
    apply simp
    apply (drule word_le_minus_one_leq)
    apply (rule disjCI)
    apply simp
   apply simp
   apply (erule word_leq_minus_one_le)
   apply fastforce
   done
 
 lemma word_minus_one_le_leq:
   "\<lbrakk> x - 1 < y \<rbrakk> \<Longrightarrow> x \<le> (y :: 'a :: len word)"
   apply (cases "x = 0")
    apply simp
   apply (simp add: word_less_nat_alt word_le_nat_alt)
   apply (subst(asm) unat_minus_one)
    apply (simp add: word_less_nat_alt)
   apply (cases "unat x")
    apply (simp add: unat_eq_zero)
   apply arith
   done
 
 lemma word_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 word_power_mod_div:
   fixes x :: "'a::len word"
   shows "\<lbrakk> n < LENGTH('a); m < LENGTH('a)\<rbrakk>
   \<Longrightarrow> x mod 2 ^ n div 2 ^ m = x div 2 ^ m mod 2 ^ (n - m)"
   apply (simp add: word_arith_nat_div unat_mod power_mod_div)
   apply (subst unat_arith_simps(3))
   apply (subst unat_mod)
   apply (subst unat_of_nat)+
   apply (simp add: mod_mod_power min.commute)
   done
 
 lemma word_range_minus_1':
   fixes a :: "'a :: len word"
   shows "a \<noteq> 0 \<Longrightarrow> {a - 1<..b} = {a..b}"
   by (simp add: greaterThanAtMost_def atLeastAtMost_def greaterThan_def atLeast_def less_1_simp)
 
 lemma word_range_minus_1:
   fixes a :: "'a :: len word"
   shows "b \<noteq> 0 \<Longrightarrow> {a..b - 1} = {a..<b}"
   apply (simp add: atLeastLessThan_def atLeastAtMost_def atMost_def lessThan_def)
   apply (rule arg_cong [where f = "\<lambda>x. {a..} \<inter> x"])
   apply rule
    apply clarsimp
    apply (erule contrapos_pp)
    apply (simp add: linorder_not_less linorder_not_le word_must_wrap)
   apply (clarsimp)
   apply (drule word_le_minus_one_leq)
   apply (auto simp: word_less_sub_1)
   done
 
 lemma ucast_nat_def:
   "of_nat (unat x) = (ucast :: 'a :: len word \<Rightarrow> 'b :: len word) x"
   by transfer simp
 
 lemma overflow_plus_one_self:
   "(1 + p \<le> p) = (p = (-1 :: 'a :: len word))"
   apply rule
   apply (rule ccontr)
    apply (drule plus_one_helper2)
    apply (rule notI)
    apply (drule arg_cong[where f="\<lambda>x. x - 1"])
    apply simp
    apply (simp add: field_simps)
   apply simp
   done
 
 lemma plus_1_less:
   "(x + 1 \<le> (x :: 'a :: len word)) = (x = -1)"
   apply (rule iffI)
    apply (rule ccontr)
    apply (cut_tac plus_one_helper2[where x=x, OF order_refl])
     apply simp
    apply clarsimp
    apply (drule arg_cong[where f="\<lambda>x. x - 1"])
    apply simp
   apply simp
   done
 
 lemma pos_mult_pos_ge:
   "[|x > (0::int); n>=0 |] ==> n * x >= n*1"
   apply (simp only: mult_left_mono)
   done
 
 lemma word_plus_strict_mono_right:
   fixes x :: "'a :: len word"
   shows "\<lbrakk>y < z; x \<le> x + z\<rbrakk> \<Longrightarrow> x + y < x + z"
   by unat_arith
 
 lemma word_div_mult:
   "0 < c \<Longrightarrow> a < b * c \<Longrightarrow> a div c < b" for a b c :: "'a::len word"
   by (rule classical)
      (use div_to_mult_word_lt [of b a c] in
       \<open>auto simp add: word_less_nat_alt word_le_nat_alt unat_div\<close>)
 
 lemma word_less_power_trans_ofnat:
   "\<lbrakk>n < 2 ^ (m - k); k \<le> m; m < LENGTH('a)\<rbrakk>
    \<Longrightarrow> of_nat n * 2 ^ k < (2::'a::len word) ^ m"
   apply (subst mult.commute)
   apply (rule word_less_power_trans)
     apply (simp_all add: word_less_nat_alt less_le_trans take_bit_eq_mod)
   done
 
 lemma word_1_le_power:
   "n < LENGTH('a) \<Longrightarrow> (1 :: 'a :: len word) \<le> 2 ^ n"
   by (rule inc_le[where i=0, simplified], erule iffD2[OF p2_gt_0])
 
 lemma unat_1_0:
   "1 \<le> (x::'a::len word) = (0 < unat x)"
   by (auto simp add: word_le_nat_alt)
 
 lemma x_less_2_0_1':
   fixes x :: "'a::len word"
   shows "\<lbrakk>LENGTH('a) \<noteq> 1; x < 2\<rbrakk> \<Longrightarrow> x = 0 \<or> x = 1"
   apply (cases \<open>2 \<le> LENGTH('a)\<close>)
   apply simp_all
   apply transfer
   apply auto
-  apply (metis add.commute add.right_neutral even_two_times_div_two mod_div_trivial mod_pos_pos_trivial mult.commute mult_zero_left not_less not_take_bit_negative odd_two_times_div_two_succ) 
+  apply (metis add.commute add.right_neutral even_two_times_div_two mod_div_trivial mod_pos_pos_trivial mult.commute mult_zero_left not_less not_take_bit_negative odd_two_times_div_two_succ)
   done
 
 lemmas word_add_le_iff2 = word_add_le_iff [folded no_olen_add_nat]
 
 lemma of_nat_power:
   shows "\<lbrakk> p < 2 ^ x; x < len_of TYPE ('a) \<rbrakk> \<Longrightarrow> of_nat p < (2 :: 'a :: len word) ^ x"
   apply (rule order_less_le_trans)
    apply (rule of_nat_mono_maybe)
     apply (erule power_strict_increasing)
     apply simp
    apply assumption
   apply (simp add: word_unat_power del: of_nat_power)
   done
 
 lemma of_nat_n_less_equal_power_2:
   "n < LENGTH('a::len) \<Longrightarrow> ((of_nat n)::'a word) < 2 ^ n"
   apply (induct n)
    apply clarsimp
   apply clarsimp
   apply (metis of_nat_power n_less_equal_power_2 of_nat_Suc power_Suc)
   done
 
 lemma eq_mask_less:
   fixes w :: "'a::len word"
   assumes eqm: "w = w AND mask n"
   and      sz: "n < len_of TYPE ('a)"
   shows "w < (2::'a word) ^ n"
   by (subst eqm, rule and_mask_less' [OF sz])
 
 lemma of_nat_mono_maybe':
   fixes Y :: "nat"
   assumes xlt: "x < 2 ^ len_of TYPE ('a)"
   assumes ylt: "y < 2 ^ len_of TYPE ('a)"
   shows   "(y < x) = (of_nat y < (of_nat x :: 'a :: len word))"
   apply (subst word_less_nat_alt)
   apply (subst unat_of_nat)+
   apply (subst mod_less)
    apply (rule ylt)
   apply (subst mod_less)
    apply (rule xlt)
   apply simp
   done
 
 lemma of_nat_mono_maybe_le:
   "\<lbrakk>x < 2 ^ LENGTH('a); y < 2 ^ LENGTH('a)\<rbrakk> \<Longrightarrow>
   (y \<le> x) = ((of_nat y :: 'a :: len word) \<le> of_nat x)"
   apply (clarsimp simp: le_less)
   apply (rule disj_cong)
    apply (rule of_nat_mono_maybe', assumption+)
   apply auto
   using of_nat_inj apply blast
   done
 
 lemma mask_AND_NOT_mask:
   "(w AND NOT (mask n)) AND mask n = 0"
   for w :: \<open>'a::len word\<close>
   by (rule bit_word_eqI) (simp add: bit_simps)
 
 lemma AND_NOT_mask_plus_AND_mask_eq:
   "(w AND NOT (mask n)) + (w AND mask n) = w"
   for w :: \<open>'a::len word\<close>
   apply (subst disjunctive_add)
   apply (auto simp add: bit_simps)
   apply (rule bit_word_eqI)
   apply (auto simp add: bit_simps)
   done
 
 lemma mask_eqI:
   fixes x :: "'a :: len word"
   assumes m1: "x AND mask n = y AND mask n"
   and     m2: "x AND NOT (mask n) = y AND NOT (mask n)"
   shows "x = y"
 proof -
   have *: \<open>x = x AND mask n OR x AND NOT (mask n)\<close> for x :: \<open>'a word\<close>
     by (rule bit_word_eqI) (auto simp add: bit_simps)
   from assms * [of x] * [of y] show ?thesis
     by simp
 qed
 
 lemma neq_0_no_wrap:
   fixes x :: "'a :: len word"
   shows "\<lbrakk> x \<le> x + y; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x + y \<noteq> 0"
   by clarsimp
 
 lemma unatSuc2:
   fixes n :: "'a :: len word"
   shows "n + 1 \<noteq> 0 \<Longrightarrow> unat (n + 1) = Suc (unat n)"
   by (simp add: add.commute unatSuc)
 
 lemma word_of_nat_le:
   "n \<le> unat x \<Longrightarrow> of_nat n \<le> x"
   apply (simp add: word_le_nat_alt unat_of_nat)
   apply (erule order_trans[rotated])
   apply (simp add: take_bit_eq_mod)
   done
 
 lemma word_unat_less_le:
   "a \<le> of_nat b \<Longrightarrow> unat a \<le> b"
   by (metis eq_iff le_cases le_unat_uoi word_of_nat_le)
 
 lemma mask_Suc_0 : "mask (Suc 0) = (1 :: 'a::len word)"
   by (simp add: mask_eq_decr_exp)
 
 lemma bool_mask':
   fixes x :: "'a :: len word"
   shows "2 < LENGTH('a) \<Longrightarrow> (0 < x AND 1) = (x AND 1 = 1)"
   by (simp add: and_one_eq mod_2_eq_odd)
 
 lemma ucast_ucast_add:
   fixes x :: "'a :: len word"
   fixes y :: "'b :: len word"
   shows
   "LENGTH('b) \<ge> LENGTH('a) \<Longrightarrow>
     ucast (ucast x + y) = x + ucast y"
   apply transfer
   apply simp
   apply (subst (2) take_bit_add [symmetric])
   apply (subst take_bit_add [symmetric])
   apply simp
   done
 
 lemma lt1_neq0:
   fixes x :: "'a :: len word"
   shows "(1 \<le> x) = (x \<noteq> 0)" by unat_arith
 
 lemma word_plus_one_nonzero:
   fixes x :: "'a :: len word"
   shows "\<lbrakk>x \<le> x + y; y \<noteq> 0\<rbrakk> \<Longrightarrow> x + 1 \<noteq> 0"
   apply (subst lt1_neq0 [symmetric])
   apply (subst olen_add_eqv [symmetric])
   apply (erule word_random)
   apply (simp add: lt1_neq0)
   done
 
 lemma word_sub_plus_one_nonzero:
   fixes n :: "'a :: len word"
   shows "\<lbrakk>n' \<le> n; n' \<noteq> 0\<rbrakk> \<Longrightarrow> (n - n') + 1 \<noteq> 0"
   apply (subst lt1_neq0 [symmetric])
   apply (subst olen_add_eqv [symmetric])
   apply (rule word_random [where x' = n'])
    apply simp
    apply (erule word_sub_le)
   apply (simp add: lt1_neq0)
   done
 
 lemma word_le_minus_mono_right:
   fixes x :: "'a :: len word"
   shows "\<lbrakk> z \<le> y; y \<le> x; z \<le> x \<rbrakk> \<Longrightarrow> x - y \<le> x - z"
   apply (rule word_sub_mono)
      apply simp
     apply assumption
    apply (erule word_sub_le)
   apply (erule word_sub_le)
   done
 
 lemma word_0_sle_from_less:
   \<open>0 \<le>s x\<close> if \<open>x < 2 ^ (LENGTH('a) - 1)\<close> for x :: \<open>'a::len word\<close>
   using that
   apply transfer
   apply (cases \<open>LENGTH('a)\<close>)
    apply simp_all
   apply (metis bit_take_bit_iff min_def nat_less_le not_less_eq take_bit_int_eq_self_iff take_bit_take_bit)
   done
 
 lemma ucast_sub_ucast:
   fixes x :: "'a::len word"
   assumes "y \<le> x"
   assumes T: "LENGTH('a) \<le> LENGTH('b)"
   shows "ucast (x - y) = (ucast x - ucast y :: 'b::len word)"
 proof -
   from T
   have P: "unat x < 2 ^ LENGTH('b)" "unat y < 2 ^ LENGTH('b)"
     by (fastforce intro!: less_le_trans[OF unat_lt2p])+
   then show ?thesis
     by (simp add: unat_arith_simps unat_ucast assms[simplified unat_arith_simps])
 qed
 
 lemma word_1_0:
   "\<lbrakk>a + (1::('a::len) word) \<le> b; a < of_nat x\<rbrakk> \<Longrightarrow> a < b"
   apply transfer
   apply (subst (asm) take_bit_incr_eq)
    apply (auto simp add: diff_less_eq)
   using take_bit_int_less_exp le_less_trans by blast
 
 lemma unat_of_nat_less:"\<lbrakk> a < b; unat b = c \<rbrakk> \<Longrightarrow> a < of_nat c"
   by fastforce
 
 lemma word_le_plus_1: "\<lbrakk> (y::('a::len) word) < y + n; a < n \<rbrakk> \<Longrightarrow> y + a \<le> y + a + 1"
   by unat_arith
 
 lemma word_le_plus:"\<lbrakk>(a::('a::len) word) < a + b; c < b\<rbrakk> \<Longrightarrow> a \<le> a + c"
   by (metis order_less_imp_le word_random)
 
 lemma sint_minus1 [simp]: "(sint x = -1) = (x = -1)"
   apply (cases \<open>LENGTH('a)\<close>)
    apply simp_all
   apply transfer
   apply (simp flip: signed_take_bit_eq_iff_take_bit_eq)
   done
 
 lemma sint_0 [simp]: "(sint x = 0) = (x = 0)"
   by (fact signed_eq_0_iff)
 
 (* It is not always that case that "sint 1 = 1", because of 1-bit word sizes.
  * This lemma produces the different cases. *)
 lemma sint_1_cases:
   P if \<open>\<lbrakk> len_of TYPE ('a::len) = 1; (a::'a word) = 0; sint a = 0 \<rbrakk> \<Longrightarrow> P\<close>
      \<open>\<lbrakk> len_of TYPE ('a) = 1; a = 1; sint (1 :: 'a word) = -1 \<rbrakk> \<Longrightarrow> P\<close>
      \<open>\<lbrakk> len_of TYPE ('a) > 1; sint (1 :: 'a word) = 1 \<rbrakk> \<Longrightarrow> P\<close>
 proof (cases \<open>LENGTH('a) = 1\<close>)
   case True
   then have \<open>a = 0 \<or> a = 1\<close>
     by transfer auto
   with True that show ?thesis
     by auto
 next
   case False
   with that show ?thesis
     by (simp add: less_le Suc_le_eq)
 qed
 
 lemma sint_int_min:
   "sint (- (2 ^ (LENGTH('a) - Suc 0)) :: ('a::len) word) = - (2 ^ (LENGTH('a) - Suc 0))"
   apply (cases \<open>LENGTH('a)\<close>)
    apply simp_all
   apply transfer
   apply (simp add: signed_take_bit_int_eq_self)
   done
 
 lemma sint_int_max_plus_1:
   "sint (2 ^ (LENGTH('a) - Suc 0) :: ('a::len) word) = - (2 ^ (LENGTH('a) - Suc 0))"
   apply (cases \<open>LENGTH('a)\<close>)
    apply simp_all
   apply (subst word_of_int_2p [symmetric])
   apply (subst int_word_sint)
   apply simp
   done
 
 lemma uint_range':
   \<open>0 \<le> uint x \<and> uint x < 2 ^ LENGTH('a)\<close>
   for x :: \<open>'a::len word\<close>
   by transfer simp
 
 lemma sint_of_int_eq:
   "\<lbrakk> - (2 ^ (LENGTH('a) - 1)) \<le> x; x < 2 ^ (LENGTH('a) - 1) \<rbrakk> \<Longrightarrow> sint (of_int x :: ('a::len) word) = x"
   by (simp add: signed_take_bit_int_eq_self)
 
 lemma of_int_sint:
   "of_int (sint a) = a"
   by simp
 
 lemma sint_ucast_eq_uint:
     "\<lbrakk> \<not> is_down (ucast :: ('a::len word \<Rightarrow> 'b::len word)) \<rbrakk>
             \<Longrightarrow> sint ((ucast :: ('a::len word \<Rightarrow> 'b::len word)) x) = uint x"
   apply transfer
   apply (simp add: signed_take_bit_take_bit)
   done
 
 lemma word_less_nowrapI':
   "(x :: 'a :: len word) \<le> z - k \<Longrightarrow> k \<le> z \<Longrightarrow> 0 < k \<Longrightarrow> x < x + k"
   by uint_arith
 
 lemma mask_plus_1:
   "mask n + 1 = (2 ^ n :: 'a::len word)"
   by (clarsimp simp: mask_eq_decr_exp)
 
 lemma unat_inj: "inj unat"
   by (metis eq_iff injI word_le_nat_alt)
 
 lemma unat_ucast_upcast:
   "is_up (ucast :: 'b word \<Rightarrow> 'a word)
       \<Longrightarrow> unat (ucast x :: ('a::len) word) = unat (x :: ('b::len) word)"
   unfolding ucast_eq unat_eq_nat_uint
   apply transfer
   apply simp
   done
 
 lemma ucast_mono:
   "\<lbrakk> (x :: 'b :: len word) < y; y < 2 ^ LENGTH('a) \<rbrakk>
    \<Longrightarrow> ucast x < ((ucast y) :: 'a :: len word)"
   apply (simp only: flip: ucast_nat_def)
   apply (rule of_nat_mono_maybe)
   apply (rule unat_less_helper)
   apply simp
   apply (simp add: word_less_nat_alt)
   done
 
 lemma ucast_mono_le:
   "\<lbrakk>x \<le> y; y < 2 ^ LENGTH('b)\<rbrakk> \<Longrightarrow> (ucast (x :: 'a :: len word) :: 'b :: len word) \<le> ucast y"
   apply (simp only: flip: ucast_nat_def)
   apply (subst of_nat_mono_maybe_le[symmetric])
     apply (rule unat_less_helper)
     apply simp
    apply (rule unat_less_helper)
    apply (erule le_less_trans)
   apply (simp_all add: word_le_nat_alt)
   done
 
 lemma ucast_mono_le':
   "\<lbrakk> unat y < 2 ^ LENGTH('b); LENGTH('b::len) < LENGTH('a::len); x \<le> y \<rbrakk>
    \<Longrightarrow> ucast x \<le> (ucast y :: 'b word)" for x y :: \<open>'a::len word\<close>
   by (auto simp: word_less_nat_alt intro: ucast_mono_le)
 
 lemma neg_mask_add_mask:
   "((x:: 'a :: len word) AND NOT (mask n)) + (2 ^ n - 1) = x OR mask n"
   unfolding mask_2pm1 [symmetric]
   apply (subst word_plus_and_or_coroll; rule bit_word_eqI)
    apply (auto simp add: bit_simps)
   done
 
 lemma le_step_down_word:"\<lbrakk>(i::('a::len) word) \<le> n; i = n \<longrightarrow> P; i \<le> n - 1 \<longrightarrow> P\<rbrakk> \<Longrightarrow> P"
   by unat_arith
 
 lemma le_step_down_word_2:
   fixes x :: "'a::len word"
   shows "\<lbrakk>x \<le>  y; x \<noteq> y\<rbrakk> \<Longrightarrow> x \<le> y - 1"
   by (subst (asm) word_le_less_eq,
       clarsimp,
       simp add: word_le_minus_one_leq)
 
 lemma NOT_mask_AND_mask[simp]: "(w AND mask n) AND NOT (mask n) = 0"
   by (clarsimp simp add: mask_eq_decr_exp Parity.bit_eq_iff bit_and_iff bit_not_iff bit_mask_iff)
 
 lemma and_and_not[simp]:"(a AND b) AND NOT b = 0"
   for a b :: \<open>'a::len word\<close>
   apply (subst word_bw_assocs(1))
   apply clarsimp
   done
 
 lemma ex_mask_1[simp]: "(\<exists>x. mask x = (1 :: 'a::len word))"
   apply (rule_tac x=1 in exI)
   apply (simp add:mask_eq_decr_exp)
   done
 
 lemma not_switch:"NOT a = x \<Longrightarrow> a = NOT x"
   by auto
 
 end
diff --git a/thys/Word_Lib/More_Word_Operations.thy b/thys/Word_Lib/More_Word_Operations.thy
--- a/thys/Word_Lib/More_Word_Operations.thy
+++ b/thys/Word_Lib/More_Word_Operations.thy
@@ -1,1007 +1,1012 @@
+(*
+ * Copyright Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
 
 section \<open>Misc word operations\<close>
 
 theory More_Word_Operations
   imports
     "HOL-Library.Word"
     Aligned
     Reversed_Bit_Lists
     More_Misc
     Signed_Words
 begin
 
 definition
   ptr_add :: "'a :: len word \<Rightarrow> nat \<Rightarrow> 'a word" where
   "ptr_add ptr n \<equiv> ptr + of_nat n"
 
 definition
   alignUp :: "'a::len word \<Rightarrow> nat \<Rightarrow> 'a word" where
  "alignUp x n \<equiv> x + 2 ^ n - 1 AND NOT (2 ^ n - 1)"
 
 lemma alignUp_unfold:
   \<open>alignUp w n = (w + mask n) AND NOT (mask n)\<close>
   by (simp add: alignUp_def mask_eq_exp_minus_1 add_mask_fold)
 
 (* standard notation for blocks of 2^n-1 words, usually aligned;
    abbreviation so it simplifies directly *)
 abbreviation mask_range :: "'a::len word \<Rightarrow> nat \<Rightarrow> 'a word set" where
   "mask_range p n \<equiv> {p .. p + mask n}"
 
 definition
   w2byte :: "'a :: len word \<Rightarrow> 8 word" where
   "w2byte \<equiv> ucast"
 
 (* Count leading zeros  *)
 definition
   word_clz :: "'a::len word \<Rightarrow> nat"
 where
   "word_clz w \<equiv> length (takeWhile Not (to_bl w))"
 
 (* Count trailing zeros  *)
 definition
   word_ctz :: "'a::len word \<Rightarrow> nat"
 where
   "word_ctz w \<equiv> length (takeWhile Not (rev (to_bl w)))"
 
 lemma word_ctz_le:
   "word_ctz (w :: ('a::len word)) \<le> LENGTH('a)"
   apply (clarsimp simp: word_ctz_def)
   using length_takeWhile_le apply (rule order_trans)
   apply simp
   done
 
 lemma word_ctz_less:
   "w \<noteq> 0 \<Longrightarrow> word_ctz (w :: ('a::len word)) < LENGTH('a)"
   apply (clarsimp simp: word_ctz_def eq_zero_set_bl)
   using length_takeWhile_less apply (rule less_le_trans)
   apply auto
   done
 
 lemma take_bit_word_ctz_eq [simp]:
   \<open>take_bit LENGTH('a) (word_ctz w) = word_ctz w\<close>
   for w :: \<open>'a::len word\<close>
   apply (simp add: take_bit_nat_eq_self_iff word_ctz_def to_bl_unfold)
   using length_takeWhile_le apply (rule le_less_trans)
   apply simp
   done
 
 lemma word_ctz_not_minus_1:
   \<open>word_of_nat (word_ctz (w :: 'a :: len word)) \<noteq> (- 1 :: 'a::len word)\<close> if \<open>1 < LENGTH('a)\<close>
 proof -
   note word_ctz_le
   also from that have \<open>LENGTH('a) < mask LENGTH('a)\<close>
     by (simp add: less_mask)
   finally have \<open>word_ctz w < mask LENGTH('a)\<close> .
   then have \<open>word_of_nat (word_ctz w) < (word_of_nat (mask LENGTH('a)) :: 'a word)\<close>
     by (simp add: of_nat_word_less_iff)
   also have \<open>\<dots> = - 1\<close>
     by (rule bit_word_eqI) (simp add: bit_simps)
   finally show ?thesis
     by simp
 qed
 
 lemma unat_of_nat_ctz_mw:
   "unat (of_nat (word_ctz (w :: 'a :: len word)) :: 'a :: len word) = word_ctz w"
   by simp
 
 lemma unat_of_nat_ctz_smw:
   "unat (of_nat (word_ctz (w :: 'a :: len word)) :: 'a :: len signed word) = word_ctz w"
   by simp
 
 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 OR NOT (mask n) else w AND mask n"
 
 lemma sign_extend_eq_signed_take_bit:
   \<open>sign_extend = signed_take_bit\<close>
 proof (rule ext)+
   fix n and w :: \<open>'a::len word\<close>
   show \<open>sign_extend n w = signed_take_bit n w\<close>
   proof (rule bit_word_eqI)
     fix q
     assume \<open>q < LENGTH('a)\<close>
     then show \<open>bit (sign_extend n w) q \<longleftrightarrow> bit (signed_take_bit n w) q\<close>
       by (auto simp add: test_bit_eq_bit bit_signed_take_bit_iff
         sign_extend_def bit_and_iff bit_or_iff bit_not_iff bit_mask_iff not_less
         exp_eq_0_imp_not_bit not_le min_def)
   qed
 qed
 
 definition
   sign_extended :: "nat \<Rightarrow> 'a::len word \<Rightarrow> bool"
 where
   "sign_extended n w \<equiv> \<forall>i. n < i \<longrightarrow> i < size w \<longrightarrow> w !! i = w !! n"
 
 lemma ptr_add_0 [simp]:
   "ptr_add ref 0 = ref "
   unfolding ptr_add_def by simp
 
 lemma pop_count_0[simp]:
   "pop_count 0 = 0"
   by (clarsimp simp:pop_count_def)
 
 lemma pop_count_1[simp]:
   "pop_count 1 = 1"
   by (clarsimp simp:pop_count_def to_bl_1)
 
 lemma pop_count_0_imp_0:
   "(pop_count w = 0) = (w = 0)"
   apply (rule iffI)
    apply (clarsimp simp:pop_count_def)
    apply (subst (asm) filter_empty_conv)
    apply (clarsimp simp:eq_zero_set_bl)
    apply fast
   apply simp
   done
 
 lemma word_log2_zero_eq [simp]:
   \<open>word_log2 0 = 0\<close>
   by (simp add: word_log2_def word_clz_def word_size)
 
 lemma word_log2_unfold:
   \<open>word_log2 w = (if w = 0 then 0 else Max {n. bit w n})\<close>
   for w :: \<open>'a::len word\<close>
 proof (cases \<open>w = 0\<close>)
   case True
   then show ?thesis
     by simp
 next
   case False
   then obtain r where \<open>bit w r\<close>
     by (auto simp add: bit_eq_iff)
   then have \<open>Max {m. bit w m} = LENGTH('a) - Suc (length
     (takeWhile (Not \<circ> bit w) (rev [0..<LENGTH('a)])))\<close>
     by (subst Max_eq_length_takeWhile [of _ \<open>LENGTH('a)\<close>])
       (auto simp add: bit_imp_le_length)
   then have \<open>word_log2 w = Max {x. bit w x}\<close>
     by (simp add: word_log2_def word_clz_def word_size to_bl_unfold rev_map takeWhile_map)
   with \<open>w \<noteq> 0\<close> show ?thesis
     by simp
 qed
 
 lemma word_log2_eqI:
   \<open>word_log2 w = n\<close>
   if \<open>w \<noteq> 0\<close> \<open>bit w n\<close> \<open>\<And>m. bit w m \<Longrightarrow> m \<le> n\<close>
   for w :: \<open>'a::len word\<close>
 proof -
   from \<open>w \<noteq> 0\<close> have \<open>word_log2 w = Max {n. bit w n}\<close>
     by (simp add: word_log2_unfold)
   also have \<open>Max {n. bit w n} = n\<close>
     using that by (auto intro: Max_eqI)
   finally show ?thesis .
 qed
 
 lemma bit_word_log2:
   \<open>bit w (word_log2 w)\<close> if \<open>w \<noteq> 0\<close>
 proof -
   from \<open>w \<noteq> 0\<close> have \<open>\<exists>r. bit w r\<close>
     by (simp add: bit_eq_iff)
   then obtain r where \<open>bit w r\<close> ..
   from \<open>w \<noteq> 0\<close> have \<open>word_log2 w = Max {n. bit w n}\<close>
     by (simp add: word_log2_unfold)
   also have \<open>Max {n. bit w n} \<in> {n. bit w n}\<close>
     using \<open>bit w r\<close> by (subst Max_in) auto
   finally show ?thesis
     by simp
-qed    
+qed
 
 lemma word_log2_maximum:
   \<open>n \<le> word_log2 w\<close> if \<open>bit w n\<close>
 proof -
   have \<open>n \<le> Max {n. bit w n}\<close>
     using that by (auto intro: Max_ge)
   also from that have \<open>w \<noteq> 0\<close>
     by force
   then have \<open>Max {n. bit w n} = word_log2 w\<close>
     by (simp add: word_log2_unfold)
   finally show ?thesis .
 qed
 
 lemma word_log2_nth_same:
   "w \<noteq> 0 \<Longrightarrow> w !! word_log2 w"
   by (drule bit_word_log2) (simp add: test_bit_eq_bit)
 
 lemma word_log2_nth_not_set:
   "\<lbrakk> word_log2 w < i ; i < size w \<rbrakk> \<Longrightarrow> \<not> w !! i"
   using word_log2_maximum [of w i] by (auto simp add: test_bit_eq_bit)
 
 lemma word_log2_highest:
   assumes a: "w !! i"
   shows "i \<le> word_log2 w"
   using a by (simp add: test_bit_eq_bit word_log2_maximum)
 
 lemma word_log2_max:
   "word_log2 w < size w"
   apply (cases \<open>w = 0\<close>)
    apply (simp_all add: word_size)
   apply (drule bit_word_log2)
   apply (fact bit_imp_le_length)
   done
 
 lemma word_clz_0[simp]:
   "word_clz (0::'a::len word) = LENGTH('a)"
   unfolding word_clz_def
   by (simp add: takeWhile_replicate)
 
 lemma word_clz_minus_one[simp]:
   "word_clz (-1::'a::len word) = 0"
   unfolding word_clz_def
   by (simp add: takeWhile_replicate)
 
 lemma is_aligned_alignUp[simp]:
   "is_aligned (alignUp p n) n"
   by (simp add: alignUp_def is_aligned_mask mask_eq_decr_exp word_bw_assocs)
 
