diff --git a/thys/Berlekamp_Zassenhaus/Finite_Field_Record_Based.thy b/thys/Berlekamp_Zassenhaus/Finite_Field_Record_Based.thy
--- a/thys/Berlekamp_Zassenhaus/Finite_Field_Record_Based.thy
+++ b/thys/Berlekamp_Zassenhaus/Finite_Field_Record_Based.thy
@@ -1,1657 +1,1657 @@
 (*
     Authors:      Jose Divasón
                   Sebastiaan Joosten
                   René Thiemann
                   Akihisa Yamada
 *)
 subsection \<open>Finite Fields\<close>
 
 text \<open>We provide four implementations for $GF(p)$ -- the field with $p$ elements for some
   prime $p$ -- one by int, one by integers, one by 32-bit numbers and one 64-bit implementation. 
   Correctness of the implementations is proven by
   transfer rules to the type-based version of $GF(p)$.\<close>
 
 theory Finite_Field_Record_Based
 imports
   Finite_Field
   Arithmetic_Record_Based
   Native_Word.Uint32 
   Native_Word.Uint64
-  Native_Word.Code_Target_Bits_Int
   "HOL-Library.Code_Target_Numeral"  
+  Native_Word.Code_Target_Int_Bit
 begin
 
 (* mod on standard case which can immediately be mapped to 
    target languages without considering special cases *)
 definition mod_nonneg_pos :: "integer \<Rightarrow> integer \<Rightarrow> integer" where
   "x \<ge> 0 \<Longrightarrow> y > 0 \<Longrightarrow> mod_nonneg_pos x y = (x mod y)"
   
 code_printing \<comment> \<open>FIXME illusion of partiality\<close>
   constant mod_nonneg_pos \<rightharpoonup>
         (SML) "IntInf.mod/ ( _,/ _ )"
     and (Eval) "IntInf.mod/ ( _,/ _ )"
     and (OCaml) "Z.rem"
     and (Haskell) "Prelude.mod/ ( _ )/ ( _ )"
     and (Scala) "!((k: BigInt) => (l: BigInt) =>/ (k '% l))"
 
 definition mod_nonneg_pos_int :: "int \<Rightarrow> int \<Rightarrow> int" where
   "mod_nonneg_pos_int x y = int_of_integer (mod_nonneg_pos (integer_of_int x) (integer_of_int y))" 
 
 lemma mod_nonneg_pos_int[simp]: "x \<ge> 0 \<Longrightarrow> y > 0 \<Longrightarrow> mod_nonneg_pos_int x y = (x mod y)" 
   unfolding mod_nonneg_pos_int_def using mod_nonneg_pos_def by simp
 
 context
   fixes p :: int
 begin
 definition plus_p :: "int \<Rightarrow> int \<Rightarrow> int" where
   "plus_p x y \<equiv> let z = x + y in if z \<ge> p then z - p else z"
 
 definition minus_p :: "int \<Rightarrow> int \<Rightarrow> int" where
   "minus_p x y \<equiv> if y \<le> x then x - y else x + p - y"
 
 definition uminus_p :: "int \<Rightarrow> int" where
   "uminus_p x = (if x = 0 then 0 else p - x)"
 
 definition mult_p :: "int \<Rightarrow> int \<Rightarrow> int" where
   "mult_p x y = (mod_nonneg_pos_int (x * y) p)"
 
 fun power_p :: "int \<Rightarrow> nat \<Rightarrow> int" where
   "power_p x n = (if n = 0 then 1 else
     let (d,r) = Divides.divmod_nat n 2;
        rec = power_p (mult_p x x) d in
     if r = 0 then rec else mult_p rec x)"
 
 text \<open>In experiments with Berlekamp-factorization (where the prime $p$ is usually small),
   it turned out that taking the below implementation of inverse via exponentiation
   is faster than the one based on the extended Euclidean algorithm.\<close>
 
 definition inverse_p :: "int \<Rightarrow> int" where
   "inverse_p x = (if x = 0 then 0 else power_p x (nat (p - 2)))"
 
 definition divide_p :: "int \<Rightarrow> int \<Rightarrow> int"  where
   "divide_p x y = mult_p x (inverse_p y)"
 
 definition finite_field_ops_int :: "int arith_ops_record" where
   "finite_field_ops_int \<equiv> Arith_Ops_Record
       0
       1
       plus_p
       mult_p
       minus_p
       uminus_p
       divide_p
       inverse_p
       (\<lambda> x y . if y = 0 then x else 0)
       (\<lambda> x . if x = 0 then 0 else 1)
       (\<lambda> x . x)
       (\<lambda> x . x)
       (\<lambda> x . x)
       (\<lambda> x. 0 \<le> x \<and> x < p)"
 
 end
 
 context
   fixes p :: uint32
 begin
 definition plus_p32 :: "uint32 \<Rightarrow> uint32 \<Rightarrow> uint32" where
   "plus_p32 x y \<equiv> let z = x + y in if z \<ge> p then z - p else z"
 
 definition minus_p32 :: "uint32 \<Rightarrow> uint32 \<Rightarrow> uint32" where
   "minus_p32 x y \<equiv> if y \<le> x then x - y else (x + p) - y"
 
 definition uminus_p32 :: "uint32 \<Rightarrow> uint32" where
   "uminus_p32 x = (if x = 0 then 0 else p - x)"
 
 definition mult_p32 :: "uint32 \<Rightarrow> uint32 \<Rightarrow> uint32" where
   "mult_p32 x y = (x * y mod p)"
 
 lemma int_of_uint32_shift: "int_of_uint32 (drop_bit k n) = (int_of_uint32 n) div (2 ^ k)" 
   apply transfer
   apply transfer
   apply (simp add: take_bit_drop_bit min_def)
   apply (simp add: drop_bit_eq_div)
   done
 
 lemma int_of_uint32_0_iff: "int_of_uint32 n = 0 \<longleftrightarrow> n = 0" 
   by (transfer, rule uint_0_iff)
   
 lemma int_of_uint32_0: "int_of_uint32 0 = 0" unfolding int_of_uint32_0_iff by simp
 
 lemma int_of_uint32_ge_0: "int_of_uint32 n \<ge> 0" 
   by (transfer, auto)
 
 lemma two_32: "2 ^ LENGTH(32) = (4294967296 :: int)" by simp
 
 lemma int_of_uint32_plus: "int_of_uint32 (x + y) = (int_of_uint32 x + int_of_uint32 y) mod 4294967296" 
   by (transfer, unfold uint_word_ariths two_32, rule refl)  
 
 lemma int_of_uint32_minus: "int_of_uint32 (x - y) = (int_of_uint32 x - int_of_uint32 y) mod 4294967296" 
   by (transfer, unfold uint_word_ariths two_32, rule refl)  
 
 lemma int_of_uint32_mult: "int_of_uint32 (x * y) = (int_of_uint32 x * int_of_uint32 y) mod 4294967296" 
   by (transfer, unfold uint_word_ariths two_32, rule refl)  
 
 lemma int_of_uint32_mod: "int_of_uint32 (x mod y) = (int_of_uint32 x mod int_of_uint32 y)" 
   by (transfer, unfold uint_mod two_32, rule refl)  
 
 lemma int_of_uint32_inv: "0 \<le> x \<Longrightarrow> x < 4294967296 \<Longrightarrow> int_of_uint32 (uint32_of_int x) = x"
   by transfer (simp add: take_bit_int_eq_self unsigned_of_int)
 
 context
   includes bit_operations_syntax
 begin
 
 function power_p32 :: "uint32 \<Rightarrow> uint32 \<Rightarrow> uint32" where
   "power_p32 x n = (if n = 0 then 1 else
     let rec = power_p32 (mult_p32 x x) (drop_bit 1 n) in
     if n AND 1 = 0 then rec else mult_p32 rec x)"
   by pat_completeness auto
 
 termination 
 proof -
   {
     fix n :: uint32
     assume "n \<noteq> 0" 
     with int_of_uint32_ge_0[of n] int_of_uint32_0_iff[of n] have "int_of_uint32 n > 0" by auto
     hence "0 < int_of_uint32 n" "int_of_uint32 n div 2 < int_of_uint32 n" by auto
   } note * = this
   show ?thesis
     by (relation "measure (\<lambda> (x,n). nat (int_of_uint32 n))", auto simp: int_of_uint32_shift *) 
 qed
 
 end
 
 text \<open>In experiments with Berlekamp-factorization (where the prime $p$ is usually small),
   it turned out that taking the below implementation of inverse via exponentiation
   is faster than the one based on the extended Euclidean algorithm.\<close>
 
 definition inverse_p32 :: "uint32 \<Rightarrow> uint32" where
   "inverse_p32 x = (if x = 0 then 0 else power_p32 x (p - 2))"
 
 definition divide_p32 :: "uint32 \<Rightarrow> uint32 \<Rightarrow> uint32"  where
   "divide_p32 x y = mult_p32 x (inverse_p32 y)"
 
 definition finite_field_ops32 :: "uint32 arith_ops_record" where
   "finite_field_ops32 \<equiv> Arith_Ops_Record
       0
       1
       plus_p32
       mult_p32
       minus_p32
       uminus_p32
       divide_p32
       inverse_p32
       (\<lambda> x y . if y = 0 then x else 0)
       (\<lambda> x . if x = 0 then 0 else 1)
       (\<lambda> x . x)
       uint32_of_int
       int_of_uint32
       (\<lambda> x. 0 \<le> x \<and> x < p)"
 end 
 
 lemma shiftr_uint32_code [code_unfold]: "drop_bit 1 x = (uint32_shiftr x 1)"
   by (simp add: uint32_shiftr_def)
 
 (* ******************************************************************************** *)
 subsubsection \<open>Transfer Relation\<close>
 locale mod_ring_locale =
   fixes p :: int and ty :: "'a :: nontriv itself"
   assumes p: "p = int CARD('a)"
 begin
 lemma nat_p: "nat p = CARD('a)" unfolding p by simp
 lemma p2: "p \<ge> 2" unfolding p using nontriv[where 'a = 'a] by auto
 lemma p2_ident: "int (CARD('a) - 2) = p - 2" using p2 unfolding p by simp
 
 definition mod_ring_rel :: "int \<Rightarrow> 'a mod_ring \<Rightarrow> bool" where
   "mod_ring_rel x x' = (x = to_int_mod_ring x')"
 
 (* domain transfer rules *)
 lemma Domainp_mod_ring_rel [transfer_domain_rule]:
   "Domainp (mod_ring_rel) = (\<lambda> v. v \<in> {0 ..< p})"
 proof -
   {
     fix v :: int
     assume *: "0 \<le> v" "v < p"
     have "Domainp mod_ring_rel v"
     proof
       show "mod_ring_rel v (of_int_mod_ring v)" unfolding mod_ring_rel_def using * p by auto
     qed
   } note * = this
   show ?thesis
     by (intro ext iffI, insert range_to_int_mod_ring[where 'a = 'a] *, auto simp: mod_ring_rel_def p)
 qed
 
 (* left/right/bi-unique *)
 lemma bi_unique_mod_ring_rel [transfer_rule]:
   "bi_unique mod_ring_rel" "left_unique mod_ring_rel" "right_unique mod_ring_rel"
   unfolding mod_ring_rel_def bi_unique_def left_unique_def right_unique_def
   by auto
 
 (* left/right-total *)
 lemma right_total_mod_ring_rel [transfer_rule]: "right_total mod_ring_rel"
   unfolding mod_ring_rel_def right_total_def by simp
 
 
 (* ************************************************************************************ *)
 subsubsection \<open>Transfer Rules\<close>
 
 (* 0 / 1 *)
 lemma mod_ring_0[transfer_rule]: "mod_ring_rel 0 0" unfolding mod_ring_rel_def by simp
 lemma mod_ring_1[transfer_rule]: "mod_ring_rel 1 1" unfolding mod_ring_rel_def by simp
 
 (* addition *)
 lemma plus_p_mod_def: assumes x: "x \<in> {0 ..< p}" and y: "y \<in> {0 ..< p}"
   shows "plus_p p x y = ((x + y) mod p)"
 proof (cases "p \<le> x + y")
   case False
   thus ?thesis using x y unfolding plus_p_def Let_def by auto
 next
   case True
   from True x y have *: "p > 0" "0 \<le> x + y - p" "x + y - p < p" by auto
   from True have id: "plus_p p x y = x + y - p" unfolding plus_p_def by auto
   show ?thesis unfolding id using * using mod_pos_pos_trivial by fastforce
 qed
 
 lemma mod_ring_plus[transfer_rule]: "(mod_ring_rel ===> mod_ring_rel ===> mod_ring_rel) (plus_p p) (+)"
 proof -
   {
     fix x y :: "'a mod_ring"
     have "plus_p p (to_int_mod_ring x) (to_int_mod_ring y) = to_int_mod_ring (x + y)"
       by (transfer, subst plus_p_mod_def, auto, auto simp: p)
   } note * = this
   show ?thesis
     by (intro rel_funI, auto simp: mod_ring_rel_def *)
 qed
 
 (* subtraction *)
 lemma minus_p_mod_def: assumes x: "x \<in> {0 ..< p}" and y: "y \<in> {0 ..< p}"
   shows "minus_p p x y = ((x - y) mod p)"
 proof (cases "x - y < 0")
   case False
   thus ?thesis using x y unfolding minus_p_def Let_def by auto
 next
   case True
   from True x y have *: "p > 0" "0 \<le> x - y + p" "x - y + p < p" by auto
   from True have id: "minus_p p x y = x - y + p" unfolding minus_p_def by auto
   show ?thesis unfolding id using * using mod_pos_pos_trivial by fastforce
 qed
 
 lemma mod_ring_minus[transfer_rule]: "(mod_ring_rel ===> mod_ring_rel ===> mod_ring_rel) (minus_p p) (-)"
 proof -
   {
     fix x y :: "'a mod_ring"
     have "minus_p p (to_int_mod_ring x) (to_int_mod_ring y) = to_int_mod_ring (x - y)"
       by (transfer, subst minus_p_mod_def, auto simp: p)
   } note * = this
   show ?thesis
     by (intro rel_funI, auto simp: mod_ring_rel_def *)
 qed
 
 (* unary minus *)
 lemma mod_ring_uminus[transfer_rule]: "(mod_ring_rel ===> mod_ring_rel) (uminus_p p) uminus"
 proof -
   {
     fix x :: "'a mod_ring"
     have "uminus_p p (to_int_mod_ring x) = to_int_mod_ring (uminus x)"
       by (transfer, auto simp: uminus_p_def p)
   } note * = this
   show ?thesis
     by (intro rel_funI, auto simp: mod_ring_rel_def *)
 qed
 
 (* multiplication *)
 lemma mod_ring_mult[transfer_rule]: "(mod_ring_rel ===> mod_ring_rel ===> mod_ring_rel) (mult_p p) ((*))"
 proof -
   {
     fix x y :: "'a mod_ring"
     have "mult_p p (to_int_mod_ring x) (to_int_mod_ring y) = to_int_mod_ring (x * y)"
       by (transfer, auto simp: mult_p_def p)
   } note * = this
   show ?thesis
     by (intro rel_funI, auto simp: mod_ring_rel_def *)
 qed
 
 (* equality *)
 lemma mod_ring_eq[transfer_rule]: "(mod_ring_rel ===> mod_ring_rel ===> (=)) (=) (=)"
   by (intro rel_funI, auto simp: mod_ring_rel_def)
 
 (* power *)
 lemma mod_ring_power[transfer_rule]: "(mod_ring_rel ===> (=) ===> mod_ring_rel) (power_p p) (^)"
 proof (intro rel_funI, clarify, unfold binary_power[symmetric], goal_cases)
   fix x y n
   assume xy: "mod_ring_rel x y"
   from xy show "mod_ring_rel (power_p p x n) (binary_power y n)"
   proof (induct y n arbitrary: x rule: binary_power.induct)
     case (1 x n y)
     note 1(2)[transfer_rule]
     show ?case
     proof (cases "n = 0")
       case True
       thus ?thesis by (simp add: mod_ring_1)
     next
       case False
       obtain d r where id: "Divides.divmod_nat n 2 = (d,r)" by force
       let ?int = "power_p p (mult_p p y y) d"
       let ?gfp = "binary_power (x * x) d"
       from False have id': "?thesis = (mod_ring_rel
          (if r = 0 then ?int else mult_p p ?int y)
          (if r = 0 then ?gfp else ?gfp * x))"
         unfolding power_p.simps[of _ _ n] binary_power.simps[of _ n] Let_def id split by simp
       have [transfer_rule]: "mod_ring_rel ?int ?gfp"
         by (rule 1(1)[OF False refl id[symmetric]], transfer_prover)
       show ?thesis unfolding id' by transfer_prover
     qed
   qed
 qed
 
 declare power_p.simps[simp del]
 
 lemma ring_finite_field_ops_int: "ring_ops (finite_field_ops_int p) mod_ring_rel"
   by (unfold_locales, auto simp:
   finite_field_ops_int_def
   bi_unique_mod_ring_rel
   right_total_mod_ring_rel
   mod_ring_plus
   mod_ring_minus
   mod_ring_uminus
   mod_ring_mult
   mod_ring_eq
   mod_ring_0
   mod_ring_1
   Domainp_mod_ring_rel)
 end
 
 locale prime_field = mod_ring_locale p ty for p and ty :: "'a :: prime_card itself"
 begin
 
 lemma prime: "prime p" unfolding p using prime_card[where 'a = 'a] by simp
 
 (* mod *)
 lemma mod_ring_mod[transfer_rule]:
  "(mod_ring_rel ===> mod_ring_rel ===> mod_ring_rel) ((\<lambda> x y. if y = 0 then x else 0)) (mod)"
 proof -
   {
     fix x y :: "'a mod_ring"
     have "(if to_int_mod_ring y = 0 then to_int_mod_ring x else 0) = to_int_mod_ring (x mod y)"
       unfolding modulo_mod_ring_def by auto
   } note * = this
   show ?thesis
     by (intro rel_funI, auto simp: mod_ring_rel_def *[symmetric])
 qed
 
 (* normalize *)
 lemma mod_ring_normalize[transfer_rule]: "(mod_ring_rel ===> mod_ring_rel) ((\<lambda> x. if x = 0 then 0 else 1)) normalize"
 proof -
   {
     fix x :: "'a mod_ring"
     have "(if to_int_mod_ring x = 0 then 0 else 1) = to_int_mod_ring (normalize x)"
       unfolding normalize_mod_ring_def by auto
   } note * = this
   show ?thesis
     by (intro rel_funI, auto simp: mod_ring_rel_def *[symmetric])
 qed
 
 (* unit_factor *)
 lemma mod_ring_unit_factor[transfer_rule]: "(mod_ring_rel ===> mod_ring_rel) (\<lambda> x. x) unit_factor"
 proof -
   {
     fix x :: "'a mod_ring"
     have "to_int_mod_ring x = to_int_mod_ring (unit_factor x)"
       unfolding unit_factor_mod_ring_def by auto
   } note * = this
   show ?thesis
     by (intro rel_funI, auto simp: mod_ring_rel_def *[symmetric])
 qed
 
 (* inverse *)
 lemma mod_ring_inverse[transfer_rule]: "(mod_ring_rel ===> mod_ring_rel) (inverse_p p) inverse"
 proof (intro rel_funI)
   fix x y
   assume [transfer_rule]: "mod_ring_rel x y"
   show "mod_ring_rel (inverse_p p x) (inverse y)"
     unfolding inverse_p_def inverse_mod_ring_def
     apply (transfer_prover_start)
     apply (transfer_step)+
     apply (unfold p2_ident)
     apply (rule refl)
     done
 qed
 
 (* division *)
 lemma mod_ring_divide[transfer_rule]: "(mod_ring_rel ===> mod_ring_rel ===> mod_ring_rel)
   (divide_p p) (/)"
   unfolding divide_p_def[abs_def] divide_mod_ring_def[abs_def] inverse_mod_ring_def[symmetric]
   by transfer_prover
 
 lemma mod_ring_rel_unsafe: assumes "x < CARD('a)"
   shows "mod_ring_rel (int x) (of_nat x)" "0 < x \<Longrightarrow> of_nat x \<noteq> (0 :: 'a mod_ring)"
 proof -
   have id: "of_nat x = (of_int (int x) :: 'a mod_ring)" by simp
   show "mod_ring_rel (int x) (of_nat x)" "0 < x \<Longrightarrow> of_nat x \<noteq> (0 :: 'a mod_ring)" unfolding id
   unfolding mod_ring_rel_def
   proof (auto simp add: assms of_int_of_int_mod_ring)
     assume "0 < x" with assms
     have "of_int_mod_ring (int x) \<noteq> (0 :: 'a mod_ring)"
       by (metis (no_types) less_imp_of_nat_less less_irrefl of_nat_0_le_iff of_nat_0_less_iff to_int_mod_ring_hom.hom_zero to_int_mod_ring_of_int_mod_ring)
     thus "of_int_mod_ring (int x) = (0 :: 'a mod_ring) \<Longrightarrow> False" by blast
   qed
 qed
 
 lemma finite_field_ops_int: "field_ops (finite_field_ops_int p) mod_ring_rel"
   by (unfold_locales, auto simp:
   finite_field_ops_int_def
   bi_unique_mod_ring_rel
   right_total_mod_ring_rel
   mod_ring_divide
   mod_ring_plus
   mod_ring_minus
   mod_ring_uminus
   mod_ring_inverse
   mod_ring_mod
   mod_ring_unit_factor
   mod_ring_normalize
   mod_ring_mult
   mod_ring_eq
   mod_ring_0
   mod_ring_1
   Domainp_mod_ring_rel)
 
 end
 
 text \<open>Once we have proven the soundness of the implementation, we do not care any longer
   that @{typ "'a mod_ring"} has been defined internally via lifting. Disabling the transfer-rules
   will hide the internal definition in further applications of transfer.\<close>
 lifting_forget mod_ring.lifting
 
 text \<open>For soundness of the 32-bit implementation, we mainly prove that this implementation
   implements the int-based implementation of the mod-ring.\<close>
 context mod_ring_locale
 begin
 
 context fixes pp :: "uint32" 
   assumes ppp: "p = int_of_uint32 pp" 
   and small: "p \<le> 65535" 
 begin
 
 lemmas uint32_simps = 
   int_of_uint32_0
   int_of_uint32_plus 
   int_of_uint32_minus
   int_of_uint32_mult
   
 
 definition urel32 :: "uint32 \<Rightarrow> int \<Rightarrow> bool" where "urel32 x y = (y = int_of_uint32 x \<and> y < p)" 
 
 definition mod_ring_rel32 :: "uint32 \<Rightarrow> 'a mod_ring \<Rightarrow> bool" where
   "mod_ring_rel32 x y = (\<exists> z. urel32 x z \<and> mod_ring_rel z y)" 
 
 lemma urel32_0: "urel32 0 0" unfolding urel32_def using p2 by (simp, transfer, simp)
 
 lemma urel32_1: "urel32 1 1" unfolding urel32_def using p2 by (simp, transfer, simp)
 
 lemma le_int_of_uint32: "(x \<le> y) = (int_of_uint32 x \<le> int_of_uint32 y)" 
   by (transfer, simp add: word_le_def)
 
 lemma urel32_plus: assumes "urel32 x y" "urel32 x' y'"
   shows "urel32 (plus_p32 pp x x') (plus_p p y y')"
 proof -    
   let ?x = "int_of_uint32 x" 
   let ?x' = "int_of_uint32 x'" 
   let ?p = "int_of_uint32 pp" 
   from assms int_of_uint32_ge_0 have id: "y = ?x" "y' = ?x'" 
     and rel: "0 \<le> ?x" "?x < p" 
       "0 \<le> ?x'" "?x' \<le> p" unfolding urel32_def by auto
   have le: "(pp \<le> x + x') = (?p \<le> ?x + ?x')" unfolding le_int_of_uint32
     using rel small by (auto simp: uint32_simps)
   show ?thesis
   proof (cases "?p \<le> ?x + ?x'")
     case True
     hence True: "(?p \<le> ?x + ?x') = True" by simp
     show ?thesis unfolding id 
       using small rel unfolding plus_p32_def plus_p_def Let_def urel32_def 
       unfolding ppp le True if_True
       using True by (auto simp: uint32_simps)
   next
     case False
     hence False: "(?p \<le> ?x + ?x') = False" by simp
     show ?thesis unfolding id 
       using small rel unfolding plus_p32_def plus_p_def Let_def urel32_def 
       unfolding ppp le False if_False
       using False by (auto simp: uint32_simps)
   qed
 qed
   
 lemma urel32_minus: assumes "urel32 x y" "urel32 x' y'"
   shows "urel32 (minus_p32 pp x x') (minus_p p y y')"
 proof -    
   let ?x = "int_of_uint32 x" 
   let ?x' = "int_of_uint32 x'" 
   from assms int_of_uint32_ge_0 have id: "y = ?x" "y' = ?x'" 
     and rel: "0 \<le> ?x" "?x < p" 
       "0 \<le> ?x'" "?x' \<le> p" unfolding urel32_def by auto
   have le: "(x' \<le> x) = (?x' \<le> ?x)" unfolding le_int_of_uint32
     using rel small by (auto simp: uint32_simps)
   show ?thesis
   proof (cases "?x' \<le> ?x")
     case True
     hence True: "(?x' \<le> ?x) = True" by simp
     show ?thesis unfolding id 
       using small rel unfolding minus_p32_def minus_p_def Let_def urel32_def 
       unfolding ppp le True if_True
       using True by (auto simp: uint32_simps)
   next
     case False
     hence False: "(?x' \<le> ?x) = False" by simp
     show ?thesis unfolding id 
       using small rel unfolding minus_p32_def minus_p_def Let_def urel32_def 
       unfolding ppp le False if_False
       using False by (auto simp: uint32_simps)
   qed
 qed
 
 lemma urel32_uminus: assumes "urel32 x y"
   shows "urel32 (uminus_p32 pp x) (uminus_p p y)"
 proof -    
   let ?x = "int_of_uint32 x"  
   from assms int_of_uint32_ge_0 have id: "y = ?x" 
     and rel: "0 \<le> ?x" "?x < p" 
       unfolding urel32_def by auto
   have le: "(x = 0) = (?x = 0)" unfolding int_of_uint32_0_iff
     using rel small by (auto simp: uint32_simps)
   show ?thesis
   proof (cases "?x = 0")
     case True
     hence True: "(?x = 0) = True" by simp
     show ?thesis unfolding id 
       using small rel unfolding uminus_p32_def uminus_p_def Let_def urel32_def 
       unfolding ppp le True if_True
       using True by (auto simp: uint32_simps)
   next
     case False
     hence False: "(?x = 0) = False" by simp
     show ?thesis unfolding id 
       using small rel unfolding uminus_p32_def uminus_p_def Let_def urel32_def 
       unfolding ppp le False if_False
       using False by (auto simp: uint32_simps)
   qed
 qed
 
 lemma urel32_mult: assumes "urel32 x y" "urel32 x' y'"
   shows "urel32 (mult_p32 pp x x') (mult_p p y y')"
 proof -    
   let ?x = "int_of_uint32 x" 
   let ?x' = "int_of_uint32 x'" 
   from assms int_of_uint32_ge_0 have id: "y = ?x" "y' = ?x'" 
     and rel: "0 \<le> ?x" "?x < p" 
       "0 \<le> ?x'" "?x' < p" unfolding urel32_def by auto
   from rel have "?x * ?x' < p * p" by (metis mult_strict_mono') 
   also have "\<dots> \<le> 65536 * 65536"
     by (rule mult_mono, insert p2 small, auto)
   finally have le: "?x * ?x' < 4294967296" by simp
   show ?thesis unfolding id
       using small rel unfolding mult_p32_def mult_p_def Let_def urel32_def 
       unfolding ppp 
     by (auto simp: uint32_simps, unfold int_of_uint32_mod int_of_uint32_mult, 
         subst mod_pos_pos_trivial[of _ 4294967296], insert le, auto)
 qed
 
 lemma urel32_eq: assumes "urel32 x y" "urel32 x' y'" 
   shows "(x = x') = (y = y')" 
 proof -    
   let ?x = "int_of_uint32 x" 
   let ?x' = "int_of_uint32 x'" 
   from assms int_of_uint32_ge_0 have id: "y = ?x" "y' = ?x'" 
     unfolding urel32_def by auto
   show ?thesis unfolding id by (transfer, transfer) rule
 qed
 
 lemma urel32_normalize: 
 assumes x: "urel32 x y"
 shows "urel32 (if x = 0 then 0 else 1) (if y = 0 then 0 else 1)"
  unfolding urel32_eq[OF x urel32_0] using urel32_0 urel32_1 by auto
 
 lemma urel32_mod: 
 assumes x: "urel32 x x'" and y: "urel32 y y'" 
 shows "urel32 (if y = 0 then x else 0) (if y' = 0 then x' else 0)"
   unfolding urel32_eq[OF y urel32_0] using urel32_0 x by auto 
 
 lemma urel32_power: "urel32 x x' \<Longrightarrow> urel32 y (int y') \<Longrightarrow> urel32 (power_p32 pp x y) (power_p p x' y')"
 including bit_operations_syntax proof (induct x' y' arbitrary: x y rule: power_p.induct[of _ p])
   case (1 x' y' x y)
   note x = 1(2) note y = 1(3)
   show ?case
   proof (cases "y' = 0")
     case True
     hence y: "y = 0" using urel32_eq[OF y urel32_0] by auto
     show ?thesis unfolding y True by (simp add: power_p.simps urel32_1)
   next
     case False
     hence id: "(y = 0) = False" "(y' = 0) = False" using urel32_eq[OF y urel32_0] by auto
     from y have \<open>int y' = int_of_uint32 y\<close> \<open>int y' < p\<close>
       by (simp_all add: urel32_def)
     obtain d' r' where dr': "Divides.divmod_nat y' 2 = (d',r')" by force
     from divmod_nat_def[of y' 2, unfolded dr']
     have r': "r' = y' mod 2" and d': "d' = y' div 2" by auto
     have "urel32 (y AND 1) r'"
       using \<open>int y' < p\<close> small
       apply (simp add: urel32_def and_one_eq r')
       apply (auto simp add: ppp and_one_eq)
       apply (simp add: of_nat_mod int_of_uint32.rep_eq modulo_uint32.rep_eq uint_mod \<open>int y' = int_of_uint32 y\<close>)
       done
     from urel32_eq[OF this urel32_0]     
     have rem: "(y AND 1 = 0) = (r' = 0)" by simp
     have div: "urel32 (drop_bit 1 y) (int d')" unfolding d' using y unfolding urel32_def using small
       unfolding ppp 
       apply transfer
       apply transfer
       apply (auto simp add: drop_bit_Suc take_bit_int_eq_self)
       done
     note IH = 1(1)[OF False refl dr'[symmetric] urel32_mult[OF x x] div]
     show ?thesis unfolding power_p.simps[of _ _ "y'"] power_p32.simps[of _ _ y] dr' id if_False rem
       using IH urel32_mult[OF IH x] by (auto simp: Let_def)
   qed
 qed
   
 
 lemma urel32_inverse: assumes x: "urel32 x x'" 
   shows "urel32 (inverse_p32 pp x) (inverse_p p x')" 
 proof -
   have p: "urel32 (pp - 2) (int (nat (p - 2)))" using p2 small unfolding urel32_def unfolding ppp
     by (simp add: int_of_uint32.rep_eq minus_uint32.rep_eq uint_sub_if')
   show ?thesis
     unfolding inverse_p32_def inverse_p_def urel32_eq[OF x urel32_0] using urel32_0 urel32_power[OF x p]
     by auto
 qed
 
 lemma mod_ring_0_32: "mod_ring_rel32 0 0"
   using urel32_0 mod_ring_0 unfolding mod_ring_rel32_def by blast
 
 lemma mod_ring_1_32: "mod_ring_rel32 1 1"
   using urel32_1 mod_ring_1 unfolding mod_ring_rel32_def by blast
 
 lemma mod_ring_uminus32: "(mod_ring_rel32 ===> mod_ring_rel32) (uminus_p32 pp) uminus"
   using urel32_uminus mod_ring_uminus unfolding mod_ring_rel32_def rel_fun_def by blast
 
 lemma mod_ring_plus32: "(mod_ring_rel32 ===> mod_ring_rel32 ===> mod_ring_rel32) (plus_p32 pp) (+)"
   using urel32_plus mod_ring_plus unfolding mod_ring_rel32_def rel_fun_def by blast
 
 lemma mod_ring_minus32: "(mod_ring_rel32 ===> mod_ring_rel32 ===> mod_ring_rel32) (minus_p32 pp) (-)"
   using urel32_minus mod_ring_minus unfolding mod_ring_rel32_def rel_fun_def by blast
 
 lemma mod_ring_mult32: "(mod_ring_rel32 ===> mod_ring_rel32 ===> mod_ring_rel32) (mult_p32 pp) ((*))"
   using urel32_mult mod_ring_mult unfolding mod_ring_rel32_def rel_fun_def by blast
 
 lemma mod_ring_eq32: "(mod_ring_rel32 ===> mod_ring_rel32 ===> (=)) (=) (=)" 
   using urel32_eq mod_ring_eq unfolding mod_ring_rel32_def rel_fun_def by blast
 
 lemma urel32_inj: "urel32 x y \<Longrightarrow> urel32 x z \<Longrightarrow> y = z" 
   using urel32_eq[of x y x z] by auto
 
 lemma urel32_inj': "urel32 x z \<Longrightarrow> urel32 y z \<Longrightarrow> x = y" 
   using urel32_eq[of x z y z] by auto
 
 lemma bi_unique_mod_ring_rel32:
   "bi_unique mod_ring_rel32" "left_unique mod_ring_rel32" "right_unique mod_ring_rel32"
   using bi_unique_mod_ring_rel urel32_inj'
   unfolding mod_ring_rel32_def bi_unique_def left_unique_def right_unique_def
   by (auto simp: urel32_def)  
 
 lemma right_total_mod_ring_rel32: "right_total mod_ring_rel32"
   unfolding mod_ring_rel32_def right_total_def
 proof 
   fix y :: "'a mod_ring" 
   from right_total_mod_ring_rel[unfolded right_total_def, rule_format, of y]
   obtain z where zy: "mod_ring_rel z y" by auto  
   hence zp: "0 \<le> z" "z < p" unfolding mod_ring_rel_def p using range_to_int_mod_ring[where 'a = 'a] by auto
   hence "urel32 (uint32_of_int z) z" unfolding urel32_def using small unfolding ppp 
     by (auto simp: int_of_uint32_inv) 
   with zy show "\<exists> x z. urel32 x z \<and> mod_ring_rel z y" by blast
 qed
 
 lemma Domainp_mod_ring_rel32: "Domainp mod_ring_rel32 = (\<lambda>x. 0 \<le> x \<and> x < pp)"
 proof 
   fix x
   show "Domainp mod_ring_rel32 x = (0 \<le> x \<and> x < pp)"   
     unfolding Domainp.simps
     unfolding mod_ring_rel32_def
   proof
     let ?i = "int_of_uint32" 
     assume *: "0 \<le> x \<and> x < pp"     
     hence "0 \<le> ?i x \<and> ?i x < p" using small unfolding ppp
       by (transfer, auto simp: word_less_def)
     hence "?i x \<in> {0 ..< p}" by auto
     with Domainp_mod_ring_rel
     have "Domainp mod_ring_rel (?i x)" by auto
     from this[unfolded Domainp.simps]
     obtain b where b: "mod_ring_rel (?i x) b" by auto
     show "\<exists>a b. x = a \<and> (\<exists>z. urel32 a z \<and> mod_ring_rel z b)" 
     proof (intro exI, rule conjI[OF refl], rule exI, rule conjI[OF _ b])
       show "urel32 x (?i x)" unfolding urel32_def using small * unfolding ppp
         by (transfer, auto simp: word_less_def)
     qed
   next
     assume "\<exists>a b. x = a \<and> (\<exists>z. urel32 a z \<and> mod_ring_rel z b)" 
     then obtain b z where xz: "urel32 x z" and zb: "mod_ring_rel z b" by auto
     hence "Domainp mod_ring_rel z"  by auto
     with Domainp_mod_ring_rel have "0 \<le> z" "z < p" by auto
     with xz show "0 \<le> x \<and> x < pp" unfolding urel32_def using small unfolding ppp
       by (transfer, auto simp: word_less_def)
   qed
 qed
 