 lemma alignUp_le[simp]:
   "alignUp p n \<le> p + 2 ^ n - 1"
   unfolding alignUp_def by (rule word_and_le2)
 
 lemma alignUp_idem:
   fixes a :: "'a::len word"
   assumes "is_aligned a n" "n < LENGTH('a)"
   shows "alignUp a n = a"
   using assms unfolding alignUp_def
   by (metis add_cancel_right_right add_diff_eq and_mask_eq_iff_le_mask mask_eq_decr_exp mask_out_add_aligned order_refl word_plus_and_or_coroll2)
 
 lemma alignUp_not_aligned_eq:
   fixes a :: "'a :: len word"
   assumes al: "\<not> is_aligned a n"
   and     sz: "n < LENGTH('a)"
   shows   "alignUp a n = (a div 2 ^ n + 1) * 2 ^ n"
 proof -
   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 mask_eq_decr_exp [symmetric])
     apply (erule ssubst)
     apply (subst neg_mask_is_div)
     apply (simp add: word_arith_nat_div)
     apply (subst unat_word_ariths(1) unat_word_ariths(2))+
     apply (subst uno_simps)
     apply (subst unat_1)
     apply (subst mod_add_right_eq)
     apply simp
     apply (subst power_mod_div)
     apply (subst div_mult_self1)
      apply simp
     apply (subst um)
     apply simp
     apply (subst mod_mod_power)
     apply simp
     apply (subst word_unat_power, subst Abs_fnat_hom_mult)
     apply (subst mult_mod_left)
     apply (subst power_add [symmetric])
     apply simp
     apply (subst Abs_fnat_hom_1)
     apply (subst Abs_fnat_hom_add)
     apply (subst word_unat_power, subst Abs_fnat_hom_mult)
     apply (subst word_unat.Rep_inverse[symmetric], subst Abs_fnat_hom_mult)
     apply simp
     done
 qed
 
 lemma alignUp_ge:
   fixes a :: "'a :: len word"
   assumes sz: "n < LENGTH('a)"
   and nowrap: "alignUp a n \<noteq> 0"
   shows "a \<le> alignUp a n"
 proof (cases "is_aligned a n")
   case True
   then show ?thesis using sz
     by (subst alignUp_idem, simp_all)
 next
   case False
 
   have lt0: "unat a div 2 ^ n < 2 ^ (LENGTH('a) - n)" using sz
     by (metis 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 (simp flip: drop_bit_eq_div unat_drop_bit_eq) (metis leI le_unat_uoi unat_mono)
 
     have "alignUp a n = (a div 2 ^ n + 1) * 2 ^ n"
       by (rule alignUp_not_aligned_eq) fact+
 
     then have "\<dots> = 0" using asm by simp
     then have "2 ^ LENGTH('a) dvd 2 ^ n * (unat a div 2 ^ n + 1)"
       using sz by (simp add: unat_arith_simps ac_simps)
                   (simp add: unat_word_ariths mod_simps mod_eq_0_iff_dvd)
     with leq have "2 ^ n * (unat a div 2 ^ n + 1) = 2 ^ LENGTH('a)"
       by (force elim!: le_SucE)
     then have "unat a div 2 ^ n = 2 ^ LENGTH('a) div 2 ^ n - 1"
       by (metis (no_types, 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 zero_neq_numeral)
     then have "unat a div 2 ^ n = 2 ^ (LENGTH('a) - n) - 1"
       using sz by (simp add: power_sub)
     then have "2 ^ (LENGTH('a) - n) - 1 < k" using r
       by simp
     then have False using kv by simp
   } then show ?thesis by clarsimp
 qed
 
 lemma alignUp_ar_helper:
   fixes a :: "'a :: len word"
   assumes al: "is_aligned x n"
   and     sz: "n < LENGTH('a)"
   and    sub: "{x..x + 2 ^ n - 1} \<subseteq> {a..b}"
   and    anz: "a \<noteq> 0"
   shows "a \<le> alignUp a n \<and> alignUp a n + 2 ^ n - 1 \<le> b"
 proof
   from al have xl: "x \<le> x + 2 ^ n - 1" by (simp add: is_aligned_no_overflow)
 
   from xl sub have ax: "a \<le> x"
     by auto
 
   show "a \<le> alignUp a n"
   proof (rule alignUp_ge)
     show "alignUp a n \<noteq> 0" using al sz ax anz
       by (rule alignUp_is_aligned_nz)
   qed fact+
 
   show "alignUp a n + 2 ^ n - 1 \<le> b"
   proof (rule order_trans)
     from xl show tp: "x + 2 ^ n - 1 \<le> b" using sub
       by auto
 
     from ax have "alignUp a n \<le> x"
       by (rule alignUp_le_greater_al) fact+
     then have "alignUp a n + (2 ^ n - 1) \<le> x + (2 ^ n - 1)"
       using xl al is_aligned_no_overflow' olen_add_eqv word_plus_mcs_3 by blast
     then show "alignUp a n + 2 ^ n - 1 \<le> x + 2 ^ n - 1"
       by (simp add: field_simps)
   qed
 qed
 
 lemma alignUp_def2:
   "alignUp a sz = a + 2 ^ sz - 1 AND NOT (mask sz)"
   by (simp add: alignUp_def flip: mask_eq_decr_exp)
 
 lemma alignUp_def3:
   "alignUp a sz = 2^ sz + (a - 1 AND NOT (mask sz))"
   by (simp add: alignUp_def2 is_aligned_triv field_simps mask_out_add_aligned)
 
 lemma  alignUp_plus:
   "is_aligned w us \<Longrightarrow> alignUp (w + a) us  = w + alignUp a us"
   by (clarsimp simp: alignUp_def2 mask_out_add_aligned field_simps)
 
 lemma alignUp_distance:
   "alignUp (q :: 'a :: len word) sz - q \<le> mask sz"
   by (metis (no_types) add.commute add_diff_cancel_left alignUp_def2 diff_add_cancel
                        mask_2pm1 subtract_mask(2) word_and_le1 word_sub_le_iff)
 
 lemma is_aligned_diff_neg_mask:
   "is_aligned p sz \<Longrightarrow> (p - q AND NOT (mask sz)) = (p - ((alignUp q sz) AND NOT (mask sz)))"
   apply (clarsimp simp only:word_and_le2 diff_conv_add_uminus)
   apply (subst mask_out_add_aligned[symmetric]; simp)
   apply (simp add: eq_neg_iff_add_eq_0)
   apply (subst add.commute)
   apply (simp add: alignUp_distance is_aligned_neg_mask_eq mask_out_add_aligned and_mask_eq_iff_le_mask flip: mask_eq_x_eq_0)
   done
 
 lemma word_clz_max:
   "word_clz w \<le> size (w::'a::len word)"
   unfolding word_clz_def
   by (metis length_takeWhile_le word_size_bl)
 
 lemma word_clz_nonzero_max:
   fixes w :: "'a::len word"
   assumes nz: "w \<noteq> 0"
   shows "word_clz w < size (w::'a::len word)"
 proof -
   {
     assume a: "word_clz w = size (w::'a::len word)"
     hence "length (takeWhile Not (to_bl w)) = length (to_bl w)"
       by (simp add: word_clz_def word_size)
     hence allj: "\<forall>j\<in>set(to_bl w). \<not> j"
       by (metis a length_takeWhile_less less_irrefl_nat word_clz_def)
     hence "to_bl w = replicate (length (to_bl w)) False"
       by (auto simp add: to_bl_unfold rev_map simp flip: map_replicate_trivial)
         (metis allj eq_zero_set_bl nz)
     hence "w = 0"
       by (metis to_bl_0 word_bl.Rep_eqD word_bl_Rep')
     with nz have False by simp
   }
   thus ?thesis using word_clz_max
     by (fastforce intro: le_neq_trans)
 qed
 
 (* Sign extension from bit n. *)
 
 lemma 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_if'  [word_eqI_simps]:
   \<open>i < LENGTH('a) \<Longrightarrow> sign_extend e w !! i \<longleftrightarrow> (if i < e then w !! i else w !! e)\<close>
   for w :: \<open>'a::len word\<close>
   using sign_extend_bitwise_if [of i w e] by (simp add: word_size)
 
 lemma sign_extend_bitwise_disj:
   "i < size w \<Longrightarrow> 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_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 OR NOT (mask (Suc n)) else w AND mask (Suc n))"
   by (rule bit_word_eqI) (auto simp add: bit_simps sign_extend_eq_signed_take_bit min_def test_bit_eq_bit less_Suc_eq_le)
 
 lemma sign_extended_sign_extend:
   "sign_extended n (sign_extend n w)"
   by (clarsimp simp: sign_extended_def word_size sign_extend_bitwise_if)
 
 lemma sign_extended_iff_sign_extend:
   "sign_extended n w \<longleftrightarrow> sign_extend n w = w"
   apply auto
    apply (auto simp add: bit_eq_iff)
     apply (simp_all add: bit_simps sign_extend_eq_signed_take_bit not_le min_def sign_extended_def test_bit_eq_bit word_size split: if_splits)
   using le_imp_less_or_eq apply auto[1]
    apply (metis bit_imp_le_length nat_less_le)
   apply (metis Suc_leI Suc_n_not_le_n le_trans nat_less_le)
   done
 
 lemma sign_extended_weaken:
   "sign_extended n w \<Longrightarrow> n \<le> m \<Longrightarrow> sign_extended m w"
   unfolding sign_extended_def by (cases "n < m") auto
 
 lemma sign_extend_sign_extend_eq:
   "sign_extend m (sign_extend n w) = sign_extend (min m n) w"
   by (rule bit_word_eqI) (simp add: sign_extend_eq_signed_take_bit bit_simps)
 
 lemma sign_extended_high_bits:
   "\<lbrakk> sign_extended e p; j < size p; e \<le> i; i < j \<rbrakk> \<Longrightarrow> p !! i = p !! j"
   by (drule (1) sign_extended_weaken; simp add: sign_extended_def)
 
 lemma sign_extend_eq:
   "w AND mask (Suc n) = v AND mask (Suc n) \<Longrightarrow> sign_extend n w = sign_extend n v"
   by (simp flip: take_bit_eq_mask add: sign_extend_eq_signed_take_bit signed_take_bit_eq_iff_take_bit_eq)
 
 lemma sign_extended_add:
   assumes p: "is_aligned p n"
   assumes f: "f < 2 ^ n"
   assumes e: "n \<le> e"
   assumes "sign_extended e p"
   shows "sign_extended e (p + f)"
 proof (cases "e < size p")
   case True
   note and_or = is_aligned_add_or[OF p f]
   have "\<not> 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 AND mask e = f"
     by (fastforce intro: subst[where P="\<lambda>f. f AND mask e = f", OF less_mask_eq[OF f]]
                   simp: mask_twice e)
   show ?thesis
     using assms
      apply (simp add: sign_extended_iff_sign_extend sign_extend_def i)
      apply (simp add: and_or word_bw_comms[of p f])
      apply (clarsimp simp: word_ao_dist fm word_bw_assocs split: if_splits)
     done
 next
   case False thus ?thesis
     by (simp add: sign_extended_def word_size)
 qed
 
 lemma sign_extended_neq_mask:
   "\<lbrakk>sign_extended n ptr; m \<le> n\<rbrakk> \<Longrightarrow> sign_extended n (ptr AND NOT (mask m))"
   by (fastforce simp: sign_extended_def word_size neg_mask_test_bit)
 
 definition
   "limited_and (x :: 'a :: len word) y \<longleftrightarrow> (x AND y = x)"
 
 lemma limited_and_eq_0:
   "\<lbrakk> limited_and x z; y AND NOT z = y \<rbrakk> \<Longrightarrow> x AND y = 0"
   unfolding limited_and_def
   apply (subst arg_cong2[where f="(AND)"])
     apply (erule sym)+
   apply (simp(no_asm) add: word_bw_assocs word_bw_comms word_bw_lcs)
   done
 
 lemma limited_and_eq_id:
   "\<lbrakk> limited_and x z; y AND z = z \<rbrakk> \<Longrightarrow> x AND y = x"
   unfolding limited_and_def
   by (erule subst, fastforce simp: word_bw_lcs word_bw_assocs word_bw_comms)
 
 lemma lshift_limited_and:
   "limited_and x z \<Longrightarrow> limited_and (x << n) (z << n)"
   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_eq_decr_exp, folded limited_and_def]
 
 lemmas limited_and_simps = limited_and_simps1
        limited_and_simps1[OF is_aligned_limited_and]
        limited_and_simps1[OF lshift_limited_and]
        limited_and_simps1[OF rshift_limited_and]
        limited_and_simps1[OF rshift_limited_and, OF is_aligned_limited_and]
        not_one shiftl_shiftr1[unfolded word_size mask_eq_decr_exp]
        shiftl_shiftr2[unfolded word_size mask_eq_decr_exp]
 
 definition
   from_bool :: "bool \<Rightarrow> 'a::len word" where
   "from_bool b \<equiv> case b of True \<Rightarrow> of_nat 1
                          | False \<Rightarrow> of_nat 0"
 
 lemma from_bool_eq:
   \<open>from_bool = of_bool\<close>
   by (simp add: fun_eq_iff from_bool_def)
 
 lemma from_bool_0:
   "(from_bool x = 0) = (\<not> x)"
   by (simp add: from_bool_def split: bool.split)
 
 lemma from_bool_eq_if':
   "((if P then 1 else 0) = from_bool Q) = (P = Q)"
   by (cases Q) (simp_all add: from_bool_def)
 
 definition
   to_bool :: "'a::len word \<Rightarrow> bool" where
   "to_bool \<equiv> (\<noteq>) 0"
 
 lemma to_bool_and_1:
   "to_bool (x AND 1) = (x !! 0)"
   by (simp add: test_bit_word_eq to_bool_def and_one_eq mod_2_eq_odd)
 
 lemma to_bool_from_bool [simp]:
   "to_bool (from_bool r) = r"
   unfolding from_bool_def to_bool_def
   by (simp split: bool.splits)
 
 lemma from_bool_neq_0 [simp]:
   "(from_bool b \<noteq> 0) = b"
   by (simp add: from_bool_def split: bool.splits)
 
 lemma from_bool_mask_simp [simp]:
   "(from_bool r :: 'a::len word) AND 1 = from_bool r"
   unfolding from_bool_def
   by (clarsimp split: bool.splits)
 
 lemma from_bool_1 [simp]:
   "(from_bool P = 1) = P"
   by (simp add: from_bool_def split: bool.splits)
 
 lemma ge_0_from_bool [simp]:
   "(0 < from_bool P) = P"
   by (simp add: from_bool_def split: bool.splits)
 
 lemma limited_and_from_bool:
   "limited_and (from_bool b) 1"
   by (simp add: from_bool_def limited_and_def split: bool.split)
 
 lemma to_bool_1 [simp]: "to_bool 1" by (simp add: to_bool_def)
 lemma to_bool_0 [simp]: "\<not>to_bool 0" by (simp add: to_bool_def)
 
 lemma from_bool_eq_if:
   "(from_bool Q = (if P then 1 else 0)) = (P = Q)"
   by (cases Q) (simp_all add: from_bool_def)
 
 lemma to_bool_eq_0:
   "(\<not> to_bool x) = (x = 0)"
   by (simp add: to_bool_def)
 
 lemma to_bool_neq_0:
   "(to_bool x) = (x \<noteq> 0)"
   by (simp add: to_bool_def)
 
 lemma from_bool_all_helper:
   "(\<forall>bool. from_bool bool = val \<longrightarrow> P bool)
       = ((\<exists>bool. from_bool bool = val) \<longrightarrow> P (val \<noteq> 0))"
   by (auto simp: from_bool_0)
 
 lemma fold_eq_0_to_bool:
   "(v = 0) = (\<not> to_bool v)"
   by (simp add: to_bool_def)
 
 lemma from_bool_to_bool_iff:
   "w = from_bool b \<longleftrightarrow> to_bool w = b \<and> (w = 0 \<or> w = 1)"
   by (cases b) (auto simp: from_bool_def to_bool_def)
 
 lemma from_bool_eqI:
   "from_bool x = from_bool y \<Longrightarrow> x = y"
   unfolding from_bool_def
   by (auto split: bool.splits)
 
 lemma neg_mask_in_mask_range:
   "is_aligned ptr bits \<Longrightarrow> (ptr' AND NOT(mask bits) = ptr) = (ptr' \<in> mask_range ptr bits)"
   apply (erule is_aligned_get_word_bits)
    apply (rule iffI)
     apply (drule sym)
     apply (simp add: word_and_le2)
     apply (subst word_plus_and_or_coroll, word_eqI_solve)
     apply (metis bit.disj_ac(2) bit.disj_conj_distrib2 le_word_or2 word_and_max word_or_not)
    apply clarsimp
    apply (smt add.right_neutral eq_iff is_aligned_neg_mask_eq mask_out_add_aligned neg_mask_mono_le
               word_and_not)
   apply (simp add: power_overflow mask_eq_decr_exp)
   done
 
 lemma aligned_offset_in_range:
   "\<lbrakk> is_aligned (x :: 'a :: len word) m; y < 2 ^ m; is_aligned p n; n \<ge> m; n < LENGTH('a) \<rbrakk>
    \<Longrightarrow> (x + y \<in> {p .. p + mask n}) = (x \<in> mask_range p n)"
   apply (subst disjunctive_add)
    apply (simp add: bit_simps)
   apply (erule is_alignedE')
    apply (auto simp add: bit_simps not_le)[1]
    apply (metis less_2p_is_upper_bits_unset test_bit_eq_bit)
   apply (simp only: is_aligned_add_or word_ao_dist flip: neg_mask_in_mask_range)
   apply (subgoal_tac \<open>y AND NOT (mask n) = 0\<close>)
    apply simp
   apply (metis (full_types) is_aligned_mask is_aligned_neg_mask less_mask_eq word_bw_comms(1) word_bw_lcs(1))
   done
 
 lemma mask_range_to_bl':
   "\<lbrakk> is_aligned (ptr :: 'a :: len word) bits; bits < LENGTH('a) \<rbrakk>
    \<Longrightarrow> mask_range ptr bits
        = {x. take (LENGTH('a) - bits) (to_bl x) = take (LENGTH('a) - bits) (to_bl ptr)}"
   apply (rule set_eqI, rule iffI)
    apply clarsimp
    apply (subgoal_tac "\<exists>y. x = ptr + y \<and> y < 2 ^ bits")
     apply clarsimp
     apply (subst is_aligned_add_conv)
        apply assumption
       apply simp
     apply simp
    apply (rule_tac x="x - ptr" in exI)
    apply (simp add: add_diff_eq[symmetric])
    apply (simp only: word_less_sub_le[symmetric])
    apply (rule word_diff_ls')
     apply (simp add: field_simps mask_eq_decr_exp)
    apply assumption
   apply simp
   apply (subgoal_tac "\<exists>y. y < 2 ^ bits \<and> to_bl (ptr + y) = to_bl x")
    apply clarsimp
    apply (rule conjI)
     apply (erule(1) is_aligned_no_wrap')
    apply (simp only: add_diff_eq[symmetric] mask_eq_decr_exp)
    apply (rule word_plus_mono_right)
     apply simp
    apply (erule is_aligned_no_wrap')
    apply simp
   apply (rule_tac x="of_bl (drop (LENGTH('a) - bits) (to_bl x))" in exI)
   apply (rule context_conjI)
    apply (rule order_less_le_trans [OF of_bl_length])
     apply simp
    apply simp
   apply (subst is_aligned_add_conv)
      apply assumption
     apply simp
   apply (drule sym)
   apply (simp add: word_rep_drop)
   done
 
 lemma mask_range_to_bl:
   "is_aligned (ptr :: 'a :: len word) bits
    \<Longrightarrow> mask_range ptr bits
         = {x. take (LENGTH('a) - bits) (to_bl x) = take (LENGTH('a) - bits) (to_bl ptr)}"
   apply (erule is_aligned_get_word_bits)
    apply (erule(1) mask_range_to_bl')
   apply (rule set_eqI)
   apply (simp add: power_overflow mask_eq_decr_exp)
   done
 
 lemma aligned_mask_range_cases:
   "\<lbrakk> is_aligned (p :: 'a :: len word) n; is_aligned (p' :: 'a :: len word) n' \<rbrakk>
    \<Longrightarrow> mask_range p n \<inter> mask_range p' n' = {} \<or>
        mask_range p n \<subseteq> mask_range p' n' \<or>
        mask_range p n \<supseteq> mask_range p' n'"
   apply (simp add: mask_range_to_bl)
   apply (rule Meson.disj_comm, rule disjCI)
   apply auto
   apply (subgoal_tac "(\<exists>n''. LENGTH('a) - n = (LENGTH('a) - n') + n'')
                     \<or> (\<exists>n''. LENGTH('a) - n' = (LENGTH('a) - n) + n'')")
    apply (fastforce simp: take_add)
   apply arith
   done
 
 lemma aligned_mask_range_offset_subset:
   assumes al: "is_aligned (ptr :: 'a :: len word) sz" and al': "is_aligned x sz'"
   and szv: "sz' \<le> sz"
   and xsz: "x < 2 ^ sz"
   shows "mask_range (ptr+x) sz' \<subseteq> mask_range ptr sz"
   using al
 proof (rule is_aligned_get_word_bits)
   assume p0: "ptr = 0" and szv': "LENGTH ('a) \<le> sz"
   then have "(2 ::'a word) ^ sz = 0" by simp
   show ?thesis using p0
     by (simp add: \<open>2 ^ sz = 0\<close> mask_eq_decr_exp)
 next
   assume szv': "sz < LENGTH('a)"
 
   hence blah: "2 ^ (sz - sz') < (2 :: nat) ^ LENGTH('a)"
     using szv by auto
   show ?thesis using szv szv'
     apply auto
     using al assms(4) is_aligned_no_wrap' apply blast
     apply (simp only: flip: add_diff_eq add_mask_fold)
     apply (subst add.assoc, rule word_plus_mono_right)
      using al' is_aligned_add_less_t2n xsz
      apply fastforce
     apply (simp add: field_simps szv al is_aligned_no_overflow)
     done
 qed
 
 lemma aligned_mask_ranges_disjoint:
   "\<lbrakk> is_aligned (p :: 'a :: len word) n; is_aligned (p' :: 'a :: len word) n';
      p AND NOT(mask n') \<noteq> p'; p' AND NOT(mask n) \<noteq> p \<rbrakk>
    \<Longrightarrow> mask_range p n \<inter> mask_range p' n' = {}"
   using aligned_mask_range_cases
   by (auto simp: neg_mask_in_mask_range)
 
 lemma aligned_mask_ranges_disjoint2:
   "\<lbrakk> is_aligned p n; is_aligned ptr bits; n \<ge> m; n < size p; m \<le> bits;
      (\<forall>y < 2 ^ (n - m). p + (y << m) \<notin> mask_range ptr bits) \<rbrakk>
    \<Longrightarrow> mask_range p n \<inter> mask_range ptr bits = {}"
   apply safe
   apply (simp only: flip: neg_mask_in_mask_range)
   apply (drule_tac x="x AND mask n >> m" in spec)
   apply (clarsimp simp: and_mask_less_size wsst_TYs shiftr_less_t2n multiple_mask_trivia neg_mask_twice
                         word_bw_assocs max_absorb2 shiftr_shiftl1)
   done
 
 lemma word_clz_sint_upper[simp]:
   "LENGTH('a) \<ge> 3 \<Longrightarrow> sint (of_nat (word_clz (w :: 'a :: len word)) :: 'a sword) \<le> int (LENGTH('a))"
   using word_clz_max [of w]
   apply (simp add: word_size)
   apply (subst signed_take_bit_int_eq_self)
     apply simp_all
    apply (metis negative_zle of_nat_numeral semiring_1_class.of_nat_power)
   apply (drule small_powers_of_2)
   apply (erule le_less_trans)
   apply simp
   done
 
 lemma word_clz_sint_lower[simp]:
   "LENGTH('a) \<ge> 3
    \<Longrightarrow> - sint (of_nat (word_clz (w :: 'a :: len word)) :: 'a signed word) \<le> int (LENGTH('a))"
   apply (subst sint_eq_uint)
   using word_clz_max [of w]
    apply (simp_all add: word_size)
   apply (rule not_msb_from_less)
   apply (simp add: word_less_nat_alt)
   apply (subst take_bit_nat_eq_self)
    apply (simp add: le_less_trans)
   apply (drule small_powers_of_2)
   apply (erule le_less_trans)
   apply simp
   done
 
 lemma mask_range_subsetD:
   "\<lbrakk> p' \<in> mask_range p n; x' \<in> mask_range p' n'; n' \<le> n; is_aligned p n; is_aligned p' n' \<rbrakk> \<Longrightarrow>
    x' \<in> mask_range p n"
   using aligned_mask_step by fastforce
 
 lemma 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 add_mult_in_mask_range:
   "\<lbrakk> is_aligned (base :: 'a :: len word) n; n < LENGTH('a); bits \<le> n; x < 2 ^ (n - bits) \<rbrakk>
    \<Longrightarrow> base + x * 2^bits \<in> mask_range base n"
   by (simp add: is_aligned_no_wrap' mask_2pm1 nasty_split_lt word_less_power_trans2
                 word_plus_mono_right)
 
 lemma from_to_bool_last_bit:
   "from_bool (to_bool (x AND 1)) = x AND 1"
   by (metis from_bool_to_bool_iff word_and_1)
 
 lemma sint_ctz:
   "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> int (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)
   using small_powers_of_2 [of \<open>LENGTH('a)\<close>] by simp
 
 lemma unat_of_nat_word_log2:
   "LENGTH('a) < 2 ^ LENGTH('b)
    \<Longrightarrow> unat (of_nat (word_log2 (n :: 'a :: len word)) :: 'b :: len word) = word_log2 n"
   by (metis less_trans unat_of_nat_eq word_log2_max word_size)
 
 lemma aligned_mask_diff:
   "\<lbrakk> is_aligned (dest :: 'a :: len word) bits; is_aligned (ptr :: 'a :: len word) sz;
      bits \<le> sz; sz < LENGTH('a); dest < ptr \<rbrakk>
    \<Longrightarrow> mask bits + dest < ptr"
   apply (frule_tac p' = ptr in aligned_mask_range_cases, assumption)
   apply (elim disjE)
     apply (drule_tac is_aligned_no_overflow_mask, simp)+
     apply (simp add: algebra_split_simps word_le_not_less)
    apply (drule is_aligned_no_overflow_mask; fastforce)
   apply (simp add: is_aligned_weaken algebra_split_simps)
   apply (auto simp add: not_le)
   using is_aligned_no_overflow_mask leD apply blast
   apply (meson aligned_add_mask_less_eq is_aligned_weaken le_less_trans)
   done
 
 end
\ No newline at end of file
diff --git a/thys/Word_Lib/Most_significant_bit.thy b/thys/Word_Lib/Most_significant_bit.thy
--- a/thys/Word_Lib/Most_significant_bit.thy
+++ b/thys/Word_Lib/Most_significant_bit.thy
@@ -1,198 +1,203 @@
-(*  Author:     Jeremy Dawson, NICTA
-*)
+(*
+ * Copyright Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
+
+(* Author: Jeremy Dawson, NICTA *)
 
 section \<open>Dedicated operation for the most significant bit\<close>
 
 theory Most_significant_bit
   imports
     "HOL-Library.Word"
     Bits_Int
     Traditional_Infix_Syntax
     More_Arithmetic
 begin
 
 class msb =
   fixes msb :: \<open>'a \<Rightarrow> bool\<close>
 
 instantiation int :: msb
 begin
 
 definition \<open>msb x \<longleftrightarrow> x < 0\<close> for x :: int
 
 instance ..
 
 end
 
 lemma msb_conv_bin_sign: "msb x \<longleftrightarrow> bin_sign x = -1"
 by(simp add: bin_sign_def not_le msb_int_def)
 
 lemma msb_bin_rest [simp]: "msb (x div 2) = msb x"
   for x :: int
   by (simp add: msb_int_def)
 
 lemma int_msb_and [simp]: "msb ((x :: int) AND y) \<longleftrightarrow> msb x \<and> msb y"
 by(simp add: msb_int_def)
 
 lemma int_msb_or [simp]: "msb ((x :: int) OR y) \<longleftrightarrow> msb x \<or> msb y"
 by(simp add: msb_int_def)
 
 lemma int_msb_xor [simp]: "msb ((x :: int) XOR y) \<longleftrightarrow> msb x \<noteq> msb y"
 by(simp add: msb_int_def)
 
 lemma int_msb_not [simp]: "msb (NOT (x :: int)) \<longleftrightarrow> \<not> msb x"
 by(simp add: msb_int_def not_less)
 
 lemma msb_shiftl [simp]: "msb ((x :: int) << n) \<longleftrightarrow> msb x"
 by(simp add: msb_int_def)
 
 lemma msb_shiftr [simp]: "msb ((x :: int) >> r) \<longleftrightarrow> msb x"
 by(simp add: msb_int_def)
 
 lemma msb_bin_sc [simp]: "msb (bin_sc n b x) \<longleftrightarrow> msb x"
 by(simp add: msb_conv_bin_sign)
 
 lemma msb_0 [simp]: "msb (0 :: int) = False"
 by(simp add: msb_int_def)
 
 lemma msb_1 [simp]: "msb (1 :: int) = False"
 by(simp add: msb_int_def)
 
 lemma msb_numeral [simp]:
   "msb (numeral n :: int) = False"
   "msb (- numeral n :: int) = True"
 by(simp_all add: msb_int_def)
 
 instantiation word :: (len) msb
 begin
 
 definition msb_word :: \<open>'a word \<Rightarrow> bool\<close>
   where \<open>msb a \<longleftrightarrow> bin_sign (sbintrunc (LENGTH('a) - 1) (uint a)) = - 1\<close>
 
 lemma msb_word_eq:
   \<open>msb w \<longleftrightarrow> bit w (LENGTH('a) - 1)\<close> for w :: \<open>'a::len word\<close>
   by (simp add: msb_word_def bin_sign_lem bit_uint_iff)
 
 instance ..
 
 end
 
 lemma msb_word_iff_bit:
   \<open>msb w \<longleftrightarrow> bit w (LENGTH('a) - Suc 0)\<close>
   for w :: \<open>'a::len word\<close>
   by (simp add: msb_word_def bin_sign_def bit_uint_iff)
 
 lemma word_msb_def:
   "msb a \<longleftrightarrow> bin_sign (sint a) = - 1"
   by (simp add: msb_word_def sint_uint)
 
 lemma word_msb_sint: "msb w \<longleftrightarrow> sint w < 0"
   by (simp add: msb_word_eq bit_last_iff)
 
 lemma msb_word_iff_sless_0:
   \<open>msb w \<longleftrightarrow> w <s 0\<close>
   by (simp add: word_msb_sint word_sless_alt)
 
 lemma msb_word_of_int: "msb (word_of_int x::'a::len word) = bin_nth x (LENGTH('a) - 1)"
   by (simp add: word_msb_def bin_sign_lem)
 
 lemma word_msb_numeral [simp]:
   "msb (numeral w::'a::len word) = bin_nth (numeral w) (LENGTH('a) - 1)"
   unfolding word_numeral_alt by (rule msb_word_of_int)
 
 lemma word_msb_neg_numeral [simp]:
   "msb (- numeral w::'a::len word) = bin_nth (- numeral w) (LENGTH('a) - 1)"
   unfolding word_neg_numeral_alt by (rule msb_word_of_int)
 
 lemma word_msb_0 [simp]: "\<not> msb (0::'a::len word)"
   by (simp add: word_msb_def bin_sign_def sint_uint sbintrunc_eq_take_bit)
 
 lemma word_msb_1 [simp]: "msb (1::'a::len word) \<longleftrightarrow> LENGTH('a) = 1"
   unfolding word_1_wi msb_word_of_int eq_iff [where 'a=nat]
   by (simp add: Suc_le_eq)
 
 lemma word_msb_nth: "msb w = bin_nth (uint w) (LENGTH('a) - 1)"
   for w :: "'a::len word"
   by (simp add: word_msb_def sint_uint bin_sign_lem)
 
 lemma msb_nth: "msb w = w !! (LENGTH('a) - 1)"
   for w :: "'a::len word"
   by (simp add: word_msb_nth word_test_bit_def)
 
 lemma word_msb_n1 [simp]: "msb (-1::'a::len word)"
   by (simp add: msb_word_eq not_le)
 
 lemma msb_shift: "msb w \<longleftrightarrow> w >> (LENGTH('a) - 1) \<noteq> 0"
   for w :: "'a::len word"
   by (simp add: msb_word_eq shiftr_word_eq bit_iff_odd_drop_bit drop_bit_eq_zero_iff_not_bit_last)
 
 lemmas word_ops_msb = msb1 [unfolded msb_nth [symmetric, unfolded One_nat_def]]
 
 lemma word_sint_msb_eq: "sint x = uint x - (if msb x then 2 ^ size x else 0)"
   apply (cases \<open>LENGTH('a)\<close>)
    apply (simp_all add: msb_word_def bin_sign_def bit_simps word_size)
   apply transfer
   apply (auto simp add: take_bit_Suc_from_most signed_take_bit_eq_if_positive signed_take_bit_eq_if_negative minus_exp_eq_not_mask ac_simps)
   apply (subst disjunctive_add)
    apply (simp_all add: bit_simps)
   done
 
 lemma word_sle_msb_le: "x <=s y \<longleftrightarrow> (msb y \<longrightarrow> msb x) \<and> ((msb x \<and> \<not> msb y) \<or> x \<le> y)"
   apply (simp add: word_sle_eq word_sint_msb_eq word_size word_le_def)
   apply safe
    apply (rule order_trans[OF _ uint_ge_0])
    apply (simp add: order_less_imp_le)
   apply (erule notE[OF leD])
   apply (rule order_less_le_trans[OF _ uint_ge_0])
   apply simp
   done
 
 lemma word_sless_msb_less: "x <s y \<longleftrightarrow> (msb y \<longrightarrow> msb x) \<and> ((msb x \<and> \<not> msb y) \<or> x < y)"
   by (auto simp add: word_sless_eq word_sle_msb_le)
 
 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 sint_eq_uint:
   "\<not> msb x \<Longrightarrow> sint x = uint x"
   apply (simp add: msb_word_eq)
   apply transfer
   apply auto
   apply (smt One_nat_def bintrunc_bintrunc_l bintrunc_sbintrunc' diff_le_self len_gt_0 signed_take_bit_eq_if_positive)
   done
 
 lemma scast_eq_ucast:
   "\<not> msb x \<Longrightarrow> scast x = ucast x"
   apply (cases \<open>LENGTH('a)\<close>)
   apply simp
   apply (rule bit_word_eqI)
   apply (auto simp add: bit_signed_iff bit_unsigned_iff min_def msb_word_eq)
   apply (erule notE)
   apply (metis le_less_Suc_eq test_bit_bin test_bit_word_eq)
   done
 
 lemma msb_ucast_eq:
     "LENGTH('a) = LENGTH('b) \<Longrightarrow>
          msb (ucast x :: ('a::len) word) = msb (x :: ('b::len) word)"
   by (simp add: msb_word_eq bit_simps)
 
 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 AND 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
 
 end
diff --git a/thys/Word_Lib/Next_and_Prev.thy b/thys/Word_Lib/Next_and_Prev.thy
--- a/thys/Word_Lib/Next_and_Prev.thy
+++ b/thys/Word_Lib/Next_and_Prev.thy
@@ -1,48 +1,51 @@
-(* SPDX-License-Identifier: BSD-3-Clause *)
+(*
+ * Copyright Julius Michaelis, Cornelius Diekmann
+ * SPDX-License-Identifier: BSD-3-Clause
+ *)
 
 section\<open>Increment and Decrement Machine Words Without Wrap-Around\<close>
 
 theory Next_and_Prev
 imports
   Aligned
 begin
 
 text \<open>Previous and next words addresses, without wrap around.\<close>
 
 lift_definition word_next :: \<open>'a::len word \<Rightarrow> 'a word\<close>
   is \<open>\<lambda>k. if 2 ^ LENGTH('a) dvd k + 1 then - 1 else k + 1\<close>
   by (simp flip: take_bit_eq_0_iff) (metis take_bit_add)
 
 lift_definition word_prev :: \<open>'a::len word \<Rightarrow> 'a word\<close>
   is \<open>\<lambda>k. if 2 ^ LENGTH('a) dvd k then 0 else k - 1\<close>
   by (simp flip: take_bit_eq_0_iff) (metis take_bit_diff)
 
 lemma word_next_unfold:
   \<open>word_next w = (if w = - 1 then - 1 else w + 1)\<close>
   by transfer (simp add: take_bit_minus_one_eq_mask flip: take_bit_eq_mask_iff_exp_dvd)
 
 lemma word_prev_unfold:
   \<open>word_prev w = (if w = 0 then 0 else w - 1)\<close>
   by transfer (simp flip: take_bit_eq_0_iff)
 
 lemma [code]:
   \<open>Word.the_int (word_next w :: 'a::len word) =
     (if w = - 1 then Word.the_int w else Word.the_int w + 1)\<close>
   by transfer
-    (simp add: take_bit_minus_one_eq_mask mask_eq_exp_minus_1 take_bit_incr_eq flip: take_bit_eq_mask_iff_exp_dvd) 
+    (simp add: take_bit_minus_one_eq_mask mask_eq_exp_minus_1 take_bit_incr_eq flip: take_bit_eq_mask_iff_exp_dvd)
 
 lemma [code]:
   \<open>Word.the_int (word_prev w :: 'a::len word) =
     (if w = 0 then Word.the_int w else Word.the_int w - 1)\<close>
   by transfer (simp add: take_bit_eq_0_iff take_bit_decr_eq)
 
 lemma word_adjacent_union:
   "word_next e = s' \<Longrightarrow> s \<le> e \<Longrightarrow> s' \<le> e' \<Longrightarrow> {s..e} \<union> {s'..e'} = {s .. e'}"
   apply (simp add: word_next_unfold ivl_disj_un_two_touch split: if_splits)
   apply (drule sym)
   apply simp
   apply (subst word_atLeastLessThan_Suc_atLeastAtMost_union)
      apply (simp_all add: word_Suc_le)
   done
 
 end
diff --git a/thys/Word_Lib/Reversed_Bit_Lists.thy b/thys/Word_Lib/Reversed_Bit_Lists.thy
--- a/thys/Word_Lib/Reversed_Bit_Lists.thy
+++ b/thys/Word_Lib/Reversed_Bit_Lists.thy
@@ -1,2191 +1,2196 @@
-(*  Author:     Jeremy Dawson, NICTA
-*)
+(*
+ * Copyright Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
+
+(* Author: Jeremy Dawson, NICTA *)
 
 section \<open>Bit values as reversed lists of bools\<close>
 
 theory Reversed_Bit_Lists
   imports
     "HOL-Library.Word"
     Typedef_Morphisms
     Least_significant_bit
     Most_significant_bit
     Even_More_List
     "HOL-Library.Sublist"
     Aligned
 begin
 
 lemma horner_sum_of_bool_2_concat:
   \<open>horner_sum of_bool 2 (concat (map (\<lambda>x. map (bit x) [0..<LENGTH('a)]) ws)) = horner_sum uint (2 ^ LENGTH('a)) ws\<close>
   for ws :: \<open>'a::len word list\<close>
 proof (induction ws)
   case Nil
   then show ?case
     by simp
 next
   case (Cons w ws)
   moreover have \<open>horner_sum of_bool 2 (map (bit w) [0..<LENGTH('a)]) = uint w\<close>
   proof transfer
     fix k :: int
     have \<open>map (\<lambda>n. n < LENGTH('a) \<and> bit k n) [0..<LENGTH('a)]
       = map (bit k) [0..<LENGTH('a)]\<close>
       by simp
     then show \<open>horner_sum of_bool 2 (map (\<lambda>n. n < LENGTH('a) \<and> bit k n) [0..<LENGTH('a)])
       = take_bit LENGTH('a) k\<close>
       by (simp only: horner_sum_bit_eq_take_bit)
   qed
   ultimately show ?case
     by (simp add: horner_sum_append)
 qed
 
 
 subsection \<open>Implicit augmentation of list prefixes\<close>
 
 primrec takefill :: "'a \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
 where
     Z: "takefill fill 0 xs = []"
   | Suc: "takefill fill (Suc n) xs =
       (case xs of
         [] \<Rightarrow> fill # takefill fill n xs
       | y # ys \<Rightarrow> y # takefill fill n ys)"
 
 lemma nth_takefill: "m < n \<Longrightarrow> takefill fill n l ! m = (if m < length l then l ! m else fill)"
   apply (induct n arbitrary: m l)
    apply clarsimp
   apply clarsimp
   apply (case_tac m)
    apply (simp split: list.split)
   apply (simp split: list.split)
   done
 
 lemma takefill_alt: "takefill fill n l = take n l @ replicate (n - length l) fill"
   by (induct n arbitrary: l) (auto split: list.split)
 
 lemma takefill_replicate [simp]: "takefill fill n (replicate m fill) = replicate n fill"
   by (simp add: takefill_alt replicate_add [symmetric])
 
 lemma takefill_le': "n = m + k \<Longrightarrow> takefill x m (takefill x n l) = takefill x m l"
   by (induct m arbitrary: l n) (auto split: list.split)
 
 lemma length_takefill [simp]: "length (takefill fill n l) = n"
   by (simp add: takefill_alt)
 
 lemma take_takefill': "n = k + m \<Longrightarrow> take k (takefill fill n w) = takefill fill k w"
   by (induct k arbitrary: w n) (auto split: list.split)
 
 lemma drop_takefill: "drop k (takefill fill (m + k) w) = takefill fill m (drop k w)"
   by (induct k arbitrary: w) (auto split: list.split)
 
 lemma takefill_le [simp]: "m \<le> n \<Longrightarrow> takefill x m (takefill x n l) = takefill x m l"
   by (auto simp: le_iff_add takefill_le')
 
 lemma take_takefill [simp]: "m \<le> n \<Longrightarrow> take m (takefill fill n w) = takefill fill m w"
   by (auto simp: le_iff_add take_takefill')
 
 lemma takefill_append: "takefill fill (m + length xs) (xs @ w) = xs @ (takefill fill m w)"
   by (induct xs) auto
 
 lemma takefill_same': "l = length xs \<Longrightarrow> takefill fill l xs = xs"
   by (induct xs arbitrary: l) auto
 
 lemmas takefill_same [simp] = takefill_same' [OF refl]
 
 lemma tf_rev:
   "n + k = m + length bl \<Longrightarrow> takefill x m (rev (takefill y n bl)) =
     rev (takefill y m (rev (takefill x k (rev bl))))"
   apply (rule nth_equalityI)
    apply (auto simp add: nth_takefill rev_nth)
   apply (rule_tac f = "\<lambda>n. bl ! n" in arg_cong)
   apply arith
   done
 
 lemma takefill_minus: "0 < n \<Longrightarrow> takefill fill (Suc (n - 1)) w = takefill fill n w"
   by auto
 
 lemmas takefill_Suc_cases =
   list.cases [THEN takefill.Suc [THEN trans]]
 
 lemmas takefill_Suc_Nil = takefill_Suc_cases (1)
 lemmas takefill_Suc_Cons = takefill_Suc_cases (2)
 
 lemmas takefill_minus_simps = takefill_Suc_cases [THEN [2]
   takefill_minus [symmetric, THEN trans]]
 
 lemma takefill_numeral_Nil [simp]:
   "takefill fill (numeral k) [] = fill # takefill fill (pred_numeral k) []"
   by (simp add: numeral_eq_Suc)
 
 lemma takefill_numeral_Cons [simp]:
   "takefill fill (numeral k) (x # xs) = x # takefill fill (pred_numeral k) xs"
   by (simp add: numeral_eq_Suc)
 
 
 subsection \<open>Range projection\<close>
 
 definition bl_of_nth :: "nat \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> 'a list"
   where "bl_of_nth n f = map f (rev [0..<n])"
 
 lemma bl_of_nth_simps [simp, code]:
   "bl_of_nth 0 f = []"
   "bl_of_nth (Suc n) f = f n # bl_of_nth n f"
   by (simp_all add: bl_of_nth_def)
 
 lemma length_bl_of_nth [simp]: "length (bl_of_nth n f) = n"
   by (simp add: bl_of_nth_def)
 
 lemma nth_bl_of_nth [simp]: "m < n \<Longrightarrow> rev (bl_of_nth n f) ! m = f m"
   by (simp add: bl_of_nth_def rev_map)
 
 lemma bl_of_nth_inj: "(\<And>k. k < n \<Longrightarrow> f k = g k) \<Longrightarrow> bl_of_nth n f = bl_of_nth n g"
   by (simp add: bl_of_nth_def)
 
 lemma bl_of_nth_nth_le: "n \<le> length xs \<Longrightarrow> bl_of_nth n (nth (rev xs)) = drop (length xs - n) xs"
   apply (induct n arbitrary: xs)
    apply clarsimp
   apply clarsimp
   apply (rule trans [OF _ hd_Cons_tl])
    apply (frule Suc_le_lessD)
    apply (simp add: rev_nth trans [OF drop_Suc drop_tl, symmetric])
    apply (subst hd_drop_conv_nth)
     apply force
    apply simp_all
   apply (rule_tac f = "\<lambda>n. drop n xs" in arg_cong)
   apply simp
   done
 
 lemma bl_of_nth_nth [simp]: "bl_of_nth (length xs) ((!) (rev xs)) = xs"
   by (simp add: bl_of_nth_nth_le)
 
 
 subsection \<open>More\<close>
 
 definition rotater1 :: "'a list \<Rightarrow> 'a list"
   where "rotater1 ys =
     (case ys of [] \<Rightarrow> [] | x # xs \<Rightarrow> last ys # butlast ys)"
 
 definition rotater :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
   where "rotater n = rotater1 ^^ n"
 
 lemmas rotater_0' [simp] = rotater_def [where n = "0", simplified]
 
 lemma rotate1_rl': "rotater1 (l @ [a]) = a # l"
   by (cases l) (auto simp: rotater1_def)
 
 lemma rotate1_rl [simp] : "rotater1 (rotate1 l) = l"
   apply (unfold rotater1_def)
   apply (cases "l")
    apply (case_tac [2] "list")
     apply auto
   done
 
 lemma rotate1_lr [simp] : "rotate1 (rotater1 l) = l"
   by (cases l) (auto simp: rotater1_def)
 
 lemma rotater1_rev': "rotater1 (rev xs) = rev (rotate1 xs)"
   by (cases "xs") (simp add: rotater1_def, simp add: rotate1_rl')
 
 lemma rotater_rev': "rotater n (rev xs) = rev (rotate n xs)"
   by (induct n) (auto simp: rotater_def intro: rotater1_rev')
 
 lemma rotater_rev: "rotater n ys = rev (rotate n (rev ys))"
   using rotater_rev' [where xs = "rev ys"] by simp
 
 lemma rotater_drop_take:
   "rotater n xs =
     drop (length xs - n mod length xs) xs @
     take (length xs - n mod length xs) xs"
   by (auto simp: rotater_rev rotate_drop_take rev_take rev_drop)
 
 lemma rotater_Suc [simp]: "rotater (Suc n) xs = rotater1 (rotater n xs)"
   unfolding rotater_def by auto
 
 lemma nth_rotater:
   \<open>rotater m xs ! n = xs ! ((n + (length xs - m mod length xs)) mod length xs)\<close> if \<open>n < length xs\<close>
   using that by (simp add: rotater_drop_take nth_append not_less less_diff_conv ac_simps le_mod_geq)
 
 lemma nth_rotater1:
   \<open>rotater1 xs ! n = xs ! ((n + (length xs - 1)) mod length xs)\<close> if \<open>n < length xs\<close>
   using that nth_rotater [of n xs 1] by simp
 
 lemma rotate_inv_plus [rule_format]:
   "\<forall>k. k = m + n \<longrightarrow> rotater k (rotate n xs) = rotater m xs \<and>
     rotate k (rotater n xs) = rotate m xs \<and>
     rotater n (rotate k xs) = rotate m xs \<and>
     rotate n (rotater k xs) = rotater m xs"
   by (induct n) (auto simp: rotater_def rotate_def intro: funpow_swap1 [THEN trans])
 
 lemmas rotate_inv_rel = le_add_diff_inverse2 [symmetric, THEN rotate_inv_plus]
 
 lemmas rotate_inv_eq = order_refl [THEN rotate_inv_rel, simplified]
 
 lemmas rotate_lr [simp] = rotate_inv_eq [THEN conjunct1]
 lemmas rotate_rl [simp] = rotate_inv_eq [THEN conjunct2, THEN conjunct1]
 
 lemma rotate_gal: "rotater n xs = ys \<longleftrightarrow> rotate n ys = xs"
   by auto
 
 lemma rotate_gal': "ys = rotater n xs \<longleftrightarrow> xs = rotate n ys"
   by auto
 
 lemma length_rotater [simp]: "length (rotater n xs) = length xs"
   by (simp add : rotater_rev)
 
 lemma rotate_eq_mod: "m mod length xs = n mod length xs \<Longrightarrow> rotate m xs = rotate n xs"
   apply (rule box_equals)
     defer
     apply (rule rotate_conv_mod [symmetric])+
   apply simp
   done
 
 lemma restrict_to_left: "x = y \<Longrightarrow> x = z \<longleftrightarrow> y = z"
   by simp
 
 lemmas rotate_eqs =
   trans [OF rotate0 [THEN fun_cong] id_apply]
   rotate_rotate [symmetric]
   rotate_id
   rotate_conv_mod
   rotate_eq_mod
 
 lemmas rrs0 = rotate_eqs [THEN restrict_to_left,
   simplified rotate_gal [symmetric] rotate_gal' [symmetric]]
 lemmas rrs1 = rrs0 [THEN refl [THEN rev_iffD1]]
 lemmas rotater_eqs = rrs1 [simplified length_rotater]
 lemmas rotater_0 = rotater_eqs (1)
 lemmas rotater_add = rotater_eqs (2)
 
 lemma butlast_map: "xs \<noteq> [] \<Longrightarrow> butlast (map f xs) = map f (butlast xs)"
   by (induct xs) auto
 
 lemma rotater1_map: "rotater1 (map f xs) = map f (rotater1 xs)"
   by (cases xs) (auto simp: rotater1_def last_map butlast_map)
 
 lemma rotater_map: "rotater n (map f xs) = map f (rotater n xs)"
   by (induct n) (auto simp: rotater_def rotater1_map)
 
 lemma but_last_zip [rule_format] :
   "\<forall>ys. length xs = length ys \<longrightarrow> xs \<noteq> [] \<longrightarrow>
     last (zip xs ys) = (last xs, last ys) \<and>
     butlast (zip xs ys) = zip (butlast xs) (butlast ys)"
   apply (induct xs)
    apply auto
      apply ((case_tac ys, auto simp: neq_Nil_conv)[1])+
   done
 
 lemma but_last_map2 [rule_format] :
   "\<forall>ys. length xs = length ys \<longrightarrow> xs \<noteq> [] \<longrightarrow>
     last (map2 f xs ys) = f (last xs) (last ys) \<and>
     butlast (map2 f xs ys) = map2 f (butlast xs) (butlast ys)"
   apply (induct xs)
    apply auto
      apply ((case_tac ys, auto simp: neq_Nil_conv)[1])+
   done
 
 lemma rotater1_zip:
   "length xs = length ys \<Longrightarrow>
     rotater1 (zip xs ys) = zip (rotater1 xs) (rotater1 ys)"
   apply (unfold rotater1_def)
   apply (cases xs)
    apply auto
    apply ((case_tac ys, auto simp: neq_Nil_conv but_last_zip)[1])+
   done
 
 lemma rotater1_map2:
   "length xs = length ys \<Longrightarrow>
     rotater1 (map2 f xs ys) = map2 f (rotater1 xs) (rotater1 ys)"
   by (simp add: rotater1_map rotater1_zip)
 
 lemmas lrth =
   box_equals [OF asm_rl length_rotater [symmetric]
                  length_rotater [symmetric],
               THEN rotater1_map2]
 
 lemma rotater_map2:
   "length xs = length ys \<Longrightarrow>
     rotater n (map2 f xs ys) = map2 f (rotater n xs) (rotater n ys)"
   by (induct n) (auto intro!: lrth)
 
 lemma rotate1_map2:
   "length xs = length ys \<Longrightarrow>
     rotate1 (map2 f xs ys) = map2 f (rotate1 xs) (rotate1 ys)"
   by (cases xs; cases ys) auto
 
 lemmas lth = box_equals [OF asm_rl length_rotate [symmetric]
   length_rotate [symmetric], THEN rotate1_map2]
 
 lemma rotate_map2:
   "length xs = length ys \<Longrightarrow>
     rotate n (map2 f xs ys) = map2 f (rotate n xs) (rotate n ys)"
   by (induct n) (auto intro!: lth)
 
 
 subsection \<open>Explicit bit representation of \<^typ>\<open>int\<close>\<close>
 
 primrec bl_to_bin_aux :: "bool list \<Rightarrow> int \<Rightarrow> int"
   where
     Nil: "bl_to_bin_aux [] w = w"
   | Cons: "bl_to_bin_aux (b # bs) w = bl_to_bin_aux bs (of_bool b + 2 * w)"
 
 definition bl_to_bin :: "bool list \<Rightarrow> int"
   where "bl_to_bin bs = bl_to_bin_aux bs 0"
 
 primrec bin_to_bl_aux :: "nat \<Rightarrow> int \<Rightarrow> bool list \<Rightarrow> bool list"
   where
     Z: "bin_to_bl_aux 0 w bl = bl"
   | Suc: "bin_to_bl_aux (Suc n) w bl = bin_to_bl_aux n (bin_rest w) ((bin_last w) # bl)"
 
 definition bin_to_bl :: "nat \<Rightarrow> int \<Rightarrow> bool list"
   where "bin_to_bl n w = bin_to_bl_aux n w []"
 
 lemma bin_to_bl_aux_zero_minus_simp [simp]:
   "0 < n \<Longrightarrow> bin_to_bl_aux n 0 bl = bin_to_bl_aux (n - 1) 0 (False # bl)"
   by (cases n) auto
 
 lemma bin_to_bl_aux_minus1_minus_simp [simp]:
   "0 < n \<Longrightarrow> bin_to_bl_aux n (- 1) bl = bin_to_bl_aux (n - 1) (- 1) (True # bl)"
   by (cases n) auto
 
 lemma bin_to_bl_aux_one_minus_simp [simp]:
   "0 < n \<Longrightarrow> bin_to_bl_aux n 1 bl = bin_to_bl_aux (n - 1) 0 (True # bl)"
   by (cases n) auto
 
 lemma bin_to_bl_aux_Bit0_minus_simp [simp]:
   "0 < n \<Longrightarrow>
     bin_to_bl_aux n (numeral (Num.Bit0 w)) bl = bin_to_bl_aux (n - 1) (numeral w) (False # bl)"
   by (cases n) simp_all
 
 lemma bin_to_bl_aux_Bit1_minus_simp [simp]:
   "0 < n \<Longrightarrow>
     bin_to_bl_aux n (numeral (Num.Bit1 w)) bl = bin_to_bl_aux (n - 1) (numeral w) (True # bl)"
   by (cases n) simp_all
 
 lemma bl_to_bin_aux_append: "bl_to_bin_aux (bs @ cs) w = bl_to_bin_aux cs (bl_to_bin_aux bs w)"
   by (induct bs arbitrary: w) auto
 
 lemma bin_to_bl_aux_append: "bin_to_bl_aux n w bs @ cs = bin_to_bl_aux n w (bs @ cs)"
   by (induct n arbitrary: w bs) auto
 
 lemma bl_to_bin_append: "bl_to_bin (bs @ cs) = bl_to_bin_aux cs (bl_to_bin bs)"
   unfolding bl_to_bin_def by (rule bl_to_bin_aux_append)
 
 lemma bin_to_bl_aux_alt: "bin_to_bl_aux n w bs = bin_to_bl n w @ bs"
   by (simp add: bin_to_bl_def bin_to_bl_aux_append)
 
 lemma bin_to_bl_0 [simp]: "bin_to_bl 0 bs = []"
   by (auto simp: bin_to_bl_def)
 
 lemma size_bin_to_bl_aux: "length (bin_to_bl_aux n w bs) = n + length bs"
   by (induct n arbitrary: w bs) auto
 
 lemma size_bin_to_bl [simp]: "length (bin_to_bl n w) = n"
   by (simp add: bin_to_bl_def size_bin_to_bl_aux)
 
 lemma bl_bin_bl': "bin_to_bl (n + length bs) (bl_to_bin_aux bs w) = bin_to_bl_aux n w bs"
   apply (induct bs arbitrary: w n)
    apply auto
     apply (simp_all only: add_Suc [symmetric])
     apply (auto simp add: bin_to_bl_def)
   done
 
 lemma bl_bin_bl [simp]: "bin_to_bl (length bs) (bl_to_bin bs) = bs"
   unfolding bl_to_bin_def
   apply (rule box_equals)
     apply (rule bl_bin_bl')
    prefer 2
    apply (rule bin_to_bl_aux.Z)
   apply simp
   done
 
 lemma bl_to_bin_inj: "bl_to_bin bs = bl_to_bin cs \<Longrightarrow> length bs = length cs \<Longrightarrow> bs = cs"
   apply (rule_tac box_equals)
     defer
     apply (rule bl_bin_bl)
    apply (rule bl_bin_bl)
   apply simp
   done
 
 lemma bl_to_bin_False [simp]: "bl_to_bin (False # bl) = bl_to_bin bl"
   by (auto simp: bl_to_bin_def)
 
 lemma bl_to_bin_Nil [simp]: "bl_to_bin [] = 0"
   by (auto simp: bl_to_bin_def)
 
 lemma bin_to_bl_zero_aux: "bin_to_bl_aux n 0 bl = replicate n False @ bl"
   by (induct n arbitrary: bl) (auto simp: replicate_app_Cons_same)
 
 lemma bin_to_bl_zero: "bin_to_bl n 0 = replicate n False"
   by (simp add: bin_to_bl_def bin_to_bl_zero_aux)
 
 lemma bin_to_bl_minus1_aux: "bin_to_bl_aux n (- 1) bl = replicate n True @ bl"
   by (induct n arbitrary: bl) (auto simp: replicate_app_Cons_same)
 
 lemma bin_to_bl_minus1: "bin_to_bl n (- 1) = replicate n True"
   by (simp add: bin_to_bl_def bin_to_bl_minus1_aux)
 
 
 subsection \<open>Semantic interpretation of \<^typ>\<open>bool list\<close> as \<^typ>\<open>int\<close>\<close>
 
 lemma bin_bl_bin': "bl_to_bin (bin_to_bl_aux n w bs) = bl_to_bin_aux bs (bintrunc n w)"
   by (induct n arbitrary: w bs) (auto simp: bl_to_bin_def take_bit_Suc ac_simps mod_2_eq_odd)
 
 lemma bin_bl_bin [simp]: "bl_to_bin (bin_to_bl n w) = bintrunc n w"
   by (auto simp: bin_to_bl_def bin_bl_bin')
 
 lemma bl_to_bin_rep_F: "bl_to_bin (replicate n False @ bl) = bl_to_bin bl"
   by (simp add: bin_to_bl_zero_aux [symmetric] bin_bl_bin') (simp add: bl_to_bin_def)
 
 lemma bin_to_bl_trunc [simp]: "n \<le> m \<Longrightarrow> bin_to_bl n (bintrunc m w) = bin_to_bl n w"
   by (auto intro: bl_to_bin_inj)
 
 lemma bin_to_bl_aux_bintr:
   "bin_to_bl_aux n (bintrunc m bin) bl =
     replicate (n - m) False @ bin_to_bl_aux (min n m) bin bl"
   apply (induct n arbitrary: m bin bl)
    apply clarsimp
   apply clarsimp
   apply (case_tac "m")
    apply (clarsimp simp: bin_to_bl_zero_aux)
    apply (erule thin_rl)
    apply (induct_tac n)
     apply (auto simp add: take_bit_Suc)
   done
 
 lemma bin_to_bl_bintr:
   "bin_to_bl n (bintrunc m bin) = replicate (n - m) False @ bin_to_bl (min n m) bin"
   unfolding bin_to_bl_def by (rule bin_to_bl_aux_bintr)
 
 lemma bl_to_bin_rep_False: "bl_to_bin (replicate n False) = 0"
   by (induct n) auto
 
 lemma len_bin_to_bl_aux: "length (bin_to_bl_aux n w bs) = n + length bs"
   by (fact size_bin_to_bl_aux)
 
 lemma len_bin_to_bl: "length (bin_to_bl n w) = n"
   by (fact size_bin_to_bl) (* FIXME: duplicate *)
 
 lemma sign_bl_bin': "bin_sign (bl_to_bin_aux bs w) = bin_sign w"
   by (induction bs arbitrary: w) (simp_all add: bin_sign_def)
 
 lemma sign_bl_bin: "bin_sign (bl_to_bin bs) = 0"
   by (simp add: bl_to_bin_def sign_bl_bin')
 
 lemma bl_sbin_sign_aux: "hd (bin_to_bl_aux (Suc n) w bs) = (bin_sign (sbintrunc n w) = -1)"
   by (induction n arbitrary: w bs) (auto simp add: bin_sign_def even_iff_mod_2_eq_zero bit_Suc)
 