 lemma ring_finite_field_ops32: "ring_ops (finite_field_ops32 pp) mod_ring_rel32"
   by (unfold_locales, auto simp:
   finite_field_ops32_def
   bi_unique_mod_ring_rel32
   right_total_mod_ring_rel32
   mod_ring_plus32
   mod_ring_minus32
   mod_ring_uminus32
   mod_ring_mult32
   mod_ring_eq32
   mod_ring_0_32
   mod_ring_1_32
   Domainp_mod_ring_rel32)
 end
 end
 
 context prime_field
 begin
 context fixes pp :: "uint32" 
   assumes *: "p = int_of_uint32 pp" "p \<le> 65535" 
 begin
 
 lemma mod_ring_normalize32: "(mod_ring_rel32 ===> mod_ring_rel32) (\<lambda>x. if x = 0 then 0 else 1) normalize" 
   using urel32_normalize[OF *] mod_ring_normalize unfolding mod_ring_rel32_def[OF *] rel_fun_def by blast
 
 lemma mod_ring_mod32: "(mod_ring_rel32 ===> mod_ring_rel32 ===> mod_ring_rel32) (\<lambda>x y. if y = 0 then x else 0) (mod)" 
   using urel32_mod[OF *] mod_ring_mod unfolding mod_ring_rel32_def[OF *] rel_fun_def by blast
 
 lemma mod_ring_unit_factor32: "(mod_ring_rel32 ===> mod_ring_rel32) (\<lambda>x. x) unit_factor" 
   using mod_ring_unit_factor unfolding mod_ring_rel32_def[OF *] rel_fun_def by blast
 
 lemma mod_ring_inverse32: "(mod_ring_rel32 ===> mod_ring_rel32) (inverse_p32 pp) inverse"
   using urel32_inverse[OF *] mod_ring_inverse unfolding mod_ring_rel32_def[OF *] rel_fun_def by blast
 
 lemma mod_ring_divide32: "(mod_ring_rel32 ===> mod_ring_rel32 ===> mod_ring_rel32) (divide_p32 pp) (/)"
   using mod_ring_inverse32 mod_ring_mult32[OF *]
   unfolding divide_p32_def divide_mod_ring_def inverse_mod_ring_def[symmetric]
     rel_fun_def by blast
 
 lemma finite_field_ops32: "field_ops (finite_field_ops32 pp) mod_ring_rel32"
   by (unfold_locales, insert ring_finite_field_ops32[OF *], auto simp:
   ring_ops_def
   finite_field_ops32_def
   mod_ring_divide32
   mod_ring_inverse32
   mod_ring_mod32
   mod_ring_normalize32)
 
 end
 end
 
 (* now there is 64-bit time *)
 context
   fixes p :: uint64
 begin
 definition plus_p64 :: "uint64 \<Rightarrow> uint64 \<Rightarrow> uint64" where
   "plus_p64 x y \<equiv> let z = x + y in if z \<ge> p then z - p else z"
 
 definition minus_p64 :: "uint64 \<Rightarrow> uint64 \<Rightarrow> uint64" where
   "minus_p64 x y \<equiv> if y \<le> x then x - y else (x + p) - y"
 
 definition uminus_p64 :: "uint64 \<Rightarrow> uint64" where
   "uminus_p64 x = (if x = 0 then 0 else p - x)"
 
 definition mult_p64 :: "uint64 \<Rightarrow> uint64 \<Rightarrow> uint64" where
   "mult_p64 x y = (x * y mod p)"
 
 lemma int_of_uint64_shift: "int_of_uint64 (drop_bit k n) = (int_of_uint64 n) div (2 ^ k)" 
   apply transfer
   apply transfer
   apply (simp add: take_bit_drop_bit min_def)
   apply (simp add: drop_bit_eq_div)
   done
 
 lemma int_of_uint64_0_iff: "int_of_uint64 n = 0 \<longleftrightarrow> n = 0" 
   by (transfer, rule uint_0_iff)
   
 lemma int_of_uint64_0: "int_of_uint64 0 = 0" unfolding int_of_uint64_0_iff by simp
 
 lemma int_of_uint64_ge_0: "int_of_uint64 n \<ge> 0" 
   by (transfer, auto)
 
 lemma two_64: "2 ^ LENGTH(64) = (18446744073709551616 :: int)" by simp
 
 lemma int_of_uint64_plus: "int_of_uint64 (x + y) = (int_of_uint64 x + int_of_uint64 y) mod 18446744073709551616" 
   by (transfer, unfold uint_word_ariths two_64, rule refl)  
 
 lemma int_of_uint64_minus: "int_of_uint64 (x - y) = (int_of_uint64 x - int_of_uint64 y) mod 18446744073709551616" 
   by (transfer, unfold uint_word_ariths two_64, rule refl)  
 
 lemma int_of_uint64_mult: "int_of_uint64 (x * y) = (int_of_uint64 x * int_of_uint64 y) mod 18446744073709551616" 
   by (transfer, unfold uint_word_ariths two_64, rule refl)  
 
 lemma int_of_uint64_mod: "int_of_uint64 (x mod y) = (int_of_uint64 x mod int_of_uint64 y)" 
   by (transfer, unfold uint_mod two_64, rule refl)  
 
 lemma int_of_uint64_inv: "0 \<le> x \<Longrightarrow> x < 18446744073709551616 \<Longrightarrow> int_of_uint64 (uint64_of_int x) = x"
   by transfer (simp add: take_bit_int_eq_self unsigned_of_int)
 
 context
   includes bit_operations_syntax 
 begin
 
 function power_p64 :: "uint64 \<Rightarrow> uint64 \<Rightarrow> uint64" where
   "power_p64 x n = (if n = 0 then 1 else
     let rec = power_p64 (mult_p64 x x) (drop_bit 1 n) in
     if n AND 1 = 0 then rec else mult_p64 rec x)"
   by pat_completeness auto
 
 termination 
 proof -
   {
     fix n :: uint64
     assume "n \<noteq> 0" 
     with int_of_uint64_ge_0[of n] int_of_uint64_0_iff[of n] have "int_of_uint64 n > 0" by auto
     hence "0 < int_of_uint64 n" "int_of_uint64 n div 2 < int_of_uint64 n" by auto
   } note * = this
   show ?thesis
     by (relation "measure (\<lambda> (x,n). nat (int_of_uint64 n))", auto simp: int_of_uint64_shift *) 
 qed
 
 end
 
 text \<open>In experiments with Berlekamp-factorization (where the prime $p$ is usually small),
   it turned out that taking the below implementation of inverse via exponentiation
   is faster than the one based on the extended Euclidean algorithm.\<close>
 
 definition inverse_p64 :: "uint64 \<Rightarrow> uint64" where
   "inverse_p64 x = (if x = 0 then 0 else power_p64 x (p - 2))"
 
 definition divide_p64 :: "uint64 \<Rightarrow> uint64 \<Rightarrow> uint64"  where
   "divide_p64 x y = mult_p64 x (inverse_p64 y)"
 
 definition finite_field_ops64 :: "uint64 arith_ops_record" where
   "finite_field_ops64 \<equiv> Arith_Ops_Record
       0
       1
       plus_p64
       mult_p64
       minus_p64
       uminus_p64
       divide_p64
       inverse_p64
       (\<lambda> x y . if y = 0 then x else 0)
       (\<lambda> x . if x = 0 then 0 else 1)
       (\<lambda> x . x)
       uint64_of_int
       int_of_uint64
       (\<lambda> x. 0 \<le> x \<and> x < p)"
 end 
 
 lemma shiftr_uint64_code [code_unfold]: "drop_bit 1 x = (uint64_shiftr x 1)"
   by (simp add: uint64_shiftr_def)
 
 text \<open>For soundness of the 64-bit implementation, we mainly prove that this implementation
   implements the int-based implementation of GF(p).\<close>
 context mod_ring_locale
 begin
 
 context fixes pp :: "uint64" 
   assumes ppp: "p = int_of_uint64 pp" 
   and small: "p \<le> 4294967295" 
 begin
 
 lemmas uint64_simps = 
   int_of_uint64_0
   int_of_uint64_plus 
   int_of_uint64_minus
   int_of_uint64_mult
   
 
 definition urel64 :: "uint64 \<Rightarrow> int \<Rightarrow> bool" where "urel64 x y = (y = int_of_uint64 x \<and> y < p)" 
 
 definition mod_ring_rel64 :: "uint64 \<Rightarrow> 'a mod_ring \<Rightarrow> bool" where
   "mod_ring_rel64 x y = (\<exists> z. urel64 x z \<and> mod_ring_rel z y)" 
 
 lemma urel64_0: "urel64 0 0" unfolding urel64_def using p2 by (simp, transfer, simp)
 
 lemma urel64_1: "urel64 1 1" unfolding urel64_def using p2 by (simp, transfer, simp)
 
 lemma le_int_of_uint64: "(x \<le> y) = (int_of_uint64 x \<le> int_of_uint64 y)" 
   by (transfer, simp add: word_le_def)
 
 lemma urel64_plus: assumes "urel64 x y" "urel64 x' y'"
   shows "urel64 (plus_p64 pp x x') (plus_p p y y')"
 proof -    
   let ?x = "int_of_uint64 x" 
   let ?x' = "int_of_uint64 x'" 
   let ?p = "int_of_uint64 pp" 
   from assms int_of_uint64_ge_0 have id: "y = ?x" "y' = ?x'" 
     and rel: "0 \<le> ?x" "?x < p" 
       "0 \<le> ?x'" "?x' \<le> p" unfolding urel64_def by auto
   have le: "(pp \<le> x + x') = (?p \<le> ?x + ?x')" unfolding le_int_of_uint64
     using rel small by (auto simp: uint64_simps)
   show ?thesis
   proof (cases "?p \<le> ?x + ?x'")
     case True
     hence True: "(?p \<le> ?x + ?x') = True" by simp
     show ?thesis unfolding id 
       using small rel unfolding plus_p64_def plus_p_def Let_def urel64_def 
       unfolding ppp le True if_True
       using True by (auto simp: uint64_simps)
   next
     case False
     hence False: "(?p \<le> ?x + ?x') = False" by simp
     show ?thesis unfolding id 
       using small rel unfolding plus_p64_def plus_p_def Let_def urel64_def 
       unfolding ppp le False if_False
       using False by (auto simp: uint64_simps)
   qed
 qed
   
 lemma urel64_minus: assumes "urel64 x y" "urel64 x' y'"
   shows "urel64 (minus_p64 pp x x') (minus_p p y y')"
 proof -    
   let ?x = "int_of_uint64 x" 
   let ?x' = "int_of_uint64 x'" 
   from assms int_of_uint64_ge_0 have id: "y = ?x" "y' = ?x'" 
     and rel: "0 \<le> ?x" "?x < p" 
       "0 \<le> ?x'" "?x' \<le> p" unfolding urel64_def by auto
   have le: "(x' \<le> x) = (?x' \<le> ?x)" unfolding le_int_of_uint64
     using rel small by (auto simp: uint64_simps)
   show ?thesis
   proof (cases "?x' \<le> ?x")
     case True
     hence True: "(?x' \<le> ?x) = True" by simp
     show ?thesis unfolding id 
       using small rel unfolding minus_p64_def minus_p_def Let_def urel64_def 
       unfolding ppp le True if_True
       using True by (auto simp: uint64_simps)
   next
     case False
     hence False: "(?x' \<le> ?x) = False" by simp
     show ?thesis unfolding id 
       using small rel unfolding minus_p64_def minus_p_def Let_def urel64_def 
       unfolding ppp le False if_False
       using False by (auto simp: uint64_simps)
   qed
 qed
 
 lemma urel64_uminus: assumes "urel64 x y"
   shows "urel64 (uminus_p64 pp x) (uminus_p p y)"
 proof -    
   let ?x = "int_of_uint64 x"  
   from assms int_of_uint64_ge_0 have id: "y = ?x" 
     and rel: "0 \<le> ?x" "?x < p" 
       unfolding urel64_def by auto
   have le: "(x = 0) = (?x = 0)" unfolding int_of_uint64_0_iff
     using rel small by (auto simp: uint64_simps)
   show ?thesis
   proof (cases "?x = 0")
     case True
     hence True: "(?x = 0) = True" by simp
     show ?thesis unfolding id 
       using small rel unfolding uminus_p64_def uminus_p_def Let_def urel64_def 
       unfolding ppp le True if_True
       using True by (auto simp: uint64_simps)
   next
     case False
     hence False: "(?x = 0) = False" by simp
     show ?thesis unfolding id 
       using small rel unfolding uminus_p64_def uminus_p_def Let_def urel64_def 
       unfolding ppp le False if_False
       using False by (auto simp: uint64_simps)
   qed
 qed
 
 lemma urel64_mult: assumes "urel64 x y" "urel64 x' y'"
   shows "urel64 (mult_p64 pp x x') (mult_p p y y')"
 proof -    
   let ?x = "int_of_uint64 x" 
   let ?x' = "int_of_uint64 x'" 
   from assms int_of_uint64_ge_0 have id: "y = ?x" "y' = ?x'" 
     and rel: "0 \<le> ?x" "?x < p" 
       "0 \<le> ?x'" "?x' < p" unfolding urel64_def by auto
   from rel have "?x * ?x' < p * p" by (metis mult_strict_mono') 
   also have "\<dots> \<le> 4294967296 * 4294967296"
     by (rule mult_mono, insert p2 small, auto)
   finally have le: "?x * ?x' < 18446744073709551616" by simp
   show ?thesis unfolding id
       using small rel unfolding mult_p64_def mult_p_def Let_def urel64_def 
       unfolding ppp 
     by (auto simp: uint64_simps, unfold int_of_uint64_mod int_of_uint64_mult, 
         subst mod_pos_pos_trivial[of _ 18446744073709551616], insert le, auto)
 qed
 
 lemma urel64_eq: assumes "urel64 x y" "urel64 x' y'" 
   shows "(x = x') = (y = y')" 
 proof -    
   let ?x = "int_of_uint64 x" 
   let ?x' = "int_of_uint64 x'" 
   from assms int_of_uint64_ge_0 have id: "y = ?x" "y' = ?x'" 
     unfolding urel64_def by auto
   show ?thesis unfolding id by (transfer, transfer) rule
 qed
 
 lemma urel64_normalize: 
 assumes x: "urel64 x y"
 shows "urel64 (if x = 0 then 0 else 1) (if y = 0 then 0 else 1)"
  unfolding urel64_eq[OF x urel64_0] using urel64_0 urel64_1 by auto
 
 lemma urel64_mod: 
 assumes x: "urel64 x x'" and y: "urel64 y y'" 
 shows "urel64 (if y = 0 then x else 0) (if y' = 0 then x' else 0)"
   unfolding urel64_eq[OF y urel64_0] using urel64_0 x by auto 
 
 lemma urel64_power: "urel64 x x' \<Longrightarrow> urel64 y (int y') \<Longrightarrow> urel64 (power_p64 pp x y) (power_p p x' y')"
 including bit_operations_syntax proof (induct x' y' arbitrary: x y rule: power_p.induct[of _ p])
   case (1 x' y' x y)
   note x = 1(2) note y = 1(3)
   show ?case
   proof (cases "y' = 0")
     case True
     hence y: "y = 0" using urel64_eq[OF y urel64_0] by auto
     show ?thesis unfolding y True by (simp add: power_p.simps urel64_1)
   next
     case False
     hence id: "(y = 0) = False" "(y' = 0) = False" using urel64_eq[OF y urel64_0] by auto
     from y have \<open>int y' = int_of_uint64 y\<close> \<open>int y' < p\<close>
       by (simp_all add: urel64_def)
     obtain d' r' where dr': "Divides.divmod_nat y' 2 = (d',r')" by force
     from divmod_nat_def[of y' 2, unfolded dr']
     have r': "r' = y' mod 2" and d': "d' = y' div 2" by auto
     have "urel64 (y AND 1) r'"
       using \<open>int y' < p\<close> small
       apply (simp add: urel64_def and_one_eq r')
       apply (auto simp add: ppp and_one_eq)
       apply (simp add: of_nat_mod int_of_uint64.rep_eq modulo_uint64.rep_eq uint_mod \<open>int y' = int_of_uint64 y\<close>)
       done
     from urel64_eq[OF this urel64_0]     
     have rem: "(y AND 1 = 0) = (r' = 0)" by simp
     have div: "urel64 (drop_bit 1 y) (int d')" unfolding d' using y unfolding urel64_def using small
       unfolding ppp
       apply transfer
       apply transfer
       apply (auto simp add: drop_bit_Suc take_bit_int_eq_self)
       done
     note IH = 1(1)[OF False refl dr'[symmetric] urel64_mult[OF x x] div]
     show ?thesis unfolding power_p.simps[of _ _ "y'"] power_p64.simps[of _ _ y] dr' id if_False rem
       using IH urel64_mult[OF IH x] by (auto simp: Let_def)
   qed
 qed
   
 
 lemma urel64_inverse: assumes x: "urel64 x x'" 
   shows "urel64 (inverse_p64 pp x) (inverse_p p x')" 
 proof -
   have p: "urel64 (pp - 2) (int (nat (p - 2)))" using p2 small unfolding urel64_def unfolding ppp
     by (simp add: int_of_uint64.rep_eq minus_uint64.rep_eq uint_sub_if')
   show ?thesis
     unfolding inverse_p64_def inverse_p_def urel64_eq[OF x urel64_0] using urel64_0 urel64_power[OF x p]
     by auto
 qed
 
 lemma mod_ring_0_64: "mod_ring_rel64 0 0"
   using urel64_0 mod_ring_0 unfolding mod_ring_rel64_def by blast
 
 lemma mod_ring_1_64: "mod_ring_rel64 1 1"
   using urel64_1 mod_ring_1 unfolding mod_ring_rel64_def by blast
 
 lemma mod_ring_uminus64: "(mod_ring_rel64 ===> mod_ring_rel64) (uminus_p64 pp) uminus"
   using urel64_uminus mod_ring_uminus unfolding mod_ring_rel64_def rel_fun_def by blast
 
 lemma mod_ring_plus64: "(mod_ring_rel64 ===> mod_ring_rel64 ===> mod_ring_rel64) (plus_p64 pp) (+)"
   using urel64_plus mod_ring_plus unfolding mod_ring_rel64_def rel_fun_def by blast
 
 lemma mod_ring_minus64: "(mod_ring_rel64 ===> mod_ring_rel64 ===> mod_ring_rel64) (minus_p64 pp) (-)"
   using urel64_minus mod_ring_minus unfolding mod_ring_rel64_def rel_fun_def by blast
 
 lemma mod_ring_mult64: "(mod_ring_rel64 ===> mod_ring_rel64 ===> mod_ring_rel64) (mult_p64 pp) ((*))"
   using urel64_mult mod_ring_mult unfolding mod_ring_rel64_def rel_fun_def by blast
 
 lemma mod_ring_eq64: "(mod_ring_rel64 ===> mod_ring_rel64 ===> (=)) (=) (=)" 
   using urel64_eq mod_ring_eq unfolding mod_ring_rel64_def rel_fun_def by blast
 
 lemma urel64_inj: "urel64 x y \<Longrightarrow> urel64 x z \<Longrightarrow> y = z" 
   using urel64_eq[of x y x z] by auto
 
 lemma urel64_inj': "urel64 x z \<Longrightarrow> urel64 y z \<Longrightarrow> x = y" 
   using urel64_eq[of x z y z] by auto
 
 lemma bi_unique_mod_ring_rel64:
   "bi_unique mod_ring_rel64" "left_unique mod_ring_rel64" "right_unique mod_ring_rel64"
   using bi_unique_mod_ring_rel urel64_inj'
   unfolding mod_ring_rel64_def bi_unique_def left_unique_def right_unique_def
   by (auto simp: urel64_def)  
 
 lemma right_total_mod_ring_rel64: "right_total mod_ring_rel64"
   unfolding mod_ring_rel64_def right_total_def
 proof 
   fix y :: "'a mod_ring" 
   from right_total_mod_ring_rel[unfolded right_total_def, rule_format, of y]
   obtain z where zy: "mod_ring_rel z y" by auto  
   hence zp: "0 \<le> z" "z < p" unfolding mod_ring_rel_def p using range_to_int_mod_ring[where 'a = 'a] by auto
   hence "urel64 (uint64_of_int z) z" unfolding urel64_def using small unfolding ppp 
     by (auto simp: int_of_uint64_inv) 
   with zy show "\<exists> x z. urel64 x z \<and> mod_ring_rel z y" by blast
 qed
 
 lemma Domainp_mod_ring_rel64: "Domainp mod_ring_rel64 = (\<lambda>x. 0 \<le> x \<and> x < pp)"
 proof 
   fix x
   show "Domainp mod_ring_rel64 x = (0 \<le> x \<and> x < pp)"   
     unfolding Domainp.simps
     unfolding mod_ring_rel64_def
   proof
     let ?i = "int_of_uint64" 
     assume *: "0 \<le> x \<and> x < pp"     
     hence "0 \<le> ?i x \<and> ?i x < p" using small unfolding ppp
       by (transfer, auto simp: word_less_def)
     hence "?i x \<in> {0 ..< p}" by auto
     with Domainp_mod_ring_rel
     have "Domainp mod_ring_rel (?i x)" by auto
     from this[unfolded Domainp.simps]
     obtain b where b: "mod_ring_rel (?i x) b" by auto
     show "\<exists>a b. x = a \<and> (\<exists>z. urel64 a z \<and> mod_ring_rel z b)" 
     proof (intro exI, rule conjI[OF refl], rule exI, rule conjI[OF _ b])
       show "urel64 x (?i x)" unfolding urel64_def using small * unfolding ppp
         by (transfer, auto simp: word_less_def)
     qed
   next
     assume "\<exists>a b. x = a \<and> (\<exists>z. urel64 a z \<and> mod_ring_rel z b)" 
     then obtain b z where xz: "urel64 x z" and zb: "mod_ring_rel z b" by auto
     hence "Domainp mod_ring_rel z"  by auto
     with Domainp_mod_ring_rel have "0 \<le> z" "z < p" by auto
     with xz show "0 \<le> x \<and> x < pp" unfolding urel64_def using small unfolding ppp
       by (transfer, auto simp: word_less_def)
   qed
 qed
 
 lemma ring_finite_field_ops64: "ring_ops (finite_field_ops64 pp) mod_ring_rel64"
   by (unfold_locales, auto simp:
   finite_field_ops64_def
   bi_unique_mod_ring_rel64
   right_total_mod_ring_rel64
   mod_ring_plus64
   mod_ring_minus64
   mod_ring_uminus64
   mod_ring_mult64
   mod_ring_eq64
   mod_ring_0_64
   mod_ring_1_64
   Domainp_mod_ring_rel64)
 end
 end
 
 context prime_field
 begin
 context fixes pp :: "uint64" 
   assumes *: "p = int_of_uint64 pp" "p \<le> 4294967295" 
 begin
 
 lemma mod_ring_normalize64: "(mod_ring_rel64 ===> mod_ring_rel64) (\<lambda>x. if x = 0 then 0 else 1) normalize" 
   using urel64_normalize[OF *] mod_ring_normalize unfolding mod_ring_rel64_def[OF *] rel_fun_def by blast
 
 lemma mod_ring_mod64: "(mod_ring_rel64 ===> mod_ring_rel64 ===> mod_ring_rel64) (\<lambda>x y. if y = 0 then x else 0) (mod)" 
   using urel64_mod[OF *] mod_ring_mod unfolding mod_ring_rel64_def[OF *] rel_fun_def by blast
 
 lemma mod_ring_unit_factor64: "(mod_ring_rel64 ===> mod_ring_rel64) (\<lambda>x. x) unit_factor" 
   using mod_ring_unit_factor unfolding mod_ring_rel64_def[OF *] rel_fun_def by blast
 
 lemma mod_ring_inverse64: "(mod_ring_rel64 ===> mod_ring_rel64) (inverse_p64 pp) inverse"
   using urel64_inverse[OF *] mod_ring_inverse unfolding mod_ring_rel64_def[OF *] rel_fun_def by blast
 
 lemma mod_ring_divide64: "(mod_ring_rel64 ===> mod_ring_rel64 ===> mod_ring_rel64) (divide_p64 pp) (/)"
   using mod_ring_inverse64 mod_ring_mult64[OF *]
   unfolding divide_p64_def divide_mod_ring_def inverse_mod_ring_def[symmetric]
     rel_fun_def by blast
 
 lemma finite_field_ops64: "field_ops (finite_field_ops64 pp) mod_ring_rel64"
   by (unfold_locales, insert ring_finite_field_ops64[OF *], auto simp:
   ring_ops_def
   finite_field_ops64_def
   mod_ring_divide64
   mod_ring_inverse64
   mod_ring_mod64
   mod_ring_normalize64)
 end
 end
 
 (* and a final implementation via integer *)
 context
   fixes p :: integer
 begin
 definition plus_p_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer" where
   "plus_p_integer x y \<equiv> let z = x + y in if z \<ge> p then z - p else z"
 
 definition minus_p_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer" where
   "minus_p_integer x y \<equiv> if y \<le> x then x - y else (x + p) - y"
 
 definition uminus_p_integer :: "integer \<Rightarrow> integer" where
   "uminus_p_integer x = (if x = 0 then 0 else p - x)"
 
 definition mult_p_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer" where
   "mult_p_integer x y = (x * y mod p)"
 
 lemma int_of_integer_0_iff: "int_of_integer n = 0 \<longleftrightarrow> n = 0"
   using integer_eqI by auto
   
 lemma int_of_integer_0: "int_of_integer 0 = 0" unfolding int_of_integer_0_iff by simp
 
 lemma int_of_integer_plus: "int_of_integer (x + y) = (int_of_integer x + int_of_integer y)" 
   by simp
 
 lemma int_of_integer_minus: "int_of_integer (x - y) = (int_of_integer x - int_of_integer y)"
   by simp
 
 lemma int_of_integer_mult: "int_of_integer (x * y) = (int_of_integer x * int_of_integer y)" 
   by simp  
 
 lemma int_of_integer_mod: "int_of_integer (x mod y) = (int_of_integer x mod int_of_integer y)" 
   by simp  
 
 lemma int_of_integer_inv: "int_of_integer (integer_of_int x) = x" by simp
 
 lemma int_of_integer_shift: "int_of_integer (drop_bit k n) = (int_of_integer n) div (2 ^ k)" 
   by transfer (simp add: int_of_integer_pow shiftr_integer_conv_div_pow2)
 
 context
   includes bit_operations_syntax
 begin
 
 function power_p_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer" where
   "power_p_integer x n = (if n \<le> 0 then 1 else
     let rec = power_p_integer (mult_p_integer x x) (drop_bit 1 n) in
     if n AND 1 = 0 then rec else mult_p_integer rec x)"
   by pat_completeness auto
 
 termination 
 proof -
   {
     fix n :: integer
     assume "\<not> (n \<le> 0)" 
     hence "n > 0" by auto
     hence "int_of_integer n > 0"
       by (simp add: less_integer.rep_eq)
     hence "0 < int_of_integer n" "int_of_integer n div 2 < int_of_integer n" by auto
   } note * = this
   show ?thesis
     by (relation "measure (\<lambda> (x,n). nat (int_of_integer n))", auto simp: * int_of_integer_shift) 
 qed
 
 end
 
 text \<open>In experiments with Berlekamp-factorization (where the prime $p$ is usually small),
   it turned out that taking the below implementation of inverse via exponentiation
   is faster than the one based on the extended Euclidean algorithm.\<close>
 
 definition inverse_p_integer :: "integer \<Rightarrow> integer" where
   "inverse_p_integer x = (if x = 0 then 0 else power_p_integer x (p - 2))"
 
 definition divide_p_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"  where
   "divide_p_integer x y = mult_p_integer x (inverse_p_integer y)"
 
 definition finite_field_ops_integer :: "integer arith_ops_record" where
   "finite_field_ops_integer \<equiv> Arith_Ops_Record
       0
       1
       plus_p_integer
       mult_p_integer
       minus_p_integer
       uminus_p_integer
       divide_p_integer
       inverse_p_integer
       (\<lambda> x y . if y = 0 then x else 0)
       (\<lambda> x . if x = 0 then 0 else 1)
       (\<lambda> x . x)
       integer_of_int
       int_of_integer
       (\<lambda> x. 0 \<le> x \<and> x < p)"
 end 
 
 lemma shiftr_integer_code [code_unfold]: "drop_bit 1 x = (integer_shiftr x 1)"
   unfolding shiftr_integer_code using integer_of_nat_1 by auto
 
 text \<open>For soundness of the integer implementation, we mainly prove that this implementation
   implements the int-based implementation of GF(p).\<close>
 context mod_ring_locale
 begin
 
 context fixes pp :: "integer" 
   assumes ppp: "p = int_of_integer pp" 
 begin
 
 lemmas integer_simps = 
   int_of_integer_0
   int_of_integer_plus 
   int_of_integer_minus
   int_of_integer_mult
   
 
 definition urel_integer :: "integer \<Rightarrow> int \<Rightarrow> bool" where "urel_integer x y = (y = int_of_integer x \<and> y \<ge> 0 \<and> y < p)" 
 
 definition mod_ring_rel_integer :: "integer \<Rightarrow> 'a mod_ring \<Rightarrow> bool" where
   "mod_ring_rel_integer x y = (\<exists> z. urel_integer x z \<and> mod_ring_rel z y)" 
 
 lemma urel_integer_0: "urel_integer 0 0" unfolding urel_integer_def using p2 by simp
 
 lemma urel_integer_1: "urel_integer 1 1" unfolding urel_integer_def using p2 by simp
 
 lemma le_int_of_integer: "(x \<le> y) = (int_of_integer x \<le> int_of_integer y)" 
   by (rule less_eq_integer.rep_eq)
 
 lemma urel_integer_plus: assumes "urel_integer x y" "urel_integer x' y'"
   shows "urel_integer (plus_p_integer pp x x') (plus_p p y y')"
 proof -    
   let ?x = "int_of_integer x" 
   let ?x' = "int_of_integer x'" 
   let ?p = "int_of_integer pp" 
   from assms have id: "y = ?x" "y' = ?x'" 
     and rel: "0 \<le> ?x" "?x < p" 
       "0 \<le> ?x'" "?x' \<le> p" unfolding urel_integer_def by auto
   have le: "(pp \<le> x + x') = (?p \<le> ?x + ?x')" unfolding le_int_of_integer
     using rel by auto
   show ?thesis
   proof (cases "?p \<le> ?x + ?x'")
     case True
     hence True: "(?p \<le> ?x + ?x') = True" by simp
     show ?thesis unfolding id 
       using rel unfolding plus_p_integer_def plus_p_def Let_def urel_integer_def 
       unfolding ppp le True if_True
       using True by auto
   next
     case False
     hence False: "(?p \<le> ?x + ?x') = False" by simp
     show ?thesis unfolding id 
       using rel unfolding plus_p_integer_def plus_p_def Let_def urel_integer_def 
       unfolding ppp le False if_False
       using False by auto
   qed
 qed
   
 lemma urel_integer_minus: assumes "urel_integer x y" "urel_integer x' y'"
   shows "urel_integer (minus_p_integer pp x x') (minus_p p y y')"
 proof -    
   let ?x = "int_of_integer x" 
   let ?x' = "int_of_integer x'" 
   from assms have id: "y = ?x" "y' = ?x'" 
     and rel: "0 \<le> ?x" "?x < p" 
       "0 \<le> ?x'" "?x' \<le> p" unfolding urel_integer_def by auto
   have le: "(x' \<le> x) = (?x' \<le> ?x)" unfolding le_int_of_integer
     using rel by auto
   show ?thesis
   proof (cases "?x' \<le> ?x")
     case True
     hence True: "(?x' \<le> ?x) = True" by simp
     show ?thesis unfolding id 
       using rel unfolding minus_p_integer_def minus_p_def Let_def urel_integer_def 
       unfolding ppp le True if_True
       using True by auto
   next
     case False
     hence False: "(?x' \<le> ?x) = False" by simp
     show ?thesis unfolding id 
       using rel unfolding minus_p_integer_def minus_p_def Let_def urel_integer_def 
       unfolding ppp le False if_False
       using False by auto
   qed
 qed
 
 lemma urel_integer_uminus: assumes "urel_integer x y"
   shows "urel_integer (uminus_p_integer pp x) (uminus_p p y)"
 proof -    
   let ?x = "int_of_integer x"  
   from assms have id: "y = ?x" 
     and rel: "0 \<le> ?x" "?x < p" 
       unfolding urel_integer_def by auto
   have le: "(x = 0) = (?x = 0)" unfolding int_of_integer_0_iff
     using rel by auto
   show ?thesis
   proof (cases "?x = 0")
     case True
     hence True: "(?x = 0) = True" by simp
     show ?thesis unfolding id 
       using rel unfolding uminus_p_integer_def uminus_p_def Let_def urel_integer_def 
       unfolding ppp le True if_True
       using True by auto
   next
     case False
     hence False: "(?x = 0) = False" by simp
     show ?thesis unfolding id 
       using rel unfolding uminus_p_integer_def uminus_p_def Let_def urel_integer_def 
       unfolding ppp le False if_False
       using False by auto
   qed
 qed
 
 lemma pp_pos: "int_of_integer pp > 0" 
   using ppp nontriv[where 'a = 'a]  unfolding p
   by (simp add: less_integer.rep_eq)
 
 lemma urel_integer_mult: assumes "urel_integer x y" "urel_integer x' y'"
   shows "urel_integer (mult_p_integer pp x x') (mult_p p y y')"
 proof -    
   let ?x = "int_of_integer x" 
   let ?x' = "int_of_integer x'" 
   from assms  have id: "y = ?x" "y' = ?x'" 
     and rel: "0 \<le> ?x" "?x < p" 
       "0 \<le> ?x'" "?x' < p" unfolding urel_integer_def by auto
   from rel(1,3) have xx: "0 \<le> ?x * ?x'" by simp
   show ?thesis unfolding id
     using rel unfolding mult_p_integer_def mult_p_def Let_def urel_integer_def     
     unfolding ppp mod_nonneg_pos_int[OF xx pp_pos] using xx pp_pos by simp        
 qed
 
 lemma urel_integer_eq: assumes "urel_integer x y" "urel_integer x' y'" 
   shows "(x = x') = (y = y')" 
 proof -    
   let ?x = "int_of_integer x" 
   let ?x' = "int_of_integer x'" 
   from assms have id: "y = ?x" "y' = ?x'" 
     unfolding urel_integer_def by auto
   show ?thesis unfolding id integer_eq_iff ..
 qed
 
 lemma urel_integer_normalize: 
 assumes x: "urel_integer x y"
 shows "urel_integer (if x = 0 then 0 else 1) (if y = 0 then 0 else 1)"
  unfolding urel_integer_eq[OF x urel_integer_0] using urel_integer_0 urel_integer_1 by auto
 
 lemma urel_integer_mod: 
 assumes x: "urel_integer x x'" and y: "urel_integer y y'" 
 shows "urel_integer (if y = 0 then x else 0) (if y' = 0 then x' else 0)"
   unfolding urel_integer_eq[OF y urel_integer_0] using urel_integer_0 x by auto 
 
 lemma urel_integer_power: "urel_integer x x' \<Longrightarrow> urel_integer y (int y') \<Longrightarrow> urel_integer (power_p_integer pp x y) (power_p p x' y')"
 including bit_operations_syntax proof (induct x' y' arbitrary: x y rule: power_p.induct[of _ p])
   case (1 x' y' x y)
   note x = 1(2) note y = 1(3)
   show ?case
   proof (cases "y' \<le> 0")
     case True
     hence y: "y = 0" "y' = 0" using urel_integer_eq[OF y urel_integer_0] by auto
     show ?thesis unfolding y True by (simp add: power_p.simps urel_integer_1)
   next
     case False
     hence id: "(y \<le> 0) = False" "(y' = 0) = False" using False y
       by (auto simp add: urel_integer_def not_le) (metis of_int_integer_of of_int_of_nat_eq of_nat_0_less_iff)
     obtain d' r' where dr': "Divides.divmod_nat y' 2 = (d',r')" by force
     from divmod_nat_def[of y' 2, unfolded dr']
     have r': "r' = y' mod 2" and d': "d' = y' div 2" by auto
     have aux: "\<And> y'. int (y' mod 2) = int y' mod 2" by presburger
     have "urel_integer (y AND 1) r'" unfolding r' using y unfolding urel_integer_def 
       unfolding ppp
       apply (auto simp add: and_one_eq)
       apply (simp add: of_nat_mod)
       done
     from urel_integer_eq[OF this urel_integer_0]     
     have rem: "(y AND 1 = 0) = (r' = 0)" by simp
     have div: "urel_integer (drop_bit 1 y) (int d')" unfolding d' using y unfolding urel_integer_def
       unfolding ppp shiftr_integer_conv_div_pow2 by auto
     from id have "y' \<noteq> 0" by auto
     note IH = 1(1)[OF this refl dr'[symmetric] urel_integer_mult[OF x x] div]
     show ?thesis unfolding power_p.simps[of _ _ "y'"] power_p_integer.simps[of _ _ y] dr' id if_False rem
       using IH urel_integer_mult[OF IH x] by (auto simp: Let_def)
   qed
 qed
   