 lemma bl_sbin_sign: "hd (bin_to_bl (Suc n) w) = (bin_sign (sbintrunc n w) = -1)"
   unfolding bin_to_bl_def by (rule bl_sbin_sign_aux)
 
 lemma bin_nth_of_bl_aux:
   "bin_nth (bl_to_bin_aux bl w) n =
     (n < size bl \<and> rev bl ! n \<or> n \<ge> length bl \<and> bin_nth w (n - size bl))"
   apply (induction bl arbitrary: w)
    apply simp_all
   apply safe
                       apply (simp_all add: not_le nth_append bit_double_iff even_bit_succ_iff split: if_splits)
   done
 
 lemma bin_nth_of_bl: "bin_nth (bl_to_bin bl) n = (n < length bl \<and> rev bl ! n)"
   by (simp add: bl_to_bin_def bin_nth_of_bl_aux)
 
 lemma bin_nth_bl: "n < m \<Longrightarrow> bin_nth w n = nth (rev (bin_to_bl m w)) n"
   apply (induct n arbitrary: m w)
    apply clarsimp
    apply (case_tac m, clarsimp)
    apply (clarsimp simp: bin_to_bl_def)
    apply (simp add: bin_to_bl_aux_alt)
   apply (case_tac m, clarsimp)
   apply (clarsimp simp: bin_to_bl_def)
   apply (simp add: bin_to_bl_aux_alt bit_Suc)
   done
 
 lemma nth_bin_to_bl_aux:
   "n < m + length bl \<Longrightarrow> (bin_to_bl_aux m w bl) ! n =
     (if n < m then bit w (m - 1 - n) else bl ! (n - m))"
   apply (induction bl arbitrary: w)
    apply simp_all
    apply (simp add: bin_nth_bl [of \<open>m - Suc n\<close> m] rev_nth flip: bin_to_bl_def)
    apply (metis One_nat_def Suc_pred add_diff_cancel_left'
      add_diff_cancel_right' bin_to_bl_aux_alt bin_to_bl_def
      diff_Suc_Suc diff_is_0_eq diff_zero less_Suc_eq_0_disj
      less_antisym less_imp_Suc_add list.size(3) nat_less_le nth_append size_bin_to_bl_aux)
   done
 
 lemma nth_bin_to_bl: "n < m \<Longrightarrow> (bin_to_bl m w) ! n = bin_nth w (m - Suc n)"
   by (simp add: bin_to_bl_def nth_bin_to_bl_aux)
 
 lemma takefill_bintrunc: "takefill False n bl = rev (bin_to_bl n (bl_to_bin (rev bl)))"
   apply (rule nth_equalityI)
    apply simp
   apply (clarsimp simp: nth_takefill rev_nth nth_bin_to_bl bin_nth_of_bl)
   done
 
 lemma bl_bin_bl_rtf: "bin_to_bl n (bl_to_bin bl) = rev (takefill False n (rev bl))"
   by (simp add: takefill_bintrunc)
 
 lemma bl_to_bin_lt2p_aux: "bl_to_bin_aux bs w < (w + 1) * (2 ^ length bs)"
 proof (induction bs arbitrary: w)
   case Nil
   then show ?case
     by simp
 next
   case (Cons b bs)
   from Cons.IH [of \<open>1 + 2 * w\<close>] Cons.IH [of \<open>2 * w\<close>]
   show ?case
     apply (auto simp add: algebra_simps)
     apply (subst mult_2 [of \<open>2 ^ length bs\<close>])
     apply (simp only: add.assoc)
     apply (rule pos_add_strict)
      apply simp_all
     done
 qed
 
 lemma bl_to_bin_lt2p_drop: "bl_to_bin bs < 2 ^ length (dropWhile Not bs)"
 proof (induct bs)
   case Nil
   then show ?case by simp
 next
   case (Cons b bs)
   with bl_to_bin_lt2p_aux[where w=1] show ?case
     by (simp add: bl_to_bin_def)
 qed
 
 lemma bl_to_bin_lt2p: "bl_to_bin bs < 2 ^ length bs"
   by (metis bin_bl_bin bintr_lt2p bl_bin_bl)
 
 lemma bl_to_bin_ge2p_aux: "bl_to_bin_aux bs w \<ge> w * (2 ^ length bs)"
 proof (induction bs arbitrary: w)
   case Nil
   then show ?case
     by simp
 next
   case (Cons b bs)
   from Cons.IH [of \<open>1 + 2 * w\<close>] Cons.IH [of \<open>2 * w\<close>]
   show ?case
     apply (auto simp add: algebra_simps)
     apply (rule add_le_imp_le_left [of \<open>2 ^ length bs\<close>])
     apply (rule add_increasing)
     apply simp_all
     done
 qed
 
 lemma bl_to_bin_ge0: "bl_to_bin bs \<ge> 0"
   apply (unfold bl_to_bin_def)
   apply (rule xtrans(4))
    apply (rule bl_to_bin_ge2p_aux)
   apply simp
   done
 
 lemma butlast_rest_bin: "butlast (bin_to_bl n w) = bin_to_bl (n - 1) (bin_rest w)"
   apply (unfold bin_to_bl_def)
   apply (cases n, clarsimp)
   apply clarsimp
   apply (auto simp add: bin_to_bl_aux_alt)
   done
 
 lemma butlast_bin_rest: "butlast bl = bin_to_bl (length bl - Suc 0) (bin_rest (bl_to_bin bl))"
   using butlast_rest_bin [where w="bl_to_bin bl" and n="length bl"] by simp
 
 lemma butlast_rest_bl2bin_aux:
   "bl \<noteq> [] \<Longrightarrow> bl_to_bin_aux (butlast bl) w = bin_rest (bl_to_bin_aux bl w)"
   by (induct bl arbitrary: w) auto
 
 lemma butlast_rest_bl2bin: "bl_to_bin (butlast bl) = bin_rest (bl_to_bin bl)"
   by (cases bl) (auto simp: bl_to_bin_def butlast_rest_bl2bin_aux)
 
 lemma trunc_bl2bin_aux:
   "bintrunc m (bl_to_bin_aux bl w) =
     bl_to_bin_aux (drop (length bl - m) bl) (bintrunc (m - length bl) w)"
 proof (induct bl arbitrary: w)
   case Nil
   show ?case by simp
 next
   case (Cons b bl)
   show ?case
   proof (cases "m - length bl")
     case 0
     then have "Suc (length bl) - m = Suc (length bl - m)" by simp
     with Cons show ?thesis by simp
   next
     case (Suc n)
     then have "m - Suc (length bl) = n" by simp
     with Cons Suc show ?thesis by (simp add: take_bit_Suc ac_simps)
   qed
 qed
 
 lemma trunc_bl2bin: "bintrunc m (bl_to_bin bl) = bl_to_bin (drop (length bl - m) bl)"
   by (simp add: bl_to_bin_def trunc_bl2bin_aux)
 
 lemma trunc_bl2bin_len [simp]: "bintrunc (length bl) (bl_to_bin bl) = bl_to_bin bl"
   by (simp add: trunc_bl2bin)
 
 lemma bl2bin_drop: "bl_to_bin (drop k bl) = bintrunc (length bl - k) (bl_to_bin bl)"
   apply (rule trans)
    prefer 2
    apply (rule trunc_bl2bin [symmetric])
   apply (cases "k \<le> length bl")
    apply auto
   done
 
 lemma take_rest_power_bin: "m \<le> n \<Longrightarrow> take m (bin_to_bl n w) = bin_to_bl m ((bin_rest ^^ (n - m)) w)"
   apply (rule nth_equalityI)
    apply simp
   apply (clarsimp simp add: nth_bin_to_bl nth_rest_power_bin)
   done
 
 lemma last_bin_last': "size xs > 0 \<Longrightarrow> last xs \<longleftrightarrow> bin_last (bl_to_bin_aux xs w)"
   by (induct xs arbitrary: w) auto
 
 lemma last_bin_last: "size xs > 0 \<Longrightarrow> last xs \<longleftrightarrow> bin_last (bl_to_bin xs)"
   unfolding bl_to_bin_def by (erule last_bin_last')
 
 lemma bin_last_last: "bin_last w \<longleftrightarrow> last (bin_to_bl (Suc n) w)"
   by (simp add: bin_to_bl_def) (auto simp: bin_to_bl_aux_alt)
 
 lemma drop_bin2bl_aux:
   "drop m (bin_to_bl_aux n bin bs) =
     bin_to_bl_aux (n - m) bin (drop (m - n) bs)"
   apply (induction n arbitrary: m bin bs)
    apply auto
   apply (case_tac "m \<le> n")
    apply (auto simp add: not_le Suc_diff_le)
   apply (case_tac "m - n")
    apply auto
   apply (use Suc_diff_Suc in fastforce)
   done
 
 lemma drop_bin2bl: "drop m (bin_to_bl n bin) = bin_to_bl (n - m) bin"
   by (simp add: bin_to_bl_def drop_bin2bl_aux)
 
 lemma take_bin2bl_lem1: "take m (bin_to_bl_aux m w bs) = bin_to_bl m w"
   apply (induct m arbitrary: w bs)
    apply clarsimp
   apply clarsimp
   apply (simp add: bin_to_bl_aux_alt)
   apply (simp add: bin_to_bl_def)
   apply (simp add: bin_to_bl_aux_alt)
   done
 
 lemma take_bin2bl_lem: "take m (bin_to_bl_aux (m + n) w bs) = take m (bin_to_bl (m + n) w)"
   by (induct n arbitrary: w bs) (simp_all (no_asm) add: bin_to_bl_def take_bin2bl_lem1, simp)
 
 lemma bin_split_take: "bin_split n c = (a, b) \<Longrightarrow> bin_to_bl m a = take m (bin_to_bl (m + n) c)"
   apply (induct n arbitrary: b c)
    apply clarsimp
   apply (clarsimp simp: Let_def split: prod.split_asm)
   apply (simp add: bin_to_bl_def)
   apply (simp add: take_bin2bl_lem drop_bit_Suc)
   done
 
 lemma bin_to_bl_drop_bit:
   "k = m + n \<Longrightarrow> bin_to_bl m (drop_bit n c) = take m (bin_to_bl k c)"
   using bin_split_take by simp
 
 lemma bin_split_take1:
   "k = m + n \<Longrightarrow> bin_split n c = (a, b) \<Longrightarrow> bin_to_bl m a = take m (bin_to_bl k c)"
   using bin_split_take by simp
 
 lemma bl_bin_bl_rep_drop:
   "bin_to_bl n (bl_to_bin bl) =
     replicate (n - length bl) False @ drop (length bl - n) bl"
   by (simp add: bl_to_bin_inj bl_to_bin_rep_F trunc_bl2bin)
 
 lemma bl_to_bin_aux_cat:
   "bl_to_bin_aux bs (bin_cat w nv v) =
     bin_cat w (nv + length bs) (bl_to_bin_aux bs v)"
   by (rule bit_eqI)
     (auto simp add: bin_nth_of_bl_aux bin_nth_cat algebra_simps)
 
 lemma bin_to_bl_aux_cat:
   "bin_to_bl_aux (nv + nw) (bin_cat v nw w) bs =
     bin_to_bl_aux nv v (bin_to_bl_aux nw w bs)"
   by (induction nw arbitrary: w bs) (simp_all add: concat_bit_Suc)
 
 lemma bl_to_bin_aux_alt: "bl_to_bin_aux bs w = bin_cat w (length bs) (bl_to_bin bs)"
   using bl_to_bin_aux_cat [where nv = "0" and v = "0"]
   by (simp add: bl_to_bin_def [symmetric])
 
 lemma bin_to_bl_cat:
   "bin_to_bl (nv + nw) (bin_cat v nw w) =
     bin_to_bl_aux nv v (bin_to_bl nw w)"
   by (simp add: bin_to_bl_def bin_to_bl_aux_cat)
 
 lemmas bl_to_bin_aux_app_cat =
   trans [OF bl_to_bin_aux_append bl_to_bin_aux_alt]
 
 lemmas bin_to_bl_aux_cat_app =
   trans [OF bin_to_bl_aux_cat bin_to_bl_aux_alt]
 
 lemma bl_to_bin_app_cat:
   "bl_to_bin (bsa @ bs) = bin_cat (bl_to_bin bsa) (length bs) (bl_to_bin bs)"
   by (simp only: bl_to_bin_aux_app_cat bl_to_bin_def)
 
 lemma bin_to_bl_cat_app:
   "bin_to_bl (n + nw) (bin_cat w nw wa) = bin_to_bl n w @ bin_to_bl nw wa"
   by (simp only: bin_to_bl_def bin_to_bl_aux_cat_app)
 
 text \<open>\<open>bl_to_bin_app_cat_alt\<close> and \<open>bl_to_bin_app_cat\<close> are easily interderivable.\<close>
 lemma bl_to_bin_app_cat_alt: "bin_cat (bl_to_bin cs) n w = bl_to_bin (cs @ bin_to_bl n w)"
   by (simp add: bl_to_bin_app_cat)
 
 lemma mask_lem: "(bl_to_bin (True # replicate n False)) = bl_to_bin (replicate n True) + 1"
   apply (unfold bl_to_bin_def)
   apply (induct n)
    apply simp
   apply (simp only: Suc_eq_plus1 replicate_add append_Cons [symmetric] bl_to_bin_aux_append)
   apply simp
   done
 
 lemma bin_exhaust:
   "(\<And>x b. bin = of_bool b + 2 * x \<Longrightarrow> Q) \<Longrightarrow> Q" for bin :: int
   apply (cases \<open>even bin\<close>)
    apply (auto elim!: evenE oddE)
    apply fastforce
   apply fastforce
   done
 
 primrec rbl_succ :: "bool list \<Rightarrow> bool list"
   where
     Nil: "rbl_succ Nil = Nil"
   | Cons: "rbl_succ (x # xs) = (if x then False # rbl_succ xs else True # xs)"
 
 primrec rbl_pred :: "bool list \<Rightarrow> bool list"
   where
     Nil: "rbl_pred Nil = Nil"
   | Cons: "rbl_pred (x # xs) = (if x then False # xs else True # rbl_pred xs)"
 
 primrec rbl_add :: "bool list \<Rightarrow> bool list \<Rightarrow> bool list"
   where \<comment> \<open>result is length of first arg, second arg may be longer\<close>
     Nil: "rbl_add Nil x = Nil"
   | Cons: "rbl_add (y # ys) x =
       (let ws = rbl_add ys (tl x)
        in (y \<noteq> hd x) # (if hd x \<and> y then rbl_succ ws else ws))"
 
 primrec rbl_mult :: "bool list \<Rightarrow> bool list \<Rightarrow> bool list"
   where \<comment> \<open>result is length of first arg, second arg may be longer\<close>
     Nil: "rbl_mult Nil x = Nil"
   | Cons: "rbl_mult (y # ys) x =
       (let ws = False # rbl_mult ys x
        in if y then rbl_add ws x else ws)"
 
 lemma size_rbl_pred: "length (rbl_pred bl) = length bl"
   by (induct bl) auto
 
 lemma size_rbl_succ: "length (rbl_succ bl) = length bl"
   by (induct bl) auto
 
 lemma size_rbl_add: "length (rbl_add bl cl) = length bl"
   by (induct bl arbitrary: cl) (auto simp: Let_def size_rbl_succ)
 
 lemma size_rbl_mult: "length (rbl_mult bl cl) = length bl"
   by (induct bl arbitrary: cl) (auto simp add: Let_def size_rbl_add)
 
 lemmas rbl_sizes [simp] =
   size_rbl_pred size_rbl_succ size_rbl_add size_rbl_mult
 
 lemmas rbl_Nils =
   rbl_pred.Nil rbl_succ.Nil rbl_add.Nil rbl_mult.Nil
 
 lemma rbl_add_app2: "length blb \<ge> length bla \<Longrightarrow> rbl_add bla (blb @ blc) = rbl_add bla blb"
   apply (induct bla arbitrary: blb)
    apply simp
   apply clarsimp
   apply (case_tac blb, clarsimp)
   apply (clarsimp simp: Let_def)
   done
 
 lemma rbl_add_take2:
   "length blb \<ge> length bla \<Longrightarrow> rbl_add bla (take (length bla) blb) = rbl_add bla blb"
   apply (induct bla arbitrary: blb)
    apply simp
   apply clarsimp
   apply (case_tac blb, clarsimp)
   apply (clarsimp simp: Let_def)
   done
 
 lemma rbl_mult_app2: "length blb \<ge> length bla \<Longrightarrow> rbl_mult bla (blb @ blc) = rbl_mult bla blb"
   apply (induct bla arbitrary: blb)
    apply simp
   apply clarsimp
   apply (case_tac blb, clarsimp)
   apply (clarsimp simp: Let_def rbl_add_app2)
   done
 
 lemma rbl_mult_take2:
   "length blb \<ge> length bla \<Longrightarrow> rbl_mult bla (take (length bla) blb) = rbl_mult bla blb"
   apply (rule trans)
    apply (rule rbl_mult_app2 [symmetric])
    apply simp
   apply (rule_tac f = "rbl_mult bla" in arg_cong)
   apply (rule append_take_drop_id)
   done
 
 lemma rbl_add_split:
   "P (rbl_add (y # ys) (x # xs)) =
     (\<forall>ws. length ws = length ys \<longrightarrow> ws = rbl_add ys xs \<longrightarrow>
       (y \<longrightarrow> ((x \<longrightarrow> P (False # rbl_succ ws)) \<and> (\<not> x \<longrightarrow> P (True # ws)))) \<and>
       (\<not> y \<longrightarrow> P (x # ws)))"
   by (cases y) (auto simp: Let_def)
 
 lemma rbl_mult_split:
   "P (rbl_mult (y # ys) xs) =
     (\<forall>ws. length ws = Suc (length ys) \<longrightarrow> ws = False # rbl_mult ys xs \<longrightarrow>
       (y \<longrightarrow> P (rbl_add ws xs)) \<and> (\<not> y \<longrightarrow> P ws))"
   by (auto simp: Let_def)
 
 lemma rbl_pred: "rbl_pred (rev (bin_to_bl n bin)) = rev (bin_to_bl n (bin - 1))"
 proof (unfold bin_to_bl_def, induction n arbitrary: bin)
   case 0
   then show ?case
     by simp
 next
   case (Suc n)
   obtain b k where \<open>bin = of_bool b + 2 * k\<close>
     using bin_exhaust by blast
   moreover have \<open>(2 * k - 1) div 2 = k - 1\<close>
-    using even_succ_div_2 [of \<open>2 * (k - 1)\<close>] 
+    using even_succ_div_2 [of \<open>2 * (k - 1)\<close>]
     by simp
   ultimately show ?case
     using Suc [of \<open>bin div 2\<close>]
     by simp (simp add: bin_to_bl_aux_alt)
 qed
 
 lemma rbl_succ: "rbl_succ (rev (bin_to_bl n bin)) = rev (bin_to_bl n (bin + 1))"
   apply (unfold bin_to_bl_def)
   apply (induction n arbitrary: bin)
    apply simp_all
   apply (case_tac bin rule: bin_exhaust)
   apply simp
   apply (simp add: bin_to_bl_aux_alt ac_simps)
   done
 
 lemma rbl_add:
   "\<And>bina binb. rbl_add (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb)) =
     rev (bin_to_bl n (bina + binb))"
   apply (unfold bin_to_bl_def)
   apply (induct n)
    apply simp
   apply clarsimp
   apply (case_tac bina rule: bin_exhaust)
   apply (case_tac binb rule: bin_exhaust)
   apply (case_tac b)
    apply (case_tac [!] "ba")
      apply (auto simp: rbl_succ bin_to_bl_aux_alt Let_def ac_simps)
   done
 
 lemma rbl_add_long:
   "m \<ge> n \<Longrightarrow> rbl_add (rev (bin_to_bl n bina)) (rev (bin_to_bl m binb)) =
     rev (bin_to_bl n (bina + binb))"
   apply (rule box_equals [OF _ rbl_add_take2 rbl_add])
    apply (rule_tac f = "rbl_add (rev (bin_to_bl n bina))" in arg_cong)
    apply (rule rev_swap [THEN iffD1])
    apply (simp add: rev_take drop_bin2bl)
   apply simp
   done
 
 lemma rbl_mult_gt1:
   "m \<ge> length bl \<Longrightarrow>
     rbl_mult bl (rev (bin_to_bl m binb)) =
     rbl_mult bl (rev (bin_to_bl (length bl) binb))"
   apply (rule trans)
    apply (rule rbl_mult_take2 [symmetric])
    apply simp_all
   apply (rule_tac f = "rbl_mult bl" in arg_cong)
   apply (rule rev_swap [THEN iffD1])
   apply (simp add: rev_take drop_bin2bl)
   done
 
 lemma rbl_mult_gt:
   "m > n \<Longrightarrow>
     rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl m binb)) =
     rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb))"
   by (auto intro: trans [OF rbl_mult_gt1])
 
 lemmas rbl_mult_Suc = lessI [THEN rbl_mult_gt]
 
 lemma rbbl_Cons: "b # rev (bin_to_bl n x) = rev (bin_to_bl (Suc n) (of_bool b + 2 * x))"
   by (simp add: bin_to_bl_def) (simp add: bin_to_bl_aux_alt)
 
 lemma rbl_mult:
   "rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb)) =
     rev (bin_to_bl n (bina * binb))"
   apply (induct n arbitrary: bina binb)
    apply simp_all
   apply (unfold bin_to_bl_def)
   apply clarsimp
   apply (case_tac bina rule: bin_exhaust)
   apply (case_tac binb rule: bin_exhaust)
   apply simp
   apply (simp add: bin_to_bl_aux_alt)
   apply (simp add: rbbl_Cons rbl_mult_Suc rbl_add algebra_simps)
   done
 
 lemma sclem: "size (concat (map (bin_to_bl n) xs)) = length xs * n"
   by (induct xs) auto
 
 lemma bin_cat_foldl_lem:
   "foldl (\<lambda>u. bin_cat u n) x xs =
     bin_cat x (size xs * n) (foldl (\<lambda>u. bin_cat u n) y xs)"
   apply (induct xs arbitrary: x)
    apply simp
   apply (simp (no_asm))
   apply (frule asm_rl)
   apply (drule meta_spec)
   apply (erule trans)
   apply (drule_tac x = "bin_cat y n a" in meta_spec)
   apply (simp add: bin_cat_assoc_sym min.absorb2)
   done
 
 lemma bin_rcat_bl: "bin_rcat n wl = bl_to_bin (concat (map (bin_to_bl n) wl))"
   apply (unfold bin_rcat_eq_foldl)
   apply (rule sym)
   apply (induct wl)
    apply (auto simp add: bl_to_bin_append)
   apply (simp add: bl_to_bin_aux_alt sclem)
   apply (simp add: bin_cat_foldl_lem [symmetric])
   done
 
 lemma bin_last_bl_to_bin: "bin_last (bl_to_bin bs) \<longleftrightarrow> bs \<noteq> [] \<and> last bs"
 by(cases "bs = []")(auto simp add: bl_to_bin_def last_bin_last'[where w=0])
 
 lemma bin_rest_bl_to_bin: "bin_rest (bl_to_bin bs) = bl_to_bin (butlast bs)"
 by(cases "bs = []")(simp_all add: bl_to_bin_def butlast_rest_bl2bin_aux)
 
 lemma bl_xor_aux_bin:
   "map2 (\<lambda>x y. x \<noteq> y) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) =
     bin_to_bl_aux n (v XOR w) (map2 (\<lambda>x y. x \<noteq> y) bs cs)"
   apply (induction n arbitrary: v w bs cs)
    apply auto
   apply (case_tac v rule: bin_exhaust)
   apply (case_tac w rule: bin_exhaust)
   apply clarsimp
   done
 
 lemma bl_or_aux_bin:
   "map2 (\<or>) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) =
     bin_to_bl_aux n (v OR w) (map2 (\<or>) bs cs)"
   by (induct n arbitrary: v w bs cs) simp_all
 
 lemma bl_and_aux_bin:
   "map2 (\<and>) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) =
     bin_to_bl_aux n (v AND w) (map2 (\<and>) bs cs)"
   by (induction n arbitrary: v w bs cs) simp_all
 
 lemma bl_not_aux_bin: "map Not (bin_to_bl_aux n w cs) = bin_to_bl_aux n (NOT w) (map Not cs)"
   by (induct n arbitrary: w cs) auto
 
 lemma bl_not_bin: "map Not (bin_to_bl n w) = bin_to_bl n (NOT w)"
   by (simp add: bin_to_bl_def bl_not_aux_bin)
 
 lemma bl_and_bin: "map2 (\<and>) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v AND w)"
   by (simp add: bin_to_bl_def bl_and_aux_bin)
 
 lemma bl_or_bin: "map2 (\<or>) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v OR w)"
   by (simp add: bin_to_bl_def bl_or_aux_bin)
 
 lemma bl_xor_bin: "map2 (\<noteq>) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v XOR w)"
   using bl_xor_aux_bin by (simp add: bin_to_bl_def)
 
 
 subsection \<open>Type \<^typ>\<open>'a word\<close>\<close>
 
 lift_definition of_bl :: \<open>bool list \<Rightarrow> 'a::len word\<close>
   is bl_to_bin .
 
 lift_definition to_bl :: \<open>'a::len word \<Rightarrow> bool list\<close>
   is \<open>bin_to_bl LENGTH('a)\<close>
   by (simp add: bl_to_bin_inj)
 
 lemma to_bl_eq:
   \<open>to_bl w = bin_to_bl (LENGTH('a)) (uint w)\<close>
   for w :: \<open>'a::len word\<close>
   by transfer simp
 
 lemma bit_of_bl_iff [bit_simps]:
   \<open>bit (of_bl bs :: 'a word) n \<longleftrightarrow> rev bs ! n \<and> n < LENGTH('a::len) \<and> n < length bs\<close>
   by transfer (simp add: bin_nth_of_bl ac_simps)
 
 lemma rev_to_bl_eq:
   \<open>rev (to_bl w) = map (bit w) [0..<LENGTH('a)]\<close>
   for w :: \<open>'a::len word\<close>
   apply (rule nth_equalityI)
    apply (simp add: to_bl.rep_eq)
   apply (simp add: bin_nth_bl bit_word.rep_eq to_bl.rep_eq)
   done
 
 lemma to_bl_eq_rev:
   \<open>to_bl w = map (bit w) (rev [0..<LENGTH('a)])\<close>
   for w :: \<open>'a::len word\<close>
   using rev_to_bl_eq [of w]
   apply (subst rev_is_rev_conv [symmetric])
   apply (simp add: rev_map)
   done
 
 lemma of_bl_rev_eq:
   \<open>of_bl (rev bs) = horner_sum of_bool 2 bs\<close>
   apply (rule bit_word_eqI)
   apply (simp add: bit_of_bl_iff)
   apply transfer
   apply (simp add: bit_horner_sum_bit_iff ac_simps)
   done
 
 lemma of_bl_eq:
   \<open>of_bl bs = horner_sum of_bool 2 (rev bs)\<close>
   using of_bl_rev_eq [of \<open>rev bs\<close>] by simp
 
 lemma bshiftr1_eq:
   \<open>bshiftr1 b w = of_bl (b # butlast (to_bl w))\<close>
   apply transfer
   apply auto
    apply (subst bl_to_bin_app_cat [of \<open>[True]\<close>, simplified])
    apply simp
    apply (metis One_nat_def add.commute bin_bl_bin bin_last_bl_to_bin bin_rest_bl_to_bin butlast_bin_rest concat_bit_eq last.simps list.distinct(1) list.size(3) list.size(4) odd_iff_mod_2_eq_one plus_1_eq_Suc power_Suc0_right push_bit_of_1 size_bin_to_bl take_bit_eq_mod trunc_bl2bin_len)
   apply (simp add: butlast_rest_bl2bin)
   done
 
 lemma length_to_bl_eq:
   \<open>length (to_bl w) = LENGTH('a)\<close>
   for w :: \<open>'a::len word\<close>
   by transfer simp
 
 lemma word_rotr_eq:
   \<open>word_rotr n w = of_bl (rotater n (to_bl w))\<close>
   apply (rule bit_word_eqI)
   subgoal for n
     apply (cases \<open>n < LENGTH('a)\<close>)
      apply (simp_all add: bit_word_rotr_iff bit_of_bl_iff rotater_rev length_to_bl_eq nth_rotate rev_to_bl_eq ac_simps)
     done
   done
 
 lemma word_rotl_eq:
   \<open>word_rotl n w = of_bl (rotate n (to_bl w))\<close>
 proof -
   have \<open>rotate n (to_bl w) = rev (rotater n (rev (to_bl w)))\<close>
     by (simp add: rotater_rev')
   then show ?thesis
     apply (simp add: word_rotl_eq_word_rotr bit_of_bl_iff length_to_bl_eq rev_to_bl_eq)
     apply (rule bit_word_eqI)
     subgoal for n
       apply (cases \<open>n < LENGTH('a)\<close>)
        apply (simp_all add: bit_word_rotr_iff bit_of_bl_iff nth_rotater)
       done
     done
 qed
 
 lemma to_bl_def': "(to_bl :: 'a::len word \<Rightarrow> bool list) = bin_to_bl (LENGTH('a)) \<circ> uint"
   by transfer (simp add: fun_eq_iff)
 