 
 lemma urel_integer_inverse: assumes x: "urel_integer x x'" 
   shows "urel_integer (inverse_p_integer pp x) (inverse_p p x')" 
 proof -
   have p: "urel_integer (pp - 2) (int (nat (p - 2)))" using p2 unfolding urel_integer_def unfolding ppp
     by auto
   show ?thesis
     unfolding inverse_p_integer_def inverse_p_def urel_integer_eq[OF x urel_integer_0] using urel_integer_0 urel_integer_power[OF x p]
     by auto
 qed
 
 lemma mod_ring_0__integer: "mod_ring_rel_integer 0 0"
   using urel_integer_0 mod_ring_0 unfolding mod_ring_rel_integer_def by blast
 
 lemma mod_ring_1__integer: "mod_ring_rel_integer 1 1"
   using urel_integer_1 mod_ring_1 unfolding mod_ring_rel_integer_def by blast
 
 lemma mod_ring_uminus_integer: "(mod_ring_rel_integer ===> mod_ring_rel_integer) (uminus_p_integer pp) uminus"
   using urel_integer_uminus mod_ring_uminus unfolding mod_ring_rel_integer_def rel_fun_def by blast
 
 lemma mod_ring_plus_integer: "(mod_ring_rel_integer ===> mod_ring_rel_integer ===> mod_ring_rel_integer) (plus_p_integer pp) (+)"
   using urel_integer_plus mod_ring_plus unfolding mod_ring_rel_integer_def rel_fun_def by blast
 
 lemma mod_ring_minus_integer: "(mod_ring_rel_integer ===> mod_ring_rel_integer ===> mod_ring_rel_integer) (minus_p_integer pp) (-)"
   using urel_integer_minus mod_ring_minus unfolding mod_ring_rel_integer_def rel_fun_def by blast
 
 lemma mod_ring_mult_integer: "(mod_ring_rel_integer ===> mod_ring_rel_integer ===> mod_ring_rel_integer) (mult_p_integer pp) ((*))"
   using urel_integer_mult mod_ring_mult unfolding mod_ring_rel_integer_def rel_fun_def by blast
 
 lemma mod_ring_eq_integer: "(mod_ring_rel_integer ===> mod_ring_rel_integer ===> (=)) (=) (=)" 
   using urel_integer_eq mod_ring_eq unfolding mod_ring_rel_integer_def rel_fun_def by blast
 
 lemma urel_integer_inj: "urel_integer x y \<Longrightarrow> urel_integer x z \<Longrightarrow> y = z" 
   using urel_integer_eq[of x y x z] by auto
 
 lemma urel_integer_inj': "urel_integer x z \<Longrightarrow> urel_integer y z \<Longrightarrow> x = y" 
   using urel_integer_eq[of x z y z] by auto
 
 lemma bi_unique_mod_ring_rel_integer:
   "bi_unique mod_ring_rel_integer" "left_unique mod_ring_rel_integer" "right_unique mod_ring_rel_integer"
   using bi_unique_mod_ring_rel urel_integer_inj'
   unfolding mod_ring_rel_integer_def bi_unique_def left_unique_def right_unique_def
   by (auto simp: urel_integer_def)  
 
 lemma right_total_mod_ring_rel_integer: "right_total mod_ring_rel_integer"
   unfolding mod_ring_rel_integer_def right_total_def
 proof 
   fix y :: "'a mod_ring" 
   from right_total_mod_ring_rel[unfolded right_total_def, rule_format, of y]
   obtain z where zy: "mod_ring_rel z y" by auto  
   hence zp: "0 \<le> z" "z < p" unfolding mod_ring_rel_def p using range_to_int_mod_ring[where 'a = 'a] by auto
   hence "urel_integer (integer_of_int z) z" unfolding urel_integer_def unfolding ppp 
     by auto 
   with zy show "\<exists> x z. urel_integer x z \<and> mod_ring_rel z y" by blast
 qed
 
 lemma Domainp_mod_ring_rel_integer: "Domainp mod_ring_rel_integer = (\<lambda>x. 0 \<le> x \<and> x < pp)"
 proof 
   fix x
   show "Domainp mod_ring_rel_integer x = (0 \<le> x \<and> x < pp)"   
     unfolding Domainp.simps
     unfolding mod_ring_rel_integer_def
   proof
     let ?i = "int_of_integer" 
     assume *: "0 \<le> x \<and> x < pp"     
     hence "0 \<le> ?i x \<and> ?i x < p" unfolding ppp 
       by (simp add: le_int_of_integer less_integer.rep_eq)
     hence "?i x \<in> {0 ..< p}" by auto
     with Domainp_mod_ring_rel
     have "Domainp mod_ring_rel (?i x)" by auto
     from this[unfolded Domainp.simps]
     obtain b where b: "mod_ring_rel (?i x) b" by auto
     show "\<exists>a b. x = a \<and> (\<exists>z. urel_integer a z \<and> mod_ring_rel z b)" 
     proof (intro exI, rule conjI[OF refl], rule exI, rule conjI[OF _ b])
       show "urel_integer x (?i x)" unfolding urel_integer_def using * unfolding ppp
         by (simp add: le_int_of_integer less_integer.rep_eq)
     qed
   next
     assume "\<exists>a b. x = a \<and> (\<exists>z. urel_integer a z \<and> mod_ring_rel z b)" 
     then obtain b z where xz: "urel_integer x z" and zb: "mod_ring_rel z b" by auto
     hence "Domainp mod_ring_rel z"  by auto
     with Domainp_mod_ring_rel have "0 \<le> z" "z < p" by auto
     with xz show "0 \<le> x \<and> x < pp" unfolding urel_integer_def unfolding ppp
       by (simp add: le_int_of_integer less_integer.rep_eq)
   qed
 qed
 
 lemma ring_finite_field_ops_integer: "ring_ops (finite_field_ops_integer pp) mod_ring_rel_integer"
   by (unfold_locales, auto simp:
   finite_field_ops_integer_def
   bi_unique_mod_ring_rel_integer
   right_total_mod_ring_rel_integer
   mod_ring_plus_integer
   mod_ring_minus_integer
   mod_ring_uminus_integer
   mod_ring_mult_integer
   mod_ring_eq_integer
   mod_ring_0__integer
   mod_ring_1__integer
   Domainp_mod_ring_rel_integer)
 end
 end
 
 context prime_field
 begin
 context fixes pp :: "integer" 
   assumes *: "p = int_of_integer pp"  
 begin
 
 lemma mod_ring_normalize_integer: "(mod_ring_rel_integer ===> mod_ring_rel_integer) (\<lambda>x. if x = 0 then 0 else 1) normalize" 
   using urel_integer_normalize[OF *] mod_ring_normalize unfolding mod_ring_rel_integer_def[OF *] rel_fun_def by blast
 
 lemma mod_ring_mod_integer: "(mod_ring_rel_integer ===> mod_ring_rel_integer ===> mod_ring_rel_integer) (\<lambda>x y. if y = 0 then x else 0) (mod)" 
   using urel_integer_mod[OF *] mod_ring_mod unfolding mod_ring_rel_integer_def[OF *] rel_fun_def by blast
 
 lemma mod_ring_unit_factor_integer: "(mod_ring_rel_integer ===> mod_ring_rel_integer) (\<lambda>x. x) unit_factor" 
   using mod_ring_unit_factor unfolding mod_ring_rel_integer_def[OF *] rel_fun_def by blast
 
 lemma mod_ring_inverse_integer: "(mod_ring_rel_integer ===> mod_ring_rel_integer) (inverse_p_integer pp) inverse"
   using urel_integer_inverse[OF *] mod_ring_inverse unfolding mod_ring_rel_integer_def[OF *] rel_fun_def by blast
 
 lemma mod_ring_divide_integer: "(mod_ring_rel_integer ===> mod_ring_rel_integer ===> mod_ring_rel_integer) (divide_p_integer pp) (/)"
   using mod_ring_inverse_integer mod_ring_mult_integer[OF *]
   unfolding divide_p_integer_def divide_mod_ring_def inverse_mod_ring_def[symmetric]
     rel_fun_def by blast
 
 lemma finite_field_ops_integer: "field_ops (finite_field_ops_integer pp) mod_ring_rel_integer"
   by (unfold_locales, insert ring_finite_field_ops_integer[OF *], auto simp:
   ring_ops_def
   finite_field_ops_integer_def
   mod_ring_divide_integer
   mod_ring_inverse_integer
   mod_ring_mod_integer
   mod_ring_normalize_integer)
 end
 end
 
 context prime_field
 begin
  (* four implementations of modular integer arithmetic for finite fields *)
 thm 
   finite_field_ops64
   finite_field_ops32
   finite_field_ops_integer
   finite_field_ops_int
 end
 
 context mod_ring_locale
 begin
  (* four implementations of modular integer arithmetic for finite rings *)
 thm 
   ring_finite_field_ops64
   ring_finite_field_ops32
   ring_finite_field_ops_integer
   ring_finite_field_ops_int
 end
 
 end
diff --git a/thys/Collections/GenCF/Impl/Impl_Bit_Set.thy b/thys/Collections/GenCF/Impl/Impl_Bit_Set.thy
--- a/thys/Collections/GenCF/Impl/Impl_Bit_Set.thy
+++ b/thys/Collections/GenCF/Impl/Impl_Bit_Set.thy
@@ -1,384 +1,384 @@
 section "Bitvector based Sets of Naturals"
 theory Impl_Bit_Set
 imports 
   "../../Iterator/Iterator" 
   "../Intf/Intf_Set" 
-  Native_Word.Bits_Integer
+  Native_Word.Code_Target_Integer_Bit
 begin
   text \<open>
     Based on the Native-Word library, using bit-operations on arbitrary
     precision integers. Fast for sets of small numbers, 
     direct and fast implementations of equal, union, inter, diff.
 
     Note: On Poly/ML 5.5.1, bit-operations on arbitrary precision integers are 
       rather inefficient. Use MLton instead, here they are efficiently implemented.
 \<close>
 
   type_synonym bitset = integer
 
   definition bs_\<alpha> :: "bitset \<Rightarrow> nat set" where "bs_\<alpha> s \<equiv> { n . bit s n}"
 
 
   context
     includes integer.lifting bit_operations_syntax
   begin
 
   definition bs_empty :: "unit \<Rightarrow> bitset" where "bs_empty \<equiv> \<lambda>_. 0"
 
 
   lemma bs_empty_correct: "bs_\<alpha> (bs_empty ()) = {}"
     unfolding bs_\<alpha>_def bs_empty_def 
     apply transfer
     by auto
 
   definition bs_isEmpty :: "bitset \<Rightarrow> bool" where "bs_isEmpty s \<equiv> s=0"
 
   lemma bs_isEmpty_correct: "bs_isEmpty s \<longleftrightarrow> bs_\<alpha> s = {}"
     unfolding bs_isEmpty_def bs_\<alpha>_def 
     by transfer (auto simp: bit_eq_iff) 
     
   term set_bit
   definition bs_insert :: "nat \<Rightarrow> bitset \<Rightarrow> bitset" where
     "bs_insert i s \<equiv> set_bit s i True"
 
   lemma bs_insert_correct: "bs_\<alpha> (bs_insert i s) = insert i (bs_\<alpha> s)"
     unfolding bs_\<alpha>_def bs_insert_def
     by transfer (auto simp add: bit_simps)
 
   definition bs_delete :: "nat \<Rightarrow> bitset \<Rightarrow> bitset" where
     "bs_delete i s \<equiv> set_bit s i False"
 
   lemma bs_delete_correct: "bs_\<alpha> (bs_delete i s) = (bs_\<alpha> s) - {i}"
     unfolding bs_\<alpha>_def bs_delete_def
     by transfer (auto simp add: bit_simps split: if_splits)
   
   definition bs_mem :: "nat \<Rightarrow> bitset \<Rightarrow> bool" where
     "bs_mem i s \<equiv> bit s i"
 
   lemma bs_mem_correct: "bs_mem i s \<longleftrightarrow> i\<in>bs_\<alpha> s"
     unfolding bs_mem_def bs_\<alpha>_def by transfer auto
 
 
   definition bs_eq :: "bitset \<Rightarrow> bitset \<Rightarrow> bool" where 
     "bs_eq s1 s2 \<equiv> (s1=s2)"
 
   lemma bs_eq_correct: "bs_eq s1 s2 \<longleftrightarrow> bs_\<alpha> s1 = bs_\<alpha> s2"
     unfolding bs_eq_def bs_\<alpha>_def
     including integer.lifting
     by transfer (simp add: bit_eq_iff set_eq_iff)
 
   definition bs_subset_eq :: "bitset \<Rightarrow> bitset \<Rightarrow> bool" where
     "bs_subset_eq s1 s2 \<equiv> s1 AND NOT s2 = 0"
   
   lemma bs_subset_eq_correct: "bs_subset_eq s1 s2 \<longleftrightarrow> bs_\<alpha> s1 \<subseteq> bs_\<alpha> s2"
     unfolding bs_\<alpha>_def bs_subset_eq_def
     by transfer (simp add: bit_eq_iff, auto simp add: bit_simps)
 
   definition bs_disjoint :: "bitset \<Rightarrow> bitset \<Rightarrow> bool" where
     "bs_disjoint s1 s2 \<equiv> s1 AND s2 = 0"
   
   lemma bs_disjoint_correct: "bs_disjoint s1 s2 \<longleftrightarrow> bs_\<alpha> s1 \<inter> bs_\<alpha> s2 = {}"
     unfolding bs_\<alpha>_def bs_disjoint_def
     by transfer (simp add: bit_eq_iff, auto simp add: bit_simps)
 
   definition bs_union :: "bitset \<Rightarrow> bitset \<Rightarrow> bitset" where
     "bs_union s1 s2 = s1 OR s2"
 
   lemma bs_union_correct: "bs_\<alpha> (bs_union s1 s2) = bs_\<alpha> s1 \<union> bs_\<alpha> s2"
     unfolding bs_\<alpha>_def bs_union_def
     by transfer (simp add: bit_eq_iff, auto simp add: bit_simps)
 
   definition bs_inter :: "bitset \<Rightarrow> bitset \<Rightarrow> bitset" where
     "bs_inter s1 s2 = s1 AND s2"
 
   lemma bs_inter_correct: "bs_\<alpha> (bs_inter s1 s2) = bs_\<alpha> s1 \<inter> bs_\<alpha> s2"
     unfolding bs_\<alpha>_def bs_inter_def
     by transfer (simp add: bit_eq_iff, auto simp add: bit_simps)
 
   definition bs_diff :: "bitset \<Rightarrow> bitset \<Rightarrow> bitset" where
     "bs_diff s1 s2 = s1 AND NOT s2"
 
   lemma bs_diff_correct: "bs_\<alpha> (bs_diff s1 s2) = bs_\<alpha> s1 - bs_\<alpha> s2"
     unfolding bs_\<alpha>_def bs_diff_def
     by transfer (simp add: bit_eq_iff, auto simp add: bit_simps)
 
   definition bs_UNIV :: "unit \<Rightarrow> bitset" where "bs_UNIV \<equiv> \<lambda>_. -1"
 
   lemma bs_UNIV_correct: "bs_\<alpha> (bs_UNIV ()) = UNIV"
     unfolding bs_\<alpha>_def bs_UNIV_def
     by transfer (auto)
 
   definition bs_complement :: "bitset \<Rightarrow> bitset" where
     "bs_complement s = NOT s"
 
   lemma bs_complement_correct: "bs_\<alpha> (bs_complement s) = - bs_\<alpha> s"
     unfolding bs_\<alpha>_def bs_complement_def
     by transfer (simp add: bit_eq_iff, auto simp add: bit_simps)
 
 end
 
   lemmas bs_correct[simp] = 
     bs_empty_correct
     bs_isEmpty_correct
     bs_insert_correct
     bs_delete_correct
     bs_mem_correct
     bs_eq_correct
     bs_subset_eq_correct
     bs_disjoint_correct
     bs_union_correct
     bs_inter_correct
     bs_diff_correct
     bs_UNIV_correct
     bs_complement_correct
 
 
 subsection \<open>Autoref Setup\<close>
 
 definition bs_set_rel_def_internal: 
   "bs_set_rel Rk \<equiv> 
     if Rk=nat_rel then br bs_\<alpha> (\<lambda>_. True) else {}"
 lemma bs_set_rel_def: 
   "\<langle>nat_rel\<rangle>bs_set_rel \<equiv> br bs_\<alpha> (\<lambda>_. True)" 
   unfolding bs_set_rel_def_internal relAPP_def by simp
 
 lemmas [autoref_rel_intf] = REL_INTFI[of "bs_set_rel" i_set]
 
 lemma bs_set_rel_sv[relator_props]: "single_valued (\<langle>nat_rel\<rangle>bs_set_rel)"
   unfolding bs_set_rel_def by auto
 
 
 term bs_empty
 
 lemma [autoref_rules]: "(bs_empty (),{})\<in>\<langle>nat_rel\<rangle>bs_set_rel"
   by (auto simp: bs_set_rel_def br_def)
 
 lemma [autoref_rules]: "(bs_UNIV (),UNIV)\<in>\<langle>nat_rel\<rangle>bs_set_rel"
   by (auto simp: bs_set_rel_def br_def)
 
 lemma [autoref_rules]: "(bs_isEmpty,op_set_isEmpty)\<in>\<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> bool_rel"
   by (auto simp: bs_set_rel_def br_def)
 
 term insert
 lemma [autoref_rules]: "(bs_insert,insert)\<in>nat_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel"
   by (auto simp: bs_set_rel_def br_def)
 
 term op_set_delete
 lemma [autoref_rules]: "(bs_delete,op_set_delete)\<in>nat_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel"
   by (auto simp: bs_set_rel_def br_def)
 
 lemma [autoref_rules]: "(bs_mem,(\<in>))\<in>nat_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> bool_rel"
   by (auto simp: bs_set_rel_def br_def)
 
 lemma [autoref_rules]: "(bs_eq,(=))\<in>\<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> bool_rel"
   by (auto simp: bs_set_rel_def br_def)
 
 lemma [autoref_rules]: "(bs_subset_eq,(\<subseteq>))\<in>\<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> bool_rel"
   by (auto simp: bs_set_rel_def br_def)
 
 lemma [autoref_rules]: "(bs_union,(\<union>))\<in>\<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel"
   by (auto simp: bs_set_rel_def br_def)
 
 lemma [autoref_rules]: "(bs_inter,(\<inter>))\<in>\<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel"
   by (auto simp: bs_set_rel_def br_def)
 
 lemma [autoref_rules]: "(bs_diff,(-))\<in>\<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel"
   by (auto simp: bs_set_rel_def br_def)
 
 lemma [autoref_rules]: "(bs_complement,uminus)\<in>\<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel"
   by (auto simp: bs_set_rel_def br_def)
 
 lemma [autoref_rules]: "(bs_disjoint,op_set_disjoint)\<in>\<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel \<rightarrow> bool_rel"
   by (auto simp: bs_set_rel_def br_def)
 
 
 export_code
     bs_empty
     bs_isEmpty
     bs_insert
     bs_delete
     bs_mem
     bs_eq
     bs_subset_eq
     bs_disjoint
     bs_union
     bs_inter
     bs_diff
     bs_UNIV
     bs_complement
  in SML
 
 (*
 
     TODO: Iterator
 
   definition "maxbi s \<equiv> GREATEST i. s!!i"
 
   lemma cmp_BIT_append_conv[simp]: "i < i BIT b \<longleftrightarrow> ((i\<ge>0 \<and> b=1) \<or> i>0)"
     by (cases b) (auto simp: Bit_B0 Bit_B1)
 
   lemma BIT_append_cmp_conv[simp]: "i BIT b < i \<longleftrightarrow> ((i<0 \<and> (i=-1 \<longrightarrow> b=0)))"
     by (cases b) (auto simp: Bit_B0 Bit_B1)
 
   lemma BIT_append_eq[simp]: fixes i :: int shows "i BIT b = i \<longleftrightarrow> (i=0 \<and> b=0) \<or> (i=-1 \<and> b=1)"
     by (cases b) (auto simp: Bit_B0 Bit_B1)
 
   lemma int_no_bits_eq_zero[simp]:
     fixes s::int shows "(\<forall>i. \<not>s!!i) \<longleftrightarrow> s=0"
     apply clarsimp
     by (metis bin_eqI bin_nth_code(1))
 
   lemma int_obtain_bit:
     fixes s::int
     assumes "s\<noteq>0"
     obtains i where "s!!i"
     by (metis assms int_no_bits_eq_zero)
     
   lemma int_bit_bound:
     fixes s::int
     assumes "s\<ge>0" and "s!!i"
     shows "i \<le> Bits_Integer.log2 s"
   proof (rule ccontr)
     assume "\<not>i\<le>Bits_Integer.log2 s"
     hence "i>Bits_Integer.log2 s" by simp
     hence "i - 1 \<ge> Bits_Integer.log2 s" by simp
     hence "s AND bin_mask (i - 1) = s" by (simp add: int_and_mask `s\<ge>0`)
     hence "\<not> (s!!i)"  
       by clarsimp (metis Nat.diff_le_self bin_nth_mask bin_nth_ops(1) leD)
     thus False using `s!!i` ..
   qed
 
   lemma int_bit_bound':
     fixes s::int
     assumes "s\<ge>0" and "s!!i"
     shows "i < Bits_Integer.log2 s + 1"
     using assms int_bit_bound by smt
 
   lemma int_obtain_bit_pos:
     fixes s::int
     assumes "s>0"
     obtains i where "s!!i" "i < Bits_Integer.log2 s + 1"
     by (metis assms int_bit_bound' int_no_bits_eq_zero less_imp_le less_irrefl)
 
   lemma maxbi_set: fixes s::int shows "s>0 \<Longrightarrow> s!!maxbi s"
     unfolding maxbi_def
     apply (rule int_obtain_bit_pos, assumption)
     apply (rule GreatestI_nat, assumption)
     apply (intro allI impI)
     apply (rule int_bit_bound'[rotated], assumption)
     by auto
 
   lemma maxbi_max: fixes s::int shows "i>maxbi s \<Longrightarrow> \<not> s!!i"
     oops
 
   function get_maxbi :: "nat \<Rightarrow> int \<Rightarrow> nat" where
     "get_maxbi n s = (let
         b = 1<<n
       in
         if b\<le>s then get_maxbi (n+1) s
         else n
     )"
     by pat_completeness auto
 
   termination
     apply (rule "termination"[of "measure (\<lambda>(n,s). nat (s + 1 - (1<<n)))"])
     apply simp
     apply auto
     by (smt bin_mask_ge0 bin_mask_p1_conv_shift)
 
 
   partial_function (tailrec) 
     bs_iterate_aux :: "nat \<Rightarrow> bitset \<Rightarrow> ('\<sigma> \<Rightarrow> bool) \<Rightarrow> (nat \<Rightarrow> '\<sigma> \<Rightarrow> '\<sigma>) \<Rightarrow> '\<sigma> \<Rightarrow> '\<sigma>"
     where "bs_iterate_aux i s c f \<sigma> = (
     if s < 1 << i then \<sigma>
     else if \<not>c \<sigma> then \<sigma>
     else if test_bit s i then bs_iterate_aux (i+1) s c f (f i \<sigma>)
     else bs_iterate_aux (i+1) s c f \<sigma>
   )"
 
   definition bs_iteratei :: "bitset \<Rightarrow> (nat,'\<sigma>) set_iterator" where 
     "bs_iteratei s = bs_iterate_aux 0 s"
 
 
   definition bs_set_rel_def_internal: 
     "bs_set_rel Rk \<equiv> 
       if Rk=nat_rel then br bs_\<alpha> (\<lambda>_. True) else {}"
   lemma bs_set_rel_def: 
     "\<langle>nat_rel\<rangle>bs_set_rel \<equiv> br bs_\<alpha> (\<lambda>_. True)" 
     unfolding bs_set_rel_def_internal relAPP_def by simp
 
 
   definition "bs_to_list \<equiv> it_to_list bs_iteratei"
 
   lemma "(1::int)<<i = 2^i"
     by (simp add: shiftl_int_def)
 
   lemma 
     fixes s :: int
     assumes "s\<ge>0"  
     shows "s < 1<<i \<longleftrightarrow> Bits_Integer.log2 s \<le> i"
     using assms
   proof (induct i arbitrary: s)
     case 0 thus ?case by auto
   next
     case (Suc i)
     note GE=`0\<le>s`
     show ?case proof
       assume "s < 1 << Suc i"
 
       have "s \<le> (s >> 1) BIT 1"
 
       hence "(s >> 1) < (1<<i)" using GE apply auto
       with Suc.hyps[of "s div 2"]
 
 
     apply auto
     
 
 
   lemma "distinct (bs_to_list s)"
     unfolding bs_to_list_def it_to_list_def bs_iteratei_def[abs_def]
   proof -
     {
       fix l i
       assume "distinct l"
       show "distinct (bs_iterate_aux 0 s (\<lambda>_. True) (\<lambda>x l. l @ [x]) [])"
 
     }
 
 
     apply auto
     
 
 
 
     lemma "set (bs_to_list s) = bs_\<alpha> s"
 
 
   lemma autoref_iam_is_iterator[autoref_ga_rules]: 
     shows "is_set_to_list nat_rel bs_set_rel bs_to_list"
     unfolding is_set_to_list_def is_set_to_sorted_list_def
     apply clarsimp
     unfolding it_to_sorted_list_def
     apply (refine_rcg refine_vcg)
     apply (simp_all add: bs_set_rel_def br_def)
 
   proof (clarsimp)
 
 
 
   definition 
 
 "iterate s c f \<sigma> \<equiv> let
     i=0;
     b=0;
     (_,_,s) = while 
   in
 
   end"
 
 
 *)
 
 
 end
diff --git a/thys/Collections/Lib/Code_Target_ICF.thy b/thys/Collections/Lib/Code_Target_ICF.thy
--- a/thys/Collections/Lib/Code_Target_ICF.thy
+++ b/thys/Collections/Lib/Code_Target_ICF.thy
@@ -1,8 +1,8 @@
 section \<open>Default Code Generator Setup for the Isabelle Collection Framework\<close>
 theory Code_Target_ICF
 imports   
   "HOL-Library.Code_Target_Numeral"
-  Native_Word.Code_Target_Bits_Int
+  Native_Word.Code_Target_Int_Bit
 begin
 
 end
diff --git a/thys/Iptables_Semantics/Primitive_Matchers/Code_Interface.thy b/thys/Iptables_Semantics/Primitive_Matchers/Code_Interface.thy
--- a/thys/Iptables_Semantics/Primitive_Matchers/Code_Interface.thy
+++ b/thys/Iptables_Semantics/Primitive_Matchers/Code_Interface.thy
@@ -1,208 +1,208 @@
 theory Code_Interface
 imports 
   Common_Primitive_toString
   IP_Addresses.IP_Address_Parser
   "../Call_Return_Unfolding"
   Transform
   No_Spoof
   "../Simple_Firewall/SimpleFw_Compliance"
   Simple_Firewall.SimpleFw_toString
   Simple_Firewall.Service_Matrix
   "../Semantics_Ternary/Optimizing" (*do we use this?*)
   "../Semantics_Goto"
-  Native_Word.Code_Target_Bits_Int
   "HOL-Library.Code_Target_Nat"
   "HOL-Library.Code_Target_Int"
+  Native_Word.Code_Target_Int_Bit
 begin
 
 (*Note: common_primitive_match_expr_ipv4_toString can be really slow*)
 
 section\<open>Code Interface\<close>
 
 text\<open>HACK: rewrite quotes such that they are better printable by Isabelle\<close>
 definition quote_rewrite :: "string \<Rightarrow> string" where
   "quote_rewrite \<equiv> map (\<lambda>c. if c = char_of_nat 34 then CHR ''~'' else c)"
 
 lemma "quote_rewrite (''foo''@[char_of_nat 34]) = ''foo~''" by eval
 
 text\<open>The parser returns the @{typ "'i::len common_primitive ruleset"} not as a map but as an association list.
       This function converts it\<close>
 
 (*this is only to tighten the types*)
 definition map_of_string_ipv4
   :: "(string \<times> 32 common_primitive rule list) list \<Rightarrow> string \<rightharpoonup> 32 common_primitive rule list" where
   "map_of_string_ipv4 rs = map_of rs"
 definition map_of_string_ipv6
   :: "(string \<times> 128 common_primitive rule list) list \<Rightarrow> string \<rightharpoonup> 128 common_primitive rule list" where
   "map_of_string_ipv6 rs = map_of rs"
 definition map_of_string
   :: "(string \<times> 'i common_primitive rule list) list \<Rightarrow> string \<rightharpoonup> 'i common_primitive rule list" where
   "map_of_string rs = map_of rs"
 
 
 definition unfold_ruleset_CHAIN_safe :: "string \<Rightarrow> action \<Rightarrow> 'i::len common_primitive ruleset \<Rightarrow> 'i common_primitive rule list option" where
 "unfold_ruleset_CHAIN_safe = unfold_optimize_ruleset_CHAIN optimize_primitive_univ"
 
 lemma "(unfold_ruleset_CHAIN_safe chain a rs = Some rs') \<Longrightarrow> simple_ruleset rs'"
   by(simp add: Let_def unfold_ruleset_CHAIN_safe_def unfold_optimize_ruleset_CHAIN_def split: if_split_asm)
 
 (*This is just for legacy code compatibility. Use the new _safe function instead*)
 definition unfold_ruleset_CHAIN :: "string \<Rightarrow> action \<Rightarrow> 'i::len common_primitive ruleset \<Rightarrow> 'i common_primitive rule list" where
   "unfold_ruleset_CHAIN chain default_action rs = the (unfold_ruleset_CHAIN_safe chain default_action rs)"
 
 
 definition unfold_ruleset_FORWARD :: "action \<Rightarrow> 'i::len common_primitive ruleset \<Rightarrow> 'i::len common_primitive rule list" where
   "unfold_ruleset_FORWARD = unfold_ruleset_CHAIN ''FORWARD''"
 
 definition unfold_ruleset_INPUT :: "action \<Rightarrow> 'i::len common_primitive ruleset \<Rightarrow> 'i::len common_primitive rule list" where
   "unfold_ruleset_INPUT = unfold_ruleset_CHAIN ''INPUT''"
 
 definition unfold_ruleset_OUTPUT :: "action \<Rightarrow> 'i::len common_primitive ruleset \<Rightarrow> 'i::len common_primitive rule list" where
   "unfold_ruleset_OUTPUT \<equiv> unfold_ruleset_CHAIN ''OUTPUT''"
 
 
 lemma "let fw = [''FORWARD'' \<mapsto> []] in
   unfold_ruleset_FORWARD action.Drop fw
   = [Rule (MatchAny :: 32 common_primitive match_expr) action.Drop]" by eval
 
 
 (* only used for ML/Haskell code to convert types *)
 definition nat_to_8word :: "nat \<Rightarrow> 8 word" where
   "nat_to_8word i \<equiv> of_nat i"
 
 definition nat_to_16word :: "nat \<Rightarrow> 16 word" where
   "nat_to_16word i \<equiv> of_nat i"
 
 definition integer_to_16word :: "integer \<Rightarrow> 16 word" where
   "integer_to_16word i \<equiv> nat_to_16word (nat_of_integer i)"
 
 
 
 
 context
 begin
   private definition is_pos_Extra :: "'i::len common_primitive negation_type \<Rightarrow> bool" where
     "is_pos_Extra a \<equiv> (case a of Pos (Extra _) \<Rightarrow> True | _ \<Rightarrow> False)"
   private definition get_pos_Extra :: "'i::len common_primitive negation_type \<Rightarrow> string" where
     "get_pos_Extra a \<equiv> (case a of Pos (Extra e) \<Rightarrow> e | _ \<Rightarrow> undefined)"
   
   fun compress_parsed_extra
     :: "'i::len common_primitive negation_type list \<Rightarrow> 'i common_primitive negation_type list" where
     "compress_parsed_extra [] = []" |
     "compress_parsed_extra (a1#a2#as) = (if is_pos_Extra a1 \<and> is_pos_Extra a2
         then compress_parsed_extra (Pos (Extra (get_pos_Extra a1@'' ''@get_pos_Extra a2))#as)
         else a1#compress_parsed_extra (a2#as)
         )" |
     "compress_parsed_extra (a#as) = a#compress_parsed_extra as"
   
   lemma "compress_parsed_extra
     (map Pos [Extra ''-m'', (Extra ''recent'' :: 32 common_primitive),
               Extra ''--update'', Extra ''--seconds'', Extra ''60'',
               IIface (Iface ''foobar''),
               Extra ''--name'', Extra ''DEFAULT'', Extra ''--rsource'']) =
      map Pos [Extra ''-m recent --update --seconds 60'',
               IIface (Iface ''foobar''),
               Extra ''--name DEFAULT --rsource'']" by eval
   
   private lemma eval_ternary_And_Unknown_Unkown:
     "eval_ternary_And TernaryUnknown (eval_ternary_And TernaryUnknown tv) =
         eval_ternary_And TernaryUnknown tv"
     by(cases tv) (simp_all)
   
   private lemma is_pos_Extra_alist_and:
     "is_pos_Extra a \<Longrightarrow> alist_and (a#as) = MatchAnd (Match (Extra (get_pos_Extra a))) (alist_and as)"
     apply(cases a)
      apply(simp_all add: get_pos_Extra_def is_pos_Extra_def)
     apply(rename_tac e)
     by(case_tac e)(simp_all)
   
   private lemma compress_parsed_extra_matchexpr_helper:
     "ternary_ternary_eval (map_match_tac common_matcher p (alist_and (compress_parsed_extra as))) =
          ternary_ternary_eval (map_match_tac common_matcher p (alist_and as))"
    proof(induction as rule: compress_parsed_extra.induct)
    case 1 thus ?case by(simp)
    next
    case (2 a1 a2) thus ?case
      apply(simp add: is_pos_Extra_alist_and)
      apply(cases a1)
       apply(simp_all add: eval_ternary_And_Unknown_Unkown)
      done
    next
    case 3 thus ?case by(simp)
    qed
   
   text\<open>This lemma justifies that it is okay to fold together the parsed unknown tokens\<close>
   lemma compress_parsed_extra_matchexpr:
     "matches (common_matcher, \<alpha>) (alist_and (compress_parsed_extra as)) =
         matches (common_matcher, \<alpha>) (alist_and as)"
     apply(simp add: fun_eq_iff)
     apply(intro allI)
     apply(rule matches_iff_apply_f)
     apply(simp add: compress_parsed_extra_matchexpr_helper)
     done
 end
 