 \<comment> \<open>type definitions theorem for in terms of equivalent bool list\<close>
 lemma td_bl:
   "type_definition
     (to_bl :: 'a::len word \<Rightarrow> bool list)
     of_bl
     {bl. length bl = LENGTH('a)}"
   apply (standard; transfer)
   apply (auto dest: sym)
   done
 
 interpretation word_bl:
   type_definition
     "to_bl :: 'a::len word \<Rightarrow> bool list"
     of_bl
     "{bl. length bl = LENGTH('a::len)}"
   by (fact td_bl)
 
 lemmas word_bl_Rep' = word_bl.Rep [unfolded mem_Collect_eq, iff]
 
 lemma word_size_bl: "size w = size (to_bl w)"
   by (auto simp: word_size)
 
 lemma to_bl_use_of_bl: "to_bl w = bl \<longleftrightarrow> w = of_bl bl \<and> length bl = length (to_bl w)"
   by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq])
 
 lemma length_bl_gt_0 [iff]: "0 < length (to_bl x)"
   for x :: "'a::len word"
   unfolding word_bl_Rep' by (rule len_gt_0)
 
 lemma bl_not_Nil [iff]: "to_bl x \<noteq> []"
   for x :: "'a::len word"
   by (fact length_bl_gt_0 [unfolded length_greater_0_conv])
 
 lemma length_bl_neq_0 [iff]: "length (to_bl x) \<noteq> 0"
   for x :: "'a::len word"
   by (fact length_bl_gt_0 [THEN gr_implies_not0])
 
 lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = -1)"
   apply transfer
   apply (auto simp add: bin_sign_def)
   using bin_sign_lem bl_sbin_sign apply fastforce
   using bin_sign_lem bl_sbin_sign apply force
   done
 
 lemma of_bl_drop':
   "lend = length bl - LENGTH('a::len) \<Longrightarrow>
     of_bl (drop lend bl) = (of_bl bl :: 'a word)"
   by transfer (simp flip: trunc_bl2bin)
 
 lemma test_bit_of_bl:
   "(of_bl bl::'a::len word) !! n = (rev bl ! n \<and> n < LENGTH('a) \<and> n < length bl)"
   by transfer (simp add: bin_nth_of_bl ac_simps)
 
 lemma no_of_bl: "(numeral bin ::'a::len word) = of_bl (bin_to_bl (LENGTH('a)) (numeral bin))"
   by transfer simp
 
 lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)"
   by transfer simp
 
 lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w"
   by (simp add: uint_bl word_size)
 
 lemma to_bl_of_bin: "to_bl (word_of_int bin::'a::len word) = bin_to_bl (LENGTH('a)) bin"
   by (auto simp: uint_bl word_ubin.eq_norm word_size)
 
 lemma to_bl_numeral [simp]:
   "to_bl (numeral bin::'a::len word) =
     bin_to_bl (LENGTH('a)) (numeral bin)"
   unfolding word_numeral_alt by (rule to_bl_of_bin)
 
 lemma to_bl_neg_numeral [simp]:
   "to_bl (- numeral bin::'a::len word) =
     bin_to_bl (LENGTH('a)) (- numeral bin)"
   unfolding word_neg_numeral_alt by (rule to_bl_of_bin)
 
 lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w"
   by (simp add: uint_bl word_size)
 
 lemma uint_bl_bin: "bl_to_bin (bin_to_bl (LENGTH('a)) (uint x)) = uint x"
   for x :: "'a::len word"
   by (rule trans [OF bin_bl_bin word_ubin.norm_Rep])
 
 lemma ucast_bl: "ucast w = of_bl (to_bl w)"
   by transfer simp
 
 lemma ucast_down_bl:
   \<open>(ucast :: 'a::len word \<Rightarrow> 'b::len word) (of_bl bl) = of_bl bl\<close>
     if \<open>is_down (ucast :: 'a::len word \<Rightarrow> 'b::len word)\<close>
   using that by transfer simp
 
 lemma of_bl_append_same: "of_bl (X @ to_bl w) = w"
-  by transfer (simp add: bl_to_bin_app_cat) 
+  by transfer (simp add: bl_to_bin_app_cat)
 
 lemma ucast_of_bl_up:
   \<open>ucast (of_bl bl :: 'a::len word) = of_bl bl\<close>
   if \<open>size bl \<le> size (of_bl bl :: 'a::len word)\<close>
   using that
   apply transfer
   apply (rule bit_eqI)
   apply (auto simp add: bit_take_bit_iff)
   apply (subst (asm) trunc_bl2bin_len [symmetric])
   apply (auto simp only: bit_take_bit_iff)
   done
 
 lemma word_rev_tf:
   "to_bl (of_bl bl::'a::len word) =
     rev (takefill False (LENGTH('a)) (rev bl))"
   by transfer (simp add: bl_bin_bl_rtf)
 
 lemma word_rep_drop:
   "to_bl (of_bl bl::'a::len word) =
     replicate (LENGTH('a) - length bl) False @
     drop (length bl - LENGTH('a)) bl"
   by (simp add: word_rev_tf takefill_alt rev_take)
 
 lemma to_bl_ucast:
   "to_bl (ucast (w::'b::len word) ::'a::len word) =
     replicate (LENGTH('a) - LENGTH('b)) False @
     drop (LENGTH('b) - LENGTH('a)) (to_bl w)"
   apply (unfold ucast_bl)
   apply (rule trans)
    apply (rule word_rep_drop)
   apply simp
   done
 
 lemma ucast_up_app:
   \<open>to_bl (ucast w :: 'b::len word) = replicate n False @ (to_bl w)\<close>
     if \<open>source_size (ucast :: 'a word \<Rightarrow> 'b word) + n = target_size (ucast :: 'a word \<Rightarrow> 'b word)\<close>
     for w :: \<open>'a::len word\<close>
   using that
   by (auto simp add : source_size target_size to_bl_ucast)
 
 lemma ucast_down_drop [OF refl]:
   "uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow>
     to_bl (uc w) = drop n (to_bl w)"
   by (auto simp add : source_size target_size to_bl_ucast)
 
 lemma scast_down_drop [OF refl]:
   "sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow>
     to_bl (sc w) = drop n (to_bl w)"
   apply (subgoal_tac "sc = ucast")
    apply safe
    apply simp
    apply (erule ucast_down_drop)
   apply (rule down_cast_same [symmetric])
   apply (simp add : source_size target_size is_down)
   done
 
 lemma word_0_bl [simp]: "of_bl [] = 0"
   by transfer simp
 
 lemma word_1_bl: "of_bl [True] = 1"
   by transfer (simp add: bl_to_bin_def)
 
 lemma of_bl_0 [simp]: "of_bl (replicate n False) = 0"
   by transfer (simp add: bl_to_bin_rep_False)
 
 lemma to_bl_0 [simp]: "to_bl (0::'a::len word) = replicate (LENGTH('a)) False"
   by (simp add: uint_bl word_size bin_to_bl_zero)
 
 \<comment> \<open>links with \<open>rbl\<close> operations\<close>
 lemma word_succ_rbl: "to_bl w = bl \<Longrightarrow> to_bl (word_succ w) = rev (rbl_succ (rev bl))"
   by transfer (simp add: rbl_succ)
 
 lemma word_pred_rbl: "to_bl w = bl \<Longrightarrow> to_bl (word_pred w) = rev (rbl_pred (rev bl))"
   by transfer (simp add: rbl_pred)
 
 lemma word_add_rbl:
   "to_bl v = vbl \<Longrightarrow> to_bl w = wbl \<Longrightarrow>
     to_bl (v + w) = rev (rbl_add (rev vbl) (rev wbl))"
   apply transfer
   apply (drule sym)
   apply (drule sym)
   apply (simp add: rbl_add)
   done
 
 lemma word_mult_rbl:
   "to_bl v = vbl \<Longrightarrow> to_bl w = wbl \<Longrightarrow>
     to_bl (v * w) = rev (rbl_mult (rev vbl) (rev wbl))"
   apply transfer
   apply (drule sym)
   apply (drule sym)
   apply (simp add: rbl_mult)
   done
 
 lemma rtb_rbl_ariths:
   "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_succ w)) = rbl_succ ys"
   "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_pred w)) = rbl_pred ys"
   "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v * w)) = rbl_mult ys xs"
   "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v + w)) = rbl_add ys xs"
   by (auto simp: rev_swap [symmetric] word_succ_rbl word_pred_rbl word_mult_rbl word_add_rbl)
 
 lemma of_bl_length_less:
   \<open>(of_bl x :: 'a::len word) < 2 ^ k\<close>
     if \<open>length x = k\<close> \<open>k < LENGTH('a)\<close>
 proof -
   from that have \<open>length x < LENGTH('a)\<close>
     by simp
   then have \<open>(of_bl x :: 'a::len word) < 2 ^ length x\<close>
     apply (simp add: of_bl_eq)
     apply transfer
     apply (simp add: take_bit_horner_sum_bit_eq)
     apply (subst length_rev [symmetric])
     apply (simp only: horner_sum_of_bool_2_less)
     done
   with that show ?thesis
     by simp
 qed
 
 lemma word_eq_rbl_eq: "x = y \<longleftrightarrow> rev (to_bl x) = rev (to_bl y)"
   by simp
 
 lemma bl_word_not: "to_bl (NOT w) = map Not (to_bl w)"
   by transfer (simp add: bl_not_bin)
 
 lemma bl_word_xor: "to_bl (v XOR w) = map2 (\<noteq>) (to_bl v) (to_bl w)"
   by transfer (simp flip: bl_xor_bin)
 
 lemma bl_word_or: "to_bl (v OR w) = map2 (\<or>) (to_bl v) (to_bl w)"
   by transfer (simp flip: bl_or_bin)
 
 lemma bl_word_and: "to_bl (v AND w) = map2 (\<and>) (to_bl v) (to_bl w)"
   by transfer (simp flip: bl_and_bin)
 
 lemma bin_nth_uint': "bin_nth (uint w) n \<longleftrightarrow> rev (bin_to_bl (size w) (uint w)) ! n \<and> n < size w"
   apply (unfold word_size)
   apply (safe elim!: bin_nth_uint_imp)
    apply (frule bin_nth_uint_imp)
    apply (fast dest!: bin_nth_bl)+
   done
 
 lemmas bin_nth_uint = bin_nth_uint' [unfolded word_size]
 
 lemma test_bit_bl: "w !! n \<longleftrightarrow> rev (to_bl w) ! n \<and> n < size w"
   by transfer (auto simp add: bin_nth_bl)
 
 lemma to_bl_nth: "n < size w \<Longrightarrow> to_bl w ! n = w !! (size w - Suc n)"
   by (simp add: word_size rev_nth test_bit_bl)
 
 lemma map_bit_interval_eq:
   \<open>map (bit w) [0..<n] = takefill False n (rev (to_bl w))\<close> for w :: \<open>'a::len word\<close>
 proof (rule nth_equalityI)
   show \<open>length (map (bit w) [0..<n]) =
     length (takefill False n (rev (to_bl w)))\<close>
     by simp
   fix m
   assume \<open>m < length (map (bit w) [0..<n])\<close>
   then have \<open>m < n\<close>
     by simp
   then have \<open>bit w m \<longleftrightarrow> takefill False n (rev (to_bl w)) ! m\<close>
     by (auto simp add: nth_takefill not_less rev_nth to_bl_nth word_size test_bit_word_eq dest: bit_imp_le_length)
   with \<open>m < n \<close>show \<open>map (bit w) [0..<n] ! m \<longleftrightarrow> takefill False n (rev (to_bl w)) ! m\<close>
     by simp
 qed
 
 lemma to_bl_unfold:
   \<open>to_bl w = rev (map (bit w) [0..<LENGTH('a)])\<close> for w :: \<open>'a::len word\<close>
   by (simp add: map_bit_interval_eq takefill_bintrunc to_bl_def flip: bin_to_bl_def)
 
 lemma nth_rev_to_bl:
   \<open>rev (to_bl w) ! n \<longleftrightarrow> bit w n\<close>
   if \<open>n < LENGTH('a)\<close> for w :: \<open>'a::len word\<close>
   using that by (simp add: to_bl_unfold)
 
 lemma nth_to_bl:
   \<open>to_bl w ! n \<longleftrightarrow> bit w (LENGTH('a) - Suc n)\<close>
   if \<open>n < LENGTH('a)\<close> for w :: \<open>'a::len word\<close>
   using that by (simp add: to_bl_unfold rev_nth)
 
 lemma of_bl_rep_False: "of_bl (replicate n False @ bs) = of_bl bs"
   by (auto simp: of_bl_def bl_to_bin_rep_F)
 
 lemma [code abstract]:
   \<open>Word.the_int (of_bl bs :: 'a word) = horner_sum of_bool 2 (take LENGTH('a::len) (rev bs))\<close>
   apply (simp add: of_bl_eq flip: take_bit_horner_sum_bit_eq)
   apply transfer
   apply simp
   done
 
 lemma [code]:
   \<open>to_bl w = map (bit w) (rev [0..<LENGTH('a::len)])\<close>
   for w :: \<open>'a::len word\<close>
   by (fact to_bl_eq_rev)
 
 lemma word_reverse_eq_of_bl_rev_to_bl:
   \<open>word_reverse w = of_bl (rev (to_bl w))\<close>
   by (rule bit_word_eqI)
     (auto simp add: bit_word_reverse_iff bit_of_bl_iff nth_to_bl)
 
 lemmas word_reverse_no_def [simp] =
   word_reverse_eq_of_bl_rev_to_bl [of "numeral w"] for w
 
 lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)"
   by (rule nth_equalityI) (simp_all add: nth_rev_to_bl word_reverse_def word_rep_drop flip: of_bl_eq)
 
 lemma to_bl_n1 [simp]: "to_bl (-1::'a::len word) = replicate (LENGTH('a)) True"
   apply (rule word_bl.Abs_inverse')
    apply simp
   apply (rule word_eqI)
   apply (clarsimp simp add: word_size)
   apply (auto simp add: word_bl.Abs_inverse test_bit_bl word_size)
   done
 
 lemma rbl_word_or: "rev (to_bl (x OR y)) = map2 (\<or>) (rev (to_bl x)) (rev (to_bl y))"
   by (simp add: zip_rev bl_word_or rev_map)
 
 lemma rbl_word_and: "rev (to_bl (x AND y)) = map2 (\<and>) (rev (to_bl x)) (rev (to_bl y))"
   by (simp add: zip_rev bl_word_and rev_map)
 
 lemma rbl_word_xor: "rev (to_bl (x XOR y)) = map2 (\<noteq>) (rev (to_bl x)) (rev (to_bl y))"
   by (simp add: zip_rev bl_word_xor rev_map)
 
 lemma rbl_word_not: "rev (to_bl (NOT x)) = map Not (rev (to_bl x))"
   by (simp add: bl_word_not rev_map)
 
 lemma bshiftr1_numeral [simp]:
   \<open>bshiftr1 b (numeral w :: 'a word) = of_bl (b # butlast (bin_to_bl LENGTH('a::len) (numeral w)))\<close>
   by (simp add: bshiftr1_eq)
 
 lemma bshiftr1_bl: "to_bl (bshiftr1 b w) = b # butlast (to_bl w)"
   unfolding bshiftr1_eq by (rule word_bl.Abs_inverse) simp
 
 lemma shiftl1_of_bl: "shiftl1 (of_bl bl) = of_bl (bl @ [False])"
   by transfer (simp add: bl_to_bin_append)
 
 lemma shiftl1_bl: "shiftl1 w = of_bl (to_bl w @ [False])"
   for w :: "'a::len word"
 proof -
   have "shiftl1 w = shiftl1 (of_bl (to_bl w))"
     by simp
   also have "\<dots> = of_bl (to_bl w @ [False])"
     by (rule shiftl1_of_bl)
   finally show ?thesis .
 qed
 
 lemma bl_shiftl1: "to_bl (shiftl1 w) = tl (to_bl w) @ [False]"
   for w :: "'a::len word"
   by (simp add: shiftl1_bl word_rep_drop drop_Suc drop_Cons') (fast intro!: Suc_leI)
 
 \<comment> \<open>Generalized version of \<open>bl_shiftl1\<close>. Maybe this one should replace it?\<close>
 lemma bl_shiftl1': "to_bl (shiftl1 w) = tl (to_bl w @ [False])"
   by (simp add: shiftl1_bl word_rep_drop drop_Suc del: drop_append)
 
 lemma shiftr1_bl:
   \<open>shiftr1 w = of_bl (butlast (to_bl w))\<close>
 proof (rule bit_word_eqI)
   fix n
   assume \<open>n < LENGTH('a)\<close>
   show \<open>bit (shiftr1 w) n \<longleftrightarrow> bit (of_bl (butlast (to_bl w)) :: 'a word) n\<close>
   proof (cases \<open>n = LENGTH('a) - 1\<close>)
     case True
     then show ?thesis
       by (simp add: bit_shiftr1_iff bit_of_bl_iff)
   next
     case False
     with \<open>n < LENGTH('a)\<close>
     have \<open>n < LENGTH('a) - 1\<close>
       by simp
-    with \<open>n < LENGTH('a)\<close> show ?thesis 
+    with \<open>n < LENGTH('a)\<close> show ?thesis
       by (simp add: bit_shiftr1_iff bit_of_bl_iff rev_nth nth_butlast
         word_size test_bit_word_eq to_bl_nth)
   qed
 qed
 
 lemma bl_shiftr1: "to_bl (shiftr1 w) = False # butlast (to_bl w)"
   for w :: "'a::len word"
   by (simp add: shiftr1_bl word_rep_drop len_gt_0 [THEN Suc_leI])
 
 \<comment> \<open>Generalized version of \<open>bl_shiftr1\<close>. Maybe this one should replace it?\<close>
 lemma bl_shiftr1': "to_bl (shiftr1 w) = butlast (False # to_bl w)"
   apply (rule word_bl.Abs_inverse')
    apply (simp del: butlast.simps)
   apply (simp add: shiftr1_bl of_bl_def)
   done
 
 lemma bl_sshiftr1: "to_bl (sshiftr1 w) = hd (to_bl w) # butlast (to_bl w)"
   for w :: "'a::len word"
 proof (rule nth_equalityI)
   fix n
   assume \<open>n < length (to_bl (sshiftr1 w))\<close>
   then have \<open>n < LENGTH('a)\<close>
     by simp
   then show \<open>to_bl (sshiftr1 w) ! n \<longleftrightarrow> (hd (to_bl w) # butlast (to_bl w)) ! n\<close>
     apply (cases n)
      apply (simp_all add: to_bl_nth word_size hd_conv_nth test_bit_eq_bit bit_sshiftr1_iff nth_butlast Suc_diff_Suc nth_to_bl)
     done
 qed simp
 
 lemma drop_shiftr: "drop n (to_bl (w >> n)) = take (size w - n) (to_bl w)"
   for w :: "'a::len word"
   apply (unfold shiftr_def)
   apply (induct n)
    prefer 2
    apply (simp add: drop_Suc bl_shiftr1 butlast_drop [symmetric])
    apply (rule butlast_take [THEN trans])
     apply (auto simp: word_size)
   done
 
 lemma drop_sshiftr: "drop n (to_bl (w >>> n)) = take (size w - n) (to_bl w)"
   for w :: "'a::len word"
    apply (simp_all add: word_size sshiftr_eq)
   apply (rule nth_equalityI)
    apply (simp_all add: word_size nth_to_bl bit_signed_drop_bit_iff)
   done
 
 lemma take_shiftr: "n \<le> size w \<Longrightarrow> take n (to_bl (w >> n)) = replicate n False"
   apply (unfold shiftr_def)
   apply (induct n)
    prefer 2
    apply (simp add: bl_shiftr1' length_0_conv [symmetric] word_size)
    apply (rule take_butlast [THEN trans])
     apply (auto simp: word_size)
   done
 
 lemma take_sshiftr':
   "n \<le> size w \<Longrightarrow> hd (to_bl (w >>> n)) = hd (to_bl w) \<and>
     take n (to_bl (w >>> n)) = replicate n (hd (to_bl w))"
   for w :: "'a::len word"
   apply (auto simp add: sshiftr_eq hd_bl_sign_sint bin_sign_def not_le word_size sint_signed_drop_bit_eq)
   apply (rule nth_equalityI)
     apply (auto simp add: nth_to_bl bit_signed_drop_bit_iff bit_last_iff)
   apply (rule nth_equalityI)
    apply (auto simp add: nth_to_bl bit_signed_drop_bit_iff bit_last_iff)
   done
 
 lemmas hd_sshiftr = take_sshiftr' [THEN conjunct1]
 lemmas take_sshiftr = take_sshiftr' [THEN conjunct2]
 
 lemma atd_lem: "take n xs = t \<Longrightarrow> drop n xs = d \<Longrightarrow> xs = t @ d"
   by (auto intro: append_take_drop_id [symmetric])
 
 lemmas bl_shiftr = atd_lem [OF take_shiftr drop_shiftr]
 lemmas bl_sshiftr = atd_lem [OF take_sshiftr drop_sshiftr]
 
 lemma shiftl_of_bl: "of_bl bl << n = of_bl (bl @ replicate n False)"
   by (induct n) (auto simp: shiftl_def shiftl1_of_bl replicate_app_Cons_same)
 
 lemma shiftl_bl: "w << n = of_bl (to_bl w @ replicate n False)"
   for w :: "'a::len word"
 proof -
   have "w << n = of_bl (to_bl w) << n"
     by simp
   also have "\<dots> = of_bl (to_bl w @ replicate n False)"
     by (rule shiftl_of_bl)
   finally show ?thesis .
 qed
 
 lemma bl_shiftl: "to_bl (w << n) = drop n (to_bl w) @ replicate (min (size w) n) False"
   by (simp add: shiftl_bl word_rep_drop word_size)
 
 lemma shiftr1_bl_of:
   "length bl \<le> LENGTH('a) \<Longrightarrow>
     shiftr1 (of_bl bl::'a::len word) = of_bl (butlast bl)"
   by transfer (simp add: butlast_rest_bl2bin trunc_bl2bin)
 
 lemma shiftr_bl_of:
   "length bl \<le> LENGTH('a) \<Longrightarrow>
     (of_bl bl::'a::len word) >> n = of_bl (take (length bl - n) bl)"
   apply (unfold shiftr_def)
   apply (induct n)
    apply clarsimp
   apply clarsimp
   apply (subst shiftr1_bl_of)
    apply simp
   apply (simp add: butlast_take)
   done
 
 lemma shiftr_bl: "x >> n \<equiv> of_bl (take (LENGTH('a) - n) (to_bl x))"
   for x :: "'a::len word"
   using shiftr_bl_of [where 'a='a, of "to_bl x"] by simp
 
 lemma aligned_bl_add_size [OF refl]:
   "size x - n = m \<Longrightarrow> n \<le> size x \<Longrightarrow> drop m (to_bl x) = replicate n False \<Longrightarrow>
     take m (to_bl y) = replicate m False \<Longrightarrow>
     to_bl (x + y) = take m (to_bl x) @ drop m (to_bl y)" for x :: \<open>'a::len word\<close>
   apply (subgoal_tac "x AND y = 0")
    prefer 2
    apply (rule word_bl.Rep_eqD)
    apply (simp add: bl_word_and)
    apply (rule align_lem_and [THEN trans])
        apply (simp_all add: word_size)[5]
    apply simp
   apply (subst word_plus_and_or [symmetric])
   apply (simp add : bl_word_or)
   apply (rule align_lem_or)
      apply (simp_all add: word_size)
   done
 
 lemma mask_bl: "mask n = of_bl (replicate n True)"
   by (auto simp add : test_bit_of_bl word_size intro: word_eqI)
 
 lemma bl_and_mask':
   "to_bl (w AND mask n :: 'a::len word) =
     replicate (LENGTH('a) - n) False @
     drop (LENGTH('a) - n) (to_bl w)"
   apply (rule nth_equalityI)
    apply simp
   apply (clarsimp simp add: to_bl_nth word_size)
   apply (auto simp add: word_size test_bit_bl nth_append rev_nth)
   done
 
 lemma slice1_eq_of_bl:
   \<open>(slice1 n w :: 'b::len word) = of_bl (takefill False n (to_bl w))\<close>
   for w :: \<open>'a::len word\<close>
 proof (rule bit_word_eqI)
   fix m
   assume \<open>m < LENGTH('b)\<close>
   show \<open>bit (slice1 n w :: 'b::len word) m \<longleftrightarrow> bit (of_bl (takefill False n (to_bl w)) :: 'b word) m\<close>
     by (cases \<open>m \<ge> n\<close>; cases \<open>LENGTH('a) \<ge> n\<close>)
       (auto simp add: bit_slice1_iff bit_of_bl_iff not_less rev_nth not_le nth_takefill nth_to_bl algebra_simps)
 qed
 
 lemma slice1_no_bin [simp]:
   "slice1 n (numeral w :: 'b word) = of_bl (takefill False n (bin_to_bl (LENGTH('b::len)) (numeral w)))"
   by (simp add: slice1_eq_of_bl) (* TODO: neg_numeral *)
 
 lemma slice_no_bin [simp]:
   "slice n (numeral w :: 'b word) = of_bl (takefill False (LENGTH('b::len) - n)
     (bin_to_bl (LENGTH('b::len)) (numeral w)))"
   by (simp add: slice_def) (* TODO: neg_numeral *)
 
 lemma slice_take': "slice n w = of_bl (take (size w - n) (to_bl w))"
   by (simp add: slice_def word_size slice1_eq_of_bl takefill_alt)
 
 lemmas slice_take = slice_take' [unfolded word_size]
 
 \<comment> \<open>shiftr to a word of the same size is just slice,
     slice is just shiftr then ucast\<close>
 lemmas shiftr_slice = trans [OF shiftr_bl [THEN meta_eq_to_obj_eq] slice_take [symmetric]]
 
 lemma slice1_down_alt':
   "sl = slice1 n w \<Longrightarrow> fs = size sl \<Longrightarrow> fs + k = n \<Longrightarrow>
     to_bl sl = takefill False fs (drop k (to_bl w))"
   apply (simp add: slice1_eq_of_bl)
   apply transfer
   apply (simp add: bl_bin_bl_rep_drop)
   using drop_takefill
   apply force
   done
 
 lemma slice1_up_alt':
   "sl = slice1 n w \<Longrightarrow> fs = size sl \<Longrightarrow> fs = n + k \<Longrightarrow>
     to_bl sl = takefill False fs (replicate k False @ (to_bl w))"
   apply (simp add: slice1_eq_of_bl)
   apply transfer
   apply (simp add: bl_bin_bl_rep_drop flip: takefill_append)
   apply (metis diff_add_inverse)
   done
 
 lemmas sd1 = slice1_down_alt' [OF refl refl, unfolded word_size]
 lemmas su1 = slice1_up_alt' [OF refl refl, unfolded word_size]
 lemmas slice1_down_alt = le_add_diff_inverse [THEN sd1]
 lemmas slice1_up_alts =
   le_add_diff_inverse [symmetric, THEN su1]
   le_add_diff_inverse2 [symmetric, THEN su1]
 
 lemma slice1_tf_tf':
   "to_bl (slice1 n w :: 'a::len word) =
     rev (takefill False (LENGTH('a)) (rev (takefill False n (to_bl w))))"
   unfolding slice1_eq_of_bl by (rule word_rev_tf)
 
 lemmas slice1_tf_tf = slice1_tf_tf' [THEN word_bl.Rep_inverse', symmetric]
 
 lemma revcast_eq_of_bl:
   \<open>(revcast w :: 'b::len word) = of_bl (takefill False (LENGTH('b)) (to_bl w))\<close>
   for w :: \<open>'a::len word\<close>
   by (simp add: revcast_def slice1_eq_of_bl)
 
 lemmas revcast_no_def [simp] = revcast_eq_of_bl [where w="numeral w", unfolded word_size] for w
 
 lemma to_bl_revcast:
   "to_bl (revcast w :: 'a::len word) = takefill False (LENGTH('a)) (to_bl w)"
   apply (rule nth_equalityI)
   apply simp
   apply (cases \<open>LENGTH('a) \<le> LENGTH('b)\<close>)
    apply (auto simp add: nth_to_bl nth_takefill bit_revcast_iff)
   done
 
 lemma word_cat_bl: "word_cat a b = of_bl (to_bl a @ to_bl b)"
   apply (rule bit_word_eqI)
   apply (simp add: bit_word_cat_iff bit_of_bl_iff nth_append not_less nth_rev_to_bl)
   apply (meson bit_word.rep_eq less_diff_conv2 nth_rev_to_bl)
   done
 
 lemma of_bl_append:
   "(of_bl (xs @ ys) :: 'a::len word) = of_bl xs * 2^(length ys) + of_bl ys"
   apply transfer
   apply (simp add: bl_to_bin_app_cat bin_cat_num)
   done
 
 lemma of_bl_False [simp]: "of_bl (False#xs) = of_bl xs"
   by (rule word_eqI) (auto simp: test_bit_of_bl nth_append)
 
 lemma of_bl_True [simp]: "(of_bl (True # xs) :: 'a::len word) = 2^length xs + of_bl xs"
   by (subst of_bl_append [where xs="[True]", simplified]) (simp add: word_1_bl)
 
 lemma of_bl_Cons: "of_bl (x#xs) = of_bool x * 2^length xs + of_bl xs"
   by (cases x) simp_all
 
 lemma word_split_bl':
   "std = size c - size b \<Longrightarrow> (word_split c = (a, b)) \<Longrightarrow>
     (a = of_bl (take std (to_bl c)) \<and> b = of_bl (drop std (to_bl c)))"
   apply (simp add: word_split_def)
   apply transfer
   apply (cases \<open>LENGTH('b) \<le> LENGTH('a)\<close>)
    apply (auto simp add: drop_bit_take_bit drop_bin2bl bin_to_bl_drop_bit [symmetric, of \<open>LENGTH('a)\<close> \<open>LENGTH('a) - LENGTH('b)\<close> \<open>LENGTH('b)\<close>] min_absorb2)
   done
 
 lemma word_split_bl: "std = size c - size b \<Longrightarrow>
     (a = of_bl (take std (to_bl c)) \<and> b = of_bl (drop std (to_bl c))) \<longleftrightarrow>
     word_split c = (a, b)"
   apply (rule iffI)
    defer
    apply (erule (1) word_split_bl')
   apply (case_tac "word_split c")
   apply (auto simp add: word_size)
   apply (frule word_split_bl' [rotated])
    apply (auto simp add: word_size)
   done
 
 lemma word_split_bl_eq:
   "(word_split c :: ('c::len word \<times> 'd::len word)) =
     (of_bl (take (LENGTH('a::len) - LENGTH('d::len)) (to_bl c)),
      of_bl (drop (LENGTH('a) - LENGTH('d)) (to_bl c)))"
   for c :: "'a::len word"
   apply (rule word_split_bl [THEN iffD1])
    apply (unfold word_size)
    apply (rule refl conjI)+
   done
 
 lemma word_rcat_bl:
   \<open>word_rcat wl = of_bl (concat (map to_bl wl))\<close>
 proof -
   define ws where \<open>ws = rev wl\<close>
   moreover have \<open>word_rcat (rev ws) = of_bl (concat (map to_bl (rev ws)))\<close>
     apply (simp add: word_rcat_def of_bl_eq rev_concat rev_map comp_def rev_to_bl_eq flip: horner_sum_of_bool_2_concat)
     apply transfer
     apply simp
     done
   ultimately show ?thesis
     by simp
 qed
 
 lemma size_rcat_lem': "size (concat (map to_bl wl)) = length wl * size (hd wl)"
   by (induct wl) (auto simp: word_size)
 
 lemmas size_rcat_lem = size_rcat_lem' [unfolded word_size]
 
 lemma nth_rcat_lem:
   "n < length (wl::'a word list) * LENGTH('a::len) \<Longrightarrow>
     rev (concat (map to_bl wl)) ! n =
     rev (to_bl (rev wl ! (n div LENGTH('a)))) ! (n mod LENGTH('a))"
   apply (induct wl)
    apply clarsimp
   apply (clarsimp simp add : nth_append size_rcat_lem)
   apply (simp flip: mult_Suc minus_div_mult_eq_mod add: less_Suc_eq_le not_less)
   apply (metis (no_types, lifting) diff_is_0_eq div_le_mono len_not_eq_0 less_Suc_eq less_mult_imp_div_less nonzero_mult_div_cancel_right not_le nth_Cons_0)
   done
 
 lemma foldl_eq_foldr: "foldl (+) x xs = foldr (+) (x # xs) 0"
   for x :: "'a::comm_monoid_add"
   by (induct xs arbitrary: x) (auto simp: add.assoc)
 
 lemmas word_cat_bl_no_bin [simp] =
   word_cat_bl [where a="numeral a" and b="numeral b", unfolded to_bl_numeral]
   for a b (* FIXME: negative numerals, 0 and 1 *)
 
 lemmas word_split_bl_no_bin [simp] =
   word_split_bl_eq [where c="numeral c", unfolded to_bl_numeral] for c
 
 lemmas word_rot_defs = word_roti_eq_word_rotr_word_rotl word_rotr_eq word_rotl_eq
 
 lemma to_bl_rotl: "to_bl (word_rotl n w) = rotate n (to_bl w)"
   by (simp add: word_rotl_eq to_bl_use_of_bl)
 
 lemmas blrs0 = rotate_eqs [THEN to_bl_rotl [THEN trans]]
 
 lemmas word_rotl_eqs =
   blrs0 [simplified word_bl_Rep' word_bl.Rep_inject to_bl_rotl [symmetric]]
 
 lemma to_bl_rotr: "to_bl (word_rotr n w) = rotater n (to_bl w)"
   by (simp add: word_rotr_eq to_bl_use_of_bl)
 
 lemmas brrs0 = rotater_eqs [THEN to_bl_rotr [THEN trans]]
 
 lemmas word_rotr_eqs =
   brrs0 [simplified word_bl_Rep' word_bl.Rep_inject to_bl_rotr [symmetric]]
 
 declare word_rotr_eqs (1) [simp]
 declare word_rotl_eqs (1) [simp]
 
 lemmas abl_cong = arg_cong [where f = "of_bl"]
 
 locale word_rotate
 begin
 
 lemmas word_rot_defs' = to_bl_rotl to_bl_rotr
 
 lemmas blwl_syms [symmetric] = bl_word_not bl_word_and bl_word_or bl_word_xor
 
 lemmas lbl_lbl = trans [OF word_bl_Rep' word_bl_Rep' [symmetric]]
 
 lemmas ths_map2 [OF lbl_lbl] = rotate_map2 rotater_map2
 
 lemmas ths_map [where xs = "to_bl v"] = rotate_map rotater_map for v
 
 lemmas th1s [simplified word_rot_defs' [symmetric]] = ths_map2 ths_map
 
 end
 
 lemmas bl_word_rotl_dt = trans [OF to_bl_rotl rotate_drop_take,
   simplified word_bl_Rep']
 
 lemmas bl_word_rotr_dt = trans [OF to_bl_rotr rotater_drop_take,
   simplified word_bl_Rep']
 
 lemma bl_word_roti_dt':
   "n = nat ((- i) mod int (size (w :: 'a::len word))) \<Longrightarrow>
     to_bl (word_roti i w) = drop n (to_bl w) @ take n (to_bl w)"
   apply (unfold word_roti_eq_word_rotr_word_rotl)
   apply (simp add: bl_word_rotl_dt bl_word_rotr_dt word_size)
   apply safe
    apply (simp add: zmod_zminus1_eq_if)
    apply safe
     apply (simp add: nat_mult_distrib)
    apply (simp add: nat_diff_distrib [OF pos_mod_sign pos_mod_conj
                                       [THEN conjunct2, THEN order_less_imp_le]]
                     nat_mod_distrib)
   apply (simp add: nat_mod_distrib)
   done
 
 lemmas bl_word_roti_dt = bl_word_roti_dt' [unfolded word_size]
 
 lemmas word_rotl_dt = bl_word_rotl_dt [THEN word_bl.Rep_inverse' [symmetric]]
 lemmas word_rotr_dt = bl_word_rotr_dt [THEN word_bl.Rep_inverse' [symmetric]]
 lemmas word_roti_dt = bl_word_roti_dt [THEN word_bl.Rep_inverse' [symmetric]]
 
 lemmas word_rotr_dt_no_bin' [simp] =
   word_rotr_dt [where w="numeral w", unfolded to_bl_numeral] for w
   (* FIXME: negative numerals, 0 and 1 *)
 
 lemmas word_rotl_dt_no_bin' [simp] =
   word_rotl_dt [where w="numeral w", unfolded to_bl_numeral] for w
   (* FIXME: negative numerals, 0 and 1 *)
 
 lemma max_word_bl: "to_bl (max_word::'a::len word) = replicate LENGTH('a) True"
   by (fact to_bl_n1)
 
 lemma to_bl_mask:
   "to_bl (mask n :: 'a::len word) =
   replicate (LENGTH('a) - n) False @
     replicate (min (LENGTH('a)) n) True"
   by (simp add: mask_bl word_rep_drop min_def)
 
 lemma map_replicate_True:
   "n = length xs \<Longrightarrow>
     map (\<lambda>(x,y). x \<and> y) (zip xs (replicate n True)) = xs"
   by (induct xs arbitrary: n) auto
 
 lemma map_replicate_False:
   "n = length xs \<Longrightarrow> map (\<lambda>(x,y). x \<and> y)
     (zip xs (replicate n False)) = replicate n False"
   by (induct xs arbitrary: n) auto
 
 lemma bl_and_mask:
   fixes w :: "'a::len word"
     and n :: nat
   defines "n' \<equiv> LENGTH('a) - n"
   shows "to_bl (w AND mask n) = replicate n' False @ drop n' (to_bl w)"
 proof -
   note [simp] = map_replicate_True map_replicate_False
   have "to_bl (w AND mask n) = map2 (\<and>) (to_bl w) (to_bl (mask n::'a::len word))"
     by (simp add: bl_word_and)
   also have "to_bl w = take n' (to_bl w) @ drop n' (to_bl w)"
     by simp
   also have "map2 (\<and>) \<dots> (to_bl (mask n::'a::len word)) =
       replicate n' False @ drop n' (to_bl w)"
     unfolding to_bl_mask n'_def by (subst zip_append) auto
   finally show ?thesis .
 qed
 
 lemma drop_rev_takefill:
   "length xs \<le> n \<Longrightarrow>
     drop (n - length xs) (rev (takefill False n (rev xs))) = xs"
   by (simp add: takefill_alt rev_take)
 
 declare bin_to_bl_def [simp]
 
 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 transfer
   apply (simp add: take_bit_eq_mod)
   apply (rule Divides.mod_less)
    apply (rule bl_to_bin_ge0)
   using bl_to_bin_lt2p_drop apply (rule order.strict_trans2)
   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_and_1_bl:
   fixes x::"'a::len word"
   shows "(x AND 1) = of_bl [x !! 0]"
   by (simp add: mod_2_eq_odd test_bit_word_eq and_one_eq)
 
 lemma word_1_and_bl:
   fixes x::"'a::len word"
   shows "(1 AND x) = of_bl [x !! 0]"
   by (simp add: mod_2_eq_odd test_bit_word_eq one_and_eq)
 
 lemma of_bl_drop:
   "of_bl (drop n xs) = (of_bl xs AND 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 to_bl_1:
   "to_bl (1::'a::len word) = replicate (LENGTH('a) - 1) False @ [True]"
   by (rule nth_equalityI) (auto simp add: to_bl_unfold nth_append rev_nth bit_1_iff not_less not_le)
 
 lemma eq_zero_set_bl:
   "(w = 0) = (True \<notin> set (to_bl w))"
   apply (auto simp add: to_bl_unfold)
   apply (rule bit_word_eqI)
   apply auto
   done
 
 lemma of_drop_to_bl:
   "of_bl (drop n (to_bl x)) = (x AND mask (size x - n))"
   by (simp add: of_bl_drop word_size_bl)
 
 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 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 word_msb_alt: "msb w \<longleftrightarrow> hd (to_bl w)"
   for w :: "'a::len word"
   apply (simp add: msb_word_eq)
   apply (subst hd_conv_nth)
    apply simp
   apply (subst nth_to_bl)
    apply simp
   apply simp
   done
 
 lemma word_lsb_last:
   \<open>lsb w \<longleftrightarrow> last (to_bl w)\<close>
   for w :: \<open>'a::len word\<close>
   using nth_to_bl [of \<open>LENGTH('a) - Suc 0\<close> w]
   by (simp add: lsb_odd last_conv_nth)
 
 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 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 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 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 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
 
 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 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 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) AND NOT(mask n) = a"
   apply (simp flip: push_bit_eq_mult subtract_mask(1) take_bit_eq_mask)
   apply (subst disjunctive_add)
    apply (auto simp add: bit_simps not_le not_less)
    apply (meson is_aligned_imp_not_bit is_aligned_weaken less_diff_conv2)
   apply (erule is_alignedE')
   apply (simp add: take_bit_push_bit)
   apply (rule bit_word_eqI)
   apply (auto simp add: bit_simps)
   done
 
 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 (simp flip: push_bit_eq_mult take_bit_eq_mask add: shiftr_eq_drop_bit)
   apply (subst disjunctive_add)
    apply (auto simp add: bit_simps not_le not_less unsigned_or_eq unsigned_drop_bit_eq
      unsigned_push_bit_eq bin_to_bl_or simp flip: bin_to_bl_def)
    apply (meson is_aligned_imp_not_bit is_aligned_weaken less_diff_conv2)
   apply (erule is_alignedE')
   apply (rule nth_equalityI)
    apply (auto simp add: nth_bin_to_bl bit_simps rev_nth simp flip: bin_to_bl_def)
   done
 
 (* 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)
 
 (* 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 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 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)
 
 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 of_bl_max:
   "(of_bl xs :: 'a::len word) \<le> mask (length xs)"
 proof -
   define ys where \<open>ys = rev xs\<close>
   have \<open>take_bit (length ys) (horner_sum of_bool 2 ys :: 'a word) = horner_sum of_bool 2 ys\<close>
     by transfer (simp add: take_bit_horner_sum_bit_eq min_def)
   then have \<open>(of_bl (rev ys) :: 'a word) \<le> mask (length ys)\<close>
     by (simp only: of_bl_rev_eq less_eq_mask_iff_take_bit_eq_self)
   with ys_def show ?thesis
     by simp
 qed
 
 end
diff --git a/thys/Word_Lib/Rsplit.thy b/thys/Word_Lib/Rsplit.thy
--- a/thys/Word_Lib/Rsplit.thy
+++ b/thys/Word_Lib/Rsplit.thy
@@ -1,161 +1,166 @@
-(*  Author:     Jeremy Dawson and Gerwin Klein, NICTA
-*)
+(*
+ * Copyright Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
+
+(* Author: Jeremy Dawson and Gerwin Klein, NICTA *)
 
 theory Rsplit
   imports "HOL-Library.Word" Bits_Int
 begin
 
 definition word_rsplit :: "'a::len word \<Rightarrow> 'b::len word list"
   where "word_rsplit w = map word_of_int (bin_rsplit (LENGTH('b)) (LENGTH('a), uint w))"
 
 lemma word_rsplit_no:
   "(word_rsplit (numeral bin :: 'b::len word) :: 'a word list) =
     map word_of_int (bin_rsplit (LENGTH('a::len))
       (LENGTH('b), take_bit (LENGTH('b)) (numeral bin)))"
   by (simp add: word_rsplit_def of_nat_take_bit)
 
 lemmas word_rsplit_no_cl [simp] = word_rsplit_no
   [unfolded bin_rsplitl_def bin_rsplit_l [symmetric]]
 
 text \<open>
   This odd result arises from the fact that the statement of the
   result implies that the decoded words are of the same type,
   and therefore of the same length, as the original word.\<close>
 
 lemma word_rsplit_same: "word_rsplit w = [w]"
   apply (simp add: word_rsplit_def bin_rsplit_all)
   apply transfer
   apply simp
   done
 
 lemma word_rsplit_empty_iff_size: "word_rsplit w = [] \<longleftrightarrow> size w = 0"
   by (simp add: word_rsplit_def bin_rsplit_def word_size bin_rsplit_aux_simp_alt Let_def
       split: prod.split)
 
 lemma test_bit_rsplit:
   "sw = word_rsplit w \<Longrightarrow> m < size (hd sw) \<Longrightarrow>
     k < length sw \<Longrightarrow> (rev sw ! k) !! m = w !! (k * size (hd sw) + m)"
   for sw :: "'a::len word list"
   apply (unfold word_rsplit_def word_test_bit_def)
   apply (rule trans)
    apply (rule_tac f = "\<lambda>x. bin_nth x m" in arg_cong)
    apply (rule nth_map [symmetric])
    apply simp
   apply (rule bin_nth_rsplit)
      apply simp_all
   apply (simp add : word_size rev_map)
   apply (rule trans)
    defer
    apply (rule map_ident [THEN fun_cong])
   apply (rule refl [THEN map_cong])
   apply simp
   using bin_rsplit_size_sign take_bit_int_eq_self_iff by blast
 
 lemma test_bit_rsplit_alt:
   \<open>(word_rsplit w  :: 'b::len word list) ! i !! m \<longleftrightarrow>
     w !! ((length (word_rsplit w :: 'b::len word list) - Suc i) * size (hd (word_rsplit w :: 'b::len word list)) + m)\<close>
   if \<open>i < length (word_rsplit w :: 'b::len word list)\<close> \<open>m < size (hd (word_rsplit w :: 'b::len word list))\<close> \<open>0 < length (word_rsplit w :: 'b::len word list)\<close>
   for w :: \<open>'a::len word\<close>
   apply (rule trans)
    apply (rule test_bit_cong)
    apply (rule rev_nth [of _ \<open>rev (word_rsplit w)\<close>, simplified rev_rev_ident])
   apply simp
    apply (rule that(1))
   apply simp
   apply (rule test_bit_rsplit)
     apply (rule refl)
   apply (rule asm_rl)
    apply (rule that(2))
   apply (rule diff_Suc_less)
   apply (rule that(3))
   done
 
 lemma word_rsplit_len_indep [OF refl refl refl refl]:
   "[u,v] = p \<Longrightarrow> [su,sv] = q \<Longrightarrow> word_rsplit u = su \<Longrightarrow>
     word_rsplit v = sv \<Longrightarrow> length su = length sv"
   by (auto simp: word_rsplit_def bin_rsplit_len_indep)
 
 lemma length_word_rsplit_size:
   "n = LENGTH('a::len) \<Longrightarrow>
     length (word_rsplit w :: 'a word list) \<le> m \<longleftrightarrow> size w \<le> m * n"
   by (auto simp: word_rsplit_def word_size bin_rsplit_len_le)
 
 lemmas length_word_rsplit_lt_size =
   length_word_rsplit_size [unfolded Not_eq_iff linorder_not_less [symmetric]]
 
 lemma length_word_rsplit_exp_size:
   "n = LENGTH('a::len) \<Longrightarrow>
     length (word_rsplit w :: 'a word list) = (size w + n - 1) div n"
   by (auto simp: word_rsplit_def word_size bin_rsplit_len)
 
 lemma length_word_rsplit_even_size:
   "n = LENGTH('a::len) \<Longrightarrow> size w = m * n \<Longrightarrow>
     length (word_rsplit w :: 'a word list) = m"
   by (cases \<open>LENGTH('a)\<close>) (simp_all add: length_word_rsplit_exp_size div_nat_eqI)
 
 lemmas length_word_rsplit_exp_size' = refl [THEN length_word_rsplit_exp_size]
 
 \<comment> \<open>alternative proof of \<open>word_rcat_rsplit\<close>\<close>
 lemmas tdle = times_div_less_eq_dividend
 lemmas dtle = xtrans(4) [OF tdle mult.commute]
 
 lemma word_rcat_rsplit: "word_rcat (word_rsplit w) = w"
   apply (rule word_eqI)
   apply (clarsimp simp: test_bit_rcat word_size)
   apply (subst refl [THEN test_bit_rsplit])
     apply (simp_all add: word_size
       refl [THEN length_word_rsplit_size [simplified not_less [symmetric], simplified]])
    apply safe
    apply (erule xtrans(7), rule dtle)+
   done
 
 lemma size_word_rsplit_rcat_size:
   "word_rcat ws = frcw \<Longrightarrow> size frcw = length ws * LENGTH('a)
     \<Longrightarrow> length (word_rsplit frcw::'a word list) = length ws"
   for ws :: "'a::len word list" and frcw :: "'b::len word"
   by (cases \<open>LENGTH('a)\<close>) (simp_all add: word_size length_word_rsplit_exp_size' div_nat_eqI)
 
 lemma msrevs:
   "0 < n \<Longrightarrow> (k * n + m) div n = m div n + k"
   "(k * n + m) mod n = m mod n"
   for n :: nat
   by (auto simp: add.commute)
 
 lemma word_rsplit_rcat_size [OF refl]:
   "word_rcat ws = frcw \<Longrightarrow>
     size frcw = length ws * LENGTH('a) \<Longrightarrow> word_rsplit frcw = ws"
   for ws :: "'a::len word list"
   apply (frule size_word_rsplit_rcat_size, assumption)
   apply (clarsimp simp add : word_size)
   apply (rule nth_equalityI, assumption)
   apply clarsimp
   apply (rule word_eqI [rule_format])
   apply (rule trans)
    apply (rule test_bit_rsplit_alt)
      apply (clarsimp simp: word_size)+
   apply (rule trans)
    apply (rule test_bit_rcat [OF refl refl])
   apply (simp add: word_size)
   apply (subst rev_nth)
    apply arith
   apply (simp add: le0 [THEN [2] xtrans(7), THEN diff_Suc_less])
   apply safe
   apply (simp add: diff_mult_distrib)
    apply (cases "size ws")
     apply simp_all
   done
 
 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 rev_nth field_simps)
   apply (simp add: length_word_rsplit_exp_size)
   apply transfer
   apply (metis (no_types, lifting) Nat.add_diff_assoc Suc_leI add_0_left diff_Suc_less div_less len_gt_0 msrevs(1) mult.commute)
   done
 
 end
\ No newline at end of file
diff --git a/thys/Word_Lib/Signed_Division_Word.thy b/thys/Word_Lib/Signed_Division_Word.thy
--- a/thys/Word_Lib/Signed_Division_Word.thy
+++ b/thys/Word_Lib/Signed_Division_Word.thy
@@ -1,151 +1,156 @@
+(*
+ * Copyright Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
 
 section \<open>Signed division on word\<close>
 
 theory Signed_Division_Word
   imports "HOL-Library.Signed_Division" "HOL-Library.Word"
 begin
 
 instantiation word :: (len) signed_division
 begin
 
 lift_definition signed_divide_word :: \<open>'a::len word \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
   is \<open>\<lambda>k l. signed_take_bit (LENGTH('a) - Suc 0) k sdiv signed_take_bit (LENGTH('a) - Suc 0) l\<close>
   by (simp flip: signed_take_bit_decr_length_iff)
 
 lift_definition signed_modulo_word :: \<open>'a::len word \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
   is \<open>\<lambda>k l. signed_take_bit (LENGTH('a) - Suc 0) k smod signed_take_bit (LENGTH('a) - Suc 0) l\<close>
   by (simp flip: signed_take_bit_decr_length_iff)
 
 instance ..
 
 end
 
 lemma sdiv_word_def [code]:
   \<open>v sdiv w = word_of_int (sint v sdiv sint w)\<close>
   for v w :: \<open>'a::len word\<close>
   by transfer simp
 
 lemma smod_word_def [code]:
   \<open>v smod w = word_of_int (sint v smod sint w)\<close>
   for v w :: \<open>'a::len word\<close>
   by transfer simp
 
 lemma sdiv_smod_id:
   \<open>(a sdiv b) * b + (a smod b) = a\<close>
   for a b :: \<open>'a::len word\<close>
   by (cases \<open>sint a < 0\<close>; cases \<open>sint b < 0\<close>) (simp_all add: signed_modulo_int_def sdiv_word_def smod_word_def)
 
 lemma signed_div_arith:
     "sint ((a::('a::len) word) sdiv b) = signed_take_bit (LENGTH('a) - 1) (sint a sdiv sint b)"
   apply transfer
   apply simp
   done
 
 lemma signed_mod_arith:
     "sint ((a::('a::len) word) smod b) = signed_take_bit (LENGTH('a) - 1) (sint a smod sint b)"
   apply transfer
   apply simp
   done
 
 lemma word_sdiv_div1 [simp]:
     "(a :: ('a::len) word) sdiv 1 = a"
   apply (cases \<open>LENGTH('a)\<close>)
   apply simp_all
   apply transfer
   apply simp
   apply (case_tac nat)
    apply simp_all
   apply (simp add: take_bit_signed_take_bit)
   done
 
 lemma word_sdiv_div0 [simp]:
     "(a :: ('a::len) word) sdiv 0 = 0"
   apply (auto simp: sdiv_word_def signed_divide_int_def sgn_if)
   done
 
 lemma word_sdiv_div_minus1 [simp]:
     "(a :: ('a::len) word) sdiv -1 = -a"
   apply (auto simp: sdiv_word_def signed_divide_int_def sgn_if)
   apply transfer
   apply simp
   apply (metis Suc_pred len_gt_0 signed_take_bit_eq_iff_take_bit_eq signed_take_bit_of_0 take_bit_of_0)
   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)"
   using sdiv_int_range [where a="sint a" and b="sint b"]
   apply auto
   apply (cases \<open>LENGTH('a)\<close>)
    apply simp_all
   apply transfer
   apply simp
   apply (rule order_trans)
    defer
    apply assumption
   apply simp
   apply (metis abs_le_iff add.inverse_inverse le_cases le_minus_iff not_le signed_take_bit_int_eq_self_iff signed_take_bit_minus)
   done
 
 lemmas word_sdiv_numerals_lhs = sdiv_word_def[where v="numeral x" for x]
     sdiv_word_def[where v=0] sdiv_word_def[where v=1]
 
 lemmas word_sdiv_numerals = word_sdiv_numerals_lhs[where w="numeral y" for y]
     word_sdiv_numerals_lhs[where w=0] word_sdiv_numerals_lhs[where w=1]
 
 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 (cases \<open>sint b = 0\<close>)
    apply (simp_all add: sint_less)
   apply (cases \<open>LENGTH('a)\<close>)
    apply simp_all
   using smod_int_range [where a="sint a" and b="sint b"]
   apply auto
   apply (rule less_le_trans)
    apply assumption
   apply (auto simp add: abs_le_iff)
    apply (metis diff_Suc_1 le_cases linorder_not_le sint_lt)
   apply (metis add.inverse_inverse diff_Suc_1 linorder_not_less neg_less_iff_less sint_ge)
   done
 
 lemma smod_word_min:
   "- (2 ^ (LENGTH('a::len) - Suc 0)) \<le> sint (a::'a word) smod sint (b::'a word)"
   apply (cases \<open>LENGTH('a)\<close>)
    apply simp_all
   apply (cases \<open>sint b = 0\<close>)
    apply simp_all
    apply (metis diff_Suc_1 sint_ge)
   using smod_int_range [where a="sint a" and b="sint b"]
   apply auto
   apply (rule order_trans)
    defer
    apply assumption
   apply (auto simp add: algebra_simps abs_le_iff)
    apply (metis abs_zero add.left_neutral add_mono_thms_linordered_semiring(1) diff_Suc_1 le_cases linorder_not_less sint_lt zabs_less_one_iff)
   apply (metis abs_zero add.inverse_inverse add.left_neutral add_mono_thms_linordered_semiring(1) diff_Suc_1 le_cases le_minus_iff linorder_not_less sint_ge zabs_less_one_iff)
   done
 
 lemma smod_word_alt_def:
   "(a :: ('a::len) word) smod b = a - (a sdiv b) * b"
   apply (cases \<open>a \<noteq> - (2 ^ (LENGTH('a) - 1)) \<or> b \<noteq> - 1\<close>)
    apply (clarsimp simp: smod_word_def sdiv_word_def signed_modulo_int_def
      simp flip: wi_hom_sub wi_hom_mult)
   apply (clarsimp simp: smod_word_def signed_modulo_int_def)
   done
 
 lemmas word_smod_numerals_lhs = smod_word_def[where v="numeral x" for x]
     smod_word_def[where v=0] smod_word_def[where v=1]
 
 lemmas word_smod_numerals = word_smod_numerals_lhs[where w="numeral y" for y]
     word_smod_numerals_lhs[where w=0] word_smod_numerals_lhs[where w=1]
 
 end
\ No newline at end of file
diff --git a/thys/Word_Lib/Strict_part_mono.thy b/thys/Word_Lib/Strict_part_mono.thy
--- a/thys/Word_Lib/Strict_part_mono.thy
+++ b/thys/Word_Lib/Strict_part_mono.thy
@@ -1,51 +1,56 @@
+(*
+ * Copyright Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
 
 theory Strict_part_mono
   imports "HOL-Library.Word" More_Word
 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 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
 
 end
diff --git a/thys/Word_Lib/Traditional_Infix_Syntax.thy b/thys/Word_Lib/Traditional_Infix_Syntax.thy
--- a/thys/Word_Lib/Traditional_Infix_Syntax.thy
+++ b/thys/Word_Lib/Traditional_Infix_Syntax.thy
@@ -1,1060 +1,1065 @@
-(*  Author:     Jeremy Dawson, NICTA
-*)
+(*
+ * Copyright Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
+
+(* Author: Jeremy Dawson, NICTA *)
 
 section \<open>Operation variants with traditional syntax\<close>
 
 theory Traditional_Infix_Syntax
   imports "HOL-Library.Word" More_Word Signed_Words
 begin
 
 class semiring_bit_syntax = semiring_bit_shifts
 begin
 
 definition test_bit :: \<open>'a \<Rightarrow> nat \<Rightarrow> bool\<close>  (infixl "!!" 100)
   where test_bit_eq_bit: \<open>test_bit = bit\<close>
 
 definition shiftl :: \<open>'a \<Rightarrow> nat \<Rightarrow> 'a\<close>  (infixl "<<" 55)
   where shiftl_eq_push_bit: \<open>a << n = push_bit n a\<close>
 
 definition shiftr :: \<open>'a \<Rightarrow> nat \<Rightarrow> 'a\<close>  (infixl ">>" 55)
   where shiftr_eq_drop_bit: \<open>a >> n = drop_bit n a\<close>
 
 end
 
 instance word :: (len) semiring_bit_syntax ..
 
 context
   includes lifting_syntax
 begin
 
 lemma test_bit_word_transfer [transfer_rule]:
   \<open>(pcr_word ===> (=)) (\<lambda>k n. n < LENGTH('a) \<and> bit k n) (test_bit :: 'a::len word \<Rightarrow> _)\<close>
   by (unfold test_bit_eq_bit) transfer_prover
 
 lemma shiftl_word_transfer [transfer_rule]:
   \<open>(pcr_word ===> (=) ===> pcr_word) (\<lambda>k n. push_bit n k) shiftl\<close>
   by (unfold shiftl_eq_push_bit) transfer_prover
 
 lemma shiftr_word_transfer [transfer_rule]:
   \<open>(pcr_word ===> (=) ===> pcr_word) (\<lambda>k n. (drop_bit n \<circ> take_bit LENGTH('a)) k) (shiftr :: 'a::len word \<Rightarrow> _)\<close>
   by (unfold shiftr_eq_drop_bit) transfer_prover
 
 end
 
 
 lemma test_bit_word_eq:
   \<open>test_bit = (bit :: 'a::len word \<Rightarrow> _)\<close>
   by (fact test_bit_eq_bit)
 
 lemma shiftl_word_eq:
   \<open>w << n = push_bit n w\<close> for w :: \<open>'a::len word\<close>
   by (fact shiftl_eq_push_bit)
 
 lemma shiftr_word_eq:
   \<open>w >> n = drop_bit n w\<close> for w :: \<open>'a::len word\<close>
   by (fact shiftr_eq_drop_bit)
 
 lemma test_bit_eq_iff: "test_bit u = test_bit v \<longleftrightarrow> u = v"
   for u v :: "'a::len word"
   by (simp add: bit_eq_iff test_bit_eq_bit fun_eq_iff)
 
 lemma test_bit_size: "w !! n \<Longrightarrow> n < size w"
   for w :: "'a::len word"
   by transfer simp
 
 lemma word_eq_iff: "x = y \<longleftrightarrow> (\<forall>n<LENGTH('a). x !! n = y !! n)" (is \<open>?P \<longleftrightarrow> ?Q\<close>)
   for x y :: "'a::len word"
   by transfer (auto simp add: bit_eq_iff bit_take_bit_iff)
 
 lemma word_eqI: "(\<And>n. n < size u \<longrightarrow> u !! n = v !! n) \<Longrightarrow> u = v"
   for u :: "'a::len word"
   by (simp add: word_size word_eq_iff)
 
 lemma word_eqD: "u = v \<Longrightarrow> u !! x = v !! x"
   for u v :: "'a::len word"
   by simp
 
 lemma test_bit_bin': "w !! n \<longleftrightarrow> n < size w \<and> bit (uint w) n"
   by transfer (simp add: bit_take_bit_iff)
 
 lemmas test_bit_bin = test_bit_bin' [unfolded word_size]
 