 
 
 
 subsection\<open>L4 Ports Parser Helper\<close>
 
 context
 begin
   text\<open>Replace all matches on ports with the unspecified @{term 0} protocol with the given @{typ primitive_protocol}.\<close>
   private definition fill_l4_protocol_raw
     :: "primitive_protocol \<Rightarrow> 'i::len common_primitive negation_type list \<Rightarrow> 'i common_primitive negation_type list"
   where
     "fill_l4_protocol_raw protocol \<equiv> NegPos_map
       (\<lambda> m. case m of Src_Ports (L4Ports x pts) \<Rightarrow> if x \<noteq> 0 then undefined else Src_Ports (L4Ports protocol pts)
                    |  Dst_Ports (L4Ports x pts) \<Rightarrow> if x \<noteq> 0 then undefined else Dst_Ports (L4Ports protocol pts)
                    |  MultiportPorts (L4Ports x pts) \<Rightarrow> if x \<noteq> 0 then undefined else MultiportPorts (L4Ports protocol pts)
                    |  Prot _ \<Rightarrow> undefined \<comment> \<open>there should be no more match on the protocol if it was parsed from an iptables-save line\<close>
                    | m \<Rightarrow> m
       )"
 
   lemma "fill_l4_protocol_raw TCP [Neg (Dst (IpAddrNetmask (ipv4addr_of_dotdecimal (127, 0, 0, 0)) 8)), Pos (Src_Ports (L4Ports 0 [(22,22)]))] =
           [Neg (Dst (IpAddrNetmask 0x7F000000 8)), Pos (Src_Ports (L4Ports 6 [(0x16, 0x16)]))]" by eval
 
   fun fill_l4_protocol
     :: "'i::len common_primitive negation_type list \<Rightarrow> 'i::len common_primitive negation_type list"
   where
     "fill_l4_protocol [] = []" |
     "fill_l4_protocol (Pos (Prot (Proto protocol)) # ms) = Pos (Prot (Proto protocol)) # fill_l4_protocol_raw protocol ms" |
     "fill_l4_protocol (Pos (Src_Ports _) # _) = undefined" | (*need to find proto first*)
     "fill_l4_protocol (Pos (Dst_Ports _) # _) = undefined" |
     "fill_l4_protocol (Pos (MultiportPorts _) # _) = undefined" |
     "fill_l4_protocol (Neg (Src_Ports _) # _) = undefined" |
     "fill_l4_protocol (Neg (Dst_Ports _) # _) = undefined" |
     "fill_l4_protocol (Neg (MultiportPorts _) # _) = undefined" |
     "fill_l4_protocol (m # ms) = m # fill_l4_protocol ms"
 
   lemma "fill_l4_protocol [ Neg (Dst (IpAddrNetmask (ipv4addr_of_dotdecimal (127, 0, 0, 0)) 8))
                                 , Neg (Prot (Proto UDP))
                                 , Pos (Src (IpAddrNetmask (ipv4addr_of_dotdecimal (127, 0, 0, 0)) 8))
                                 , Pos (Prot (Proto TCP))
                                 , Pos (Extra ''foo'')
                                 , Pos (Src_Ports (L4Ports 0 [(22,22)]))
                                 , Neg (Extra ''Bar'')] =
   [ Neg (Dst (IpAddrNetmask 0x7F000000 8))
   , Neg (Prot (Proto UDP))
   , Pos (Src (IpAddrNetmask 0x7F000000 8))
   , Pos (Prot (Proto TCP))
   , Pos (Extra ''foo'')
   , Pos (Src_Ports (L4Ports TCP [(0x16, 0x16)]))
   , Neg (Extra ''Bar'')]" by eval
 end
 
 
 
 (*currently unused and unverifed. may be needed for future use*)
 definition prefix_to_strange_inverse_cisco_mask:: "nat \<Rightarrow> (nat \<times> nat \<times> nat \<times> nat)" where
  "prefix_to_strange_inverse_cisco_mask n \<equiv> dotdecimal_of_ipv4addr (Bit_Operations.not (mask n << 32 - n))"
 
 lemma "prefix_to_strange_inverse_cisco_mask 8 = (0, 255, 255, 255)" by eval
 lemma "prefix_to_strange_inverse_cisco_mask 16 = (0, 0, 255, 255)" by eval
 lemma "prefix_to_strange_inverse_cisco_mask 24 = (0, 0, 0, 255)" by eval
 lemma "prefix_to_strange_inverse_cisco_mask 32 = (0, 0, 0, 0)" by eval
 
 
 end
diff --git a/thys/Mersenne_Primes/Lucas_Lehmer_Code.thy b/thys/Mersenne_Primes/Lucas_Lehmer_Code.thy
--- a/thys/Mersenne_Primes/Lucas_Lehmer_Code.thy
+++ b/thys/Mersenne_Primes/Lucas_Lehmer_Code.thy
@@ -1,268 +1,268 @@
 section \<open>Efficient code for testing Mersenne primes\<close>
 theory Lucas_Lehmer_Code
 imports
   Lucas_Lehmer
   "HOL-Library.Code_Target_Numeral"
-  "Native_Word.Code_Target_Bits_Int"
+  "Native_Word.Code_Target_Int_Bit"
 begin
 
 subsection \<open>Efficient computation of remainders modulo a Mersenne number\<close>
 
 text \<open>
   We have $k = k\ \text{mod}\ 2^n + k\ \text{div}\ 2^n\ \ (\text{mod}\ (2^n - 1))$,
   and $k\ \text{mod}\ 2^n = k\, \&\, (2^n - 1)$ and $k\ \text{div}\ 2^n = k \gg n$.
   Therefore, we can reduce $k$ modulo $2^n - 1$ using only bitwise operations, addition, and
   bit shifts. 
 \<close>
 lemma cong_mersenne_number_int:
   fixes k :: int
   shows "[k mod 2 ^ n + k div 2 ^ n = k] (mod (2 ^ n - 1))"
 proof -
   have "k = (2 ^ n - 1 + 1) * (k div 2 ^ n) + (k mod 2 ^ n)"
     by simp
   also have "[\<dots> = (0 + 1) * (k div 2 ^ n) + (k mod 2 ^ n)] (mod (2 ^ n - 1))"
     by (intro cong_add cong_mult cong_refl) (auto simp: cong_def)
   finally show ?thesis by (simp add: cong_sym add_ac)
 qed
 
 text \<open>
   We encapsulate a single reduction step in the following operation. Note, however,
   that the result is not, in general, the same as $k\ \text{mod}\ (2^n - 1)$. Multiple 
   reductions might be required in order to reduce it below $2^n$, and a multiple of $2 ^ n - 1$
   can be reduced to $2 ^ n - 1$, which is invariant to further reduction steps.
 \<close>
 definition mersenne_mod :: "int \<Rightarrow> nat \<Rightarrow> int" where
   "mersenne_mod k n = k mod 2 ^ n + k div 2 ^ n"
 
 lemma mersenne_mod_code [code]:
   "mersenne_mod k n = take_bit n k + drop_bit n k"
   by (simp add: mersenne_mod_def flip: take_bit_eq_mod drop_bit_eq_div)
 
 lemma cong_mersenne_mod: "[mersenne_mod k n = k] (mod (2 ^ n - 1))"
   unfolding mersenne_mod_def by (rule cong_mersenne_number_int)
 
 lemma mersenne_mod_nonneg [simp]: "k \<ge> 0 \<Longrightarrow> mersenne_mod k n \<ge> 0"
   unfolding mersenne_mod_def by (intro add_nonneg_nonneg) (simp_all add: pos_imp_zdiv_nonneg_iff)
 
 lemma mersenne_mod_less:
   assumes "k \<le> 2 ^ m" "m \<ge> n"
   shows   "mersenne_mod k n < 2 ^ n + 2 ^ (m - n)"
 proof -
   have "mersenne_mod k n = k mod 2 ^ n + k div 2 ^ n"
     by (simp add: mersenne_mod_def)
   also have "k mod 2 ^ n < 2 ^ n"
     by simp
   also {
     have "k div 2 ^ n * 2 ^ n + 0 \<le> k div 2 ^ n * 2 ^ n + k mod (2 ^ n)"
       by (intro add_mono) auto
     also have "\<dots> = k"
       by (subst mult.commute) auto
     also have "\<dots> \<le> 2 ^ m"
       using assms by simp
     also have "\<dots> = 2 ^ (m - n) * 2 ^ n"
       using assms by (simp flip: power_add)
     finally have "k div 2 ^ n \<le> 2 ^ (m - n)"
       by simp
   }
   finally show ?thesis by simp
 qed
 
 lemma mersenne_mod_less':
   assumes "k \<le> 5 * 2 ^ n"
   shows   "mersenne_mod k n < 2 ^ n + 5"
 proof -
   have "mersenne_mod k n = k mod 2 ^ n + k div 2 ^ n"
     by (simp add: mersenne_mod_def)
   also have "k mod 2 ^ n < 2 ^ n"
     by simp
   also {
     have "k div 2 ^ n * 2 ^ n + 0 \<le> k div 2 ^ n * 2 ^ n + k mod (2 ^ n)"
       by (intro add_mono) auto
     also have "\<dots> = k"
       by (subst mult.commute) auto
     also have "\<dots> \<le> 5 * 2 ^ n"
       using assms by simp
     finally have "k div 2 ^ n \<le> 5"
       by simp
   }
   finally show ?thesis by simp
 qed
 
 text \<open>
   It turns out that for our use case, a single reduction is not enough to reduce
   the number in question enough (or at least I was unable to prove that it is). We
   therefore perform two reduction steps, which is enough to guarantee that our numbers
   are below $2^n + 4$ before and after every step in the Lucas--Lehmer sequence.
 
   Whether one or two reductions are performed is not very important anyway, since the
   dominant step is the squaring anyway.
 \<close>
 definition mersenne_mod2 :: "int \<Rightarrow> nat \<Rightarrow> int" where
   "mersenne_mod2 k n = mersenne_mod (mersenne_mod k n) n"
 
 lemma cong_mersenne_mod2: "[mersenne_mod2 k n = k] (mod (2 ^ n - 1))"
   unfolding mersenne_mod2_def by (rule cong_trans) (rule cong_mersenne_mod)+
 
 lemma mersenne_mod2_nonneg [simp]: "k \<ge> 0 \<Longrightarrow> mersenne_mod2 k n \<ge> 0"
   unfolding mersenne_mod2_def by simp
 
 lemma mersenne_mod2_less:
   assumes "n > 2" and "k \<le> 2 ^ (2 * n + 2)"
   shows   "mersenne_mod2 k n < 2 ^ n + 5"
 proof -
   from assms have "2 ^ 3 \<le> (2 ^ n :: int)"
     by (intro power_increasing) auto
   hence "2 ^ n \<ge> (8 :: int)" by simp
   have "mersenne_mod k n < 2 ^ n + 2 ^ (2 * n + 2 - n)"
     by (rule mersenne_mod_less) (use assms in auto)
   also have "\<dots> \<le> 5 * 2 ^ n"
     by (simp add: power_add)
   finally have "mersenne_mod (mersenne_mod k n) n < 2 ^ n + 5"
     by (intro mersenne_mod_less') auto
   thus ?thesis by (simp add: mersenne_mod2_def)
 qed
 
 text \<open>
   Since we subtract 2 at one point, the intermediate results can become negative. This
   is not a problem since our reduction modulo $2 ^ p - 1$ happens to make them positive again
   immediately.
 \<close>
 lemma mersenne_mod_nonneg_strong:
   assumes "a > -(2 ^ p) + 1"
   shows   "mersenne_mod a p \<ge> 0"
 proof (cases "a < 0")
   case True
   have "eucl_rel_int a (2 ^ p) (- 1, a + 2 ^ p)"
     using assms True by (auto simp: eucl_rel_int_iff)
   hence "a div 2 ^ p = -1" and "a mod 2 ^ p = a + 2 ^ p"
     by (simp_all add: div_int_unique mod_int_unique) 
   hence "mersenne_mod a p = a + 2 ^ p - 1"
     by (simp add: mersenne_mod_def)
   also have "\<dots> > 0" using assms by simp
   finally show ?thesis by simp
 qed auto
 
 lemma mersenne_mod2_nonneg_strong:
   assumes "a > -(2 ^ p) + 1"
   shows   "mersenne_mod2 a p \<ge> 0"
   unfolding mersenne_mod2_def 
   by (rule mersenne_mod_nonneg, rule mersenne_mod_nonneg_strong) (use assms in auto)
 
 
 subsection \<open>Efficient code for the Lucas--Lehmer sequence\<close>
 
 primrec gen_lucas_lehmer_sequence'' :: "nat \<Rightarrow> int \<Rightarrow> nat \<Rightarrow> int" where
   "gen_lucas_lehmer_sequence'' p a 0 = a"
 | "gen_lucas_lehmer_sequence'' p a (Suc n) =
      gen_lucas_lehmer_sequence'' p (mersenne_mod2 (a ^ 2 - 2) p) n"
 
 lemma gen_lucas_lehmer_sequence''_correct:
   assumes "[a = a'] (mod (2 ^ p - 1))"
   shows   "[gen_lucas_lehmer_sequence'' p a n = gen_lucas_lehmer_sequence a' n] (mod (2 ^ p - 1))"
   using assms
 proof (induction n arbitrary: a a')
   case (Suc n)
   have "[mersenne_mod2 (a ^ 2 - 2) p = a ^ 2 - 2] (mod (2 ^ p - 1))"
     by (rule cong_mersenne_mod2)
   also have "[a ^ 2 - 2 = a' ^ 2 - 2] (mod (2 ^ p - 1))"
     by (intro cong_pow cong_diff Suc.prems cong_refl)
   finally have "[gen_lucas_lehmer_sequence'' p (mersenne_mod2 (a\<^sup>2 - 2) p) n =
                  gen_lucas_lehmer_sequence (a'\<^sup>2 - 2) n] (mod 2 ^ p - 1)"
     by (rule Suc.IH)
   thus ?case
     by (auto simp del: gen_lucas_lehmer_sequence.simps simp: gen_lucas_lehmer_sequence_Suc')
 qed auto
 
 lemma gen_lucas_lehmer_sequence''_bounds:
   assumes "a \<ge> 0" "a < 2 ^ p + 5" "p > 2"
   shows   "gen_lucas_lehmer_sequence'' p a n \<in> {0..<2 ^ p + 5}"
   using assms 
 proof (induction n arbitrary: a)
   case (Suc n)
   from Suc.prems have "a ^ 2 < (2 ^ p + 5) ^ 2"
     by (intro power_strict_mono Suc.prems) auto
   also have "\<dots> \<le> (2 ^ (p + 1)) ^ 2"
     using power_increasing[of 3 p "2 :: int"] \<open>p > 2\<close> by (intro power_mono) auto
   finally have "a ^ 2 - 2 < 2 ^ (2 * p + 2)"
     by (simp flip: power_mult mult_ac)
   moreover {
     from \<open>p > 2\<close> have "(2 ^ p) \<ge> (2 ^ 3 :: int)"
       by (intro power_increasing) auto
     hence "-(2 ^ p) + 1 < (-2 :: int)"
       by simp
     also have "-2 \<le> a ^ 2 - 2"
       by simp
     finally have "mersenne_mod2 (a ^ 2 - 2) p \<ge> 0"
       by (rule mersenne_mod2_nonneg_strong)
   }
   ultimately have "gen_lucas_lehmer_sequence'' p (mersenne_mod2 (a\<^sup>2 - 2) p) n \<in> {0..<2 ^ p + 5}"
     using \<open>p > 2\<close> by (intro Suc.IH mersenne_mod2_less) auto
   thus ?case by simp
 qed auto
 
 
 subsection \<open>Code for the Lucas--Lehmer test\<close>
 
 lemmas [code del] = lucas_lehmer_test_code_arithmetic
 
 lemma lucas_lehmer_test_code [code]:
   "lucas_lehmer_test p =
      (2 < p \<and> (let x = gen_lucas_lehmer_sequence'' p 4 (p - 2) in x = 0 \<or> x = (push_bit p 1) - 1))"
   unfolding lucas_lehmer_test_def
 proof (rule conj_cong)
   assume "p > 2"
   define x where "x = gen_lucas_lehmer_sequence'' p 4 (p - 2)"
   from \<open>p > 2\<close> have "2 ^ 3 \<le> (2 ^ p :: int)" by (intro power_increasing) auto
   hence "2 ^ p \<ge> (8 :: int)" by simp
   hence bounds: "x \<in> {0..<2 ^ p + 5}"
     unfolding x_def using \<open>p > 2\<close> by (intro gen_lucas_lehmer_sequence''_bounds) auto
   have "2 ^ p - 1 dvd gen_lucas_lehmer_sequence 4 (p - 2) \<longleftrightarrow> 2 ^ p - 1 dvd x"
     unfolding x_def by (intro cong_dvd_iff cong_sym[OF gen_lucas_lehmer_sequence''_correct]) auto
   also have "\<dots> \<longleftrightarrow> x \<in> {0, 2 ^ p - 1}"
   proof
     assume "2 ^ p - 1 dvd x"
     then obtain k where k: "x = (2 ^ p - 1) * k" by auto
     have "k \<ge> 0" using bounds \<open>2 ^ p \<ge> 8\<close>
       by (auto simp: k zero_le_mult_iff)
     moreover {
       have "x < 2 ^ p + 5" using bounds by simp
       also have "\<dots> \<le> (2 ^ p - 1) * 2"
         using \<open>2 ^ p \<ge> 8\<close> by simp
       finally have "(2 ^ p - 1) * k < (2 ^ p - 1) * 2"
         unfolding k .
       hence "k < 2"
         by (subst (asm) mult_less_cancel_left) auto
     }
     ultimately have "k = 0 \<or> k = 1" by auto
     thus "x \<in> {0, 2 ^ p - 1}"
       using k by auto
   qed auto
   finally show "(2 ^ p - 1 dvd gen_lucas_lehmer_sequence 4 (p - 2)) =
                 ((let x = x in x = 0 \<or> x = (push_bit p 1) - 1))"
     by (simp add: Let_def push_bit_eq_mult)
 qed auto
 
 
 subsection \<open>Examples\<close>
 
 text \<open>
   Note that for some reason, the clever bit-arithmetic version of the Lucas--Lehmer test is 
   actually much slower than the one using integer arithmetic when using PolyML, and even more so 
   when using the built-in evaluator in Isabelle (which also uses PolyML with a slightly different 
   setup).
 
   I do not quite know why this is the case, but it is likely because of inefficient implementations
   of bit arithmetic operations in PolyML and/or the code generator setup for it.
 
   When running with GHC, the bit-arithmetic version is \<^emph>\<open>much\<close> faster.
 \<close>
 
 value "filter mersenne_prime [0..<100]"
 
 lemma "prime (2 ^ 521 - 1 :: nat)"
   by (subst lucas_lehmer_correct') eval
 
 lemma "prime (2 ^ 4253 - 1 :: nat)"
   by (subst lucas_lehmer_correct') eval
 
 end
\ No newline at end of file
diff --git a/thys/Native_Word/Code_Int_Integer_Conversion.thy b/thys/Native_Word/Code_Int_Integer_Conversion.thy
--- a/thys/Native_Word/Code_Int_Integer_Conversion.thy
+++ b/thys/Native_Word/Code_Int_Integer_Conversion.thy
@@ -1,29 +1,29 @@
 (*  Title:      Code_Int_Integer_Conversion.thy
     Author:     Andreas Lochbihler, ETH Zurich
 *)
 
 chapter \<open>A special case of a conversion.\<close>
 
 theory Code_Int_Integer_Conversion
 imports
   Main
 begin
 
 text \<open>
   Use this function to convert numeral @{typ integer}s quickly into @{typ int}s.
   By default, it works only for symbolic evaluation; normally generated code raises
-  an exception at run-time. If theory \<open>Code_Target_Bits_Int\<close> is imported,
+  an exception at run-time. If theory \<^text>\<open>Code_Target_Int_Bit\<close> is imported,
   it works again, because then @{typ int} is implemented in terms of @{typ integer}
   even for symbolic evaluation.
 \<close>
 
 definition int_of_integer_symbolic :: "integer \<Rightarrow> int"
   where "int_of_integer_symbolic = int_of_integer"
 
 lemma int_of_integer_symbolic_aux_code [code nbe]:
   "int_of_integer_symbolic 0 = 0"
   "int_of_integer_symbolic (Code_Numeral.Pos n) = Int.Pos n"
   "int_of_integer_symbolic (Code_Numeral.Neg n) = Int.Neg n"
   by (simp_all add: int_of_integer_symbolic_def)
 
 end
diff --git a/thys/Native_Word/Code_Symbolic_Bits_Int.thy b/thys/Native_Word/Code_Symbolic_Int_Bit.thy
rename from thys/Native_Word/Code_Symbolic_Bits_Int.thy
rename to thys/Native_Word/Code_Symbolic_Int_Bit.thy
--- a/thys/Native_Word/Code_Symbolic_Bits_Int.thy
+++ b/thys/Native_Word/Code_Symbolic_Int_Bit.thy
@@ -1,130 +1,130 @@
-(*  Title:      Code_Symbolic_Bits_Int.thy
+(*  Title:      Code_Symbolic_Int_Bit.thy
     Author:     Andreas Lochbihler, ETH Zurich
 *)
 
 chapter \<open>Symbolic implementation of bit operations on int\<close>
 
-theory Code_Symbolic_Bits_Int
+theory Code_Symbolic_Int_Bit
 imports
   "Word_Lib.Least_significant_bit"
   "Word_Lib.Generic_set_bit"
   "Word_Lib.Bit_Comprehension"
 begin
 
 section \<open>Implementations of bit operations on \<^typ>\<open>int\<close> operating on symbolic representation\<close>
 
 context
   includes bit_operations_syntax
 begin
 
 lemma test_bit_int_code [code]:
   "bit (0::int)          n = False"
   "bit (Int.Neg num.One) n = True"
   "bit (Int.Pos num.One)      0 = True"
   "bit (Int.Pos (num.Bit0 m)) 0 = False"
   "bit (Int.Pos (num.Bit1 m)) 0 = True"
   "bit (Int.Neg (num.Bit0 m)) 0 = False"
   "bit (Int.Neg (num.Bit1 m)) 0 = True"
   "bit (Int.Pos num.One)      (Suc n) = False"
   "bit (Int.Pos (num.Bit0 m)) (Suc n) = bit (Int.Pos m) n"
   "bit (Int.Pos (num.Bit1 m)) (Suc n) = bit (Int.Pos m) n"
   "bit (Int.Neg (num.Bit0 m)) (Suc n) = bit (Int.Neg m) n"
   "bit (Int.Neg (num.Bit1 m)) (Suc n) = bit (Int.Neg (Num.inc m)) n"
   by (simp_all add: Num.add_One bit_0 bit_Suc)
 
 lemma int_not_code [code]:
   "NOT (0 :: int) = -1"
   "NOT (Int.Pos n) = Int.Neg (Num.inc n)"
   "NOT (Int.Neg n) = Num.sub n num.One"
   by (simp_all add: Num.add_One not_int_def)
 
 lemma int_and_code [code]: fixes i j :: int shows
   "0 AND j = 0"
   "i AND 0 = 0"
   "Int.Pos n AND Int.Pos m = (case and_num n m of None \<Rightarrow> 0 | Some n' \<Rightarrow> Int.Pos n')"
   "Int.Neg n AND Int.Neg m = NOT (Num.sub n num.One OR Num.sub m num.One)"
   "Int.Pos n AND Int.Neg num.One = Int.Pos n"
   "Int.Pos n AND Int.Neg (num.Bit0 m) = Num.sub (or_not_num_neg (Num.BitM m) n) num.One"
   "Int.Pos n AND Int.Neg (num.Bit1 m) = Num.sub (or_not_num_neg (num.Bit0 m) n) num.One"
   "Int.Neg num.One AND Int.Pos m = Int.Pos m"
   "Int.Neg (num.Bit0 n) AND Int.Pos m = Num.sub (or_not_num_neg (Num.BitM n) m) num.One"
   "Int.Neg (num.Bit1 n) AND Int.Pos m = Num.sub (or_not_num_neg (num.Bit0 n) m) num.One"
   apply (simp_all add: and_num_eq_None_iff [where ?'a = int] and_num_eq_Some_iff [where ?'a = int]
     split: option.split)
      apply (simp_all only: sub_one_eq_not_neg numeral_or_not_num_eq minus_minus and_not_numerals
        bit.de_Morgan_disj bit.double_compl and_not_num_eq_None_iff and_not_num_eq_Some_iff ac_simps)
      apply auto
   done
 
 lemma int_or_code [code]: fixes i j :: int shows
   "0 OR j = j"
   "i OR 0 = i"
   "Int.Pos n OR Int.Pos m = Int.Pos (or_num n m)"
   "Int.Neg n OR Int.Neg m = NOT (Num.sub n num.One AND Num.sub m num.One)"
   "Int.Pos n OR Int.Neg num.One = Int.Neg num.One"
   "Int.Pos n OR Int.Neg (num.Bit0 m) = (case and_not_num (Num.BitM m) n of None \<Rightarrow> -1 | Some n' \<Rightarrow> Int.Neg (Num.inc n'))"
   "Int.Pos n OR Int.Neg (num.Bit1 m) = (case and_not_num (num.Bit0 m) n of None \<Rightarrow> -1 | Some n' \<Rightarrow> Int.Neg (Num.inc n'))"
   "Int.Neg num.One OR Int.Pos m = Int.Neg num.One"
   "Int.Neg (num.Bit0 n) OR Int.Pos m = (case and_not_num (Num.BitM n) m of None \<Rightarrow> -1 | Some n' \<Rightarrow> Int.Neg (Num.inc n'))"
   "Int.Neg (num.Bit1 n) OR Int.Pos m = (case and_not_num (num.Bit0 n) m of None \<Rightarrow> -1 | Some n' \<Rightarrow> Int.Neg (Num.inc n'))"
   apply (auto simp add: numeral_or_num_eq split: option.splits)
          apply (simp_all only: and_not_num_eq_None_iff and_not_num_eq_Some_iff and_not_numerals
            numeral_or_not_num_eq or_int_def bit.double_compl ac_simps flip: numeral_eq_iff [where ?'a = int])
          apply simp_all
   done
 
 lemma int_xor_code [code]: fixes i j :: int shows
   "0 XOR j = j"
   "i XOR 0 = i"
   "Int.Pos n XOR Int.Pos m = (case xor_num n m of None \<Rightarrow> 0 | Some n' \<Rightarrow> Int.Pos n')"
   "Int.Neg n XOR Int.Neg m = Num.sub n num.One XOR Num.sub m num.One"
   "Int.Neg n XOR Int.Pos m = NOT (Num.sub n num.One XOR Int.Pos m)"
   "Int.Pos n XOR Int.Neg m = NOT (Int.Pos n XOR Num.sub m num.One)"
   by (simp_all add: xor_num_eq_None_iff [where ?'a = int] xor_num_eq_Some_iff [where ?'a = int] split: option.split)
 
 lemma bin_rest_code: "i div 2 = drop_bit 1 i" for i :: int
   by (simp add: drop_bit_eq_div)
   
 lemma set_bits_code [code]: 
   "set_bits = Code.abort (STR ''set_bits is unsupported on type int'') (\<lambda>_. set_bits :: _ \<Rightarrow> int)"
   by simp
 
 lemma fixes i :: int 
   shows int_set_bit_True_conv_OR [code]: "Generic_set_bit.set_bit i n True = i OR push_bit n 1"
   and int_set_bit_False_conv_NAND [code]: "Generic_set_bit.set_bit i n False = i AND NOT (push_bit n 1)"
   and int_set_bit_conv_ops: "Generic_set_bit.set_bit i n b = (if b then i OR (push_bit n 1) else i AND NOT (push_bit n 1))"
   by (simp_all add: bit_eq_iff) (auto simp add: bit_simps)
 
 declare [[code drop: \<open>drop_bit :: nat \<Rightarrow> int \<Rightarrow> int\<close>]]
 
 lemma drop_bit_int_code [code]: fixes i :: int shows
   "drop_bit 0 i = i"
   "drop_bit (Suc n) 0 = (0 :: int)"
   "drop_bit (Suc n) (Int.Pos num.One) = 0"
   "drop_bit (Suc n) (Int.Pos (num.Bit0 m)) = drop_bit n (Int.Pos m)"
   "drop_bit (Suc n) (Int.Pos (num.Bit1 m)) = drop_bit n (Int.Pos m)"
   "drop_bit (Suc n) (Int.Neg num.One) = - 1"
   "drop_bit (Suc n) (Int.Neg (num.Bit0 m)) = drop_bit n (Int.Neg m)"
   "drop_bit (Suc n) (Int.Neg (num.Bit1 m)) = drop_bit n (Int.Neg (Num.inc m))"
   by (simp_all add: drop_bit_Suc add_One)
 
 declare [[code drop: \<open>push_bit :: nat \<Rightarrow> int \<Rightarrow> int\<close>]]
 
 lemma push_bit_int_code [code]:
   "push_bit 0 i = i"
   "push_bit (Suc n) i = push_bit n (Int.dup i)"
   by (simp_all add: ac_simps)
 
 lemma int_lsb_code [code]:
   "lsb (0 :: int) = False"
   "lsb (Int.Pos num.One) = True"
   "lsb (Int.Pos (num.Bit0 w)) = False"
   "lsb (Int.Pos (num.Bit1 w)) = True"
   "lsb (Int.Neg num.One) = True"
   "lsb (Int.Neg (num.Bit0 w)) = False"
   "lsb (Int.Neg (num.Bit1 w)) = True"
   by simp_all
 
 end
 
 end
diff --git a/thys/Native_Word/Code_Target_Bits_Int.thy b/thys/Native_Word/Code_Target_Int_Bit.thy
rename from thys/Native_Word/Code_Target_Bits_Int.thy
rename to thys/Native_Word/Code_Target_Int_Bit.thy
--- a/thys/Native_Word/Code_Target_Bits_Int.thy
+++ b/thys/Native_Word/Code_Target_Int_Bit.thy
@@ -1,100 +1,100 @@
-(*  Title:      Code_Target_Bits_Int.thy
+(*  Title:      Code_Target_Int_Bit.thy
     Author:     Andreas Lochbihler, ETH Zurich
 *)
 
 chapter \<open>Implementation of bit operations on int by target language operations\<close>
 
-theory Code_Target_Bits_Int
+theory Code_Target_Int_Bit
   imports
-  Bits_Integer
+  Code_Target_Integer_Bit
   "HOL-Library.Code_Target_Int"
 begin
 
 context
   includes bit_operations_syntax
 begin
 
 declare [[code drop:
   "(AND) :: int \<Rightarrow> _" "(OR) :: int \<Rightarrow> _" "(XOR) :: int \<Rightarrow> _" "NOT :: int \<Rightarrow> _"
   "lsb :: int \<Rightarrow> _" "set_bit :: int \<Rightarrow> _" "bit :: int \<Rightarrow> _"
   "push_bit :: _ \<Rightarrow> int \<Rightarrow> _" "drop_bit :: _ \<Rightarrow> int \<Rightarrow> _"
   int_of_integer_symbolic
   ]]
 
 lemma [code_unfold]:
   \<open>of_bool (odd i) = i AND 1\<close> for i :: int
   by (simp add: and_one_eq mod2_eq_if)
 
 lemma [code_unfold]:
   \<open>bit x n \<longleftrightarrow> x AND (push_bit n 1) \<noteq> 0\<close> for x :: int
   by (fact bit_iff_and_push_bit_not_eq_0)
 
 context
 includes integer.lifting
 begin
 
 lemma bit_int_code [code]:
   "bit (int_of_integer x) n = bit x n"
   by transfer simp
 
 lemma and_int_code [code]:
   "int_of_integer i AND int_of_integer j = int_of_integer (i AND j)"
   by transfer simp
 
 lemma or_int_code [code]:
   "int_of_integer i OR int_of_integer j = int_of_integer (i OR j)"
   by transfer simp
 
 lemma xor_int_code [code]:
   "int_of_integer i XOR int_of_integer j = int_of_integer (i XOR j)"
   by transfer simp
 
 lemma not_int_code [code]:
   "NOT (int_of_integer i) = int_of_integer (NOT i)"
   by transfer simp
 
 lemma push_bit_int_code [code]:
   \<open>push_bit n (int_of_integer x) = int_of_integer (push_bit n x)\<close>
   by transfer simp
 
 lemma drop_bit_int_code [code]:
   \<open>drop_bit n (int_of_integer x) = int_of_integer (drop_bit n x)\<close>
   by transfer simp
 
 lemma take_bit_int_code [code]:
   \<open>take_bit n (int_of_integer x) = int_of_integer (take_bit n x)\<close>
   by transfer simp
 
 lemma lsb_int_code [code]:
   "lsb (int_of_integer x) = lsb x"
   by transfer simp
 
 lemma set_bit_int_code [code]:
   "set_bit (int_of_integer x) n b = int_of_integer (set_bit x n b)"
   by transfer simp
 
 lemma int_of_integer_symbolic_code [code]:
   "int_of_integer_symbolic = int_of_integer"
   by (simp add: int_of_integer_symbolic_def)
 
 context
 begin
 
 qualified definition even :: \<open>int \<Rightarrow> bool\<close>
   where [code_abbrev]: \<open>even = Parity.even\<close>
 
 end
 
 lemma [code]:
-  \<open>Code_Target_Bits_Int.even i \<longleftrightarrow> i AND 1 = 0\<close>
-  by (simp add: Code_Target_Bits_Int.even_def even_iff_mod_2_eq_zero and_one_eq)
+  \<open>Code_Target_Int_Bit.even i \<longleftrightarrow> i AND 1 = 0\<close>
+  by (simp add: Code_Target_Int_Bit.even_def even_iff_mod_2_eq_zero and_one_eq)
 
 lemma bin_rest_code:
   "int_of_integer i div 2 = int_of_integer (bin_rest_integer i)"
   by transfer simp
 
 end
 
 end
 
 end
diff --git a/thys/Native_Word/Bits_Integer.thy b/thys/Native_Word/Code_Target_Integer_Bit.thy
rename from thys/Native_Word/Bits_Integer.thy
rename to thys/Native_Word/Code_Target_Integer_Bit.thy
--- a/thys/Native_Word/Bits_Integer.thy
+++ b/thys/Native_Word/Code_Target_Integer_Bit.thy
@@ -1,680 +1,682 @@
-(*  Title:      Bits_Integer.thy
+(*  Title:      Code_Target_Integer_Bit.thy
     Author:     Andreas Lochbihler, ETH Zurich
 *)
 
 chapter \<open>Bit operations for target language integers\<close>
 
-theory Bits_Integer imports
+theory Code_Target_Integer_Bit imports
   "Word_Lib.Bit_Comprehension"
   Code_Int_Integer_Conversion
-  Code_Symbolic_Bits_Int
+  Code_Symbolic_Int_Bit
 begin
 
+text \<open>TODO: separate\<close>
+
 lemmas [transfer_rule] =
   identity_quotient
   fun_quotient
   Quotient_integer[folded integer.pcr_cr_eq]
 
 lemma undefined_transfer:
   assumes "Quotient R Abs Rep T"
   shows "T (Rep undefined) undefined"
 using assms unfolding Quotient_alt_def by blast
 
 bundle undefined_transfer = undefined_transfer[transfer_rule]
 
 section \<open>More lemmas about @{typ integer}s\<close>
 
 context
 includes integer.lifting
 begin
 
 lemma bitval_integer_transfer [transfer_rule]:
   "(rel_fun (=) pcr_integer) of_bool of_bool"
 by(auto simp add: of_bool_def integer.pcr_cr_eq cr_integer_def)
 
 lemma integer_of_nat_less_0_conv [simp]: "\<not> integer_of_nat n < 0"
 by(transfer) simp
 
 lemma int_of_integer_pow: "int_of_integer (x ^ n) = int_of_integer x ^ n"
 by(induct n) simp_all
 
 lemma pow_integer_transfer [transfer_rule]:
   "(rel_fun pcr_integer (rel_fun (=) pcr_integer)) (^) (^)"
 by(auto 4 3 simp add: integer.pcr_cr_eq cr_integer_def int_of_integer_pow)
 
 lemma sub1_lt_0_iff [simp]: "Code_Numeral.sub n num.One < 0 \<longleftrightarrow> False"
 by(cases n)(simp_all add: Code_Numeral.sub_code)
 
 lemma nat_of_integer_numeral [simp]: "nat_of_integer (numeral n) = numeral n"
 by transfer simp
 
 lemma nat_of_integer_sub1_conv_pred_numeral [simp]:
   "nat_of_integer (Code_Numeral.sub n num.One) = pred_numeral n"
 by(cases n)(simp_all add: Code_Numeral.sub_code)
 
 lemma nat_of_integer_1 [simp]: "nat_of_integer 1 = 1"
 by transfer simp
 
 lemma dup_1 [simp]: "Code_Numeral.dup 1 = 2"
 by transfer simp
 
 
 section \<open>Bit operations on @{typ integer}\<close>
 
 text \<open>Bit operations on @{typ integer} are the same as on @{typ int}\<close>
 
 lift_definition bin_rest_integer :: "integer \<Rightarrow> integer" is \<open>\<lambda>k . k div 2\<close> .
 lift_definition bin_last_integer :: "integer \<Rightarrow> bool" is odd .
 lift_definition Bit_integer :: "integer \<Rightarrow> bool \<Rightarrow> integer" is \<open>\<lambda>k b. of_bool b + 2 * k\<close> .
 
 end
 
 instantiation integer :: lsb begin
 context includes integer.lifting begin
 
 lift_definition lsb_integer :: "integer \<Rightarrow> bool" is lsb .
 
 instance
   by (standard; transfer) (fact lsb_odd)
 
 end
 end
 
 instantiation integer :: msb begin
 context includes integer.lifting begin
 
 lift_definition msb_integer :: "integer \<Rightarrow> bool" is msb .
 
 instance ..
 
 end
 end
 
 instantiation integer :: set_bit begin
 context includes integer.lifting begin
 
 lift_definition set_bit_integer :: "integer \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> integer" is set_bit .
 
 instance
   apply standard
   apply transfer
   apply (simp add: bit_simps)
   done
 
 end
 end
 
 abbreviation (input) wf_set_bits_integer
 where "wf_set_bits_integer \<equiv> wf_set_bits_int"
 
 
 section \<open>Target language implementations\<close>
 
 text \<open>
   Unfortunately, this is not straightforward,
   because these API functions have different signatures and preconditions on the parameters:
 
   \begin{description}
   \item[Standard ML] Shifts in IntInf are given as word, but not IntInf.
   \item[Haskell] In the Data.Bits.Bits type class, shifts and bit indices are given as Int rather than Integer.
   \end{description}
 
   Additional constants take only parameters of type @{typ integer} rather than @{typ nat}
   and check the preconditions as far as possible (e.g., being non-negative) in a portable way.
   Manual implementations inside code\_printing perform the remaining range checks and convert
   these @{typ integer}s into the right type.
 
   For normalisation by evaluation, we derive custom code equations, because NBE
   does not know these code\_printing serialisations and would otherwise loop.
 \<close>
 
 code_identifier code_module Bits_Integer \<rightharpoonup>
   (SML) Bits_Int and (OCaml) Bits_Int and (Haskell) Bits_Int and (Scala) Bits_Int
 
 code_printing code_module Bits_Integer \<rightharpoonup> (SML)
 \<open>structure Bits_Integer : sig
   val set_bit : IntInf.int -> IntInf.int -> bool -> IntInf.int
   val shiftl : IntInf.int -> IntInf.int -> IntInf.int
   val shiftr : IntInf.int -> IntInf.int -> IntInf.int
   val test_bit : IntInf.int -> IntInf.int -> bool
 end = struct
 
 val maxWord = IntInf.pow (2, Word.wordSize);
 
 fun set_bit x n b =
   if n < maxWord then
     if b then IntInf.orb (x, IntInf.<< (1, Word.fromLargeInt (IntInf.toLarge n)))
     else IntInf.andb (x, IntInf.notb (IntInf.<< (1, Word.fromLargeInt (IntInf.toLarge n))))
   else raise (Fail ("Bit index too large: " ^ IntInf.toString n));
 
 fun shiftl x n =
   if n < maxWord then IntInf.<< (x, Word.fromLargeInt (IntInf.toLarge n))
   else raise (Fail ("Shift operand too large: " ^ IntInf.toString n));
 
 fun shiftr x n =
   if n < maxWord then IntInf.~>> (x, Word.fromLargeInt (IntInf.toLarge n))
   else raise (Fail ("Shift operand too large: " ^ IntInf.toString n));
 
 fun test_bit x n =
   if n < maxWord then IntInf.andb (x, IntInf.<< (1, Word.fromLargeInt (IntInf.toLarge n))) <> 0
   else raise (Fail ("Bit index too large: " ^ IntInf.toString n));
 
 end; (*struct Bits_Integer*)\<close>
 code_reserved SML Bits_Integer
 
 code_printing code_module Bits_Integer \<rightharpoonup> (OCaml)
 \<open>module Bits_Integer : sig
   val shiftl : Z.t -> Z.t -> Z.t
   val shiftr : Z.t -> Z.t -> Z.t
   val test_bit : Z.t -> Z.t -> bool
 end = struct
 
 (* We do not need an explicit range checks here,
    because Big_int.int_of_big_int raises Failure
    if the argument does not fit into an int. *)
 let shiftl x n = Z.shift_left x (Z.to_int n);;
 
 let shiftr x n = Z.shift_right x (Z.to_int n);;
 
 let test_bit x n =  Z.testbit x (Z.to_int n);;
 
 end;; (*struct Bits_Integer*)\<close>
 code_reserved OCaml Bits_Integer
 
 code_printing code_module Data_Bits \<rightharpoonup> (Haskell)
 \<open>
 module Data_Bits where {
 
 import qualified Data.Bits;
 
 {-
   The ...Bounded functions assume that the Integer argument for the shift
   or bit index fits into an Int, is non-negative and (for types of fixed bit width)
   less than bitSize
 -}
 
 infixl 7 .&.;
 infixl 6 `xor`;
 infixl 5 .|.;
 
 (.&.) :: Data.Bits.Bits a => a -> a -> a;
 (.&.) = (Data.Bits..&.);
 
 xor :: Data.Bits.Bits a => a -> a -> a;
 xor = Data.Bits.xor;
 
 (.|.) :: Data.Bits.Bits a => a -> a -> a;
 (.|.) = (Data.Bits..|.);
 
 complement :: Data.Bits.Bits a => a -> a;
 complement = Data.Bits.complement;
 
 testBitUnbounded :: Data.Bits.Bits a => a -> Integer -> Bool;
 testBitUnbounded x b
   | b <= toInteger (Prelude.maxBound :: Int) = Data.Bits.testBit x (fromInteger b)
   | otherwise = error ("Bit index too large: " ++ show b)
 ;
 
 testBitBounded :: Data.Bits.Bits a => a -> Integer -> Bool;
 testBitBounded x b = Data.Bits.testBit x (fromInteger b);
 
 setBitUnbounded :: Data.Bits.Bits a => a -> Integer -> Bool -> a;
 setBitUnbounded x n b
   | n <= toInteger (Prelude.maxBound :: Int) =
     if b then Data.Bits.setBit x (fromInteger n) else Data.Bits.clearBit x (fromInteger n)
   | otherwise = error ("Bit index too large: " ++ show n)
 ;
 
 setBitBounded :: Data.Bits.Bits a => a -> Integer -> Bool -> a;
 setBitBounded x n True = Data.Bits.setBit x (fromInteger n);
 setBitBounded x n False = Data.Bits.clearBit x (fromInteger n);
 
 shiftlUnbounded :: Data.Bits.Bits a => a -> Integer -> a;
 shiftlUnbounded x n
   | n <= toInteger (Prelude.maxBound :: Int) = Data.Bits.shiftL x (fromInteger n)
   | otherwise = error ("Shift operand too large: " ++ show n)
 ;
 
 shiftlBounded :: Data.Bits.Bits a => a -> Integer -> a;
 shiftlBounded x n = Data.Bits.shiftL x (fromInteger n);
 
 shiftrUnbounded :: Data.Bits.Bits a => a -> Integer -> a;
 shiftrUnbounded x n
   | n <= toInteger (Prelude.maxBound :: Int) = Data.Bits.shiftR x (fromInteger n)
   | otherwise = error ("Shift operand too large: " ++ show n)
 ;
 
 shiftrBounded :: (Ord a, Data.Bits.Bits a) => a -> Integer -> a;
 shiftrBounded x n = Data.Bits.shiftR x (fromInteger n);
 
 }\<close>
 
   and \<comment> \<open>@{theory HOL.Quickcheck_Narrowing} maps @{typ integer} to
             Haskell's Prelude.Int type instead of Integer. For compatibility
             with the Haskell target, we nevertheless provide bounded and
             unbounded functions.\<close>
   (Haskell_Quickcheck)
 \<open>
 module Data_Bits where {
 
 import qualified Data.Bits;
 
 {-
   The functions assume that the Int argument for the shift or bit index is
   non-negative and (for types of fixed bit width) less than bitSize
 -}
 
 infixl 7 .&.;
 infixl 6 `xor`;
 infixl 5 .|.;
 
 (.&.) :: Data.Bits.Bits a => a -> a -> a;
 (.&.) = (Data.Bits..&.);
 
 xor :: Data.Bits.Bits a => a -> a -> a;
 xor = Data.Bits.xor;
 
 (.|.) :: Data.Bits.Bits a => a -> a -> a;
 (.|.) = (Data.Bits..|.);
 
 complement :: Data.Bits.Bits a => a -> a;
 complement = Data.Bits.complement;
 
 testBitUnbounded :: Data.Bits.Bits a => a -> Prelude.Int -> Bool;
 testBitUnbounded = Data.Bits.testBit;
 
 testBitBounded :: Data.Bits.Bits a => a -> Prelude.Int -> Bool;
 testBitBounded = Data.Bits.testBit;
 
 setBitUnbounded :: Data.Bits.Bits a => a -> Prelude.Int -> Bool -> a;
 setBitUnbounded x n True = Data.Bits.setBit x n;
 setBitUnbounded x n False = Data.Bits.clearBit x n;
 
 setBitBounded :: Data.Bits.Bits a => a -> Prelude.Int -> Bool -> a;
 setBitBounded x n True = Data.Bits.setBit x n;
 setBitBounded x n False = Data.Bits.clearBit x n;
 
 shiftlUnbounded :: Data.Bits.Bits a => a -> Prelude.Int -> a;
 shiftlUnbounded = Data.Bits.shiftL;
 
 shiftlBounded :: Data.Bits.Bits a => a -> Prelude.Int -> a;
 shiftlBounded = Data.Bits.shiftL;
 
 shiftrUnbounded :: Data.Bits.Bits a => a -> Prelude.Int -> a;
 shiftrUnbounded = Data.Bits.shiftR;
 
 shiftrBounded :: (Ord a, Data.Bits.Bits a) => a -> Prelude.Int -> a;
 shiftrBounded = Data.Bits.shiftR;
 
 }\<close>
 code_reserved Haskell Data_Bits
 
 code_printing code_module Bits_Integer \<rightharpoonup> (Scala)
 \<open>object Bits_Integer {
 
 def setBit(x: BigInt, n: BigInt, b: Boolean) : BigInt =
   if (n.isValidInt)
     if (b)
       x.setBit(n.toInt)
     else
       x.clearBit(n.toInt)
   else
     sys.error("Bit index too large: " + n.toString)
 
 def shiftl(x: BigInt, n: BigInt) : BigInt =
   if (n.isValidInt)
     x << n.toInt
   else
     sys.error("Shift index too large: " + n.toString)
 
 def shiftr(x: BigInt, n: BigInt) : BigInt =
   if (n.isValidInt)
     x << n.toInt
   else
     sys.error("Shift index too large: " + n.toString)
 
 def testBit(x: BigInt, n: BigInt) : Boolean =
   if (n.isValidInt)
     x.testBit(n.toInt)
   else
     sys.error("Bit index too large: " + n.toString)
 
 } /* object Bits_Integer */\<close>
 
 code_printing
   constant "Bit_Operations.and :: integer \<Rightarrow> integer \<Rightarrow> integer" \<rightharpoonup>
   (SML) "IntInf.andb ((_),/ (_))" and
   (OCaml) "Z.logand" and
   (Haskell) "((Data'_Bits..&.) :: Integer -> Integer -> Integer)" and
   (Haskell_Quickcheck) "((Data'_Bits..&.) :: Prelude.Int -> Prelude.Int -> Prelude.Int)" and
   (Scala) infixl 3 "&"
 | constant "Bit_Operations.or :: integer \<Rightarrow> integer \<Rightarrow> integer" \<rightharpoonup>
   (SML) "IntInf.orb ((_),/ (_))" and
   (OCaml) "Z.logor" and
   (Haskell) "((Data'_Bits..|.) :: Integer -> Integer -> Integer)" and
   (Haskell_Quickcheck) "((Data'_Bits..|.) :: Prelude.Int -> Prelude.Int -> Prelude.Int)" and
   (Scala) infixl 1 "|"
 | constant "Bit_Operations.xor :: integer \<Rightarrow> integer \<Rightarrow> integer" \<rightharpoonup>
   (SML) "IntInf.xorb ((_),/ (_))" and
   (OCaml) "Z.logxor" and
   (Haskell) "(Data'_Bits.xor :: Integer -> Integer -> Integer)" and
   (Haskell_Quickcheck) "(Data'_Bits.xor :: Prelude.Int -> Prelude.Int -> Prelude.Int)" and
   (Scala) infixl 2 "^"
 | constant "Bit_Operations.not :: integer \<Rightarrow> integer" \<rightharpoonup>
   (SML) "IntInf.notb" and
   (OCaml) "Z.lognot" and
   (Haskell) "(Data'_Bits.complement :: Integer -> Integer)" and
   (Haskell_Quickcheck) "(Data'_Bits.complement :: Prelude.Int -> Prelude.Int)" and
   (Scala) "_.unary'_~"
 
 code_printing constant bin_rest_integer \<rightharpoonup>
   (SML) "IntInf.div ((_), 2)" and
   (OCaml) "Z.shift'_right/ _/ 1" and
   (Haskell) "(Data'_Bits.shiftrUnbounded _ 1 :: Integer)" and
   (Haskell_Quickcheck) "(Data'_Bits.shiftrUnbounded _ 1 :: Prelude.Int)" and
   (Scala) "_ >> 1"
 
 context
   includes integer.lifting bit_operations_syntax
 begin
 
 lemma bitNOT_integer_code [code]:
   fixes i :: integer shows
   "NOT i = - i - 1"
   by transfer (simp add: not_int_def)
 
 lemma bin_rest_integer_code [code nbe]:
   "bin_rest_integer i = i div 2"
   by transfer rule
 
 lemma bin_last_integer_code [code]:
   "bin_last_integer i \<longleftrightarrow> i AND 1 \<noteq> 0"
   by transfer (simp add: and_one_eq odd_iff_mod_2_eq_one)
 
 lemma bin_last_integer_nbe [code nbe]:
   "bin_last_integer i \<longleftrightarrow> i mod 2 \<noteq> 0"
   by transfer (simp add: odd_iff_mod_2_eq_one)
 
 lemma bitval_bin_last_integer [code_unfold]:
   "of_bool (bin_last_integer i) = i AND 1"
   by transfer (simp add: and_one_eq mod_2_eq_odd)
 
 end
 
 definition integer_test_bit :: "integer \<Rightarrow> integer \<Rightarrow> bool"
   where "integer_test_bit x n = (if n < 0 then undefined x n else bit x (nat_of_integer n))"
 
 declare [[code drop: \<open>bit :: integer \<Rightarrow> nat \<Rightarrow> bool\<close>]]
 
 lemma bit_integer_code [code]:
   "bit x n \<longleftrightarrow> integer_test_bit x (integer_of_nat n)"
   by (simp add: integer_test_bit_def)
 
 lemma integer_test_bit_code [code]:
   "integer_test_bit x (Code_Numeral.Neg n) = undefined x (Code_Numeral.Neg n)"
   "integer_test_bit 0 0 = False"
   "integer_test_bit 0 (Code_Numeral.Pos n) = False"
   "integer_test_bit (Code_Numeral.Pos num.One)      0 = True"
   "integer_test_bit (Code_Numeral.Pos (num.Bit0 n)) 0 = False"
   "integer_test_bit (Code_Numeral.Pos (num.Bit1 n)) 0 = True"
   "integer_test_bit (Code_Numeral.Pos num.One)      (Code_Numeral.Pos n') = False"
   "integer_test_bit (Code_Numeral.Pos (num.Bit0 n)) (Code_Numeral.Pos n') =
    integer_test_bit (Code_Numeral.Pos n) (Code_Numeral.sub n' num.One)"
   "integer_test_bit (Code_Numeral.Pos (num.Bit1 n)) (Code_Numeral.Pos n') =
    integer_test_bit (Code_Numeral.Pos n) (Code_Numeral.sub n' num.One)"
   "integer_test_bit (Code_Numeral.Neg num.One)      0 = True"
   "integer_test_bit (Code_Numeral.Neg (num.Bit0 n)) 0 = False"
   "integer_test_bit (Code_Numeral.Neg (num.Bit1 n)) 0 = True"
   "integer_test_bit (Code_Numeral.Neg num.One)      (Code_Numeral.Pos n') = True"
   "integer_test_bit (Code_Numeral.Neg (num.Bit0 n)) (Code_Numeral.Pos n') =
    integer_test_bit (Code_Numeral.Neg n) (Code_Numeral.sub n' num.One)"
   "integer_test_bit (Code_Numeral.Neg (num.Bit1 n)) (Code_Numeral.Pos n') =
    integer_test_bit (Code_Numeral.Neg (n + num.One)) (Code_Numeral.sub n' num.One)"
   by (simp_all add: integer_test_bit_def bit_integer_def bit_0 flip: bit_not_int_iff')
 
 code_printing constant integer_test_bit \<rightharpoonup>
   (SML) "Bits'_Integer.test'_bit" and
   (OCaml) "Bits'_Integer.test'_bit" and
   (Haskell) "(Data'_Bits.testBitUnbounded :: Integer -> Integer -> Bool)" and
   (Haskell_Quickcheck) "(Data'_Bits.testBitUnbounded :: Prelude.Int -> Prelude.Int -> Bool)" and
   (Scala) "Bits'_Integer.testBit"
 
 context
   includes integer.lifting bit_operations_syntax
 begin
 
 lemma lsb_integer_code [code]:
   fixes x :: integer shows
   "lsb x = bit x 0"
 by transfer(simp add: lsb_int_def)
 
 definition integer_set_bit :: "integer \<Rightarrow> integer \<Rightarrow> bool \<Rightarrow> integer"
 where [code del]: "integer_set_bit x n b = (if n < 0 then undefined x n b else set_bit x (nat_of_integer n) b)"
 
 lemma set_bit_integer_code [code]:
   "set_bit x i b = integer_set_bit x (integer_of_nat i) b"
 by(simp add: integer_set_bit_def)
 
 lemma set_bit_integer_conv_masks:
   fixes x :: integer shows
   "set_bit x i b = (if b then x OR (push_bit i 1) else x AND NOT (push_bit i 1))"
   by transfer (simp add: int_set_bit_False_conv_NAND int_set_bit_True_conv_OR)
 
 end
 
 code_printing constant integer_set_bit \<rightharpoonup>
   (SML) "Bits'_Integer.set'_bit" and
   (Haskell) "(Data'_Bits.setBitUnbounded :: Integer -> Integer -> Bool -> Integer)" and
   (Haskell_Quickcheck) "(Data'_Bits.setBitUnbounded :: Prelude.Int -> Prelude.Int -> Bool -> Prelude.Int)" and
   (Scala) "Bits'_Integer.setBit"
 
 text \<open>
   OCaml.Big\_int does not have a method for changing an individual bit, so we emulate that with masks.
   We prefer an Isabelle implementation, because this then takes care of the signs for AND and OR.
 \<close>
 
 context
   includes bit_operations_syntax
 begin
 
 lemma integer_set_bit_code [code]:
   "integer_set_bit x n b =
   (if n < 0 then undefined x n b else
    if b then x OR (push_bit (nat_of_integer n) 1)
    else x AND NOT (push_bit (nat_of_integer n) 1))"
   by (auto simp add: integer_set_bit_def not_less set_bit_eq set_bit_def unset_bit_def)
 
 end
 
 definition integer_shiftl :: "integer \<Rightarrow> integer \<Rightarrow> integer"
 where [code del]: "integer_shiftl x n = (if n < 0 then undefined x n else push_bit (nat_of_integer n) x)"
 
 declare [[code drop: \<open>push_bit :: nat \<Rightarrow> integer \<Rightarrow> integer\<close>]]
 
 lemma shiftl_integer_code [code]:
   fixes x :: integer shows
   "push_bit n x = integer_shiftl x (integer_of_nat n)"
 by(auto simp add: integer_shiftl_def)
 
 context
 includes integer.lifting
 begin
 
 lemma shiftl_integer_conv_mult_pow2:
   fixes x :: integer shows
   "push_bit n x = x * 2 ^ n"
   by (fact push_bit_eq_mult)
 
 lemma integer_shiftl_code [code]:
   "integer_shiftl x (Code_Numeral.Neg n) = undefined x (Code_Numeral.Neg n)"
   "integer_shiftl x 0 = x"
   "integer_shiftl x (Code_Numeral.Pos n) = integer_shiftl (Code_Numeral.dup x) (Code_Numeral.sub n num.One)"
   "integer_shiftl 0 (Code_Numeral.Pos n) = 0"
      apply (simp_all add: integer_shiftl_def numeral_eq_Suc)
   apply transfer
   apply (simp add: ac_simps)
   done
 
 end
 
 code_printing constant integer_shiftl \<rightharpoonup>
   (SML) "Bits'_Integer.shiftl" and
   (OCaml) "Bits'_Integer.shiftl" and
   (Haskell) "(Data'_Bits.shiftlUnbounded :: Integer -> Integer -> Integer)" and
   (Haskell_Quickcheck) "(Data'_Bits.shiftlUnbounded :: Prelude.Int -> Prelude.Int -> Prelude.Int)" and
   (Scala) "Bits'_Integer.shiftl"
 
 definition integer_shiftr :: "integer \<Rightarrow> integer \<Rightarrow> integer"
 where [code del]: "integer_shiftr x n = (if n < 0 then undefined x n else drop_bit (nat_of_integer n) x)"
 
 declare [[code drop: \<open>drop_bit :: nat \<Rightarrow> integer \<Rightarrow> integer\<close>]]
 
 lemma shiftr_integer_conv_div_pow2:
   includes integer.lifting fixes x :: integer shows
   "drop_bit n x = x div 2 ^ n"
   by (fact drop_bit_eq_div)
 
 lemma shiftr_integer_code [code]:
   fixes x :: integer shows
   "drop_bit n x = integer_shiftr x (integer_of_nat n)"
 by(auto simp add: integer_shiftr_def)
 
 code_printing constant integer_shiftr \<rightharpoonup>
   (SML) "Bits'_Integer.shiftr" and
   (OCaml) "Bits'_Integer.shiftr" and
   (Haskell) "(Data'_Bits.shiftrUnbounded :: Integer -> Integer -> Integer)" and
   (Haskell_Quickcheck) "(Data'_Bits.shiftrUnbounded :: Prelude.Int -> Prelude.Int -> Prelude.Int)" and
   (Scala) "Bits'_Integer.shiftr"
 
 lemma integer_shiftr_code [code]:
   includes integer.lifting
   shows
   "integer_shiftr x (Code_Numeral.Neg n) = undefined x (Code_Numeral.Neg n)"
   "integer_shiftr x 0 = x"
   "integer_shiftr 0 (Code_Numeral.Pos n) = 0"
   "integer_shiftr (Code_Numeral.Pos num.One) (Code_Numeral.Pos n) = 0"
   "integer_shiftr (Code_Numeral.Pos (num.Bit0 n')) (Code_Numeral.Pos n) =
    integer_shiftr (Code_Numeral.Pos n') (Code_Numeral.sub n num.One)"
   "integer_shiftr (Code_Numeral.Pos (num.Bit1 n')) (Code_Numeral.Pos n) =
    integer_shiftr (Code_Numeral.Pos n') (Code_Numeral.sub n num.One)"
   "integer_shiftr (Code_Numeral.Neg num.One) (Code_Numeral.Pos n) = -1"
   "integer_shiftr (Code_Numeral.Neg (num.Bit0 n')) (Code_Numeral.Pos n) =
    integer_shiftr (Code_Numeral.Neg n') (Code_Numeral.sub n num.One)"
   "integer_shiftr (Code_Numeral.Neg (num.Bit1 n')) (Code_Numeral.Pos n) =
    integer_shiftr (Code_Numeral.Neg (Num.inc n')) (Code_Numeral.sub n num.One)"
           apply (simp_all add: integer_shiftr_def numeral_eq_Suc drop_bit_Suc)
     apply transfer apply simp
    apply transfer apply simp
   apply transfer apply (simp add: add_One)
   done
 
 context
 includes integer.lifting
 begin
 
 lemma Bit_integer_code [code]:
   "Bit_integer i False = push_bit 1 i"
   "Bit_integer i True = push_bit 1 i + 1"
   by (transfer; simp)+
 
 lemma msb_integer_code [code]:
   "msb (x :: integer) \<longleftrightarrow> x < 0"
   by transfer (simp add: msb_int_def)
 
 end
 
 context
   includes integer.lifting natural.lifting bit_operations_syntax
 begin
 
 lemma bitAND_integer_unfold [code]:
   "x AND y =
    (if x = 0 then 0
     else if x = - 1 then y
     else Bit_integer (bin_rest_integer x AND bin_rest_integer y) (bin_last_integer x \<and> bin_last_integer y))"
   by transfer
     (auto simp add: algebra_simps
       and_int_rec [of _ \<open>_ * 2\<close>] and_int_rec [of \<open>_ * 2\<close>] and_int_rec [of \<open>1 + _ * 2\<close>]
       elim!: evenE oddE)
 
 lemma bitOR_integer_unfold [code]:
   "x OR y =
    (if x = 0 then y
     else if x = - 1 then - 1
     else Bit_integer (bin_rest_integer x OR bin_rest_integer y) (bin_last_integer x \<or> bin_last_integer y))"
   by transfer
     (auto simp add: algebra_simps
       or_int_rec [of _ \<open>_ * 2\<close>] or_int_rec [of _ \<open>1 + _ * 2\<close>] or_int_rec [of \<open>1 + _ * 2\<close>]
       elim!: evenE oddE)
 
 lemma bitXOR_integer_unfold [code]:
   "x XOR y =
    (if x = 0 then y
     else if x = - 1 then NOT y
     else Bit_integer (bin_rest_integer x XOR bin_rest_integer y)
       (\<not> bin_last_integer x \<longleftrightarrow> bin_last_integer y))"
   by transfer
     (auto simp add: algebra_simps
       xor_int_rec [of _ \<open>_ * 2\<close>] xor_int_rec [of \<open>_ * 2\<close>] xor_int_rec [of \<open>1 + _ * 2\<close>]
       elim!: evenE oddE)
 
 end
 
 
 section \<open>Test code generator setup\<close>
 
 context
   includes bit_operations_syntax
 begin
 
 definition bit_integer_test :: "bool" where
   "bit_integer_test =
   (([ -1 AND 3, 1 AND -3, 3 AND 5, -3 AND (- 5)
     , -3 OR 1, 1 OR -3, 3 OR 5, -3 OR (- 5)
     , NOT 1, NOT (- 3)
     , -1 XOR 3, 1 XOR (- 3), 3 XOR 5, -5 XOR (- 3)
     , set_bit 5 4 True, set_bit (- 5) 2 True, set_bit 5 0 False, set_bit (- 5) 1 False
     , push_bit 2 1, push_bit 3 (- 1)
     , drop_bit 3 100, drop_bit 3 (- 100)] :: integer list)
   = [ 3, 1, 1, -7
     , -3, -3, 7, -1
     , -2, 2
     , -4, -4, 6, 6
     , 21, -1, 4, -7
     , 4, -8
     , 12, -13] \<and>
     [ bit (5 :: integer) 4, bit (5 :: integer) 2, bit (-5 :: integer) 4, bit (-5 :: integer) 2
     , lsb (5 :: integer), lsb (4 :: integer), lsb (-1 :: integer), lsb (-2 :: integer),
       msb (5 :: integer), msb (0 :: integer), msb (-1 :: integer), msb (-2 :: integer)]
   = [ False, True, True, False,
       True, False, True, False,
       False, False, True, True])"
 
 export_code bit_integer_test checking SML Haskell? Haskell_Quickcheck? OCaml? Scala
 
 notepad begin
 have bit_integer_test by eval
 have bit_integer_test by normalization
 have bit_integer_test by code_simp
 end
 ML_val \<open>val true = @{code bit_integer_test}\<close>
 
 lemma "x AND y = x OR (y :: integer)"
 quickcheck[random, expect=counterexample]
 quickcheck[exhaustive, expect=counterexample]
 oops
 
 lemma "(x :: integer) AND x = x OR x"
 quickcheck[narrowing, expect=no_counterexample]
 oops
 
 lemma "(f :: integer \<Rightarrow> unit) = g"
 quickcheck[narrowing, size=3, expect=no_counterexample]
 by(simp add: fun_eq_iff)
 
 end
 
 hide_const bit_integer_test
 hide_fact bit_integer_test_def
 
 end
diff --git a/thys/Native_Word/Code_Target_Word_Base.thy b/thys/Native_Word/Code_Target_Word_Base.thy
--- a/thys/Native_Word/Code_Target_Word_Base.thy
+++ b/thys/Native_Word/Code_Target_Word_Base.thy
@@ -1,426 +1,428 @@
 (*  Title:      Code_Target_Word_Base.thy
     Author:     Andreas Lochbihler, ETH Zurich
 *)
 
 chapter \<open>Common base for target language implementations of word types\<close>
 
 theory Code_Target_Word_Base imports
   "HOL-Library.Word"
   "Word_Lib.Signed_Division_Word"
-  Bits_Integer
+  "Word_Lib.More_Word"
+  "Word_Lib.Bit_Comprehension"
 begin
 
 text \<open>More lemmas\<close>
 
 lemma div_half_nat:
   fixes x y :: nat
   assumes "y \<noteq> 0"
   shows "(x div y, x mod y) = (let q = 2 * (x div 2 div y); r = x - q * y in if y \<le> r then (q + 1, r - y) else (q, r))"
 proof -
   let ?q = "2 * (x div 2 div y)"
   have q: "?q = x div y - x div y mod 2"
     by(metis div_mult2_eq mult.commute minus_mod_eq_mult_div [symmetric])
   let ?r = "x - ?q * y"
   have r: "?r = x mod y + x div y mod 2 * y"
     by(simp add: q diff_mult_distrib minus_mod_eq_div_mult [symmetric])(metis diff_diff_cancel mod_less_eq_dividend mod_mult2_eq add.commute mult.commute)
 
   show ?thesis
   proof(cases "y \<le> x - ?q * y")
     case True
     with assms q have "x div y mod 2 \<noteq> 0" unfolding r
       by (metis Nat.add_0_right diff_0_eq_0 diff_Suc_1 le_div_geq mod2_gr_0 mod_div_trivial mult_0 neq0_conv numeral_1_eq_Suc_0 numerals(1)) 
     hence "x div y = ?q + 1" unfolding q
       by simp
     moreover hence "x mod y = ?r - y"
       by simp(metis minus_div_mult_eq_mod [symmetric] diff_commute diff_diff_left mult_Suc)
     ultimately show ?thesis using True by(simp add: Let_def)
   next
     case False
     hence "x div y mod 2 = 0" unfolding r
       by(simp add: not_le)(metis Nat.add_0_right assms div_less div_mult_self2 mod_div_trivial mult.commute)
     hence "x div y = ?q" unfolding q by simp
     moreover hence "x mod y = ?r" by (metis minus_div_mult_eq_mod [symmetric])
     ultimately show ?thesis using False by(simp add: Let_def)
   qed
 qed
 
 lemma div_half_word:
   fixes x y :: "'a :: len word"
   assumes "y \<noteq> 0"
   shows "(x div y, x mod y) = (let q = push_bit 1 (drop_bit 1 x div y); r = x - q * y in if y \<le> r then (q + 1, r - y) else (q, r))"
 proof -
   obtain n where n: "x = of_nat n" "n < 2 ^ LENGTH('a)"
     by (rule that [of \<open>unat x\<close>]) simp_all
   moreover obtain m where m: "y = of_nat m" "m < 2 ^ LENGTH('a)"
     by (rule that [of \<open>unat y\<close>]) simp_all
   ultimately have [simp]: \<open>unat (of_nat n :: 'a word) = n\<close> \<open>unat (of_nat m :: 'a word) = m\<close>
     by (transfer, simp add: take_bit_of_nat take_bit_nat_eq_self_iff)+
   let ?q = "push_bit 1 (drop_bit 1 x div y)"
   let ?q' = "2 * (n div 2 div m)"
   have "n div 2 div m < 2 ^ LENGTH('a)"
-    using n by (metis of_nat_inverse unat_lt2p uno_simps(2))
+    using n by (metis of_nat_inverse uno_simps(2) unsigned_less)
   hence q: "?q = of_nat ?q'" using n m
     by (auto simp add: drop_bit_eq_div word_arith_nat_div uno_simps take_bit_nat_eq_self unsigned_of_nat)
   from assms have "m \<noteq> 0" using m by -(rule notI, simp)
 
   from n have "2 * (n div 2 div m) < 2 ^ LENGTH('a)"
-    by(metis mult.commute div_mult2_eq minus_mod_eq_mult_div [symmetric] less_imp_diff_less of_nat_inverse unat_lt2p uno_simps(2))
+    by (metis mult.commute div_mult2_eq minus_mod_eq_mult_div [symmetric] less_imp_diff_less of_nat_inverse unsigned_less uno_simps(2))
   moreover
   have "2 * (n div 2 div m) * m < 2 ^ LENGTH('a)" using n unfolding div_mult2_eq[symmetric]
     by(subst (2) mult.commute)(simp add: minus_mod_eq_div_mult [symmetric] diff_mult_distrib minus_mod_eq_mult_div [symmetric] div_mult2_eq)
   moreover have "2 * (n div 2 div m) * m \<le> n"
     by (simp flip: div_mult2_eq ac_simps)
   ultimately
   have r: "x - ?q * y = of_nat (n - ?q' * m)"
     and "y \<le> x - ?q * y \<Longrightarrow> of_nat (n - ?q' * m) - y = of_nat (n - ?q' * m - m)"
     using n m unfolding q
      apply (simp_all add: of_nat_diff)
     apply (subst of_nat_diff)
     apply (cases \<open>LENGTH('a) \<ge> 2\<close>)
      apply (simp_all add: word_le_nat_alt take_bit_nat_eq_self unat_sub_if' unat_word_ariths unsigned_of_nat)
     done
   then show ?thesis using n m div_half_nat [OF \<open>m \<noteq> 0\<close>, of n] unfolding q
     by (simp add: word_le_nat_alt word_div_def word_mod_def Let_def take_bit_nat_eq_self unsigned_of_nat
       flip: zdiv_int zmod_int
       split del: if_split split: if_split_asm)
 qed
 
 lemma word_test_bit_set_bits: "bit (BITS n. f n :: 'a :: len word) n \<longleftrightarrow> n < LENGTH('a) \<and> f n"
   by (fact bit_set_bits_word_iff)
 
 lemma word_of_int_conv_set_bits: "word_of_int i = (BITS n. bit i n)"
-  by (rule word_eqI) (auto simp add: word_test_bit_set_bits bit_simps)
+  by (rule bit_eqI) (auto simp add: bit_simps)
 
 context
   includes bit_operations_syntax
 begin
 
 lemma word_and_mask_or_conv_and_mask:
   "bit n index \<Longrightarrow> (n AND mask index) OR (push_bit index 1) = n AND mask (index + 1)"
   for n :: \<open>'a::len word\<close>
-by(rule word_eqI)(auto simp add: bit_simps)
+  by (rule bit_eqI) (auto simp add: bit_simps)
 
 lemma uint_and_mask_or_full:
   fixes n :: "'a :: len word"
   assumes "bit n (LENGTH('a) - 1)"
   and "mask1 = mask (LENGTH('a) - 1)"
   and "mask2 = push_bit (LENGTH('a) - 1) 1"
   shows "uint (n AND mask1) OR mask2 = uint n"
 proof -
   have "mask2 = uint (push_bit (LENGTH('a) - 1) 1 :: 'a word)" using assms
     by transfer (simp add: take_bit_push_bit)
   hence "uint (n AND mask1) OR mask2 = uint (n AND mask1 OR (push_bit (LENGTH('a) - 1) 1 :: 'a word))"
     by(simp add: uint_or)
   also have "\<dots> = uint (n AND mask (LENGTH('a) - 1 + 1))"
     using assms by(simp only: word_and_mask_or_conv_and_mask)
   also have "\<dots> = uint n" by simp
   finally show ?thesis .
 qed
 
 end
 
 
 text \<open>Division on @{typ "'a word"} is unsigned, but Scala and OCaml only have signed division and modulus.\<close>
 
 lemmas word_sdiv_def = sdiv_word_def
 lemmas word_smod_def = smod_word_def
 
 lemma [code]:
   "x sdiv y =
    (let x' = sint x; y' = sint y;
         negative = (x' < 0) \<noteq> (y' < 0);
         result = abs x' div abs y'
     in word_of_int (if negative then -result else result))"
   for x y :: \<open>'a::len word\<close>
   by (simp add: sdiv_word_def signed_divide_int_def sgn_if Let_def not_less not_le)
 
 lemma [code]:
   "x smod y =
    (let x' = sint x; y' = sint y;
         negative = (x' < 0);
         result = abs x' mod abs y'
     in word_of_int (if negative then -result else result))"
   for x y :: \<open>'a::len word\<close>
 proof -
   have *: \<open>k mod l = k - k div l * l\<close> for k l :: int
     by (simp add: minus_div_mult_eq_mod)
   show ?thesis
     by (simp add: smod_word_def signed_modulo_int_def signed_divide_int_def * sgn_if Let_def)
 qed
 
 text \<open>
   This algorithm implements unsigned division in terms of signed division.
   Taken from Hacker's Delight.
 \<close>
 
 lemma divmod_via_sdivmod:
   fixes x y :: "'a :: len word"
   assumes "y \<noteq> 0"
   shows
   "(x div y, x mod y) =
   (if push_bit (LENGTH('a) - 1) 1 \<le> y then if x < y then (0, x) else (1, x - y)
    else let q = (push_bit 1 (drop_bit 1 x sdiv y));
             r = x - q * y
         in if r \<ge> y then (q + 1, r - y) else (q, r))"
 proof(cases "push_bit (LENGTH('a) - 1) 1 \<le> y")
   case True
   note y = this
   show ?thesis
   proof(cases "x < y")
     case True
     then have "x mod y = x"
       by transfer simp
-    thus ?thesis using True y by(simp add: word_div_lt_eq_0)
+    then show ?thesis using True y
+      using bits_mod_div_trivial [of x y] by simp
   next
     case False
     obtain n where n: "y = of_nat n" "n < 2 ^ LENGTH('a)"
       by (rule that [of \<open>unat y\<close>]) simp_all
-    have "unat x < 2 ^ LENGTH('a)" by(rule unat_lt2p)
+    have "unat x < 2 ^ LENGTH('a)" by (rule unsigned_less)
     also have "\<dots> = 2 * 2 ^ (LENGTH('a) - 1)"
       by(metis Suc_pred len_gt_0 power_Suc One_nat_def)
     also have "\<dots> \<le> 2 * n" using y n
       by transfer (simp add: push_bit_of_1 take_bit_eq_mod)
     finally have div: "x div of_nat n = 1" using False n
-      by (simp add: word_div_eq_1_iff take_bit_nat_eq_self unsigned_of_nat)
+      by (simp add: take_bit_nat_eq_self unsigned_of_nat word_div_eq_1_iff)
     moreover have "x mod y = x - x div y * y"
       by (simp add: minus_div_mult_eq_mod)
     with div n have "x mod y = x - y" by simp
     ultimately show ?thesis using False y n by simp
   qed
 next
   case False
   note y = this
   obtain n where n: "x = of_nat n" "n < 2 ^ LENGTH('a)"
     by (rule that [of \<open>unat x\<close>]) simp_all
   hence "int n div 2 + 2 ^ (LENGTH('a) - Suc 0) < 2 ^ LENGTH('a)"
     by (cases \<open>LENGTH('a)\<close>)
       (auto dest: less_imp_of_nat_less [where ?'a = int])
   with y n have "sint (drop_bit 1 x) = uint (drop_bit 1 x)"
     by (cases \<open>LENGTH('a)\<close>)
       (auto simp add: sint_uint drop_bit_eq_div take_bit_nat_eq_self uint_div_distrib
         signed_take_bit_int_eq_self_iff unsigned_of_nat)
   moreover have "uint y + 2 ^ (LENGTH('a) - Suc 0) < 2 ^ LENGTH('a)"
     using y by (cases \<open>LENGTH('a)\<close>)
       (simp_all add: not_le push_bit_of_1 word_less_alt uint_power_lower)
   then have "sint y = uint y"
     apply (cases \<open>LENGTH('a)\<close>)
      apply (auto simp add: sint_uint signed_take_bit_int_eq_self_iff)
     using uint_ge_0 [of y]
     by linarith 
   ultimately show ?thesis using y
     apply (subst div_half_word [OF assms])
     apply (simp add: sdiv_word_def signed_divide_int_def flip: uint_div)
     done
 qed
 
 
 text \<open>More implementations tailored towards target-language implementations\<close>
 
 context
   includes integer.lifting
 begin
 
 lift_definition word_of_integer :: "integer \<Rightarrow> 'a :: len word" is word_of_int .
 
 lemma word_of_integer_code [code]: "word_of_integer n = word_of_int (int_of_integer n)"
   by(simp add: word_of_integer.rep_eq)
 
 end
 
 context
   includes bit_operations_syntax
 begin
 
 lemma word_of_int_code:
   "uint (word_of_int x :: 'a word) = x AND mask (LENGTH('a :: len))"
   by (simp add: unsigned_of_int take_bit_eq_mask)
 
 context
   fixes f :: "nat \<Rightarrow> bool"
 begin
 
 definition set_bits_aux :: \<open>nat \<Rightarrow> 'a word \<Rightarrow> 'a::len word\<close>
   where \<open>set_bits_aux n w = push_bit n w OR take_bit n (set_bits f)\<close>
 
 lemma bit_set_bit_aux [bit_simps]:
   \<open>bit (set_bits_aux n w) m \<longleftrightarrow> m < LENGTH('a) \<and>
     (if m < n then f m else bit w (m - n))\<close> for w :: \<open>'a::len word\<close>
   by (auto simp add: bit_simps set_bits_aux_def)
 
 corollary set_bits_conv_set_bits_aux:
   \<open>set_bits f = (set_bits_aux LENGTH('a) 0 :: 'a :: len word)\<close>
   by (rule bit_word_eqI) (simp add: bit_simps)
 
 lemma set_bits_aux_0 [simp]:
   \<open>set_bits_aux 0 w = w\<close>
   by (simp add: set_bits_aux_def)
 
 lemma set_bits_aux_Suc [simp]:
   \<open>set_bits_aux (Suc n) w = set_bits_aux n (push_bit 1 w OR (if f n then 1 else 0))\<close>
   by (rule bit_word_eqI) (auto simp add: bit_simps le_less_Suc_eq mult.commute [of _ 2])
 
 lemma set_bits_aux_simps [code]:
   \<open>set_bits_aux 0 w = w\<close>
   \<open>set_bits_aux (Suc n) w = set_bits_aux n (push_bit 1 w OR (if f n then 1 else 0))\<close>
   by simp_all
 
 lemma set_bits_aux_rec:
   \<open>set_bits_aux n w =
   (if n = 0 then w 
    else let n' = n - 1 in set_bits_aux n' (push_bit 1 w OR (if f n' then 1 else 0)))\<close>
   by (cases n) simp_all
 
 end
 
 lemma word_of_int_via_signed:
   fixes mask
   assumes mask_def: "mask = Bit_Operations.mask (LENGTH('a))"
   and shift_def: "shift = push_bit LENGTH('a) 1"
   and index_def: "index = LENGTH('a) - 1"
   and overflow_def:"overflow = push_bit (LENGTH('a) - 1) 1"
   and least_def: "least = - overflow"
   shows
   "(word_of_int i :: 'a :: len word) =
    (let i' = i AND mask
     in if bit i' index then
          if i' - shift < least \<or> overflow \<le> i' - shift then arbitrary1 i' else word_of_int (i' - shift)
        else if i' < least \<or> overflow \<le> i' then arbitrary2 i' else word_of_int i')"
 proof -
   define i' where "i' = i AND mask"
   have "shift = mask + 1" unfolding assms
     by (simp add: mask_eq_exp_minus_1 push_bit_of_1) 
   hence "i' < shift"
     by (simp add: mask_def i'_def)
   show ?thesis
   proof(cases "bit i' index")
     case True
     then have unf: "i' = overflow OR i'"
       apply (simp add: assms i'_def push_bit_of_1 flip: take_bit_eq_mask)
       apply (rule bit_eqI)
       apply (auto simp add: bit_take_bit_iff bit_or_iff bit_exp_iff)
       done
     have \<open>overflow \<le> overflow OR i'\<close>
       by (simp add: i'_def mask_def or_greater_eq)
     then have "overflow \<le> i'"
       by (subst unf)
     hence "i' - shift < least \<longleftrightarrow> False" unfolding assms
       by(cases "LENGTH('a)")(simp_all add: not_less push_bit_of_1)
     moreover
     have "overflow \<le> i' - shift \<longleftrightarrow> False" using \<open>i' < shift\<close> unfolding assms
       by(cases "LENGTH('a)")(auto simp add: not_le push_bit_of_1 elim: less_le_trans)
     moreover
     have "word_of_int (i' - shift) = (word_of_int i :: 'a word)" using \<open>i' < shift\<close>
       by (simp add: i'_def shift_def mask_def push_bit_of_1 word_of_int_eq_iff flip: take_bit_eq_mask)
     ultimately show ?thesis using True by(simp add: Let_def i'_def)
   next
     case False
     have "i' = i AND Bit_Operations.mask (LENGTH('a) - 1)"
       apply (rule bit_eqI)
       apply (use False in \<open>auto simp add: bit_simps assms i'_def\<close>)
       apply (auto simp add: less_le)
       done
     also have "\<dots> \<le> Bit_Operations.mask (LENGTH('a) - 1)"
       using AND_upper2 mask_nonnegative_int by blast
     also have "\<dots> < overflow"
       by (simp add: mask_int_def push_bit_of_1 overflow_def)
     also
     have "least \<le> 0" unfolding least_def overflow_def by simp
     have "0 \<le> i'" by (simp add: i'_def mask_def)
     hence "least \<le> i'" using \<open>least \<le> 0\<close> by simp
     moreover
     have "word_of_int i' = (word_of_int i :: 'a word)"
       by (simp add: i'_def mask_def of_int_and_eq of_int_mask_eq)
     ultimately show ?thesis using False by(simp add: Let_def i'_def)
   qed
 qed
 
 end
 
 
 text \<open>Quickcheck conversion functions\<close>
 
 context
   includes state_combinator_syntax
 begin
 
 definition qc_random_cnv ::
   "(natural \<Rightarrow> 'a::term_of) \<Rightarrow> natural \<Rightarrow> Random.seed
     \<Rightarrow> ('a \<times> (unit \<Rightarrow> Code_Evaluation.term)) \<times> Random.seed"
   where "qc_random_cnv a_of_natural i = Random.range (i + 1) \<circ>\<rightarrow> (\<lambda>k. Pair (
        let n = a_of_natural k
        in (n, \<lambda>_. Code_Evaluation.term_of n)))"
 
 end
 
 definition qc_exhaustive_cnv :: "(natural \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> (bool \<times> term list) option)
   \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
 where
   "qc_exhaustive_cnv a_of_natural f d =
    Quickcheck_Exhaustive.exhaustive (%x. f (a_of_natural x)) d"
 
 definition qc_full_exhaustive_cnv ::
   "(natural \<Rightarrow> ('a::term_of)) \<Rightarrow> ('a \<times> (unit \<Rightarrow> term) \<Rightarrow> (bool \<times> term list) option)
   \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
 where
   "qc_full_exhaustive_cnv a_of_natural f d = Quickcheck_Exhaustive.full_exhaustive
   (%(x, xt). f (a_of_natural x, %_. Code_Evaluation.term_of (a_of_natural x))) d"
 
 declare [[quickcheck_narrowing_ghc_options = "-XTypeSynonymInstances"]]
 
 definition qc_narrowing_drawn_from :: "'a list \<Rightarrow> integer \<Rightarrow> _"
 where
   "qc_narrowing_drawn_from xs =
    foldr Quickcheck_Narrowing.sum (map Quickcheck_Narrowing.cons (butlast xs)) (Quickcheck_Narrowing.cons (last xs))"
 
 locale quickcheck_narrowing_samples =
   fixes a_of_integer :: "integer \<Rightarrow> 'a \<times> 'a :: {partial_term_of, term_of}"
   and zero :: "'a"
   and tr :: "typerep"
 begin
 
 function narrowing_samples :: "integer \<Rightarrow> 'a list"
 where
   "narrowing_samples i =
    (if i > 0 then let (a, a') = a_of_integer i in narrowing_samples (i - 1) @ [a, a'] else [zero])"
 by pat_completeness auto
 termination including integer.lifting
 proof(relation "measure nat_of_integer")
   fix i :: integer
   assume "0 < i"
   thus "(i - 1, i) \<in> measure nat_of_integer"
     by simp(transfer, simp)
 qed simp
 
 definition partial_term_of_sample :: "integer \<Rightarrow> 'a"
 where
   "partial_term_of_sample i =
   (if i < 0 then undefined
    else if i = 0 then zero
    else if i mod 2 = 0 then snd (a_of_integer (i div 2))
    else fst (a_of_integer (i div 2 + 1)))"
 
 lemma partial_term_of_code:
   "partial_term_of (ty :: 'a itself) (Quickcheck_Narrowing.Narrowing_variable p t) \<equiv>
     Code_Evaluation.Free (STR ''_'') tr"
   "partial_term_of (ty :: 'a itself) (Quickcheck_Narrowing.Narrowing_constructor i []) \<equiv>
    Code_Evaluation.term_of (partial_term_of_sample i)"
 by (rule partial_term_of_anything)+
 
 end
 
 lemmas [code] =
   quickcheck_narrowing_samples.narrowing_samples.simps
   quickcheck_narrowing_samples.partial_term_of_sample_def
 
 
 text \<open>
   The separate code target \<open>SML_word\<close> collects setups for the
   code generator that PolyML does not provide.
 \<close>
 
 setup \<open>Code_Target.add_derived_target ("SML_word", [(Code_ML.target_SML, I)])\<close>
 
 code_identifier code_module Code_Target_Word_Base \<rightharpoonup>
   (SML) Word and (Haskell) Word and (OCaml) Word and (Scala) Word
 
 end
diff --git a/thys/Native_Word/Native_Word_Test_Emu.thy b/thys/Native_Word/Native_Word_Test_Emu.thy
--- a/thys/Native_Word/Native_Word_Test_Emu.thy
+++ b/thys/Native_Word/Native_Word_Test_Emu.thy
@@ -1,55 +1,55 @@
 (*  Title:      Native_Word_Test_Emu.thy
     Author:     Andreas Lochbihler, ETH Zurich
 *)
 
 theory Native_Word_Test_Emu imports
   Native_Word_Test
-  Code_Target_Bits_Int
+  Code_Target_Int_Bit
 begin
 
 section \<open>Test cases for emulation of native words\<close>
 
 subsection \<open>Tests for @{typ uint16}\<close>
 
 text \<open>
   Test that @{typ uint16} is emulated for PolyML and OCaml via @{typ "16 word"} 
-  if @{theory Native_Word.Code_Target_Bits_Int} is imported.
+  if @{theory Native_Word.Code_Target_Int_Bit} is imported.
 \<close>
 
 definition test_uint16_emulation :: bool where
   "test_uint16_emulation \<longleftrightarrow> (0xFFFFF - 0x1000 = (0xEFFF :: uint16))"
 
 export_code test_uint16_emulation checking SML OCaml?
   \<comment> \<open>test the other target languages as well\<close> Haskell? Scala
 
 notepad begin
 have test_uint16 by eval
 have test_uint16_emulation by eval
 have test_uint16_emulation by normalization
 have test_uint16_emulation by code_simp
 end
 
 ML_val \<open>
   val true = @{code test_uint16};
   val true = @{code test_uint16_emulation};
 \<close>
 
 lemma "x AND y = x OR (y :: uint16)"
 quickcheck[random, expect=counterexample]
 quickcheck[exhaustive, expect=counterexample]
 oops
 
 subsection \<open>Tests for @{typ uint8}\<close>
 
 text \<open>
   Test that @{typ uint8} is emulated for OCaml via @{typ "8 word"} 
-  if @{theory Native_Word.Code_Target_Bits_Int} is imported.
+  if @{theory Native_Word.Code_Target_Int_Bit} is imported.
 \<close>
 
 definition test_uint8_emulation :: bool where
   "test_uint8_emulation \<longleftrightarrow> (0xFFF - 0x10 = (0xEF :: uint8))"
 
 export_code test_uint8_emulation checking OCaml?
   \<comment> \<open>test the other target languages as well\<close> SML Haskell? Scala
 
 end
diff --git a/thys/Native_Word/Uint.thy b/thys/Native_Word/Uint.thy
--- a/thys/Native_Word/Uint.thy
+++ b/thys/Native_Word/Uint.thy
@@ -1,824 +1,825 @@
 (*  Title:      Uint.thy
     Author:     Peter Lammich, TU Munich
     Author:     Andreas Lochbihler, ETH Zurich
 *)
 
 chapter \<open>Unsigned words of default size\<close>
 
 theory Uint imports
-  Code_Target_Word_Base Word_Type_Copies
+  Word_Type_Copies
+  Code_Target_Integer_Bit
 begin
 
 text \<open>
   This theory provides access to words in the target languages of the code generator
   whose bit width is the default of the target language. To that end, the type \<open>uint\<close>
   models words of width \<open>dflt_size\<close>, but \<open>dflt_size\<close> is known only to be positive.
 
   Usage restrictions:
   Default-size words (type \<open>uint\<close>) cannot be used for evaluation, because 
   the results depend on the particular choice of word size in the target language
   and implementation. Symbolic evaluation has not yet been set up for \<open>uint\<close>.
 \<close>
 
 text \<open>The default size type\<close>
 typedecl dflt_size
 
 instantiation dflt_size :: typerep begin
 definition "typerep_class.typerep \<equiv>  \<lambda>_ :: dflt_size itself. Typerep.Typerep (STR ''Uint.dflt_size'') []"
 instance ..
 end
 
 consts dflt_size_aux :: "nat"
 specification (dflt_size_aux) dflt_size_aux_g0: "dflt_size_aux > 0"
   by auto
 hide_fact dflt_size_aux_def
 
 instantiation dflt_size :: len begin
 definition "len_of_dflt_size (_ :: dflt_size itself) \<equiv> dflt_size_aux"
 instance by(intro_classes)(simp add: len_of_dflt_size_def dflt_size_aux_g0)
 end
 
 abbreviation "dflt_size \<equiv> len_of (TYPE (dflt_size))"
 
 context includes integer.lifting begin
 lift_definition dflt_size_integer :: integer is "int dflt_size" .
 declare dflt_size_integer_def[code del]
   \<comment> \<open>The code generator will substitute a machine-dependent value for this constant\<close>
 
 lemma dflt_size_by_int[code]: "dflt_size = nat_of_integer dflt_size_integer"
 by transfer simp
 
 lemma dflt_size[simp]: 
   "dflt_size > 0"
   "dflt_size \<ge> Suc 0"
   "\<not> dflt_size < Suc 0"
   using len_gt_0[where 'a=dflt_size]
   by (simp_all del: len_gt_0)
 end
 
 section \<open>Type definition and primitive operations\<close>
 
 typedef uint = \<open>UNIV :: dflt_size word set\<close> ..
 
 global_interpretation uint: word_type_copy Abs_uint Rep_uint
   using type_definition_uint by (rule word_type_copy.intro)
 
 setup_lifting type_definition_uint
 
 declare uint.of_word_of [code abstype]
 
 declare Quotient_uint [transfer_rule]
 
 instantiation uint :: \<open>{comm_ring_1, semiring_modulo, equal, linorder}\<close>
 begin
 
 lift_definition zero_uint :: uint is 0 .
 lift_definition one_uint :: uint is 1 .
 lift_definition plus_uint :: \<open>uint \<Rightarrow> uint \<Rightarrow> uint\<close> is \<open>(+)\<close> .
 lift_definition uminus_uint :: \<open>uint \<Rightarrow> uint\<close> is uminus .
 lift_definition minus_uint :: \<open>uint \<Rightarrow> uint \<Rightarrow> uint\<close> is \<open>(-)\<close> .
 lift_definition times_uint :: \<open>uint \<Rightarrow> uint \<Rightarrow> uint\<close> is \<open>(*)\<close> .
 lift_definition divide_uint :: \<open>uint \<Rightarrow> uint \<Rightarrow> uint\<close> is \<open>(div)\<close> .
 lift_definition modulo_uint :: \<open>uint \<Rightarrow> uint \<Rightarrow> uint\<close> is \<open>(mod)\<close> .
 lift_definition equal_uint :: \<open>uint \<Rightarrow> uint \<Rightarrow> bool\<close> is \<open>HOL.equal\<close> .
 lift_definition less_eq_uint :: \<open>uint \<Rightarrow> uint \<Rightarrow> bool\<close> is \<open>(\<le>)\<close> .
 lift_definition less_uint :: \<open>uint \<Rightarrow> uint \<Rightarrow> bool\<close> is \<open>(<)\<close> .
 
 global_interpretation uint: word_type_copy_ring Abs_uint Rep_uint
   by standard (fact zero_uint.rep_eq one_uint.rep_eq
     plus_uint.rep_eq uminus_uint.rep_eq minus_uint.rep_eq
     times_uint.rep_eq divide_uint.rep_eq modulo_uint.rep_eq
     equal_uint.rep_eq less_eq_uint.rep_eq less_uint.rep_eq)+
 
 instance proof -
   show \<open>OFCLASS(uint, comm_ring_1_class)\<close>
     by (rule uint.of_class_comm_ring_1)
   show \<open>OFCLASS(uint, semiring_modulo_class)\<close>
     by (fact uint.of_class_semiring_modulo)
   show \<open>OFCLASS(uint, equal_class)\<close>
     by (fact uint.of_class_equal)
   show \<open>OFCLASS(uint, linorder_class)\<close>
     by (fact uint.of_class_linorder)
 qed
 
 end
 
 instantiation uint :: ring_bit_operations
 begin
 
 lift_definition bit_uint :: \<open>uint \<Rightarrow> nat \<Rightarrow> bool\<close> is bit .
 lift_definition not_uint :: \<open>uint \<Rightarrow> uint\<close> is \<open>Bit_Operations.not\<close> .
 lift_definition and_uint :: \<open>uint \<Rightarrow> uint \<Rightarrow> uint\<close> is \<open>Bit_Operations.and\<close> .
 lift_definition or_uint :: \<open>uint \<Rightarrow> uint \<Rightarrow> uint\<close> is \<open>Bit_Operations.or\<close> .
 lift_definition xor_uint :: \<open>uint \<Rightarrow> uint \<Rightarrow> uint\<close> is \<open>Bit_Operations.xor\<close> .
 lift_definition mask_uint :: \<open>nat \<Rightarrow> uint\<close> is mask .
 lift_definition push_bit_uint :: \<open>nat \<Rightarrow> uint \<Rightarrow> uint\<close> is push_bit .
 lift_definition drop_bit_uint :: \<open>nat \<Rightarrow> uint \<Rightarrow> uint\<close> is drop_bit .
 lift_definition signed_drop_bit_uint :: \<open>nat \<Rightarrow> uint \<Rightarrow> uint\<close> is signed_drop_bit .
 lift_definition take_bit_uint :: \<open>nat \<Rightarrow> uint \<Rightarrow> uint\<close> is take_bit .
 lift_definition set_bit_uint :: \<open>nat \<Rightarrow> uint \<Rightarrow> uint\<close> is Bit_Operations.set_bit .
 lift_definition unset_bit_uint :: \<open>nat \<Rightarrow> uint \<Rightarrow> uint\<close> is unset_bit .
 lift_definition flip_bit_uint :: \<open>nat \<Rightarrow> uint \<Rightarrow> uint\<close> is flip_bit .
 
 global_interpretation uint: word_type_copy_bits Abs_uint Rep_uint signed_drop_bit_uint
   by standard (fact bit_uint.rep_eq not_uint.rep_eq and_uint.rep_eq or_uint.rep_eq xor_uint.rep_eq
     mask_uint.rep_eq push_bit_uint.rep_eq drop_bit_uint.rep_eq signed_drop_bit_uint.rep_eq take_bit_uint.rep_eq
     set_bit_uint.rep_eq unset_bit_uint.rep_eq flip_bit_uint.rep_eq)+
 
 instance
   by (fact uint.of_class_ring_bit_operations)
 
 end
 
 lift_definition uint_of_nat :: \<open>nat \<Rightarrow> uint\<close>
   is word_of_nat .
 
 lift_definition nat_of_uint :: \<open>uint \<Rightarrow> nat\<close>
   is unat .
 
 lift_definition uint_of_int :: \<open>int \<Rightarrow> uint\<close>
   is word_of_int .
 
 lift_definition int_of_uint :: \<open>uint \<Rightarrow> int\<close>
   is uint .
 
 context
   includes integer.lifting
 begin
 
 lift_definition Uint :: \<open>integer \<Rightarrow> uint\<close>
   is word_of_int .
 
 lift_definition integer_of_uint :: \<open>uint \<Rightarrow> integer\<close>
   is uint .
 
 end
 
 global_interpretation uint: word_type_copy_more Abs_uint Rep_uint signed_drop_bit_uint
   uint_of_nat nat_of_uint uint_of_int int_of_uint Uint integer_of_uint
   apply standard
        apply (simp_all add: uint_of_nat.rep_eq nat_of_uint.rep_eq
          uint_of_int.rep_eq int_of_uint.rep_eq
          Uint.rep_eq integer_of_uint.rep_eq integer_eq_iff)
   done
 
 instantiation uint :: "{size, msb, lsb, set_bit, bit_comprehension}"
 begin
 
 lift_definition size_uint :: \<open>uint \<Rightarrow> nat\<close> is size .
 
 lift_definition msb_uint :: \<open>uint \<Rightarrow> bool\<close> is msb .
 lift_definition lsb_uint :: \<open>uint \<Rightarrow> bool\<close> is lsb .
 
 text \<open>Workaround: avoid name space clash by spelling out \<^text>\<open>lift_definition\<close> explicitly.\<close>
 
 definition set_bit_uint :: \<open>uint \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> uint\<close>
   where set_bit_uint_eq: \<open>set_bit_uint a n b = (if b then Bit_Operations.set_bit else unset_bit) n a\<close>
 
 context
   includes lifting_syntax
 begin
 
 lemma set_bit_uint_transfer [transfer_rule]:
   \<open>(cr_uint ===> (=) ===> (\<longleftrightarrow>) ===> cr_uint) Generic_set_bit.set_bit Generic_set_bit.set_bit\<close>
   by (simp only: set_bit_eq [abs_def] set_bit_uint_eq [abs_def]) transfer_prover
 
 end
 
 lift_definition set_bits_uint :: \<open>(nat \<Rightarrow> bool) \<Rightarrow> uint\<close> is set_bits .
 lift_definition set_bits_aux_uint :: \<open>(nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> uint \<Rightarrow> uint\<close> is set_bits_aux .
 
 global_interpretation uint: word_type_copy_misc Abs_uint Rep_uint signed_drop_bit_uint
   uint_of_nat nat_of_uint uint_of_int int_of_uint Uint integer_of_uint dflt_size set_bits_aux_uint
   by (standard; transfer) simp_all
 
 instance using uint.of_class_bit_comprehension
   uint.of_class_set_bit uint.of_class_lsb
   by simp_all standard
 
 end
 
 section \<open>Code setup\<close>
 
 code_printing code_module Uint \<rightharpoonup> (SML)
 \<open>
 structure Uint : sig
   val set_bit : Word.word -> IntInf.int -> bool -> Word.word
   val shiftl : Word.word -> IntInf.int -> Word.word
   val shiftr : Word.word -> IntInf.int -> Word.word
   val shiftr_signed : Word.word -> IntInf.int -> Word.word
   val test_bit : Word.word -> IntInf.int -> bool
 end = struct
 
 fun set_bit x n b =
   let val mask = Word.<< (0wx1, Word.fromLargeInt (IntInf.toLarge n))
   in if b then Word.orb (x, mask)
      else Word.andb (x, Word.notb mask)
   end
 
 fun shiftl x n =
   Word.<< (x, Word.fromLargeInt (IntInf.toLarge n))
 
 fun shiftr x n =
   Word.>> (x, Word.fromLargeInt (IntInf.toLarge n))
 
 fun shiftr_signed x n =
   Word.~>> (x, Word.fromLargeInt (IntInf.toLarge n))
 
 fun test_bit x n =
   Word.andb (x, Word.<< (0wx1, Word.fromLargeInt (IntInf.toLarge n))) <> Word.fromInt 0
 
 end; (* struct Uint *)\<close>
 code_reserved SML Uint
 
 code_printing code_module Uint \<rightharpoonup> (Haskell)
  \<open>module Uint(Int, Word, dflt_size) where
 
   import qualified Prelude
   import Data.Int(Int)
   import Data.Word(Word)
   import qualified Data.Bits
 
   dflt_size :: Prelude.Integer
   dflt_size = Prelude.toInteger (bitSize_aux (0::Word)) where
     bitSize_aux :: (Data.Bits.Bits a, Prelude.Bounded a) => a -> Int
     bitSize_aux = Data.Bits.bitSize\<close>
   and (Haskell_Quickcheck)
  \<open>module Uint(Int, Word, dflt_size) where
 
   import qualified Prelude
   import Data.Int(Int)
   import Data.Word(Word)
   import qualified Data.Bits
 
   dflt_size :: Prelude.Int
   dflt_size = bitSize_aux (0::Word) where
     bitSize_aux :: (Data.Bits.Bits a, Prelude.Bounded a) => a -> Int
     bitSize_aux = Data.Bits.bitSize
 \<close>
 code_reserved Haskell Uint dflt_size
 
 text \<open>
   OCaml and Scala provide only signed bit numbers, so we use these and 
   implement sign-sensitive operations like comparisons manually.
 \<close>
 
 code_printing code_module "Uint" \<rightharpoonup> (OCaml)
 \<open>module Uint : sig
   type t = int
   val dflt_size : Z.t
   val less : t -> t -> bool
   val less_eq : t -> t -> bool
   val set_bit : t -> Z.t -> bool -> t
   val shiftl : t -> Z.t -> t
   val shiftr : t -> Z.t -> t
   val shiftr_signed : t -> Z.t -> t
   val test_bit : t -> Z.t -> bool
   val int_mask : int
   val int32_mask : int32
   val int64_mask : int64
 end = struct
 
 type t = int
 
 let dflt_size = Z.of_int Sys.int_size;;
 
 (* negative numbers have their highest bit set, 
    so they are greater than positive ones *)
 let less x y =
   if x<0 then
     y<0 && x<y
   else y < 0 || x < y;;
 
 let less_eq x y =
   if x < 0 then
     y < 0 &&  x <= y
   else y < 0 || x <= y;;
 
 let set_bit x n b =
   let mask = 1 lsl (Z.to_int n)
   in if b then x lor mask
      else x land (lnot mask);;
 
 let shiftl x n = x lsl (Z.to_int n);;
 
 let shiftr x n = x lsr (Z.to_int n);;
 
 let shiftr_signed x n = x asr (Z.to_int n);;
 
 let test_bit x n = x land (1 lsl (Z.to_int n)) <> 0;;
 
 let int_mask =
   if Sys.int_size < 32 then lnot 0 else 0xFFFFFFFF;;
 
 let int32_mask = 
   if Sys.int_size < 32 then Int32.pred (Int32.shift_left Int32.one Sys.int_size)
   else Int32.of_string "0xFFFFFFFF";;
 
 let int64_mask = 
   if Sys.int_size < 64 then Int64.pred (Int64.shift_left Int64.one Sys.int_size)
   else Int64.of_string "0xFFFFFFFFFFFFFFFF";;
 
 end;; (*struct Uint*)\<close>
 code_reserved OCaml Uint
 
 code_printing code_module Uint \<rightharpoonup> (Scala)
 \<open>object Uint {
 def dflt_size : BigInt = BigInt(32)
 
 def less(x: Int, y: Int) : Boolean =
   if (x < 0) y < 0 && x < y
   else y < 0 || x < y
 
 def less_eq(x: Int, y: Int) : Boolean =
   if (x < 0) y < 0 && x <= y
   else y < 0 || x <= y
 
 def set_bit(x: Int, n: BigInt, b: Boolean) : Int =
   if (b)
     x | (1 << n.intValue)
   else
     x & (1 << n.intValue).unary_~
 
 def shiftl(x: Int, n: BigInt) : Int = x << n.intValue
 
 def shiftr(x: Int, n: BigInt) : Int = x >>> n.intValue
 
 def shiftr_signed(x: Int, n: BigInt) : Int = x >> n.intValue
 
 def test_bit(x: Int, n: BigInt) : Boolean =
   (x & (1 << n.intValue)) != 0
 
 } /* object Uint */\<close>
 code_reserved Scala Uint
 
 
 text \<open>
   OCaml's conversion from Big\_int to int demands that the value fits into a signed integer.
   The following justifies the implementation.
 \<close>
 
 context
   includes integer.lifting bit_operations_syntax
 begin
 
 definition wivs_mask :: int where "wivs_mask = 2^ dflt_size - 1"
 lift_definition wivs_mask_integer :: integer is wivs_mask .
 lemma [code]: "wivs_mask_integer = 2 ^ dflt_size - 1"
   by transfer (simp add: wivs_mask_def)
 
 definition wivs_shift :: int where "wivs_shift = 2 ^ dflt_size"
 lift_definition wivs_shift_integer :: integer is wivs_shift .
 lemma [code]: "wivs_shift_integer = 2 ^ dflt_size"
   by transfer (simp add: wivs_shift_def)
 
 definition wivs_index :: nat where "wivs_index == dflt_size - 1"
 lift_definition wivs_index_integer :: integer is "int wivs_index".
 lemma wivs_index_integer_code[code]: "wivs_index_integer = dflt_size_integer - 1"
   by transfer (simp add: wivs_index_def of_nat_diff)
 
 definition wivs_overflow :: int where "wivs_overflow == 2^ (dflt_size - 1)"
 lift_definition wivs_overflow_integer :: integer is wivs_overflow .
 lemma [code]: "wivs_overflow_integer = 2 ^ (dflt_size - 1)"
   by transfer (simp add: wivs_overflow_def)
 
 definition wivs_least :: int where "wivs_least == - wivs_overflow"
 lift_definition wivs_least_integer :: integer is wivs_least .
 lemma [code]: "wivs_least_integer = - (2 ^ (dflt_size - 1))"
   by transfer (simp add: wivs_overflow_def wivs_least_def)
 
 definition Uint_signed :: "integer \<Rightarrow> uint" where
   "Uint_signed i = (if i < wivs_least_integer \<or> wivs_overflow_integer \<le> i then undefined Uint i else Uint i)"
 
 lemma Uint_code [code]:
   "Uint i = 
   (let i' = i AND wivs_mask_integer in 
    if bit i' wivs_index then Uint_signed (i' - wivs_shift_integer) else Uint_signed i')"
   including undefined_transfer 
   unfolding Uint_signed_def
   apply transfer
   apply (subst word_of_int_via_signed)
        apply (auto simp add: push_bit_of_1 mask_eq_exp_minus_1 word_of_int_via_signed
          wivs_mask_def wivs_index_def wivs_overflow_def wivs_least_def wivs_shift_def)
   done
 
 lemma Uint_signed_code [code]:
   "Rep_uint (Uint_signed i) = 
   (if i < wivs_least_integer \<or> i \<ge> wivs_overflow_integer then Rep_uint (undefined Uint i) else word_of_int (int_of_integer_symbolic i))"
   unfolding Uint_signed_def Uint_def int_of_integer_symbolic_def word_of_integer_def
   by(simp add: Abs_uint_inverse)
 end
 
 text \<open>
   Avoid @{term Abs_uint} in generated code, use @{term Rep_uint'} instead. 
   The symbolic implementations for code\_simp use @{term Rep_uint}.
 
   The new destructor @{term Rep_uint'} is executable.
   As the simplifier is given the [code abstract] equations literally, 
   we cannot implement @{term Rep_uint} directly, because that makes code\_simp loop.
 
   If code generation raises Match, some equation probably contains @{term Rep_uint} 
   ([code abstract] equations for @{typ uint} may use @{term Rep_uint} because
   these instances will be folded away.)
 \<close>
 
 definition Rep_uint' where [simp]: "Rep_uint' = Rep_uint"
 
 lemma Rep_uint'_code [code]: "Rep_uint' x = (BITS n. bit x n)"
   unfolding Rep_uint'_def by transfer (simp add: set_bits_bit_eq)
 
 lift_definition Abs_uint' :: "dflt_size word \<Rightarrow> uint" is "\<lambda>x :: dflt_size word. x" .
 
 lemma Abs_uint'_code [code]:
   "Abs_uint' x = Uint (integer_of_int (uint x))"
 including integer.lifting by transfer simp
 
 declare [[code drop: "term_of_class.term_of :: uint \<Rightarrow> _"]]
 
 lemma term_of_uint_code [code]:
   defines "TR \<equiv> typerep.Typerep" and "bit0 \<equiv> STR ''Numeral_Type.bit0''" 
   shows
   "term_of_class.term_of x = 
    Code_Evaluation.App (Code_Evaluation.Const (STR ''Uint.uint.Abs_uint'') (TR (STR ''fun'') [TR (STR ''Word.word'')  [TR (STR ''Uint.dflt_size'') []], TR (STR ''Uint.uint'') []]))
        (term_of_class.term_of (Rep_uint' x))"
   by(simp add: term_of_anything)
 
 text \<open>Important:
   We must prevent the reflection oracle (eval-tac) to 
   use our machine-dependent type.
 \<close>
 
 code_printing
   type_constructor uint \<rightharpoonup>
   (SML) "Word.word" and
   (Haskell) "Uint.Word" and
   (OCaml) "Uint.t" and
   (Scala) "Int" and
   (Eval) "*** \"Error: Machine dependent type\" ***" and
   (Quickcheck) "Word.word" 
 | constant dflt_size_integer \<rightharpoonup>
   (SML) "(IntInf.fromLarge (Int.toLarge Word.wordSize))" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Word.wordSize" and
   (Haskell) "Uint.dflt'_size" and
   (OCaml) "Uint.dflt'_size" and
   (Scala) "Uint.dflt'_size"
 | constant Uint \<rightharpoonup>
   (SML) "Word.fromLargeInt (IntInf.toLarge _)" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Word.fromInt" and
   (Haskell) "(Prelude.fromInteger _ :: Uint.Word)" and
   (Haskell_Quickcheck) "(Prelude.fromInteger (Prelude.toInteger _) :: Uint.Word)" and
   (Scala) "_.intValue"
 | constant Uint_signed \<rightharpoonup>
   (OCaml) "Z.to'_int"
 | constant "0 :: uint" \<rightharpoonup>
   (SML) "(Word.fromInt 0)" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "(Word.fromInt 0)" and
   (Haskell) "(0 :: Uint.Word)" and
   (OCaml) "0" and
   (Scala) "0"
 | constant "1 :: uint" \<rightharpoonup>
   (SML) "(Word.fromInt 1)" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "(Word.fromInt 1)" and
   (Haskell) "(1 :: Uint.Word)" and
   (OCaml) "1" and
   (Scala) "1"
 | constant "plus :: uint \<Rightarrow> _ " \<rightharpoonup>
   (SML) "Word.+ ((_), (_))" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Word.+ ((_), (_))" and
   (Haskell) infixl 6 "+" and
   (OCaml) "Pervasives.(+)" and
   (Scala) infixl 7 "+"
 | constant "uminus :: uint \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word.~" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Word.~" and
   (Haskell) "negate" and
   (OCaml) "Pervasives.(~-)" and
   (Scala) "!(- _)"
 | constant "minus :: uint \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word.- ((_), (_))" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Word.- ((_), (_))" and
   (Haskell) infixl 6 "-" and
   (OCaml) "Pervasives.(-)" and
   (Scala) infixl 7 "-"
 | constant "times :: uint \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word.* ((_), (_))" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Word.* ((_), (_))" and
   (Haskell) infixl 7 "*" and
   (OCaml) "Pervasives.( * )" and
   (Scala) infixl 8 "*"
 | constant "HOL.equal :: uint \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
   (SML) "!((_ : Word.word) = _)" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "!((_ : Word.word) = _)" and
   (Haskell) infix 4 "==" and
   (OCaml) "(Pervasives.(=):Uint.t -> Uint.t -> bool)" and
   (Scala) infixl 5 "=="
 | class_instance uint :: equal \<rightharpoonup>
   (Haskell) -
 | constant "less_eq :: uint \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
   (SML) "Word.<= ((_), (_))" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Word.<= ((_), (_))" and
   (Haskell) infix 4 "<=" and
   (OCaml) "Uint.less'_eq" and
   (Scala) "Uint.less'_eq"
 | constant "less :: uint \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
   (SML) "Word.< ((_), (_))" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Word.< ((_), (_))" and
   (Haskell) infix 4 "<" and
   (OCaml) "Uint.less" and
   (Scala) "Uint.less"
 | constant "Bit_Operations.not :: uint \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word.notb" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Word.notb" and
   (Haskell) "Data'_Bits.complement" and
   (OCaml) "Pervasives.lnot" and
   (Scala) "_.unary'_~"
 | constant "Bit_Operations.and :: uint \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word.andb ((_),/ (_))" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Word.andb ((_),/ (_))" and
   (Haskell) infixl 7 "Data_Bits..&." and
   (OCaml) "Pervasives.(land)" and
   (Scala) infixl 3 "&"
 | constant "Bit_Operations.or :: uint \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word.orb ((_),/ (_))" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Word.orb ((_),/ (_))" and
   (Haskell) infixl 5 "Data_Bits..|." and
   (OCaml) "Pervasives.(lor)" and
   (Scala) infixl 1 "|"
 | constant "Bit_Operations.xor :: uint \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word.xorb ((_),/ (_))" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Word.xorb ((_),/ (_))" and
   (Haskell) "Data'_Bits.xor" and
   (OCaml) "Pervasives.(lxor)" and
   (Scala) infixl 2 "^"
 
 definition uint_divmod :: "uint \<Rightarrow> uint \<Rightarrow> uint \<times> uint" where
   "uint_divmod x y = 
   (if y = 0 then (undefined ((div) :: uint \<Rightarrow> _) x (0 :: uint), undefined ((mod) :: uint \<Rightarrow> _) x (0 :: uint)) 
   else (x div y, x mod y))"
 
 definition uint_div :: "uint \<Rightarrow> uint \<Rightarrow> uint" 
 where "uint_div x y = fst (uint_divmod x y)"
 
 definition uint_mod :: "uint \<Rightarrow> uint \<Rightarrow> uint" 
 where "uint_mod x y = snd (uint_divmod x y)"
 
 lemma div_uint_code [code]: "x div y = (if y = 0 then 0 else uint_div x y)"
 including undefined_transfer unfolding uint_divmod_def uint_div_def
 by transfer(simp add: word_div_def)
 
 lemma mod_uint_code [code]: "x mod y = (if y = 0 then x else uint_mod x y)"
 including undefined_transfer unfolding uint_mod_def uint_divmod_def
 by transfer(simp add: word_mod_def)
 
 definition uint_sdiv :: "uint \<Rightarrow> uint \<Rightarrow> uint"
 where [code del]:
   "uint_sdiv x y =
    (if y = 0 then undefined ((div) :: uint \<Rightarrow> _) x (0 :: uint)
     else Abs_uint (Rep_uint x sdiv Rep_uint y))"
 
 definition div0_uint :: "uint \<Rightarrow> uint"
 where [code del]: "div0_uint x = undefined ((div) :: uint \<Rightarrow> _) x (0 :: uint)"
 declare [[code abort: div0_uint]]
 
 definition mod0_uint :: "uint \<Rightarrow> uint"
 where [code del]: "mod0_uint x = undefined ((mod) :: uint \<Rightarrow> _) x (0 :: uint)"
 declare [[code abort: mod0_uint]]
 
 definition wivs_overflow_uint :: uint 
   where "wivs_overflow_uint \<equiv> push_bit (dflt_size - 1) 1"
 
 lemma Rep_uint_wivs_overflow_uint_eq:
   \<open>Rep_uint wivs_overflow_uint = 2 ^ (dflt_size - Suc 0)\<close>
   by (simp add: wivs_overflow_uint_def one_uint.rep_eq push_bit_uint.rep_eq uint.word_of_power push_bit_eq_mult)
 
 lemma wivs_overflow_uint_greater_eq_0:
   \<open>wivs_overflow_uint > 0\<close>
   apply (simp add: less_uint.rep_eq zero_uint.rep_eq Rep_uint_wivs_overflow_uint_eq)
   apply transfer
   apply (simp add: take_bit_push_bit push_bit_eq_mult)
   done
 
 lemma uint_divmod_code [code]:
   "uint_divmod x y =
   (if wivs_overflow_uint \<le> y then if x < y then (0, x) else (1, x - y)
    else if y = 0 then (div0_uint x, mod0_uint x)
    else let q = push_bit 1 (uint_sdiv (drop_bit 1 x) y);
             r = x - q * y
         in if r \<ge> y then (q + 1, r - y) else (q, r))"
 proof (cases \<open>y = 0\<close>)
   case True
   moreover have \<open>x \<ge> 0\<close>
     by transfer simp
   moreover note wivs_overflow_uint_greater_eq_0
   ultimately show ?thesis
     by (auto simp add: uint_divmod_def div0_uint_def mod0_uint_def not_less)
 next
   case False
   then show ?thesis
     including undefined_transfer 
     unfolding uint_divmod_def uint_sdiv_def div0_uint_def mod0_uint_def
       wivs_overflow_uint_def
     apply transfer
     apply (simp add: divmod_via_sdivmod push_bit_of_1)
     done
 qed
 
 lemma uint_sdiv_code [code]:
   "Rep_uint (uint_sdiv x y) =
    (if y = 0 then Rep_uint (undefined ((div) :: uint \<Rightarrow> _) x (0 :: uint))
     else Rep_uint x sdiv Rep_uint y)"
 unfolding uint_sdiv_def by(simp add: Abs_uint_inverse)
 
 text \<open>
   Note that we only need a translation for signed division, but not for the remainder
   because @{thm uint_divmod_code} computes both with division only.
 \<close>
 
 code_printing
   constant uint_div \<rightharpoonup>
   (SML) "Word.div ((_), (_))" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Word.div ((_), (_))" and
   (Haskell) "Prelude.div"
 | constant uint_mod \<rightharpoonup>
   (SML) "Word.mod ((_), (_))" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Word.mod ((_), (_))" and
   (Haskell) "Prelude.mod"
 | constant uint_divmod \<rightharpoonup>
   (Haskell) "divmod"
 | constant uint_sdiv \<rightharpoonup>
   (OCaml) "Pervasives.('/)" and
   (Scala) "_ '/ _"
 
 definition uint_test_bit :: "uint \<Rightarrow> integer \<Rightarrow> bool"
 where [code del]:
   "uint_test_bit x n =
   (if n < 0 \<or> dflt_size_integer \<le> n then undefined (bit :: uint \<Rightarrow> _) x n
    else bit x (nat_of_integer n))"
 
 lemma test_bit_uint_code [code]:
   "bit x n \<longleftrightarrow> n < dflt_size \<and> uint_test_bit x (integer_of_nat n)"
   including undefined_transfer integer.lifting unfolding uint_test_bit_def
   by (transfer, simp, transfer, simp)
 
 lemma uint_test_bit_code [code]:
   "uint_test_bit w n =
   (if n < 0 \<or> dflt_size_integer \<le> n then undefined (bit :: uint \<Rightarrow> _) w n else bit (Rep_uint w) (nat_of_integer n))"
   unfolding uint_test_bit_def by(simp add: bit_uint.rep_eq)
 
 code_printing constant uint_test_bit \<rightharpoonup>
   (SML) "Uint.test'_bit" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Uint.test'_bit" and
   (Haskell) "Data'_Bits.testBitBounded" and
   (OCaml) "Uint.test'_bit" and
   (Scala) "Uint.test'_bit"
 
 definition uint_set_bit :: "uint \<Rightarrow> integer \<Rightarrow> bool \<Rightarrow> uint"
 where [code del]:
   "uint_set_bit x n b =
   (if n < 0 \<or> dflt_size_integer \<le> n then undefined (set_bit :: uint \<Rightarrow> _) x n b
    else set_bit x (nat_of_integer n) b)"
 
 lemma set_bit_uint_code [code]:
   "set_bit x n b = (if n < dflt_size then uint_set_bit x (integer_of_nat n) b else x)"
   including undefined_transfer integer.lifting unfolding uint_set_bit_def
   by (transfer) (auto cong: conj_cong simp add: not_less set_bit_beyond word_size)
 
 lemma uint_set_bit_code [code]:
   "Rep_uint (uint_set_bit w n b) = 
   (if n < 0 \<or> dflt_size_integer \<le> n then Rep_uint (undefined (set_bit :: uint \<Rightarrow> _) w n b)
    else set_bit (Rep_uint w) (nat_of_integer n) b)"
 including undefined_transfer integer.lifting unfolding uint_set_bit_def by transfer simp
 
 code_printing constant uint_set_bit \<rightharpoonup>
   (SML) "Uint.set'_bit" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Uint.set'_bit" and
   (Haskell) "Data'_Bits.setBitBounded" and
   (OCaml) "Uint.set'_bit" and
   (Scala) "Uint.set'_bit"
 
 definition uint_shiftl :: "uint \<Rightarrow> integer \<Rightarrow> uint"
 where [code del]:
   "uint_shiftl x n = (if n < 0 \<or> dflt_size_integer \<le> n then undefined (push_bit :: nat \<Rightarrow> uint \<Rightarrow> _) x n else push_bit (nat_of_integer n) x)"
 
 lemma shiftl_uint_code [code]: "push_bit n x = (if n < dflt_size then uint_shiftl x (integer_of_nat n) else 0)"
   including undefined_transfer integer.lifting unfolding uint_shiftl_def
   by (transfer fixing: n) simp
 
 lemma uint_shiftl_code [code]:
   "Rep_uint (uint_shiftl w n) =
   (if n < 0 \<or> dflt_size_integer \<le> n then Rep_uint (undefined (push_bit :: nat \<Rightarrow> uint \<Rightarrow> _) w n) else push_bit (nat_of_integer n) (Rep_uint w))"
   including undefined_transfer integer.lifting unfolding uint_shiftl_def by transfer simp
 
 code_printing constant uint_shiftl \<rightharpoonup>
   (SML) "Uint.shiftl" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Uint.shiftl" and
   (Haskell) "Data'_Bits.shiftlBounded" and
   (OCaml) "Uint.shiftl" and
   (Scala) "Uint.shiftl"
 
 definition uint_shiftr :: "uint \<Rightarrow> integer \<Rightarrow> uint"
 where [code del]:
   "uint_shiftr x n = (if n < 0 \<or> dflt_size_integer \<le> n then undefined (drop_bit :: nat \<Rightarrow> uint \<Rightarrow> _) x n else drop_bit (nat_of_integer n) x)"
 
 lemma shiftr_uint_code [code]: "drop_bit n x = (if n < dflt_size then uint_shiftr x (integer_of_nat n) else 0)"
   including undefined_transfer integer.lifting unfolding uint_shiftr_def
   by (transfer fixing: n) simp
 
 lemma uint_shiftr_code [code]:
   "Rep_uint (uint_shiftr w n) =
   (if n < 0 \<or> dflt_size_integer \<le> n then Rep_uint (undefined (drop_bit :: nat \<Rightarrow> uint \<Rightarrow> _) w n) else drop_bit (nat_of_integer n) (Rep_uint w))"
   including undefined_transfer integer.lifting unfolding uint_shiftr_def by transfer simp
 
 code_printing constant uint_shiftr \<rightharpoonup>
   (SML) "Uint.shiftr" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Uint.shiftr" and
   (Haskell) "Data'_Bits.shiftrBounded" and
   (OCaml) "Uint.shiftr" and
   (Scala) "Uint.shiftr"
 
 definition uint_sshiftr :: "uint \<Rightarrow> integer \<Rightarrow> uint"
 where [code del]:
   "uint_sshiftr x n =
   (if n < 0 \<or> dflt_size_integer \<le> n then undefined signed_drop_bit_uint n x else signed_drop_bit_uint (nat_of_integer n) x)"
 
 lemma sshiftr_uint_code [code]:
   "signed_drop_bit_uint n x = 
   (if n < dflt_size then uint_sshiftr x (integer_of_nat n) else 
     if bit x wivs_index then -1 else 0)"
 including undefined_transfer integer.lifting unfolding uint_sshiftr_def
 by transfer(simp add: not_less signed_drop_bit_beyond word_size wivs_index_def)
 
 lemma uint_sshiftr_code [code]:
   "Rep_uint (uint_sshiftr w n) =
   (if n < 0 \<or> dflt_size_integer \<le> n then Rep_uint (undefined signed_drop_bit_uint n w) else signed_drop_bit (nat_of_integer n) (Rep_uint w))"
 including undefined_transfer integer.lifting unfolding uint_sshiftr_def by transfer simp
 
 code_printing constant uint_sshiftr \<rightharpoonup>
   (SML) "Uint.shiftr'_signed" and
   (Eval) "(raise (Fail \"Machine dependent code\"))" and
   (Quickcheck) "Uint.shiftr'_signed" and
   (Haskell) 
     "(Prelude.fromInteger (Prelude.toInteger (Data'_Bits.shiftrBounded (Prelude.fromInteger (Prelude.toInteger _) :: Uint.Int) _)) :: Uint.Word)" and
   (OCaml) "Uint.shiftr'_signed" and
   (Scala) "Uint.shiftr'_signed"
 
 lemma uint_msb_test_bit: "msb x \<longleftrightarrow> bit (x :: uint) wivs_index"
   by transfer (simp add: msb_word_iff_bit wivs_index_def)
 
 lemma msb_uint_code [code]: "msb x \<longleftrightarrow> uint_test_bit x wivs_index_integer"
   apply(simp add: uint_test_bit_def uint_msb_test_bit 
   wivs_index_integer_code dflt_size_integer_def wivs_index_def)
   by (metis (full_types) One_nat_def dflt_size(2) less_iff_diff_less_0 
     nat_of_integer_of_nat of_nat_1 of_nat_diff of_nat_less_0_iff wivs_index_def)
 
 lemma uint_of_int_code [code]: "uint_of_int i = (BITS n. bit i n)"
   by transfer (simp add: word_of_int_conv_set_bits)
 
 
 section \<open>Quickcheck setup\<close>
 
 definition uint_of_natural :: "natural \<Rightarrow> uint"
 where "uint_of_natural x \<equiv> Uint (integer_of_natural x)"
 
 instantiation uint :: "{random, exhaustive, full_exhaustive}" begin
 definition "random_uint \<equiv> qc_random_cnv uint_of_natural"
 definition "exhaustive_uint \<equiv> qc_exhaustive_cnv uint_of_natural"
 definition "full_exhaustive_uint \<equiv> qc_full_exhaustive_cnv uint_of_natural"
 instance ..
 end
 
 instantiation uint :: narrowing begin
 
 interpretation quickcheck_narrowing_samples
   "\<lambda>i. (Uint i, Uint (- i))" "0"
   "Typerep.Typerep (STR ''Uint.uint'') []" .
 
 definition "narrowing_uint d = qc_narrowing_drawn_from (narrowing_samples d) d"
 declare [[code drop: "partial_term_of :: uint itself \<Rightarrow> _"]]
 lemmas partial_term_of_uint [code] = partial_term_of_code
 
 instance ..
 end
 
 end
diff --git a/thys/Native_Word/Uint16.thy b/thys/Native_Word/Uint16.thy
--- a/thys/Native_Word/Uint16.thy
+++ b/thys/Native_Word/Uint16.thy
@@ -1,536 +1,537 @@
 (*  Title:      Uint16.thy
     Author:     Andreas Lochbihler, ETH Zurich
 *)
 
 chapter \<open>Unsigned words of 16 bits\<close>
 
 theory Uint16 imports
-  Code_Target_Word_Base Word_Type_Copies
+  Word_Type_Copies
+  Code_Target_Integer_Bit
 begin
 
 text \<open>
   Restriction for ML code generation:
   This theory assumes that the ML system provides a Word16
   implementation (mlton does, but PolyML 5.5 does not).
   Therefore, the code setup lives in the target \<open>SML_word\<close>
   rather than \<open>SML\<close>.  This ensures that code generation still
   works as long as \<open>uint16\<close> is not involved.
   For the target \<open>SML\<close> itself, no special code generation 
   for this type is set up. Nevertheless, it should work by emulation via \<^typ>\<open>16 word\<close>
-  if the theory \<open>Code_Target_Bits_Int\<close> is imported.
+  if the theory \<^text>\<open>Code_Target_Int_Bit\<close> is imported.
 
   Restriction for OCaml code generation:
   OCaml does not provide an int16 type, so no special code generation 
   for this type is set up.
 \<close>
 
 section \<open>Type definition and primitive operations\<close>
 
 typedef uint16 = \<open>UNIV :: 16 word set\<close> ..
 
 global_interpretation uint16: word_type_copy Abs_uint16 Rep_uint16
   using type_definition_uint16 by (rule word_type_copy.intro)
 
 setup_lifting type_definition_uint16
 
 declare uint16.of_word_of [code abstype]
 
 declare Quotient_uint16 [transfer_rule]
 
 instantiation uint16 :: \<open>{comm_ring_1, semiring_modulo, equal, linorder}\<close>
 begin
 
 lift_definition zero_uint16 :: uint16 is 0 .
 lift_definition one_uint16 :: uint16 is 1 .
 lift_definition plus_uint16 :: \<open>uint16 \<Rightarrow> uint16 \<Rightarrow> uint16\<close> is \<open>(+)\<close> .
 lift_definition uminus_uint16 :: \<open>uint16 \<Rightarrow> uint16\<close> is uminus .
 lift_definition minus_uint16 :: \<open>uint16 \<Rightarrow> uint16 \<Rightarrow> uint16\<close> is \<open>(-)\<close> .
 lift_definition times_uint16 :: \<open>uint16 \<Rightarrow> uint16 \<Rightarrow> uint16\<close> is \<open>(*)\<close> .
 lift_definition divide_uint16 :: \<open>uint16 \<Rightarrow> uint16 \<Rightarrow> uint16\<close> is \<open>(div)\<close> .
 lift_definition modulo_uint16 :: \<open>uint16 \<Rightarrow> uint16 \<Rightarrow> uint16\<close> is \<open>(mod)\<close> .
 lift_definition equal_uint16 :: \<open>uint16 \<Rightarrow> uint16 \<Rightarrow> bool\<close> is \<open>HOL.equal\<close> .
 lift_definition less_eq_uint16 :: \<open>uint16 \<Rightarrow> uint16 \<Rightarrow> bool\<close> is \<open>(\<le>)\<close> .
 lift_definition less_uint16 :: \<open>uint16 \<Rightarrow> uint16 \<Rightarrow> bool\<close> is \<open>(<)\<close> .
 
 global_interpretation uint16: word_type_copy_ring Abs_uint16 Rep_uint16
   by standard (fact zero_uint16.rep_eq one_uint16.rep_eq
     plus_uint16.rep_eq uminus_uint16.rep_eq minus_uint16.rep_eq
     times_uint16.rep_eq divide_uint16.rep_eq modulo_uint16.rep_eq
     equal_uint16.rep_eq less_eq_uint16.rep_eq less_uint16.rep_eq)+
 
 instance proof -
   show \<open>OFCLASS(uint16, comm_ring_1_class)\<close>
     by (rule uint16.of_class_comm_ring_1)
   show \<open>OFCLASS(uint16, semiring_modulo_class)\<close>
     by (fact uint16.of_class_semiring_modulo)
   show \<open>OFCLASS(uint16, equal_class)\<close>
     by (fact uint16.of_class_equal)
   show \<open>OFCLASS(uint16, linorder_class)\<close>
     by (fact uint16.of_class_linorder)
 qed
 
 end
 
 instantiation uint16 :: ring_bit_operations
 begin
 
 lift_definition bit_uint16 :: \<open>uint16 \<Rightarrow> nat \<Rightarrow> bool\<close> is bit .
 lift_definition not_uint16 :: \<open>uint16 \<Rightarrow> uint16\<close> is \<open>Bit_Operations.not\<close> .
 lift_definition and_uint16 :: \<open>uint16 \<Rightarrow> uint16 \<Rightarrow> uint16\<close> is \<open>Bit_Operations.and\<close> .
 lift_definition or_uint16 :: \<open>uint16 \<Rightarrow> uint16 \<Rightarrow> uint16\<close> is \<open>Bit_Operations.or\<close> .
 lift_definition xor_uint16 :: \<open>uint16 \<Rightarrow> uint16 \<Rightarrow> uint16\<close> is \<open>Bit_Operations.xor\<close> .
 lift_definition mask_uint16 :: \<open>nat \<Rightarrow> uint16\<close> is mask .
 lift_definition push_bit_uint16 :: \<open>nat \<Rightarrow> uint16 \<Rightarrow> uint16\<close> is push_bit .
 lift_definition drop_bit_uint16 :: \<open>nat \<Rightarrow> uint16 \<Rightarrow> uint16\<close> is drop_bit .
 lift_definition signed_drop_bit_uint16 :: \<open>nat \<Rightarrow> uint16 \<Rightarrow> uint16\<close> is signed_drop_bit .
 lift_definition take_bit_uint16 :: \<open>nat \<Rightarrow> uint16 \<Rightarrow> uint16\<close> is take_bit .
 lift_definition set_bit_uint16 :: \<open>nat \<Rightarrow> uint16 \<Rightarrow> uint16\<close> is Bit_Operations.set_bit .
 lift_definition unset_bit_uint16 :: \<open>nat \<Rightarrow> uint16 \<Rightarrow> uint16\<close> is unset_bit .
 lift_definition flip_bit_uint16 :: \<open>nat \<Rightarrow> uint16 \<Rightarrow> uint16\<close> is flip_bit .
 
 global_interpretation uint16: word_type_copy_bits Abs_uint16 Rep_uint16 signed_drop_bit_uint16
   by standard (fact bit_uint16.rep_eq not_uint16.rep_eq and_uint16.rep_eq or_uint16.rep_eq xor_uint16.rep_eq
     mask_uint16.rep_eq push_bit_uint16.rep_eq drop_bit_uint16.rep_eq signed_drop_bit_uint16.rep_eq take_bit_uint16.rep_eq
     set_bit_uint16.rep_eq unset_bit_uint16.rep_eq flip_bit_uint16.rep_eq)+
 
 instance
   by (fact uint16.of_class_ring_bit_operations)
 
 end
 
 lift_definition uint16_of_nat :: \<open>nat \<Rightarrow> uint16\<close>
   is word_of_nat .
 
 lift_definition nat_of_uint16 :: \<open>uint16 \<Rightarrow> nat\<close>
   is unat .
 
 lift_definition uint16_of_int :: \<open>int \<Rightarrow> uint16\<close>
   is word_of_int .
 
 lift_definition int_of_uint16 :: \<open>uint16 \<Rightarrow> int\<close>
   is uint .
 
 context
   includes integer.lifting
 begin
 
 lift_definition Uint16 :: \<open>integer \<Rightarrow> uint16\<close>
   is word_of_int .
 
 lift_definition integer_of_uint16 :: \<open>uint16 \<Rightarrow> integer\<close>
   is uint .
 
 end
 
 global_interpretation uint16: word_type_copy_more Abs_uint16 Rep_uint16 signed_drop_bit_uint16
   uint16_of_nat nat_of_uint16 uint16_of_int int_of_uint16 Uint16 integer_of_uint16
   apply standard
        apply (simp_all add: uint16_of_nat.rep_eq nat_of_uint16.rep_eq
          uint16_of_int.rep_eq int_of_uint16.rep_eq
          Uint16.rep_eq integer_of_uint16.rep_eq integer_eq_iff)
   done
 
 instantiation uint16 :: "{size, msb, lsb, set_bit, bit_comprehension}"
 begin
 
 lift_definition size_uint16 :: \<open>uint16 \<Rightarrow> nat\<close> is size .
 
 lift_definition msb_uint16 :: \<open>uint16 \<Rightarrow> bool\<close> is msb .
 lift_definition lsb_uint16 :: \<open>uint16 \<Rightarrow> bool\<close> is lsb .
 
 text \<open>Workaround: avoid name space clash by spelling out \<^text>\<open>lift_definition\<close> explicitly.\<close>
 
 definition set_bit_uint16 :: \<open>uint16 \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> uint16\<close>
   where set_bit_uint16_eq: \<open>set_bit_uint16 a n b = (if b then Bit_Operations.set_bit else unset_bit) n a\<close>
 
 context
   includes lifting_syntax
 begin
 
 lemma set_bit_uint16_transfer [transfer_rule]:
   \<open>(cr_uint16 ===> (=) ===> (\<longleftrightarrow>) ===> cr_uint16) Generic_set_bit.set_bit Generic_set_bit.set_bit\<close>
   by (simp only: set_bit_eq [abs_def] set_bit_uint16_eq [abs_def]) transfer_prover
 
 end
 
 lift_definition set_bits_uint16 :: \<open>(nat \<Rightarrow> bool) \<Rightarrow> uint16\<close> is set_bits .
 lift_definition set_bits_aux_uint16 :: \<open>(nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> uint16 \<Rightarrow> uint16\<close> is set_bits_aux .
 
 global_interpretation uint16: word_type_copy_misc Abs_uint16 Rep_uint16 signed_drop_bit_uint16
   uint16_of_nat nat_of_uint16 uint16_of_int int_of_uint16 Uint16 integer_of_uint16 16 set_bits_aux_uint16
   by (standard; transfer) simp_all
 
 instance using uint16.of_class_bit_comprehension
   uint16.of_class_set_bit uint16.of_class_lsb
   by simp_all standard
 
 end
 
 section \<open>Code setup\<close>
 
 code_printing code_module Uint16 \<rightharpoonup> (SML_word)
 \<open>(* Test that words can handle numbers between 0 and 15 *)
 val _ = if 4 <= Word.wordSize then () else raise (Fail ("wordSize less than 4"));
 
 structure Uint16 : sig
   val set_bit : Word16.word -> IntInf.int -> bool -> Word16.word
   val shiftl : Word16.word -> IntInf.int -> Word16.word
   val shiftr : Word16.word -> IntInf.int -> Word16.word
   val shiftr_signed : Word16.word -> IntInf.int -> Word16.word
   val test_bit : Word16.word -> IntInf.int -> bool
 end = struct
 
 fun set_bit x n b =
   let val mask = Word16.<< (0wx1, Word.fromLargeInt (IntInf.toLarge n))
   in if b then Word16.orb (x, mask)
      else Word16.andb (x, Word16.notb mask)
   end
 
 fun shiftl x n =
   Word16.<< (x, Word.fromLargeInt (IntInf.toLarge n))
 
 fun shiftr x n =
   Word16.>> (x, Word.fromLargeInt (IntInf.toLarge n))
 
 fun shiftr_signed x n =
   Word16.~>> (x, Word.fromLargeInt (IntInf.toLarge n))
 
 fun test_bit x n =
   Word16.andb (x, Word16.<< (0wx1, Word.fromLargeInt (IntInf.toLarge n))) <> Word16.fromInt 0
 
 end; (* struct Uint16 *)\<close>
 code_reserved SML_word Uint16
 
 code_printing code_module Uint16 \<rightharpoonup> (Haskell)
  \<open>module Uint16(Int16, Word16) where
 
   import Data.Int(Int16)
   import Data.Word(Word16)\<close>
 code_reserved Haskell Uint16
 
 text \<open>Scala provides unsigned 16-bit numbers as Char.\<close>
 
 code_printing code_module Uint16 \<rightharpoonup> (Scala)
 \<open>object Uint16 {
 
 def set_bit(x: scala.Char, n: BigInt, b: Boolean) : scala.Char =
   if (b)
     (x | (1.toChar << n.intValue)).toChar
   else
     (x & (1.toChar << n.intValue).unary_~).toChar
 
 def shiftl(x: scala.Char, n: BigInt) : scala.Char = (x << n.intValue).toChar
 
 def shiftr(x: scala.Char, n: BigInt) : scala.Char = (x >>> n.intValue).toChar
 
 def shiftr_signed(x: scala.Char, n: BigInt) : scala.Char = (x.toShort >> n.intValue).toChar
 
 def test_bit(x: scala.Char, n: BigInt) : Boolean = (x & (1.toChar << n.intValue)) != 0
 
 } /* object Uint16 */\<close>
 code_reserved Scala Uint16
 
 text \<open>
   Avoid @{term Abs_uint16} in generated code, use @{term Rep_uint16'} instead. 
   The symbolic implementations for code\_simp use @{term Rep_uint16}.
 
   The new destructor @{term Rep_uint16'} is executable.
   As the simplifier is given the [code abstract] equations literally, 
   we cannot implement @{term Rep_uint16} directly, because that makes code\_simp loop.
 
   If code generation raises Match, some equation probably contains @{term Rep_uint16} 
   ([code abstract] equations for @{typ uint16} may use @{term Rep_uint16} because
   these instances will be folded away.)
 
   To convert @{typ "16 word"} values into @{typ uint16}, use @{term "Abs_uint16'"}.
 \<close>
 
 definition Rep_uint16' where [simp]: "Rep_uint16' = Rep_uint16"
 
 lemma Rep_uint16'_transfer [transfer_rule]:
   "rel_fun cr_uint16 (=) (\<lambda>x. x) Rep_uint16'"
 unfolding Rep_uint16'_def by(rule uint16.rep_transfer)
 
 lemma Rep_uint16'_code [code]: "Rep_uint16' x = (BITS n. bit x n)"
   by transfer (simp add: set_bits_bit_eq)
 
 lift_definition Abs_uint16' :: "16 word \<Rightarrow> uint16" is "\<lambda>x :: 16 word. x" .
 
 lemma Abs_uint16'_code [code]:
   "Abs_uint16' x = Uint16 (integer_of_int (uint x))"
 including integer.lifting by transfer simp
 
 declare [[code drop: "term_of_class.term_of :: uint16 \<Rightarrow> _"]]
 
 lemma term_of_uint16_code [code]:
   defines "TR \<equiv> typerep.Typerep" and "bit0 \<equiv> STR ''Numeral_Type.bit0''" shows
   "term_of_class.term_of x = 
    Code_Evaluation.App (Code_Evaluation.Const (STR ''Uint16.uint16.Abs_uint16'') (TR (STR ''fun'') [TR (STR ''Word.word'') [TR bit0 [TR bit0 [TR bit0 [TR bit0 [TR (STR ''Numeral_Type.num1'') []]]]]], TR (STR ''Uint16.uint16'') []]))
        (term_of_class.term_of (Rep_uint16' x))"
 by(simp add: term_of_anything)
 
 lemma Uint16_code [code]: "Rep_uint16 (Uint16 i) = word_of_int (int_of_integer_symbolic i)"
 unfolding Uint16_def int_of_integer_symbolic_def by(simp add: Abs_uint16_inverse)
 
 code_printing
   type_constructor uint16 \<rightharpoonup>
   (SML_word) "Word16.word" and
   (Haskell) "Uint16.Word16" and
   (Scala) "scala.Char"
 | constant Uint16 \<rightharpoonup>
   (SML_word) "Word16.fromLargeInt (IntInf.toLarge _)" and
   (Haskell) "(Prelude.fromInteger _ :: Uint16.Word16)" and
   (Haskell_Quickcheck) "(Prelude.fromInteger (Prelude.toInteger _) :: Uint16.Word16)" and
   (Scala) "_.charValue"
 | constant "0 :: uint16" \<rightharpoonup>
   (SML_word) "(Word16.fromInt 0)" and
   (Haskell) "(0 :: Uint16.Word16)" and
   (Scala) "0"
 | constant "1 :: uint16" \<rightharpoonup>
   (SML_word) "(Word16.fromInt 1)" and
   (Haskell) "(1 :: Uint16.Word16)" and
   (Scala) "1"
 | constant "plus :: uint16 \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
   (SML_word) "Word16.+ ((_), (_))" and
   (Haskell) infixl 6 "+" and
   (Scala) "(_ +/ _).toChar"
 | constant "uminus :: uint16 \<Rightarrow> _" \<rightharpoonup>
   (SML_word) "Word16.~" and
   (Haskell) "negate" and
   (Scala) "(- _).toChar"
 | constant "minus :: uint16 \<Rightarrow> _" \<rightharpoonup>
   (SML_word) "Word16.- ((_), (_))" and
   (Haskell) infixl 6 "-" and
   (Scala) "(_ -/ _).toChar"
 | constant "times :: uint16 \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
   (SML_word) "Word16.* ((_), (_))" and
   (Haskell) infixl 7 "*" and
   (Scala) "(_ */ _).toChar"
 | constant "HOL.equal :: uint16 \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
   (SML_word) "!((_ : Word16.word) = _)" and
   (Haskell) infix 4 "==" and
   (Scala) infixl 5 "=="
 | class_instance uint16 :: equal \<rightharpoonup> (Haskell) -
 | constant "less_eq :: uint16 \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
   (SML_word) "Word16.<= ((_), (_))" and
   (Haskell) infix 4 "<=" and
   (Scala) infixl 4 "<="
 | constant "less :: uint16 \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
   (SML_word) "Word16.< ((_), (_))" and
   (Haskell) infix 4 "<" and
   (Scala) infixl 4 "<"
 | constant "Bit_Operations.not :: uint16 \<Rightarrow> _" \<rightharpoonup>
   (SML_word) "Word16.notb" and
   (Haskell) "Data'_Bits.complement" and
   (Scala) "_.unary'_~.toChar"
 | constant "Bit_Operations.and :: uint16 \<Rightarrow> _" \<rightharpoonup>
   (SML_word) "Word16.andb ((_),/ (_))" and
   (Haskell) infixl 7 "Data_Bits..&." and
   (Scala) "(_ & _).toChar"
 | constant "Bit_Operations.or :: uint16 \<Rightarrow> _" \<rightharpoonup>
   (SML_word) "Word16.orb ((_),/ (_))" and
   (Haskell) infixl 5 "Data_Bits..|." and
   (Scala) "(_ | _).toChar"
 | constant "Bit_Operations.xor :: uint16 \<Rightarrow> _" \<rightharpoonup>
   (SML_word) "Word16.xorb ((_),/ (_))" and
   (Haskell) "Data'_Bits.xor" and
   (Scala) "(_ ^ _).toChar"
 
 definition uint16_div :: "uint16 \<Rightarrow> uint16 \<Rightarrow> uint16" 
 where "uint16_div x y = (if y = 0 then undefined ((div) :: uint16 \<Rightarrow> _) x (0 :: uint16) else x div y)"
 
 definition uint16_mod :: "uint16 \<Rightarrow> uint16 \<Rightarrow> uint16" 
 where "uint16_mod x y = (if y = 0 then undefined ((mod) :: uint16 \<Rightarrow> _) x (0 :: uint16) else x mod y)"
 
 context includes undefined_transfer begin
 
 lemma div_uint16_code [code]: "x div y = (if y = 0 then 0 else uint16_div x y)"
 unfolding uint16_div_def by transfer (simp add: word_div_def)
 
 lemma mod_uint16_code [code]: "x mod y = (if y = 0 then x else uint16_mod x y)"
 unfolding uint16_mod_def by transfer (simp add: word_mod_def)
 
 lemma uint16_div_code [code]:
   "Rep_uint16 (uint16_div x y) =
   (if y = 0 then Rep_uint16 (undefined ((div) :: uint16 \<Rightarrow> _) x (0 :: uint16)) else Rep_uint16 x div Rep_uint16 y)"
 unfolding uint16_div_def by transfer simp
 
 lemma uint16_mod_code [code]:
   "Rep_uint16 (uint16_mod x y) =
   (if y = 0 then Rep_uint16 (undefined ((mod) :: uint16 \<Rightarrow> _) x (0 :: uint16)) else Rep_uint16 x mod Rep_uint16 y)"
 unfolding uint16_mod_def by transfer simp
 
 end
 
 code_printing constant uint16_div \<rightharpoonup>
   (SML_word) "Word16.div ((_), (_))" and
   (Haskell) "Prelude.div" and
   (Scala) "(_ '/ _).toChar"
 | constant uint16_mod \<rightharpoonup>
   (SML_word) "Word16.mod ((_), (_))" and
   (Haskell) "Prelude.mod" and
   (Scala) "(_ % _).toChar"
 
 definition uint16_test_bit :: "uint16 \<Rightarrow> integer \<Rightarrow> bool"
 where [code del]:
   "uint16_test_bit x n =
   (if n < 0 \<or> 15 < n then undefined (bit :: uint16 \<Rightarrow> _) x n
    else bit x (nat_of_integer n))"
 
 lemma test_bit_uint16_code [code]:
   "bit x n \<longleftrightarrow> n < 16 \<and> uint16_test_bit x (integer_of_nat n)"
   including undefined_transfer integer.lifting unfolding uint16_test_bit_def
   by (transfer, simp, transfer, simp)
 
 lemma uint16_test_bit_code [code]:
   "uint16_test_bit w n =
   (if n < 0 \<or> 15 < n then undefined (bit :: uint16 \<Rightarrow> _) w n else bit (Rep_uint16 w) (nat_of_integer n))"
   unfolding uint16_test_bit_def by (simp add: bit_uint16.rep_eq)
 
 code_printing constant uint16_test_bit \<rightharpoonup>
   (SML_word) "Uint16.test'_bit" and
   (Haskell) "Data'_Bits.testBitBounded" and
   (Scala) "Uint16.test'_bit"
 
 definition uint16_set_bit :: "uint16 \<Rightarrow> integer \<Rightarrow> bool \<Rightarrow> uint16"
 where [code del]:
   "uint16_set_bit x n b =
   (if n < 0 \<or> 15 < n then undefined (set_bit :: uint16 \<Rightarrow> _) x n b
    else set_bit x (nat_of_integer n) b)"
 
 lemma set_bit_uint16_code [code]:
   "set_bit x n b = (if n < 16 then uint16_set_bit x (integer_of_nat n) b else x)"
 including undefined_transfer integer.lifting unfolding uint16_set_bit_def
 by(transfer)(auto cong: conj_cong simp add: not_less set_bit_beyond word_size)
 
 lemma uint16_set_bit_code [code]:
   "Rep_uint16 (uint16_set_bit w n b) = 
   (if n < 0 \<or> 15 < n then Rep_uint16 (undefined (set_bit :: uint16 \<Rightarrow> _) w n b)
    else set_bit (Rep_uint16 w) (nat_of_integer n) b)"
 including undefined_transfer unfolding uint16_set_bit_def by transfer simp
 
 code_printing constant uint16_set_bit \<rightharpoonup>
   (SML_word) "Uint16.set'_bit" and
   (Haskell) "Data'_Bits.setBitBounded" and
   (Scala) "Uint16.set'_bit"
 
 definition uint16_shiftl :: "uint16 \<Rightarrow> integer \<Rightarrow> uint16"
 where [code del]:
   "uint16_shiftl x n = (if n < 0 \<or> 16 \<le> n then undefined (push_bit :: nat \<Rightarrow> uint16 \<Rightarrow> _) x n else push_bit (nat_of_integer n) x)"
 
 lemma shiftl_uint16_code [code]: "push_bit n x = (if n < 16 then uint16_shiftl x (integer_of_nat n) else 0)"
   including undefined_transfer integer.lifting unfolding uint16_shiftl_def
   by transfer simp
 
 lemma uint16_shiftl_code [code]:
   "Rep_uint16 (uint16_shiftl w n) =
   (if n < 0 \<or> 16 \<le> n then Rep_uint16 (undefined (push_bit :: nat \<Rightarrow> uint16 \<Rightarrow> _) w n)
    else push_bit (nat_of_integer n) (Rep_uint16 w))"
   including undefined_transfer unfolding uint16_shiftl_def
   by transfer simp
 
 code_printing constant uint16_shiftl \<rightharpoonup>
   (SML_word) "Uint16.shiftl" and
   (Haskell) "Data'_Bits.shiftlBounded" and
   (Scala) "Uint16.shiftl"
 
 definition uint16_shiftr :: "uint16 \<Rightarrow> integer \<Rightarrow> uint16"
 where [code del]:
   "uint16_shiftr x n = (if n < 0 \<or> 16 \<le> n then undefined (drop_bit :: nat \<Rightarrow> uint16 \<Rightarrow> _) x n else drop_bit (nat_of_integer n) x)"
 
 lemma shiftr_uint16_code [code]: "drop_bit n x = (if n < 16 then uint16_shiftr x (integer_of_nat n) else 0)"
   including undefined_transfer integer.lifting unfolding uint16_shiftr_def
   by transfer simp
 
 lemma uint16_shiftr_code [code]:
   "Rep_uint16 (uint16_shiftr w n) =
   (if n < 0 \<or> 16 \<le> n then Rep_uint16 (undefined (drop_bit :: nat \<Rightarrow> uint16 \<Rightarrow> _) w n)
    else drop_bit (nat_of_integer n) (Rep_uint16 w))"
 including undefined_transfer unfolding uint16_shiftr_def by transfer simp
 
 code_printing constant uint16_shiftr \<rightharpoonup>
   (SML_word) "Uint16.shiftr" and
   (Haskell) "Data'_Bits.shiftrBounded" and
   (Scala) "Uint16.shiftr"
 
 definition uint16_sshiftr :: "uint16 \<Rightarrow> integer \<Rightarrow> uint16"
 where [code del]:
   "uint16_sshiftr x n =
   (if n < 0 \<or> 16 \<le> n then undefined signed_drop_bit_uint16 n x else signed_drop_bit_uint16 (nat_of_integer n) x)"
 
 lemma sshiftr_uint16_code [code]:
   "signed_drop_bit_uint16 n x = 
   (if n < 16 then uint16_sshiftr x (integer_of_nat n) else if bit x 15 then -1 else 0)"
   including undefined_transfer integer.lifting unfolding uint16_sshiftr_def
   by transfer (simp add: not_less signed_drop_bit_beyond word_size)
 
 lemma uint16_sshiftr_code [code]:
   "Rep_uint16 (uint16_sshiftr w n) =
   (if n < 0 \<or> 16 \<le> n then Rep_uint16 (undefined signed_drop_bit_uint16 n w)
    else signed_drop_bit (nat_of_integer n) (Rep_uint16 w))"
   including undefined_transfer unfolding uint16_sshiftr_def
   by transfer simp
 
 code_printing constant uint16_sshiftr \<rightharpoonup>
   (SML_word) "Uint16.shiftr'_signed" and
   (Haskell) 
     "(Prelude.fromInteger (Prelude.toInteger (Data'_Bits.shiftrBounded (Prelude.fromInteger (Prelude.toInteger _) :: Uint16.Int16) _)) :: Uint16.Word16)" and
   (Scala) "Uint16.shiftr'_signed"
 
 lemma uint16_msb_test_bit: "msb x \<longleftrightarrow> bit (x :: uint16) 15"
   by transfer (simp add: msb_word_iff_bit)
 
 lemma msb_uint16_code [code]: "msb x \<longleftrightarrow> uint16_test_bit x 15"
   by (simp add: uint16_test_bit_def uint16_msb_test_bit)
 
 lemma uint16_of_int_code [code]: "uint16_of_int i = Uint16 (integer_of_int i)"
 including integer.lifting by transfer simp
 
 lemma int_of_uint16_code [code]:
   "int_of_uint16 x = int_of_integer (integer_of_uint16 x)"
   by (simp add: int_of_uint16.rep_eq integer_of_uint16_def)
 
 lemma uint16_of_nat_code [code]:
   "uint16_of_nat = uint16_of_int \<circ> int"
   by transfer (simp add: fun_eq_iff)
 
 lemma nat_of_uint16_code [code]:
   "nat_of_uint16 x = nat_of_integer (integer_of_uint16 x)"
   unfolding integer_of_uint16_def including integer.lifting by transfer simp
 
 lemma integer_of_uint16_code [code]:
   "integer_of_uint16 n = integer_of_int (uint (Rep_uint16' n))"
 unfolding integer_of_uint16_def by transfer auto
 
 code_printing
   constant "integer_of_uint16" \<rightharpoonup>
   (SML_word) "Word16.toInt _ : IntInf.int" and
   (Haskell) "Prelude.toInteger" and
   (Scala) "BigInt"
 
 section \<open>Quickcheck setup\<close>
 
 definition uint16_of_natural :: "natural \<Rightarrow> uint16"
 where "uint16_of_natural x \<equiv> Uint16 (integer_of_natural x)"
 
 instantiation uint16 :: "{random, exhaustive, full_exhaustive}" begin
 definition "random_uint16 \<equiv> qc_random_cnv uint16_of_natural"
 definition "exhaustive_uint16 \<equiv> qc_exhaustive_cnv uint16_of_natural"
 definition "full_exhaustive_uint16 \<equiv> qc_full_exhaustive_cnv uint16_of_natural"
 instance ..
 end
 
 instantiation uint16 :: narrowing begin
 
 interpretation quickcheck_narrowing_samples
   "\<lambda>i. let x = Uint16 i in (x, 0xFFFF - x)" "0"
   "Typerep.Typerep (STR ''Uint16.uint16'') []" .
 
 definition "narrowing_uint16 d = qc_narrowing_drawn_from (narrowing_samples d) d"
 declare [[code drop: "partial_term_of :: uint16 itself \<Rightarrow> _"]]
 lemmas partial_term_of_uint16 [code] = partial_term_of_code
 
 instance ..
 end
 
 end
diff --git a/thys/Native_Word/Uint32.thy b/thys/Native_Word/Uint32.thy
--- a/thys/Native_Word/Uint32.thy
+++ b/thys/Native_Word/Uint32.thy
@@ -1,691 +1,692 @@
 (*  Title:      Uint32.thy
     Author:     Andreas Lochbihler, ETH Zurich
 *)
 
 chapter \<open>Unsigned words of 32 bits\<close>
 
 theory Uint32 imports
-  Code_Target_Word_Base Word_Type_Copies
+  Word_Type_Copies
+  Code_Target_Integer_Bit
 begin
 
 section \<open>Type definition and primitive operations\<close>
 
 typedef uint32 = \<open>UNIV :: 32 word set\<close> ..
 
 global_interpretation uint32: word_type_copy Abs_uint32 Rep_uint32
   using type_definition_uint32 by (rule word_type_copy.intro)
 
 setup_lifting type_definition_uint32
 
 declare uint32.of_word_of [code abstype]
 
 declare Quotient_uint32 [transfer_rule]
 
 instantiation uint32 :: \<open>{comm_ring_1, semiring_modulo, equal, linorder}\<close>
 begin
 
 lift_definition zero_uint32 :: uint32 is 0 .
 lift_definition one_uint32 :: uint32 is 1 .
 lift_definition plus_uint32 :: \<open>uint32 \<Rightarrow> uint32 \<Rightarrow> uint32\<close> is \<open>(+)\<close> .
 lift_definition uminus_uint32 :: \<open>uint32 \<Rightarrow> uint32\<close> is uminus .
 lift_definition minus_uint32 :: \<open>uint32 \<Rightarrow> uint32 \<Rightarrow> uint32\<close> is \<open>(-)\<close> .
 lift_definition times_uint32 :: \<open>uint32 \<Rightarrow> uint32 \<Rightarrow> uint32\<close> is \<open>(*)\<close> .
 lift_definition divide_uint32 :: \<open>uint32 \<Rightarrow> uint32 \<Rightarrow> uint32\<close> is \<open>(div)\<close> .
 lift_definition modulo_uint32 :: \<open>uint32 \<Rightarrow> uint32 \<Rightarrow> uint32\<close> is \<open>(mod)\<close> .
 lift_definition equal_uint32 :: \<open>uint32 \<Rightarrow> uint32 \<Rightarrow> bool\<close> is \<open>HOL.equal\<close> .
 lift_definition less_eq_uint32 :: \<open>uint32 \<Rightarrow> uint32 \<Rightarrow> bool\<close> is \<open>(\<le>)\<close> .
 lift_definition less_uint32 :: \<open>uint32 \<Rightarrow> uint32 \<Rightarrow> bool\<close> is \<open>(<)\<close> .
 
 global_interpretation uint32: word_type_copy_ring Abs_uint32 Rep_uint32
   by standard (fact zero_uint32.rep_eq one_uint32.rep_eq
     plus_uint32.rep_eq uminus_uint32.rep_eq minus_uint32.rep_eq
     times_uint32.rep_eq divide_uint32.rep_eq modulo_uint32.rep_eq
     equal_uint32.rep_eq less_eq_uint32.rep_eq less_uint32.rep_eq)+
 
 instance proof -
   show \<open>OFCLASS(uint32, comm_ring_1_class)\<close>
     by (rule uint32.of_class_comm_ring_1)
   show \<open>OFCLASS(uint32, semiring_modulo_class)\<close>
     by (fact uint32.of_class_semiring_modulo)
   show \<open>OFCLASS(uint32, equal_class)\<close>
     by (fact uint32.of_class_equal)
   show \<open>OFCLASS(uint32, linorder_class)\<close>
     by (fact uint32.of_class_linorder)
 qed
 
 end
 
 instantiation uint32 :: ring_bit_operations
 begin
 
 lift_definition bit_uint32 :: \<open>uint32 \<Rightarrow> nat \<Rightarrow> bool\<close> is bit .
 lift_definition not_uint32 :: \<open>uint32 \<Rightarrow> uint32\<close> is \<open>Bit_Operations.not\<close> .
 lift_definition and_uint32 :: \<open>uint32 \<Rightarrow> uint32 \<Rightarrow> uint32\<close> is \<open>Bit_Operations.and\<close> .
 lift_definition or_uint32 :: \<open>uint32 \<Rightarrow> uint32 \<Rightarrow> uint32\<close> is \<open>Bit_Operations.or\<close> .
 lift_definition xor_uint32 :: \<open>uint32 \<Rightarrow> uint32 \<Rightarrow> uint32\<close> is \<open>Bit_Operations.xor\<close> .
 lift_definition mask_uint32 :: \<open>nat \<Rightarrow> uint32\<close> is mask .
 lift_definition push_bit_uint32 :: \<open>nat \<Rightarrow> uint32 \<Rightarrow> uint32\<close> is push_bit .
 lift_definition drop_bit_uint32 :: \<open>nat \<Rightarrow> uint32 \<Rightarrow> uint32\<close> is drop_bit .
 lift_definition signed_drop_bit_uint32 :: \<open>nat \<Rightarrow> uint32 \<Rightarrow> uint32\<close> is signed_drop_bit .
 lift_definition take_bit_uint32 :: \<open>nat \<Rightarrow> uint32 \<Rightarrow> uint32\<close> is take_bit .
 lift_definition set_bit_uint32 :: \<open>nat \<Rightarrow> uint32 \<Rightarrow> uint32\<close> is Bit_Operations.set_bit .
 lift_definition unset_bit_uint32 :: \<open>nat \<Rightarrow> uint32 \<Rightarrow> uint32\<close> is unset_bit .
 lift_definition flip_bit_uint32 :: \<open>nat \<Rightarrow> uint32 \<Rightarrow> uint32\<close> is flip_bit .
 
 global_interpretation uint32: word_type_copy_bits Abs_uint32 Rep_uint32 signed_drop_bit_uint32
   by standard (fact bit_uint32.rep_eq not_uint32.rep_eq and_uint32.rep_eq or_uint32.rep_eq xor_uint32.rep_eq
     mask_uint32.rep_eq push_bit_uint32.rep_eq drop_bit_uint32.rep_eq signed_drop_bit_uint32.rep_eq take_bit_uint32.rep_eq
     set_bit_uint32.rep_eq unset_bit_uint32.rep_eq flip_bit_uint32.rep_eq)+
 
 instance
   by (fact uint32.of_class_ring_bit_operations)
 
 end
 
 lift_definition uint32_of_nat :: \<open>nat \<Rightarrow> uint32\<close>
   is word_of_nat .
 
 lift_definition nat_of_uint32 :: \<open>uint32 \<Rightarrow> nat\<close>
   is unat .
 
 lift_definition uint32_of_int :: \<open>int \<Rightarrow> uint32\<close>
   is word_of_int .
 
 lift_definition int_of_uint32 :: \<open>uint32 \<Rightarrow> int\<close>
   is uint .
 
 context
   includes integer.lifting
 begin
 
 lift_definition Uint32 :: \<open>integer \<Rightarrow> uint32\<close>
   is word_of_int .
 
 lift_definition integer_of_uint32 :: \<open>uint32 \<Rightarrow> integer\<close>
   is uint .
 
 end
 
 global_interpretation uint32: word_type_copy_more Abs_uint32 Rep_uint32 signed_drop_bit_uint32
   uint32_of_nat nat_of_uint32 uint32_of_int int_of_uint32 Uint32 integer_of_uint32
   apply standard
        apply (simp_all add: uint32_of_nat.rep_eq nat_of_uint32.rep_eq
          uint32_of_int.rep_eq int_of_uint32.rep_eq
          Uint32.rep_eq integer_of_uint32.rep_eq integer_eq_iff)
   done
 
 instantiation uint32 :: "{size, msb, lsb, set_bit, bit_comprehension}"
 begin
 
 lift_definition size_uint32 :: \<open>uint32 \<Rightarrow> nat\<close> is size .
 
 lift_definition msb_uint32 :: \<open>uint32 \<Rightarrow> bool\<close> is msb .
 lift_definition lsb_uint32 :: \<open>uint32 \<Rightarrow> bool\<close> is lsb .
 
 text \<open>Workaround: avoid name space clash by spelling out \<^text>\<open>lift_definition\<close> explicitly.\<close>
 
 definition set_bit_uint32 :: \<open>uint32 \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> uint32\<close>
   where set_bit_uint32_eq: \<open>set_bit_uint32 a n b = (if b then Bit_Operations.set_bit else unset_bit) n a\<close>
 
 context
   includes lifting_syntax
 begin
 
 lemma set_bit_uint32_transfer [transfer_rule]:
   \<open>(cr_uint32 ===> (=) ===> (\<longleftrightarrow>) ===> cr_uint32) Generic_set_bit.set_bit Generic_set_bit.set_bit\<close>
   by (simp only: set_bit_eq [abs_def] set_bit_uint32_eq [abs_def]) transfer_prover
 
 end
 
 lift_definition set_bits_uint32 :: \<open>(nat \<Rightarrow> bool) \<Rightarrow> uint32\<close> is set_bits .
 lift_definition set_bits_aux_uint32 :: \<open>(nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> uint32 \<Rightarrow> uint32\<close> is set_bits_aux .
 
 global_interpretation uint32: word_type_copy_misc Abs_uint32 Rep_uint32 signed_drop_bit_uint32
   uint32_of_nat nat_of_uint32 uint32_of_int int_of_uint32 Uint32 integer_of_uint32 32 set_bits_aux_uint32
   by (standard; transfer) simp_all
 
 instance using uint32.of_class_bit_comprehension
   uint32.of_class_set_bit uint32.of_class_lsb
   by simp_all standard
 
 end
 
 section \<open>Code setup\<close>
 
 code_printing code_module Uint32 \<rightharpoonup> (SML)
 \<open>(* Test that words can handle numbers between 0 and 31 *)
 val _ = if 5 <= Word.wordSize then () else raise (Fail ("wordSize less than 5"));
 
 structure Uint32 : sig
   val set_bit : Word32.word -> IntInf.int -> bool -> Word32.word
   val shiftl : Word32.word -> IntInf.int -> Word32.word
   val shiftr : Word32.word -> IntInf.int -> Word32.word
   val shiftr_signed : Word32.word -> IntInf.int -> Word32.word
   val test_bit : Word32.word -> IntInf.int -> bool
 end = struct
 
 fun set_bit x n b =
   let val mask = Word32.<< (0wx1, Word.fromLargeInt (IntInf.toLarge n))
   in if b then Word32.orb (x, mask)
      else Word32.andb (x, Word32.notb mask)
   end
 
 fun shiftl x n =
   Word32.<< (x, Word.fromLargeInt (IntInf.toLarge n))
 
 fun shiftr x n =
   Word32.>> (x, Word.fromLargeInt (IntInf.toLarge n))
 
 fun shiftr_signed x n =
   Word32.~>> (x, Word.fromLargeInt (IntInf.toLarge n))
 
 fun test_bit x n =
   Word32.andb (x, Word32.<< (0wx1, Word.fromLargeInt (IntInf.toLarge n))) <> Word32.fromInt 0
 
 end; (* struct Uint32 *)\<close>
 code_reserved SML Uint32
 
 code_printing code_module Uint32 \<rightharpoonup> (Haskell)
  \<open>module Uint32(Int32, Word32) where
 
   import Data.Int(Int32)
   import Data.Word(Word32)\<close>
 code_reserved Haskell Uint32
 
 text \<open>
   OCaml and Scala provide only signed 32bit numbers, so we use these and 
   implement sign-sensitive operations like comparisons manually.
 \<close>
 code_printing code_module "Uint32" \<rightharpoonup> (OCaml)
 \<open>module Uint32 : sig
   val less : int32 -> int32 -> bool
   val less_eq : int32 -> int32 -> bool
   val set_bit : int32 -> Z.t -> bool -> int32
   val shiftl : int32 -> Z.t -> int32
   val shiftr : int32 -> Z.t -> int32
   val shiftr_signed : int32 -> Z.t -> int32
   val test_bit : int32 -> Z.t -> bool
 end = struct
 
 (* negative numbers have their highest bit set, 
    so they are greater than positive ones *)
 let less x y =
   if Int32.compare x Int32.zero < 0 then
     Int32.compare y Int32.zero < 0 && Int32.compare x y < 0
   else Int32.compare y Int32.zero < 0 || Int32.compare x y < 0;;
 
 let less_eq x y =
   if Int32.compare x Int32.zero < 0 then
     Int32.compare y Int32.zero < 0 && Int32.compare x y <= 0
   else Int32.compare y Int32.zero < 0 || Int32.compare x y <= 0;;
 
 let set_bit x n b =
   let mask = Int32.shift_left Int32.one (Z.to_int n)
   in if b then Int32.logor x mask
      else Int32.logand x (Int32.lognot mask);;
 
 let shiftl x n = Int32.shift_left x (Z.to_int n);;
 
 let shiftr x n = Int32.shift_right_logical x (Z.to_int n);;
 
 let shiftr_signed x n = Int32.shift_right x (Z.to_int n);;
 
 let test_bit x n =
   Int32.compare 
     (Int32.logand x (Int32.shift_left Int32.one (Z.to_int n)))
     Int32.zero
   <> 0;;
 
 end;; (*struct Uint32*)\<close>
 code_reserved OCaml Uint32
 
 code_printing code_module Uint32 \<rightharpoonup> (Scala)
 \<open>object Uint32 {
 
 def less(x: Int, y: Int) : Boolean =
   if (x < 0) y < 0 && x < y
   else y < 0 || x < y
 
 def less_eq(x: Int, y: Int) : Boolean =
   if (x < 0) y < 0 && x <= y
   else y < 0 || x <= y
 
 def set_bit(x: Int, n: BigInt, b: Boolean) : Int =
   if (b)
     x | (1 << n.intValue)
   else
     x & (1 << n.intValue).unary_~
 
 def shiftl(x: Int, n: BigInt) : Int = x << n.intValue
 
 def shiftr(x: Int, n: BigInt) : Int = x >>> n.intValue
 
 def shiftr_signed(x: Int, n: BigInt) : Int = x >> n.intValue
 
 def test_bit(x: Int, n: BigInt) : Boolean =
   (x & (1 << n.intValue)) != 0
 
 } /* object Uint32 */\<close>
 code_reserved Scala Uint32
 
 text \<open>
   OCaml's conversion from Big\_int to int32 demands that the value fits int a signed 32-bit integer.
   The following justifies the implementation.
 \<close>
 
 context
   includes bit_operations_syntax
 begin
 
 definition Uint32_signed :: "integer \<Rightarrow> uint32" 
 where "Uint32_signed i = (if i < -(0x80000000) \<or> i \<ge> 0x80000000 then undefined Uint32 i else Uint32 i)"
 
 lemma Uint32_code [code]:
   "Uint32 i = 
   (let i' = i AND 0xFFFFFFFF
    in if bit i' 31 then Uint32_signed (i' - 0x100000000) else Uint32_signed i')"
   including undefined_transfer integer.lifting unfolding Uint32_signed_def
   apply transfer
   apply (subst word_of_int_via_signed)
      apply (auto simp add: push_bit_of_1 mask_eq_exp_minus_1 word_of_int_via_signed cong del: if_cong)
   done
 
 lemma Uint32_signed_code [code]:
   "Rep_uint32 (Uint32_signed i) = 
   (if i < -(0x80000000) \<or> i \<ge> 0x80000000 then Rep_uint32 (undefined Uint32 i) else word_of_int (int_of_integer_symbolic i))"
 unfolding Uint32_signed_def Uint32_def int_of_integer_symbolic_def word_of_integer_def
 by(simp add: Abs_uint32_inverse)
 
 end
 
 text \<open>
   Avoid @{term Abs_uint32} in generated code, use @{term Rep_uint32'} instead. 
   The symbolic implementations for code\_simp use @{term Rep_uint32}.
 
   The new destructor @{term Rep_uint32'} is executable.
   As the simplifier is given the [code abstract] equations literally, 
   we cannot implement @{term Rep_uint32} directly, because that makes code\_simp loop.
 
   If code generation raises Match, some equation probably contains @{term Rep_uint32} 
   ([code abstract] equations for @{typ uint32} may use @{term Rep_uint32} because
   these instances will be folded away.)
 
   To convert @{typ "32 word"} values into @{typ uint32}, use @{term "Abs_uint32'"}.
 \<close>
 
 definition Rep_uint32' where [simp]: "Rep_uint32' = Rep_uint32"
 
 lemma Rep_uint32'_transfer [transfer_rule]:
   "rel_fun cr_uint32 (=) (\<lambda>x. x) Rep_uint32'"
 unfolding Rep_uint32'_def by(rule uint32.rep_transfer)
 
 lemma Rep_uint32'_code [code]: "Rep_uint32' x = (BITS n. bit x n)"
   by transfer (simp add: set_bits_bit_eq) 
 
 lift_definition Abs_uint32' :: "32 word \<Rightarrow> uint32" is "\<lambda>x :: 32 word. x" .
 
 lemma Abs_uint32'_code [code]:
   "Abs_uint32' x = Uint32 (integer_of_int (uint x))"
 including integer.lifting by transfer simp
 
 declare [[code drop: "term_of_class.term_of :: uint32 \<Rightarrow> _"]]
 
 lemma term_of_uint32_code [code]:
   defines "TR \<equiv> typerep.Typerep" and "bit0 \<equiv> STR ''Numeral_Type.bit0''" 
   shows
   "term_of_class.term_of x = 
    Code_Evaluation.App (Code_Evaluation.Const (STR ''Uint32.uint32.Abs_uint32'') (TR (STR ''fun'') [TR (STR ''Word.word'') [TR bit0 [TR bit0 [TR bit0 [TR bit0 [TR bit0 [TR (STR ''Numeral_Type.num1'') []]]]]]], TR (STR ''Uint32.uint32'') []]))
        (term_of_class.term_of (Rep_uint32' x))"
 by(simp add: term_of_anything)
 
 code_printing
   type_constructor uint32 \<rightharpoonup>
   (SML) "Word32.word" and
   (Haskell) "Uint32.Word32" and
   (OCaml) "int32" and
   (Scala) "Int" and
   (Eval) "Word32.word"
 | constant Uint32 \<rightharpoonup>
   (SML) "Word32.fromLargeInt (IntInf.toLarge _)" and
   (Haskell) "(Prelude.fromInteger _ :: Uint32.Word32)" and
   (Haskell_Quickcheck) "(Prelude.fromInteger (Prelude.toInteger _) :: Uint32.Word32)" and
   (Scala) "_.intValue"
 | constant Uint32_signed \<rightharpoonup>
   (OCaml) "Z.to'_int32"
 | constant "0 :: uint32" \<rightharpoonup>
   (SML) "(Word32.fromInt 0)" and
   (Haskell) "(0 :: Uint32.Word32)" and
   (OCaml) "Int32.zero" and
   (Scala) "0"
 | constant "1 :: uint32" \<rightharpoonup>
   (SML) "(Word32.fromInt 1)" and
   (Haskell) "(1 :: Uint32.Word32)" and
   (OCaml) "Int32.one" and
   (Scala) "1"
 | constant "plus :: uint32 \<Rightarrow> _ " \<rightharpoonup>
   (SML) "Word32.+ ((_), (_))" and
   (Haskell) infixl 6 "+" and
   (OCaml) "Int32.add" and
   (Scala) infixl 7 "+"
 | constant "uminus :: uint32 \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word32.~" and
   (Haskell) "negate" and
   (OCaml) "Int32.neg" and
   (Scala) "!(- _)"
 | constant "minus :: uint32 \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word32.- ((_), (_))" and
   (Haskell) infixl 6 "-" and
   (OCaml) "Int32.sub" and
   (Scala) infixl 7 "-"
 | constant "times :: uint32 \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word32.* ((_), (_))" and
   (Haskell) infixl 7 "*" and
   (OCaml) "Int32.mul" and
   (Scala) infixl 8 "*"
 | constant "HOL.equal :: uint32 \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
   (SML) "!((_ : Word32.word) = _)" and
   (Haskell) infix 4 "==" and
   (OCaml) "(Int32.compare _ _ = 0)" and
   (Scala) infixl 5 "=="
 | class_instance uint32 :: equal \<rightharpoonup>
   (Haskell) -
 | constant "less_eq :: uint32 \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
   (SML) "Word32.<= ((_), (_))" and
   (Haskell) infix 4 "<=" and
   (OCaml) "Uint32.less'_eq" and
   (Scala) "Uint32.less'_eq"
 | constant "less :: uint32 \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
   (SML) "Word32.< ((_), (_))" and
   (Haskell) infix 4 "<" and
   (OCaml) "Uint32.less" and
   (Scala) "Uint32.less"
 | constant "Bit_Operations.not :: uint32 \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word32.notb" and
   (Haskell) "Data'_Bits.complement" and
   (OCaml) "Int32.lognot" and
   (Scala) "_.unary'_~"
 | constant "Bit_Operations.and :: uint32 \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word32.andb ((_),/ (_))" and
   (Haskell) infixl 7 "Data_Bits..&." and
   (OCaml) "Int32.logand" and
   (Scala) infixl 3 "&"
 | constant "Bit_Operations.or :: uint32 \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word32.orb ((_),/ (_))" and
   (Haskell) infixl 5 "Data_Bits..|." and
   (OCaml) "Int32.logor" and
   (Scala) infixl 1 "|"
 | constant "Bit_Operations.xor :: uint32 \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word32.xorb ((_),/ (_))" and
   (Haskell) "Data'_Bits.xor" and
   (OCaml) "Int32.logxor" and
   (Scala) infixl 2 "^"
 
 definition uint32_divmod :: "uint32 \<Rightarrow> uint32 \<Rightarrow> uint32 \<times> uint32" where
   "uint32_divmod x y = 
   (if y = 0 then (undefined ((div) :: uint32 \<Rightarrow> _) x (0 :: uint32), undefined ((mod) :: uint32 \<Rightarrow> _) x (0 :: uint32)) 
   else (x div y, x mod y))"
 
 definition uint32_div :: "uint32 \<Rightarrow> uint32 \<Rightarrow> uint32" 
 where "uint32_div x y = fst (uint32_divmod x y)"
 
 definition uint32_mod :: "uint32 \<Rightarrow> uint32 \<Rightarrow> uint32" 
 where "uint32_mod x y = snd (uint32_divmod x y)"
 
 lemma div_uint32_code [code]: "x div y = (if y = 0 then 0 else uint32_div x y)"
 including undefined_transfer unfolding uint32_divmod_def uint32_div_def
 by transfer (simp add: word_div_def)
 
 lemma mod_uint32_code [code]: "x mod y = (if y = 0 then x else uint32_mod x y)"
 including undefined_transfer unfolding uint32_mod_def uint32_divmod_def
 by transfer (simp add: word_mod_def)
 
 definition uint32_sdiv :: "uint32 \<Rightarrow> uint32 \<Rightarrow> uint32"
 where [code del]:
   "uint32_sdiv x y =
    (if y = 0 then undefined ((div) :: uint32 \<Rightarrow> _) x (0 :: uint32)
     else Abs_uint32 (Rep_uint32 x sdiv Rep_uint32 y))"
 
 definition div0_uint32 :: "uint32 \<Rightarrow> uint32"
 where [code del]: "div0_uint32 x = undefined ((div) :: uint32 \<Rightarrow> _) x (0 :: uint32)"
 declare [[code abort: div0_uint32]]
 
 definition mod0_uint32 :: "uint32 \<Rightarrow> uint32"
 where [code del]: "mod0_uint32 x = undefined ((mod) :: uint32 \<Rightarrow> _) x (0 :: uint32)"
 declare [[code abort: mod0_uint32]]
 
 lemma uint32_divmod_code [code]:
   "uint32_divmod x y =
   (if 0x80000000 \<le> y then if x < y then (0, x) else (1, x - y)
    else if y = 0 then (div0_uint32 x, mod0_uint32 x)
    else let q = push_bit 1 (uint32_sdiv (drop_bit 1 x) y);
             r = x - q * y
         in if r \<ge> y then (q + 1, r - y) else (q, r))"
   including undefined_transfer unfolding uint32_divmod_def uint32_sdiv_def div0_uint32_def mod0_uint32_def
     less_eq_uint32.rep_eq
   apply transfer
   apply (simp add: divmod_via_sdivmod push_bit_eq_mult)
   done
 
 lemma uint32_sdiv_code [code]:
   "Rep_uint32 (uint32_sdiv x y) =
    (if y = 0 then Rep_uint32 (undefined ((div) :: uint32 \<Rightarrow> _) x (0 :: uint32))
     else Rep_uint32 x sdiv Rep_uint32 y)"
 unfolding uint32_sdiv_def by(simp add: Abs_uint32_inverse)
 
 text \<open>
   Note that we only need a translation for signed division, but not for the remainder