-lemma word_test_bit_def: 
+lemma word_test_bit_def:
   \<open>test_bit a = bit (uint a)\<close>
   by transfer (simp add: fun_eq_iff bit_take_bit_iff)
 
 lemmas test_bit_def' = word_test_bit_def [THEN fun_cong]
 
 lemma word_test_bit_transfer [transfer_rule]:
   "(rel_fun pcr_word (rel_fun (=) (=)))
     (\<lambda>x n. n < LENGTH('a) \<and> bit x n) (test_bit :: 'a::len word \<Rightarrow> _)"
   by (simp only: test_bit_eq_bit) transfer_prover
 
 lemma test_bit_wi [simp]:
   "(word_of_int x :: 'a::len word) !! n \<longleftrightarrow> n < LENGTH('a) \<and> bit x n"
   by transfer simp
 
 lemma word_ops_nth_size:
   "n < size x \<Longrightarrow>
     (x OR y) !! n = (x !! n | y !! n) \<and>
     (x AND y) !! n = (x !! n \<and> y !! n) \<and>
     (x XOR y) !! n = (x !! n \<noteq> y !! n) \<and>
     (NOT x) !! n = (\<not> x !! n)"
   for x :: "'a::len word"
   by transfer (simp add: bit_or_iff bit_and_iff bit_xor_iff bit_not_iff)
 
 lemma word_ao_nth:
   "(x OR y) !! n = (x !! n | y !! n) \<and>
     (x AND y) !! n = (x !! n \<and> y !! n)"
   for x :: "'a::len word"
   by transfer (auto simp add: bit_or_iff bit_and_iff)
 
 lemmas msb0 = len_gt_0 [THEN diff_Suc_less, THEN word_ops_nth_size [unfolded word_size]]
 lemmas msb1 = msb0 [where i = 0]
 
 lemma test_bit_numeral [simp]:
   "(numeral w :: 'a::len word) !! n \<longleftrightarrow>
     n < LENGTH('a) \<and> bit (numeral w :: int) n"
   by transfer (rule refl)
 
 lemma test_bit_neg_numeral [simp]:
   "(- numeral w :: 'a::len word) !! n \<longleftrightarrow>
     n < LENGTH('a) \<and> bit (- numeral w :: int) n"
   by transfer (rule refl)
 
 lemma test_bit_1 [iff]: "(1 :: 'a::len word) !! n \<longleftrightarrow> n = 0"
-  by transfer (auto simp add: bit_1_iff) 
+  by transfer (auto simp add: bit_1_iff)
 
 lemma nth_0 [simp]: "\<not> (0 :: 'a::len word) !! n"
   by transfer simp
 
 lemma nth_minus1 [simp]: "(-1 :: 'a::len word) !! n \<longleftrightarrow> n < LENGTH('a)"
   by transfer simp
 
 lemma shiftl1_code [code]:
   \<open>shiftl1 w = push_bit 1 w\<close>
   by transfer (simp add: ac_simps)
 
 lemma uint_shiftr_eq:
   \<open>uint (w >> n) = uint w div 2 ^ n\<close>
   by transfer (simp flip: drop_bit_eq_div add: drop_bit_take_bit min_def le_less less_diff_conv)
 
 lemma shiftr1_code [code]:
   \<open>shiftr1 w = drop_bit 1 w\<close>
   by transfer (simp add: drop_bit_Suc)
 
 lemma shiftl_def:
   \<open>w << n = (shiftl1 ^^ n) w\<close>
 proof -
   have \<open>push_bit n = (((*) 2 ^^ n) :: int \<Rightarrow> int)\<close> for n
     by (induction n) (simp_all add: fun_eq_iff funpow_swap1, simp add: ac_simps)
   then show ?thesis
     by transfer simp
 qed
 
 lemma shiftr_def:
   \<open>w >> n = (shiftr1 ^^ n) w\<close>
 proof -
   have \<open>shiftr1 ^^ n = (drop_bit n :: 'a word \<Rightarrow> 'a word)\<close>
     apply (induction n)
     apply simp
     apply (simp only: shiftr1_eq_div_2 [abs_def] drop_bit_eq_div [abs_def] funpow_Suc_right)
     apply (use div_exp_eq [of _ 1, where ?'a = \<open>'a word\<close>] in simp)
     done
   then show ?thesis
     by (simp add: shiftr_eq_drop_bit)
 qed
 
 lemma bit_shiftl_word_iff [bit_simps]:
   \<open>bit (w << m) n \<longleftrightarrow> m \<le> n \<and> n < LENGTH('a) \<and> bit w (n - m)\<close>
   for w :: \<open>'a::len word\<close>
   by (simp add: shiftl_word_eq bit_push_bit_iff exp_eq_zero_iff not_le)
 
 lemma bit_shiftr_word_iff [bit_simps]:
   \<open>bit (w >> m) n \<longleftrightarrow> bit w (m + n)\<close>
   for w :: \<open>'a::len word\<close>
   by (simp add: shiftr_word_eq bit_drop_bit_eq)
 
 lift_definition sshiftr :: \<open>'a::len word \<Rightarrow> nat \<Rightarrow> 'a word\<close>  (infixl \<open>>>>\<close> 55)
   is \<open>\<lambda>k n. take_bit LENGTH('a) (drop_bit n (signed_take_bit (LENGTH('a) - Suc 0) k))\<close>
   by (simp flip: signed_take_bit_decr_length_iff)
 
 lemma sshiftr_eq [code]:
   \<open>w >>> n = signed_drop_bit n w\<close>
   by transfer simp
 
 lemma sshiftr_eq_funpow_sshiftr1:
   \<open>w >>> n = (sshiftr1 ^^ n) w\<close>
   apply (rule sym)
   apply (simp add: sshiftr1_eq_signed_drop_bit_Suc_0 sshiftr_eq)
   apply (induction n)
    apply simp_all
   done
 
 lemma uint_sshiftr_eq:
   \<open>uint (w >>> n) = take_bit LENGTH('a) (sint w div 2 ^  n)\<close>
   for w :: \<open>'a::len word\<close>
   by transfer (simp flip: drop_bit_eq_div)
 
 lemma sshift1_code [code]:
   \<open>sshiftr1 w = signed_drop_bit 1 w\<close>
   by transfer (simp add: drop_bit_Suc)
 
 lemma sshiftr_0 [simp]: "0 >>> n = 0"
   by transfer simp
 
 lemma sshiftr_n1 [simp]: "-1 >>> n = -1"
   by transfer simp
 
 lemma bit_sshiftr_word_iff [bit_simps]:
   \<open>bit (w >>> m) n \<longleftrightarrow> bit w (if LENGTH('a) - m \<le> n \<and> n < LENGTH('a) then LENGTH('a) - 1 else (m + n))\<close>
   for w :: \<open>'a::len word\<close>
   apply transfer
   apply (auto simp add: bit_take_bit_iff bit_drop_bit_eq bit_signed_take_bit_iff min_def not_le simp flip: bit_Suc)
   using le_less_Suc_eq apply fastforce
   using le_less_Suc_eq apply fastforce
   done
 
 lemma nth_sshiftr :
   "(w >>> m) !! n =
     (n < size w \<and> (if n + m \<ge> size w then w !! (size w - 1) else w !! (n + m)))"
   apply transfer
   apply (auto simp add: bit_take_bit_iff bit_drop_bit_eq bit_signed_take_bit_iff min_def not_le ac_simps)
   using le_less_Suc_eq apply fastforce
   using le_less_Suc_eq apply fastforce
   done
 
 lemma sshiftr_numeral [simp]:
   \<open>(numeral k >>> numeral n :: 'a::len word) =
     word_of_int (drop_bit (numeral n) (signed_take_bit (LENGTH('a) - 1) (numeral k)))\<close>
   apply (rule word_eqI)
   apply (cases \<open>LENGTH('a)\<close>)
    apply (simp_all add: word_size bit_drop_bit_eq nth_sshiftr bit_signed_take_bit_iff min_def not_le not_less less_Suc_eq_le ac_simps)
   done
 
 setup \<open>
   Context.theory_map (fold SMT_Word.add_word_shift' [
     (\<^term>\<open>shiftl :: 'a::len word \<Rightarrow> _\<close>, "bvshl"),
     (\<^term>\<open>shiftr :: 'a::len word \<Rightarrow> _\<close>, "bvlshr"),
     (\<^term>\<open>sshiftr :: 'a::len word \<Rightarrow> _\<close>, "bvashr")
   ])
 \<close>
 
 lemma revcast_down_us [OF refl]:
   "rc = revcast \<Longrightarrow> source_size rc = target_size rc + n \<Longrightarrow> rc w = ucast (w >>> n)"
   for w :: "'a::len word"
   apply (simp add: source_size_def target_size_def)
   apply (rule bit_word_eqI)
   apply (simp add: bit_revcast_iff bit_ucast_iff bit_sshiftr_word_iff ac_simps)
   done
 
 lemma revcast_down_ss [OF refl]:
   "rc = revcast \<Longrightarrow> source_size rc = target_size rc + n \<Longrightarrow> rc w = scast (w >>> n)"
   for w :: "'a::len word"
   apply (simp add: source_size_def target_size_def)
   apply (rule bit_word_eqI)
   apply (simp add: bit_revcast_iff bit_word_scast_iff bit_sshiftr_word_iff ac_simps)
   done
 
 lemma sshiftr_div_2n: "sint (w >>> n) = sint w div 2 ^ n"
   using sint_signed_drop_bit_eq [of n w]
-  by (simp add: drop_bit_eq_div sshiftr_eq) 
+  by (simp add: drop_bit_eq_div sshiftr_eq)
 
 lemmas lsb0 = len_gt_0 [THEN word_ops_nth_size [unfolded word_size]]
 
 lemma nth_sint:
   fixes w :: "'a::len word"
   defines "l \<equiv> LENGTH('a)"
   shows "bit (sint w) n = (if n < l - 1 then w !! n else w !! (l - 1))"
   unfolding sint_uint l_def
   by (auto simp: bit_signed_take_bit_iff word_test_bit_def not_less min_def)
 
 lemma test_bit_2p: "(word_of_int (2 ^ n)::'a::len word) !! m \<longleftrightarrow> m = n \<and> m < LENGTH('a)"
   by transfer (auto simp add: bit_exp_iff)
 
 lemma nth_w2p: "((2::'a::len word) ^ n) !! m \<longleftrightarrow> m = n \<and> m < LENGTH('a::len)"
   by transfer (auto simp add: bit_exp_iff)
 
 lemma bang_is_le: "x !! m \<Longrightarrow> 2 ^ m \<le> x"
   for x :: "'a::len word"
   apply (rule xtrans(3))
    apply (rule_tac [2] y = "x" in le_word_or2)
   apply (rule word_eqI)
   apply (auto simp add: word_ao_nth nth_w2p word_size)
   done
 
 lemma mask_eq:
   \<open>mask n = (1 << n) - (1 :: 'a::len word)\<close>
-  by transfer (simp add: mask_eq_exp_minus_1 push_bit_of_1) 
+  by transfer (simp add: mask_eq_exp_minus_1 push_bit_of_1)
 
 lemma nth_ucast:
   "(ucast w::'a::len word) !! n = (w !! n \<and> n < LENGTH('a))"
   by transfer (simp add: bit_take_bit_iff ac_simps)
 
 lemma shiftl_0 [simp]: "(0::'a::len word) << n = 0"
   by transfer simp
 
 lemma shiftr_0 [simp]: "(0::'a::len word) >> n = 0"
   by transfer simp
 
 lemma nth_shiftl1: "shiftl1 w !! n \<longleftrightarrow> n < size w \<and> n > 0 \<and> w !! (n - 1)"
   by transfer (auto simp add: bit_double_iff)
 
 lemma nth_shiftl': "(w << m) !! n \<longleftrightarrow> n < size w \<and> n >= m \<and> w !! (n - m)"
   for w :: "'a::len word"
   by transfer (auto simp add: bit_push_bit_iff)
 
 lemmas nth_shiftl = nth_shiftl' [unfolded word_size]
 
 lemma nth_shiftr1: "shiftr1 w !! n = w !! Suc n"
   by transfer (auto simp add: bit_take_bit_iff simp flip: bit_Suc)
 
 lemma nth_shiftr: "(w >> m) !! n = w !! (n + m)"
   for w :: "'a::len word"
   apply (unfold shiftr_def)
   apply (induct "m" arbitrary: n)
    apply (auto simp add: nth_shiftr1)
   done
 
 lemma nth_sshiftr1: "sshiftr1 w !! n = (if n = size w - 1 then w !! n else w !! Suc n)"
   apply transfer
   apply (auto simp add: bit_take_bit_iff bit_signed_take_bit_iff min_def simp flip: bit_Suc)
   using le_less_Suc_eq apply fastforce
   using le_less_Suc_eq apply fastforce
   done
 
 lemma shiftr_div_2n: "uint (shiftr w n) = uint w div 2 ^ n"
   by (fact uint_shiftr_eq)
 
 lemma shiftl_rev: "shiftl w n = word_reverse (shiftr (word_reverse w) n)"
   by (induct n) (auto simp add: shiftl_def shiftr_def shiftl1_rev)
 
 lemma rev_shiftl: "word_reverse w << n = word_reverse (w >> n)"
   by (simp add: shiftl_rev)
 
 lemma shiftr_rev: "w >> n = word_reverse (word_reverse w << n)"
   by (simp add: rev_shiftl)
 
 lemma rev_shiftr: "word_reverse w >> n = word_reverse (w << n)"
   by (simp add: shiftr_rev)
 
 lemma shiftl_numeral [simp]:
   \<open>numeral k << numeral l = (push_bit (numeral l) (numeral k) :: 'a::len word)\<close>
   by (fact shiftl_word_eq)
 
 lemma shiftl_zero_size: "size x \<le> n \<Longrightarrow> x << n = 0"
   for x :: "'a::len word"
   apply transfer
   apply (simp add: take_bit_push_bit)
   done
 
 lemma shiftl_t2n: "shiftl w n = 2 ^ n * w"
   for w :: "'a::len word"
   by (induct n) (auto simp: shiftl_def shiftl1_2t)
 
 lemma shiftr_numeral [simp]:
   \<open>(numeral k >> numeral n :: 'a::len word) = drop_bit (numeral n) (numeral k)\<close>
   by (fact shiftr_word_eq)
 
 lemma nth_mask [simp]:
   \<open>(mask n :: 'a::len word) !! i \<longleftrightarrow> i < n \<and> i < size (mask n :: 'a word)\<close>
   by (auto simp add: test_bit_word_eq word_size Word.bit_mask_iff)
 
 lemma slice_shiftr: "slice n w = ucast (w >> n)"
   apply (rule bit_word_eqI)
   apply (cases \<open>n \<le> LENGTH('b)\<close>)
    apply (auto simp add: bit_slice_iff bit_ucast_iff bit_shiftr_word_iff ac_simps
     dest: bit_imp_le_length)
   done
 
 lemma nth_slice: "(slice n w :: 'a::len word) !! m = (w !! (m + n) \<and> m < LENGTH('a))"
   by (simp add: slice_shiftr nth_ucast nth_shiftr)
 
 lemma revcast_down_uu [OF refl]:
   "rc = revcast \<Longrightarrow> source_size rc = target_size rc + n \<Longrightarrow> rc w = ucast (w >> n)"
   for w :: "'a::len word"
   apply (simp add: source_size_def target_size_def)
   apply (rule bit_word_eqI)
   apply (simp add: bit_revcast_iff bit_ucast_iff bit_shiftr_word_iff ac_simps)
   done
 
 lemma revcast_down_su [OF refl]:
   "rc = revcast \<Longrightarrow> source_size rc = target_size rc + n \<Longrightarrow> rc w = scast (w >> n)"
   for w :: "'a::len word"
   apply (simp add: source_size_def target_size_def)
   apply (rule bit_word_eqI)
   apply (simp add: bit_revcast_iff bit_word_scast_iff bit_shiftr_word_iff ac_simps)
   done
 
 lemma cast_down_rev [OF refl]:
   "uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> uc w = revcast (w << n)"
   for w :: "'a::len word"
   apply (simp add: source_size_def target_size_def)
   apply (rule bit_word_eqI)
   apply (simp add: bit_revcast_iff bit_word_ucast_iff bit_shiftl_word_iff)
   done
 
 lemma revcast_up [OF refl]:
   "rc = revcast \<Longrightarrow> source_size rc + n = target_size rc \<Longrightarrow>
     rc w = (ucast w :: 'a::len word) << n"
   apply (simp add: source_size_def target_size_def)
   apply (rule bit_word_eqI)
   apply (simp add: bit_revcast_iff bit_word_ucast_iff bit_shiftl_word_iff)
   apply auto
   apply (metis add.commute add_diff_cancel_right)
   apply (metis diff_add_inverse2 diff_diff_add)
   done
 
 lemmas rc1 = revcast_up [THEN
   revcast_rev_ucast [symmetric, THEN trans, THEN word_rev_gal, symmetric]]
 lemmas rc2 = revcast_down_uu [THEN
   revcast_rev_ucast [symmetric, THEN trans, THEN word_rev_gal, symmetric]]
 
 lemmas ucast_up =
   rc1 [simplified rev_shiftr [symmetric] revcast_ucast [symmetric]]
 lemmas ucast_down =
   rc2 [simplified rev_shiftr revcast_ucast [symmetric]]
 
 \<comment> \<open>problem posed by TPHOLs referee:
       criterion for overflow of addition of signed integers\<close>
 
 lemma sofl_test:
   \<open>sint x + sint y = sint (x + y) \<longleftrightarrow>
     (x + y XOR x) AND (x + y XOR y) >> (size x - 1) = 0\<close>
   for x y :: \<open>'a::len word\<close>
 proof -
   obtain n where n: \<open>LENGTH('a) = Suc n\<close>
     by (cases \<open>LENGTH('a)\<close>) simp_all
   have *: \<open>sint x + sint y + 2 ^ Suc n > signed_take_bit n (sint x + sint y) \<Longrightarrow> sint x + sint y \<ge> - (2 ^ n)\<close>
     \<open>signed_take_bit n (sint x + sint y) > sint x + sint y - 2 ^ Suc n \<Longrightarrow> 2 ^ n > sint x + sint y\<close>
     using signed_take_bit_int_greater_eq [of \<open>sint x + sint y\<close> n] signed_take_bit_int_less_eq [of n \<open>sint x + sint y\<close>]
     by (auto intro: ccontr)
   have \<open>sint x + sint y = sint (x + y) \<longleftrightarrow>
     (sint (x + y) < 0 \<longleftrightarrow> sint x < 0) \<or>
     (sint (x + y) < 0 \<longleftrightarrow> sint y < 0)\<close>
     using sint_less [of x] sint_greater_eq [of x] sint_less [of y] sint_greater_eq [of y]
     signed_take_bit_int_eq_self [of \<open>LENGTH('a) - 1\<close> \<open>sint x + sint y\<close>]
     apply (auto simp add: not_less)
        apply (unfold sint_word_ariths)
        apply (subst signed_take_bit_int_eq_self)
          prefer 4
        apply (subst signed_take_bit_int_eq_self)
          prefer 7
        apply (subst signed_take_bit_int_eq_self)
          prefer 10
              apply (subst signed_take_bit_int_eq_self)
        apply (auto simp add: signed_take_bit_int_eq_self signed_take_bit_eq_take_bit_minus take_bit_Suc_from_most n not_less intro!: *)
     done
   then show ?thesis
     apply (simp only: One_nat_def word_size shiftr_word_eq drop_bit_eq_zero_iff_not_bit_last bit_and_iff bit_xor_iff)
     apply (simp add: bit_last_iff)
     done
 qed
 
 lemma shiftr_zero_size: "size x \<le> n \<Longrightarrow> x >> n = 0"
   for x :: "'a :: len word"
   by (rule word_eqI) (auto simp add: nth_shiftr dest: test_bit_size)
 
 lemma test_bit_cat [OF refl]:
   "wc = word_cat a b \<Longrightarrow> wc !! n = (n < size wc \<and>
     (if n < size b then b !! n else a !! (n - size b)))"
   apply (simp add: word_size not_less; transfer)
        apply (auto simp add: bit_concat_bit_iff bit_take_bit_iff)
   done
 
 \<comment> \<open>keep quantifiers for use in simplification\<close>
 lemma test_bit_split':
   "word_split c = (a, b) \<longrightarrow>
     (\<forall>n m.
       b !! n = (n < size b \<and> c !! n) \<and>
       a !! m = (m < size a \<and> c !! (m + size b)))"
   by (auto simp add: word_split_bin' test_bit_bin bit_unsigned_iff word_size
     bit_drop_bit_eq ac_simps exp_eq_zero_iff
     dest: bit_imp_le_length)
 
 lemma test_bit_split:
   "word_split c = (a, b) \<Longrightarrow>
     (\<forall>n::nat. b !! n \<longleftrightarrow> n < size b \<and> c !! n) \<and>
     (\<forall>m::nat. a !! m \<longleftrightarrow> m < size a \<and> c !! (m + size b))"
   by (simp add: test_bit_split')
 
 lemma test_bit_split_eq:
   "word_split c = (a, b) \<longleftrightarrow>
     ((\<forall>n::nat. b !! n = (n < size b \<and> c !! n)) \<and>
      (\<forall>m::nat. a !! m = (m < size a \<and> c !! (m + size b))))"
   apply (rule_tac iffI)
    apply (rule_tac conjI)
     apply (erule test_bit_split [THEN conjunct1])
    apply (erule test_bit_split [THEN conjunct2])
   apply (case_tac "word_split c")
   apply (frule test_bit_split)
   apply (erule trans)
   apply (fastforce intro!: word_eqI simp add: word_size)
   done
 
 lemma test_bit_rcat:
   "sw = size (hd wl) \<Longrightarrow> rc = word_rcat wl \<Longrightarrow> rc !! n =
     (n < size rc \<and> n div sw < size wl \<and> (rev wl) ! (n div sw) !! (n mod sw))"
   for wl :: "'a::len word list"
   by (simp add: word_size word_rcat_def foldl_map rev_map bit_horner_sum_uint_exp_iff)
     (simp add: test_bit_eq_bit)
 
 lemmas test_bit_cong = arg_cong [where f = "test_bit", THEN fun_cong]
 
 lemma max_test_bit: "(max_word::'a::len word) !! n \<longleftrightarrow> n < LENGTH('a)"
   by (fact nth_minus1)
 
 lemma shiftr_x_0 [iff]: "x >> 0 = x"
   for x :: "'a::len word"
   by transfer simp
 
 lemma shiftl_x_0 [simp]: "x << 0 = x"
   for x :: "'a::len word"
   by (simp add: shiftl_t2n)
 
 lemma shiftl_1 [simp]: "(1::'a::len word) << n = 2^n"
   by (simp add: shiftl_t2n)
 
 lemma shiftr_1[simp]: "(1::'a::len word) >> n = (if n = 0 then 1 else 0)"
   by (induct n) (auto simp: shiftr_def)
 
 lemma map_nth_0 [simp]: "map ((!!) (0::'a::len word)) xs = replicate (length xs) False"
   by (induct xs) auto
 
 lemma word_and_1:
   "n AND 1 = (if n !! 0 then 1 else 0)" for n :: "_ word"
   by (rule bit_word_eqI) (auto simp add: bit_and_iff test_bit_eq_bit bit_1_iff intro: gr0I)
 
 lemma test_bit_1' [simp]:
   "(1 :: 'a :: len word) !! n \<longleftrightarrow> 0 < LENGTH('a) \<and> n = 0"
   by simp
 
 lemma shiftl0:
   "x << 0 = (x :: 'a :: len word)"
   by (fact shiftl_x_0)
 
 lemma word_ops_nth [simp]:
   fixes x y :: \<open>'a::len word\<close>
   shows
   word_or_nth:  "(x OR y) !! n = (x !! n \<or> y !! n)" and
   word_and_nth: "(x AND 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])+
 
 lemma and_not_mask:
   "w AND NOT (mask n) = (w >> n) << n"
   for w :: \<open>'a::len word\<close>
   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)"
   for w :: \<open>'a::len word\<close>
   apply (rule word_eqI)
   apply (simp add : word_ops_nth_size word_size)
   apply (simp add : nth_shiftr nth_shiftl)
   by auto
 
 lemma nth_w2p_same:
   "(2^n :: 'a :: len word) !! n = (n < LENGTH('a))"
   by (simp add : nth_w2p)
 
 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 (metis uint_shiftr_eq word_of_int_uint)
   done
 
 lemma le_shiftr:
   "u \<le> v \<Longrightarrow> u >> (n :: nat) \<le> (v :: 'a :: len 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 :: 'a::len word)"
   apply (rule word_eqI)
   apply (simp add: word_size nth_shiftr)
   done
 
 lemma shiftr_mask [simp]:
   \<open>mask m >> m = (0::'a::len word)\<close>
   by (rule shiftr_mask_le) simp
-  
+
 lemma word_leI:
   "(\<And>n.  \<lbrakk>n < size (u::'a::len word); u !! n \<rbrakk> \<Longrightarrow> (v::'a::len word) !! n) \<Longrightarrow> u <= v"
   apply (rule xtrans(4))
    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)"
   for w :: \<open>'a::len word\<close>
   apply safe
    apply (rule word_le_0_iff [THEN iffD1])
    apply (rule xtrans(3))
     apply (erule_tac [2] le_shiftr)
    apply simp
   apply (rule word_leI)
   apply (rename_tac n')
   apply (drule_tac x = "n' - n" in word_eqD)
   apply (simp add : nth_shiftr word_size)
   apply (case_tac "n <= n'")
   by auto
 
 lemma and_mask_eq_iff_shiftr_0:
   "(w AND mask n = w) = (w >> n = 0)"
   for w :: \<open>'a::len word\<close>
   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 transfer
   apply (auto simp add: fun_eq_iff bit_simps)
   apply (metis add_diff_inverse_nat)
   done
 
 lemma mask_shiftl_decompose:
   "mask m << n = mask (m + n) AND NOT (mask n :: 'a::len word)"
   by (auto intro!: word_eqI simp: and_not_mask nth_shiftl nth_shiftr word_size)
 
 lemma bang_eq:
   fixes x :: "'a::len word"
   shows "(x = y) = (\<forall>n. x !! n = y !! n)"
   by (subst test_bit_eq_iff[symmetric]) fastforce
 
 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 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
 
 lemma word_shiftl_add_distrib:
   fixes x :: "'a :: len word"
   shows "(x + y) << n = (x << n) + (y << n)"
   by (simp add: shiftl_t2n ring_distribs)
 
 lemma mask_shift:
   "(x AND NOT (mask y)) >> y = x >> y"
   for x :: \<open>'a::len word\<close>
   apply (rule bit_eqI)
   apply (simp add: bit_and_iff bit_not_iff bit_shiftr_word_iff bit_mask_iff not_le)
   using bit_imp_le_length apply auto
   done
 
 lemma shiftr_div_2n':
   "unat (w >> n) = unat w div 2 ^ n"
   apply (unfold unat_eq_nat_uint)
   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_eq_unatI)
   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 transfer (simp add: take_bit_push_bit)
 
 lemma word_shift_nonzero:
   "\<lbrakk> (x::'a::len word) \<le> 2 ^ m; m + n < LENGTH('a::len); x \<noteq> 0\<rbrakk>
    \<Longrightarrow> x << n \<noteq> 0"
   apply (simp only: word_neq_0_conv word_less_nat_alt
                     shiftl_t2n mod_0 unat_word_ariths
                     unat_power_lower word_le_nat_alt)
   apply (subst mod_less)
    apply (rule order_le_less_trans)
     apply (erule mult_le_mono2)
    apply (subst power_add[symmetric])
    apply (rule power_strict_increasing)
     apply simp
    apply simp
   apply simp
   done
 
 lemma word_shiftr_lt:
   fixes w :: "'a::len word"
   shows "unat (w >> n) < (2 ^ (LENGTH('a) - n))"
   apply (subst shiftr_div_2n')
   apply transfer
   apply (simp flip: drop_bit_eq_div add: drop_bit_nat_eq drop_bit_take_bit)
   done
 
 lemma neg_mask_test_bit:
   "(NOT(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 upper_bits_unset_is_l2p:
   \<open>(\<forall>n' \<ge> n. n' < LENGTH('a) \<longrightarrow> \<not> p !! n') \<longleftrightarrow> (p < 2 ^ n)\<close> (is \<open>?P \<longleftrightarrow> ?Q\<close>)
     if \<open>n < LENGTH('a)\<close>
     for p :: "'a :: len word"
 proof
   assume ?Q
   then show ?P
     by (meson bang_is_le le_less_trans not_le word_power_increasing)
 next
   assume ?P
   have \<open>take_bit n p = p\<close>
   proof (rule bit_word_eqI)
     fix q
     assume \<open>q < LENGTH('a)\<close>
     show \<open>bit (take_bit n p) q \<longleftrightarrow> bit p q\<close>
     proof (cases \<open>q < n\<close>)
       case True
-      then show ?thesis 
+      then show ?thesis
         by (auto simp add: bit_simps)
     next
       case False
       then have \<open>n \<le> q\<close>
         by simp
       with \<open>?P\<close> \<open>q < LENGTH('a)\<close> have \<open>\<not> bit p q\<close>
         by (simp add: test_bit_eq_bit)
       then show ?thesis
         by (simp add: bit_simps)
     qed
   qed
   with that show ?Q
     using take_bit_word_eq_self_iff [of n p] by auto
 qed
 
 lemma less_2p_is_upper_bits_unset:
   "p < 2 ^ n \<longleftrightarrow> n < LENGTH('a) \<and> (\<forall>n' \<ge> n. n' < LENGTH('a) \<longrightarrow> \<not> p !! n')" for p :: "'a :: len word"
   by (meson le_less_trans le_mask_iff_lt_2n upper_bits_unset_is_l2p word_zero_le)
 
 lemma test_bit_over:
   "n \<ge> size (x::'a::len word) \<Longrightarrow> (x !! n) = False"
   by transfer auto
 
 lemma le_mask_high_bits:
   "w \<le> mask n \<longleftrightarrow> (\<forall>i \<in> {n ..< size w}. \<not> w !! i)"
   for w :: \<open>'a::len word\<close>
   by (auto simp: word_size and_mask_eq_iff_le_mask[symmetric] word_eq_iff)
 
 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:
   "(NOT 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)
 
 lemma shiftr_less_t2n':
   "\<lbrakk> x AND mask (n + m) = x; m < LENGTH('a) \<rbrakk> \<Longrightarrow> x >> n < 2 ^ m" for x :: "'a :: len word"
   apply (simp add: word_size mask_eq_iff_w2p [symmetric] flip: take_bit_eq_mask)
   apply transfer
   apply (simp add: take_bit_drop_bit ac_simps)
   done
 
 lemma shiftr_less_t2n:
   "x < 2 ^ (n + m) \<Longrightarrow> x >> n < 2 ^ m" for x :: "'a :: len word"
   apply (rule shiftr_less_t2n')
    apply (erule less_mask_eq)
   apply (rule ccontr)
   apply (simp add: not_less)
   apply (subst (asm) p2_eq_0[symmetric])
   apply (simp add: power_add)
   done
 
 lemma shiftr_eq_0:
   "n \<ge> LENGTH('a) \<Longrightarrow> ((w::'a::len word) >> n) = 0"
   apply (cut_tac shiftr_less_t2n'[of w n 0], simp)
    apply (simp add: mask_eq_iff)
   apply (simp add: lt2p_lem)
   apply simp
   done
 
 lemma shiftr_not_mask_0:
   "n+m \<ge> LENGTH('a :: len) \<Longrightarrow> ((w::'a::len word) >> n) AND NOT (mask m) = 0"
   by (rule bit_word_eqI) (auto simp add: bit_simps dest: bit_imp_le_length)
 
 lemma shiftl_less_t2n:
   fixes x :: "'a :: len word"
   shows "\<lbrakk> x < (2 ^ (m - n)); m < LENGTH('a) \<rbrakk> \<Longrightarrow> (x << n) < 2 ^ m"
   apply (simp add: word_size mask_eq_iff_w2p [symmetric] flip: take_bit_eq_mask)
   apply transfer
   apply (simp add: take_bit_push_bit)
   done
 
 lemma shiftl_less_t2n':
   "(x::'a::len word) < 2 ^ m \<Longrightarrow> m+n < LENGTH('a) \<Longrightarrow> x << n < 2 ^ (m + n)"
   by (rule shiftl_less_t2n) simp_all
 
 lemma nth_w2p_scast [simp]:
   "((scast ((2::'a::len signed word) ^ n) :: 'a word) !! m)
          \<longleftrightarrow> ((((2::'a::len  word) ^ n) :: 'a word) !! m)"
   by transfer (auto simp add: bit_simps)
 
 lemma scast_bit_test [simp]:
     "scast ((1 :: 'a::len signed word) << n) = (1 :: 'a word) << n"
   by (clarsimp simp: word_eq_iff)
 
 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 nth_bounded:
   "\<lbrakk>(x :: 'a :: len word) !! n; x < 2 ^ m; m \<le> len_of TYPE ('a)\<rbrakk> \<Longrightarrow> n < m"
   apply (rule ccontr)
   apply (auto simp add: not_less test_bit_word_eq)
   apply (meson bit_imp_le_length bit_uint_iff less_2p_is_upper_bits_unset test_bit_bin)
   done
 
 lemma shiftl_mask_is_0[simp]:
   "(x << n) AND mask n = 0"
   for x :: \<open>'a::len word\<close>
   by (simp flip: take_bit_eq_mask add: shiftl_eq_push_bit take_bit_push_bit)
 
 lemma rshift_sub_mask_eq:
   "(a >> (size a - b)) AND mask b = a >> (size a - b)"
   for a :: \<open>'a::len word\<close>
   using shiftl_shiftr2[where a=a and b=0 and c="size a - b"]
   apply (cases "b < size a")
    apply simp
   apply (simp add: linorder_not_less mask_eq_decr_exp word_size
                    p2_eq_0[THEN iffD2])
   done
 
 lemma shiftl_shiftr3:
   "b \<le> c \<Longrightarrow> a << b >> c = (a >> c - b) AND mask (size a - c)"
   for a :: \<open>'a::len word\<close>
   apply (cases "b = c")
    apply (simp add: shiftl_shiftr1)
   apply (simp add: shiftl_shiftr2)
   done
 
 lemma and_mask_shiftr_comm:
   "m \<le> size w \<Longrightarrow> (w AND mask m) >> n = (w >> n) AND mask (m-n)"
   for w :: \<open>'a::len word\<close>
   by (simp add: and_mask shiftr_shiftr) (simp add: word_size shiftl_shiftr3)
 
 lemma and_mask_shiftl_comm:
   "m+n \<le> size w \<Longrightarrow> (w AND mask m) << n = (w << n) AND mask (m+n)"
   for w :: \<open>'a::len word\<close>
   by (simp add: and_mask word_size shiftl_shiftl) (simp add: shiftl_shiftr1)
 
 lemma le_mask_shiftl_le_mask: "s = m + n \<Longrightarrow> x \<le> mask n \<Longrightarrow> x << m \<le> mask s"
   for x :: \<open>'a::len word\<close>
   by (simp add: le_mask_iff shiftl_shiftr3)
 
 lemma word_and_1_shiftl:
   "x AND (1 << n) = (if x !! n then (1 << n) else 0)" for x :: "'a :: len word"
   apply (rule bit_word_eqI; transfer)
   apply (auto simp add: bit_simps not_le ac_simps)
   done
 
 lemmas word_and_1_shiftls'
     = word_and_1_shiftl[where n=0]
       word_and_1_shiftl[where n=1]
       word_and_1_shiftl[where n=2]
 
 lemmas word_and_1_shiftls = word_and_1_shiftls' [simplified]
 
 lemma word_and_mask_shiftl:
   "x AND (mask n << m) = ((x >> m) AND mask n) << m"
   for x :: \<open>'a::len word\<close>
   apply (rule bit_word_eqI; transfer)
   apply (auto simp add: bit_simps not_le ac_simps)
   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 of_bool_nth:
   "of_bool (x !! v) = (x >> v) AND 1"
   for x :: \<open>'a::len word\<close>
   by (simp add: test_bit_word_eq shiftr_word_eq bit_eq_iff)
     (auto simp add: bit_1_iff bit_and_iff bit_drop_bit_eq intro: ccontr)
 
 lemma shiftr_mask_eq:
   "(x >> n) AND mask (size x - n) = x >> n" for x :: "'a :: len word"
   apply (simp flip: take_bit_eq_mask)
   apply transfer
   apply (simp add: take_bit_drop_bit)
   done
 
 lemma shiftr_mask_eq':
   "m = (size x - n) \<Longrightarrow> (x >> n) AND mask m = x >> n" for x :: "'a :: len word"
   by (simp add: shiftr_mask_eq)
 
 lemma and_eq_0_is_nth:
   fixes x :: "'a :: len word"
   shows "y = 1 << n \<Longrightarrow> ((x AND y) = 0) = (\<not> (x !! n))"
   apply safe
    apply (drule_tac u="(x AND (1 << n))" and x=n in word_eqD)
    apply (simp add: nth_w2p)
    apply (simp add: test_bit_bin)
   apply (rule bit_word_eqI)
   apply (auto simp add: bit_simps test_bit_eq_bit)
   done
 
 lemma and_neq_0_is_nth:
   \<open>x AND y \<noteq> 0 \<longleftrightarrow> x !! n\<close> if \<open>y = 2 ^ n\<close> for x y :: \<open>'a::len word\<close>
   apply (simp add: bit_eq_iff bit_simps)
   using that apply (simp add: bit_simps not_le)
   apply transfer
   apply auto
   done
 
 lemma nth_is_and_neq_0:
   "(x::'a::len word) !! n = (x AND 2 ^ n \<noteq> 0)"
   by (subst and_neq_0_is_nth; rule refl)
 
 lemma word_shift_zero:
   "\<lbrakk> x << n = 0; x \<le> 2^m; m + n < LENGTH('a)\<rbrakk> \<Longrightarrow> (x::'a::len word) = 0"
   apply (rule ccontr)
   apply (drule (2) word_shift_nonzero)
   apply simp
   done
 
 lemma mask_shift_and_negate[simp]:"(w AND mask n << m) AND NOT (mask n << m) = 0"
   for w :: \<open>'a::len word\<close>
   by (clarsimp simp add: mask_eq_decr_exp Parity.bit_eq_iff bit_and_iff bit_not_iff shiftl_word_eq bit_push_bit_iff)
 
 end
diff --git a/thys/Word_Lib/Typedef_Morphisms.thy b/thys/Word_Lib/Typedef_Morphisms.thy
--- a/thys/Word_Lib/Typedef_Morphisms.thy
+++ b/thys/Word_Lib/Typedef_Morphisms.thy
@@ -1,362 +1,368 @@
 (*
-  Author:     Jeremy Dawson and Gerwin Klein, NICTA
+ * Copyright Data61, CSIRO (ABN 41 687 119 230)
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
+
+(*
+  Author: Jeremy Dawson and Gerwin Klein, NICTA
 
   Consequences of type definition theorems, and of extended type definition.
 *)
 
 section \<open>Type Definition Theorems\<close>
 
 theory Typedef_Morphisms
   imports Main "HOL-Library.Word" Bit_Comprehension Bits_Int
 begin
 
 subsection "More lemmas about normal type definitions"
 
 lemma tdD1: "type_definition Rep Abs A \<Longrightarrow> \<forall>x. Rep x \<in> A"
   and tdD2: "type_definition Rep Abs A \<Longrightarrow> \<forall>x. Abs (Rep x) = x"
   and tdD3: "type_definition Rep Abs A \<Longrightarrow> \<forall>y. y \<in> A \<longrightarrow> Rep (Abs y) = y"
   by (auto simp: type_definition_def)
 
 lemma td_nat_int: "type_definition int nat (Collect ((\<le>) 0))"
   unfolding type_definition_def by auto
 
 context type_definition
 begin
 
 declare Rep [iff] Rep_inverse [simp] Rep_inject [simp]
 
 lemma Abs_eqD: "Abs x = Abs y \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x = y"
   by (simp add: Abs_inject)
 
 lemma Abs_inverse': "r \<in> A \<Longrightarrow> Abs r = a \<Longrightarrow> Rep a = r"
   by (safe elim!: Abs_inverse)
 
 lemma Rep_comp_inverse: "Rep \<circ> f = g \<Longrightarrow> Abs \<circ> g = f"
   using Rep_inverse by auto
 
 lemma Rep_eqD [elim!]: "Rep x = Rep y \<Longrightarrow> x = y"
   by simp
 
 lemma Rep_inverse': "Rep a = r \<Longrightarrow> Abs r = a"
   by (safe intro!: Rep_inverse)
 
 lemma comp_Abs_inverse: "f \<circ> Abs = g \<Longrightarrow> g \<circ> Rep = f"
   using Rep_inverse by auto
 
 lemma set_Rep: "A = range Rep"
 proof (rule set_eqI)
   show "x \<in> A \<longleftrightarrow> x \<in> range Rep" for x
     by (auto dest: Abs_inverse [of x, symmetric])
 qed
 
 lemma set_Rep_Abs: "A = range (Rep \<circ> Abs)"
 proof (rule set_eqI)
   show "x \<in> A \<longleftrightarrow> x \<in> range (Rep \<circ> Abs)" for x
     by (auto dest: Abs_inverse [of x, symmetric])
 qed
 
 lemma Abs_inj_on: "inj_on Abs A"
   unfolding inj_on_def
   by (auto dest: Abs_inject [THEN iffD1])
 
 lemma image: "Abs ` A = UNIV"
   by (fact Abs_image)
 
 lemmas td_thm = type_definition_axioms
 
 lemma fns1: "Rep \<circ> fa = fr \<circ> Rep \<or> fa \<circ> Abs = Abs \<circ> fr \<Longrightarrow> Abs \<circ> fr \<circ> Rep = fa"
   by (auto dest: Rep_comp_inverse elim: comp_Abs_inverse simp: o_assoc)
 
 lemmas fns1a = disjI1 [THEN fns1]
 lemmas fns1b = disjI2 [THEN fns1]
 
 lemma fns4: "Rep \<circ> fa \<circ> Abs = fr \<Longrightarrow> Rep \<circ> fa = fr \<circ> Rep \<and> fa \<circ> Abs = Abs \<circ> fr"
   by auto
 
 end
 
 interpretation nat_int: type_definition int nat "Collect ((\<le>) 0)"
   by (rule td_nat_int)
 
 declare
   nat_int.Rep_cases [cases del]
   nat_int.Abs_cases [cases del]
   nat_int.Rep_induct [induct del]
   nat_int.Abs_induct [induct del]
 
 
 subsection "Extended form of type definition predicate"
 
 lemma td_conds:
   "norm \<circ> norm = norm \<Longrightarrow>
     fr \<circ> norm = norm \<circ> fr \<longleftrightarrow> norm \<circ> fr \<circ> norm = fr \<circ> norm \<and> norm \<circ> fr \<circ> norm = norm \<circ> fr"
   apply safe
     apply (simp_all add: comp_assoc)
    apply (simp_all add: o_assoc)
   done
 
 lemma fn_comm_power: "fa \<circ> tr = tr \<circ> fr \<Longrightarrow> fa ^^ n \<circ> tr = tr \<circ> fr ^^ n"
   apply (rule ext)
   apply (induct n)
    apply (auto dest: fun_cong)
   done
 
 lemmas fn_comm_power' =
   ext [THEN fn_comm_power, THEN fun_cong, unfolded o_def]
 
 
 locale td_ext = type_definition +
   fixes norm
   assumes eq_norm: "\<And>x. Rep (Abs x) = norm x"
 begin
 
 lemma Abs_norm [simp]: "Abs (norm x) = Abs x"
   using eq_norm [of x] by (auto elim: Rep_inverse')
 
 lemma td_th: "g \<circ> Abs = f \<Longrightarrow> f (Rep x) = g x"
   by (drule comp_Abs_inverse [symmetric]) simp
 
 lemma eq_norm': "Rep \<circ> Abs = norm"
   by (auto simp: eq_norm)
 
 lemma norm_Rep [simp]: "norm (Rep x) = Rep x"
   by (auto simp: eq_norm' intro: td_th)
 
 lemmas td = td_thm
 
 lemma set_iff_norm: "w \<in> A \<longleftrightarrow> w = norm w"
   by (auto simp: set_Rep_Abs eq_norm' eq_norm [symmetric])
 
 lemma inverse_norm: "Abs n = w \<longleftrightarrow> Rep w = norm n"
   apply (rule iffI)
    apply (clarsimp simp add: eq_norm)
   apply (simp add: eq_norm' [symmetric])
   done
 
 lemma norm_eq_iff: "norm x = norm y \<longleftrightarrow> Abs x = Abs y"
   by (simp add: eq_norm' [symmetric])
 
 lemma norm_comps:
   "Abs \<circ> norm = Abs"
   "norm \<circ> Rep = Rep"
   "norm \<circ> norm = norm"
   by (auto simp: eq_norm' [symmetric] o_def)
 
 lemmas norm_norm [simp] = norm_comps
 
 lemma fns5: "Rep \<circ> fa \<circ> Abs = fr \<Longrightarrow> fr \<circ> norm = fr \<and> norm \<circ> fr = fr"
   by (fold eq_norm') auto
 
 text \<open>
   following give conditions for converses to \<open>td_fns1\<close>
   \<^item> the condition \<open>norm \<circ> fr \<circ> norm = fr \<circ> norm\<close> says that
     \<open>fr\<close> takes normalised arguments to normalised results
   \<^item> \<open>norm \<circ> fr \<circ> norm = norm \<circ> fr\<close> says that \<open>fr\<close>
     takes norm-equivalent arguments to norm-equivalent results
   \<^item> \<open>fr \<circ> norm = fr\<close> says that \<open>fr\<close>
     takes norm-equivalent arguments to the same result
   \<^item> \<open>norm \<circ> fr = fr\<close> says that \<open>fr\<close> takes any argument to a normalised result
 \<close>
 lemma fns2: "Abs \<circ> fr \<circ> Rep = fa \<Longrightarrow> norm \<circ> fr \<circ> norm = fr \<circ> norm \<longleftrightarrow> Rep \<circ> fa = fr \<circ> Rep"
   apply (fold eq_norm')
   apply safe
    prefer 2
    apply (simp add: o_assoc)
   apply (rule ext)
   apply (drule_tac x="Rep x" in fun_cong)
   apply auto
   done
 
 lemma fns3: "Abs \<circ> fr \<circ> Rep = fa \<Longrightarrow> norm \<circ> fr \<circ> norm = norm \<circ> fr \<longleftrightarrow> fa \<circ> Abs = Abs \<circ> fr"
   apply (fold eq_norm')
   apply safe
    prefer 2
    apply (simp add: comp_assoc)
   apply (rule ext)
   apply (drule_tac f="a \<circ> b" for a b in fun_cong)
   apply simp
   done
 
 lemma fns: "fr \<circ> norm = norm \<circ> fr \<Longrightarrow> fa \<circ> Abs = Abs \<circ> fr \<longleftrightarrow> Rep \<circ> fa = fr \<circ> Rep"
   apply safe
    apply (frule fns1b)
    prefer 2
    apply (frule fns1a)
    apply (rule fns3 [THEN iffD1])
      prefer 3
      apply (rule fns2 [THEN iffD1])
        apply (simp_all add: comp_assoc)
    apply (simp_all add: o_assoc)
   done
 
 lemma range_norm: "range (Rep \<circ> Abs) = A"
   by (simp add: set_Rep_Abs)
 
 end
 
 lemmas td_ext_def' =
   td_ext_def [unfolded type_definition_def td_ext_axioms_def]
 
 
 subsection \<open>Type-definition locale instantiations\<close>
 
 definition uints :: "nat \<Rightarrow> int set"
   \<comment> \<open>the sets of integers representing the words\<close>
   where "uints n = range (take_bit n)"
 
 definition sints :: "nat \<Rightarrow> int set"
   where "sints n = range (signed_take_bit (n - 1))"
 
 lemma uints_num: "uints n = {i. 0 \<le> i \<and> i < 2 ^ n}"
   by (simp add: uints_def range_bintrunc)
 
 lemma sints_num: "sints n = {i. - (2 ^ (n - 1)) \<le> i \<and> i < 2 ^ (n - 1)}"
   by (simp add: sints_def range_sbintrunc)
 
 definition unats :: "nat \<Rightarrow> nat set"
   where "unats n = {i. i < 2 ^ n}"
 
 \<comment> \<open>naturals\<close>
 lemma uints_unats: "uints n = int ` unats n"
   apply (unfold unats_def uints_num)
   apply safe
     apply (rule_tac image_eqI)
      apply (erule_tac nat_0_le [symmetric])
   by auto
 
 lemma unats_uints: "unats n = nat ` uints n"
   by (auto simp: uints_unats image_iff)
 
 lemma td_ext_uint:
   "td_ext (uint :: 'a word \<Rightarrow> int) word_of_int (uints (LENGTH('a::len)))
     (\<lambda>w::int. w mod 2 ^ LENGTH('a))"
   apply (unfold td_ext_def')
   apply transfer
   apply (simp add: uints_num take_bit_eq_mod)
   done
 
 interpretation word_uint:
   td_ext
     "uint::'a::len word \<Rightarrow> int"
     word_of_int
     "uints (LENGTH('a::len))"
     "\<lambda>w. w mod 2 ^ LENGTH('a::len)"
   by (fact td_ext_uint)
 
 lemmas td_uint = word_uint.td_thm
 lemmas int_word_uint = word_uint.eq_norm
 
 lemma td_ext_ubin:
   "td_ext (uint :: 'a word \<Rightarrow> int) word_of_int (uints (LENGTH('a::len)))
     (take_bit (LENGTH('a)))"
   apply standard
   apply transfer
   apply simp
   done
 
 interpretation word_ubin:
   td_ext
     "uint::'a::len word \<Rightarrow> int"
     word_of_int
     "uints (LENGTH('a::len))"
     "take_bit (LENGTH('a::len))"
   by (fact td_ext_ubin)
 
 lemma td_ext_unat [OF refl]:
   "n = LENGTH('a::len) \<Longrightarrow>
     td_ext (unat :: 'a word \<Rightarrow> nat) of_nat (unats n) (\<lambda>i. i mod 2 ^ n)"
   apply (standard; transfer)
      apply (simp_all add: unats_def take_bit_of_nat take_bit_nat_eq_self_iff
       flip: take_bit_eq_mod)
   done
 
 lemmas unat_of_nat = td_ext_unat [THEN td_ext.eq_norm]
 
 interpretation word_unat:
   td_ext
     "unat::'a::len word \<Rightarrow> nat"
     of_nat
     "unats (LENGTH('a::len))"
     "\<lambda>i. i mod 2 ^ LENGTH('a::len)"
   by (rule td_ext_unat)
 
 lemmas td_unat = word_unat.td_thm
 
 lemma unat_le: "y \<le> unat z \<Longrightarrow> y \<in> unats (LENGTH('a))"
   for z :: "'a::len word"
   apply (unfold unats_def)
   apply clarsimp
   apply (rule xtrans, rule unat_lt2p, assumption)
   done
 
 lemma td_ext_sbin:
   "td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (LENGTH('a::len)))
     (signed_take_bit (LENGTH('a) - 1))"
   by (standard; transfer) (auto simp add: sints_def)
 
 lemma td_ext_sint:
   "td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (LENGTH('a::len)))
      (\<lambda>w. (w + 2 ^ (LENGTH('a) - 1)) mod 2 ^ LENGTH('a) -
          2 ^ (LENGTH('a) - 1))"
   using td_ext_sbin [where ?'a = 'a] by (simp add: no_sbintr_alt2)
 
 text \<open>
   We do \<open>sint\<close> before \<open>sbin\<close>, before \<open>sint\<close> is the user version
   and interpretations do not produce thm duplicates. I.e.
   we get the name \<open>word_sint.Rep_eqD\<close>, but not \<open>word_sbin.Req_eqD\<close>,
   because the latter is the same thm as the former.
 \<close>
 interpretation word_sint:
   td_ext
     "sint ::'a::len word \<Rightarrow> int"
     word_of_int
     "sints (LENGTH('a::len))"
     "\<lambda>w. (w + 2^(LENGTH('a::len) - 1)) mod 2^LENGTH('a::len) -
       2 ^ (LENGTH('a::len) - 1)"
   by (rule td_ext_sint)
 
 interpretation word_sbin:
   td_ext
     "sint ::'a::len word \<Rightarrow> int"
     word_of_int
     "sints (LENGTH('a::len))"
     "signed_take_bit (LENGTH('a::len) - 1)"
   by (rule td_ext_sbin)
 
 lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm]
 
 lemmas td_sint = word_sint.td
 
 lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)"
   by (fact uints_def [unfolded no_bintr_alt1])
 
 lemmas bintr_num =
   word_ubin.norm_eq_iff [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
 lemmas sbintr_num =
   word_sbin.norm_eq_iff [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
 
 lemmas uint_div_alt = word_div_def [THEN trans [OF uint_cong int_word_uint]]
 lemmas uint_mod_alt = word_mod_def [THEN trans [OF uint_cong int_word_uint]]
 
 interpretation test_bit:
   td_ext
     "(!!) :: 'a::len word \<Rightarrow> nat \<Rightarrow> bool"
     set_bits
     "{f. \<forall>i. f i \<longrightarrow> i < LENGTH('a::len)}"
     "(\<lambda>h i. h i \<and> i < LENGTH('a::len))"
   by standard (auto simp add: test_bit_word_eq bit_imp_le_length bit_set_bits_word_iff set_bits_bit_eq)
 
 lemmas td_nth = test_bit.td_thm
 
 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
 
 end
diff --git a/thys/Word_Lib/Word_32.thy b/thys/Word_Lib/Word_32.thy
--- a/thys/Word_Lib/Word_32.thy
+++ b/thys/Word_Lib/Word_32.thy
@@ -1,346 +1,345 @@
 (*
  * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
  *
  * SPDX-License-Identifier: BSD-2-Clause
  *)
 
 section "Words of Length 32"
 
 theory Word_32
   imports
-    Word_Lemmas
     Word_8
     Word_16
     Word_Syntax
     Rsplit
     More_Word_Operations
     Bitwise
 begin
 
 type_synonym word32 = "32 word"
 lemma len32: "len_of (x :: 32 itself) = 32" by simp
 
 type_synonym sword32 = "32 sword"
 
 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
 
 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_eq)
   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_eq)
     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"
   by transfer simp
 
 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]"
   by (simp add: word_rsplit_def bin_rsplit_def)
 
 lemma unat_ucast_10_32 :
   fixes x :: "10 word"
   shows "unat (ucast x :: word32) = unat x"
   by transfer simp
 
 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 (simp_all add: of_nat_take_bit take_bit_word_eq_self)
 
 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 transfer (simp add: take_bit_drop_bit drop_bit_Suc)
 
 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"
   by word_bitwise auto
 
 lemma mask_step_down_32:
   \<open>\<exists>x. mask x = b\<close> if \<open>b && 1 = 1\<close>
     and \<open>\<exists>x. x < 32 \<and> mask x = b >> 1\<close> for b :: \<open>32word\<close>
 proof -
   from \<open>b && 1 = 1\<close> have \<open>odd b\<close>
     by (auto simp add: mod_2_eq_odd and_one_eq)
   then have \<open>b mod 2 = 1\<close>
     using odd_iff_mod_2_eq_one by blast
   from \<open>\<exists>x. x < 32 \<and> mask x = b >> 1\<close> obtain x where \<open>x < 32\<close> \<open>mask x = b >> 1\<close> by blast
   then have \<open>mask x = b div 2\<close>
     using shiftr1_is_div_2 [of b] by simp
   with \<open>b mod 2 = 1\<close> have \<open>2 * mask x + 1 = 2 * (b div 2) + b mod 2\<close>
-    by (simp only:) 
+    by (simp only:)
   also have \<open>\<dots> = b\<close>
     by (simp add: mult_div_mod_eq)
   finally have \<open>2 * mask x + 1 = b\<close> .
   moreover have \<open>mask (Suc x) = 2 * mask x + (1 :: 'a::len word)\<close>
     by (simp add: mask_Suc_rec)
   ultimately show ?thesis
     by auto
 qed
 
 lemma unat_of_int_32:
   "\<lbrakk>i \<ge> 0; i \<le>2 ^ 31\<rbrakk> \<Longrightarrow> (unat ((of_int i)::sword32)) = nat i"
   unfolding unat_eq_nat_uint
   apply (subst eq_nat_nat_iff)
     apply (auto simp add: take_bit_int_eq_self)
   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
 
 lemma word32_and_max_simp:
   \<open>x AND 0xFFFFFFFF = x\<close> for x :: \<open>32 word\<close>
   using word_and_full_mask_simp [of x]
   by (simp add: numeral_eq_Suc mask_Suc_exp)
 
 end
diff --git a/thys/Word_Lib/Word_Lib_Sumo.thy b/thys/Word_Lib/Word_Lib_Sumo.thy
--- a/thys/Word_Lib/Word_Lib_Sumo.thy
+++ b/thys/Word_Lib/Word_Lib_Sumo.thy
@@ -1,123 +1,128 @@
+(*
+ * Copyright Florian Haftmann
+ *
+ * SPDX-License-Identifier: BSD-2-Clause
+ *)
 
 section \<open>Ancient comprehensive Word Library\<close>
 
 theory Word_Lib_Sumo
 imports
   "HOL-Library.Word"
   Aligned
   Ancient_Numeral
   Bit_Comprehension
   Bits_Int
   Bitwise_Signed
   Bitwise
   Enumeration_Word
   Generic_set_bit
   Hex_Words
   Least_significant_bit
   More_Arithmetic
   More_Divides
   More_Sublist
   Even_More_List
   More_Misc
   Strict_part_mono
   Legacy_Aliases
   Most_significant_bit
   Next_and_Prev
   Norm_Words
   Reversed_Bit_Lists
   Rsplit
   Signed_Words
   Traditional_Infix_Syntax
   Typedef_Morphisms
   Type_Syntax
   Word_EqI
   Word_Lemmas
   Word_8
   Word_16
   Word_32
   Word_64
   Word_Syntax
   Signed_Division_Word
   More_Word_Operations
   Many_More
 begin
 
 declare signed_take_bit_Suc [simp]
 
 lemmas bshiftr1_def = bshiftr1_eq
 lemmas is_down_def = is_down_eq
 lemmas is_up_def = is_up_eq
 lemmas mask_def = mask_eq_decr_exp
 lemmas scast_def = scast_eq
 lemmas shiftl1_def = shiftl1_eq
 lemmas shiftr1_def = shiftr1_eq
 lemmas sshiftr1_def = sshiftr1_eq
 lemmas sshiftr_def = sshiftr_eq_funpow_sshiftr1
 lemmas to_bl_def = to_bl_eq
 lemmas ucast_def = ucast_eq
 lemmas unat_def = unat_eq_nat_uint
 lemmas word_cat_def = word_cat_eq
 lemmas word_reverse_def = word_reverse_eq_of_bl_rev_to_bl
 lemmas word_roti_def = word_roti_eq_word_rotr_word_rotl
 lemmas word_rotl_def = word_rotl_eq
 lemmas word_rotr_def = word_rotr_eq
 lemmas word_sle_def = word_sle_eq
 lemmas word_sless_def = word_sless_eq
 
 lemmas uint_0 = uint_nonnegative
 lemmas uint_lt = uint_bounded
 lemmas uint_mod_same = uint_idem
 lemmas of_nth_def = word_set_bits_def
 
 lemmas of_nat_word_eq_iff = word_of_nat_eq_iff
 lemmas of_nat_word_eq_0_iff = word_of_nat_eq_0_iff
 lemmas of_int_word_eq_iff = word_of_int_eq_iff
 lemmas of_int_word_eq_0_iff = word_of_int_eq_0_iff
 
 lemmas word_next_def = word_next_unfold
 
 lemmas word_prev_def = word_prev_unfold
 
 lemmas is_aligned_def = is_aligned_iff_dvd_nat
 
 lemma shiftl_transfer [transfer_rule]:
   includes lifting_syntax
   shows "(pcr_word ===> (=) ===> pcr_word) (<<) (<<)"
   by (unfold shiftl_eq_push_bit) transfer_prover
 
 lemmas word_and_max_simps =
   word8_and_max_simp
   word16_and_max_simp
   word32_and_max_simp
   word64_and_max_simp
 
 lemma distinct_lemma: "f x \<noteq> f y \<Longrightarrow> x \<noteq> y" by auto
 
 lemmas and_bang = word_and_nth
 
 lemmas sdiv_int_def = signed_divide_int_def
 lemmas smod_int_def = signed_modulo_int_def
 
 (* shortcut for some specific lengths *)
 lemma word_fixed_sint_1[simp]:
   "sint (1::8 word) = 1"
   "sint (1::16 word) = 1"
   "sint (1::32 word) = 1"
   "sint (1::64 word) = 1"
   by (auto simp: sint_word_ariths)
 
 declare of_nat_diff [simp]
 
 (* Haskellish names/syntax *)
 notation (input)
   test_bit ("testBit")
 
 lemmas cast_simps = cast_simps ucast_down_bl
 
 (* shadows the slightly weaker Word.nth_ucast *)
 lemma nth_ucast:
   "(ucast (w::'a::len word)::'b::len word) !! n =
    (w !! n \<and> n < min LENGTH('a) LENGTH('b))"
   by transfer (simp add: bit_take_bit_iff ac_simps)
 
 end
diff --git a/thys/Word_Lib/document/root.tex b/thys/Word_Lib/document/root.tex
--- a/thys/Word_Lib/document/root.tex
+++ b/thys/Word_Lib/document/root.tex
@@ -1,45 +1,51 @@
+%
+% Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
+%
+% SPDX-License-Identifier: CC-BY-SA-4.0
+%
+
 \documentclass[11pt,a4paper]{article}
 \usepackage{isabelle,isabellesym}
 
 % this should be the last package used
 \usepackage{pdfsetup}
 
 % urls in roman style, theory text in math-similar italics
 \urlstyle{rm}
 \isabellestyle{tt}
 
 
 \begin{document}
 
 \title{Finite Machine Word Library}
 \author{Joel Beeren, Sascha Böhme, Matthew Fernandez, Xin Gao, Gerwin Klein, Rafal Kolanski,\\
        Japheth Lim, Corey Lewis, Daniel Matichuk, Thomas Sewell}
 \maketitle
 
 \begin{abstract}
   This entry contains an extension to the Isabelle library for fixed-width
   machine words. In particular, the entry adds
   printing as hexadecimals, additional operations, reasoning about alignment,
   signed words, enumerations of words, normalisation of word numerals, and an
   extensive library of properties about generic fixed-width words, as well as
   an instantiation of many of these to the commonly used 32 and 64-bit bases.
 
   In addition to the listed authors, the entry contains contributions by
   Nelson Billing, Andrew Boyton, Matthew Brecknell, Cornelius Diekmann, Peter
   Gammie, Gianpaolo Gioiosa, David Greenaway, Lars Noschinski, Sean Seefried,
   and Simon Winwood.
 \end{abstract}
 
 \tableofcontents
 
 \parindent 0pt\parskip 0.5ex
 
 % generated text of all theories
 \input{session}
 
 \end{document}
 
 %%% Local Variables:
 %%% mode: latex
 %%% TeX-master: t
 %%% End: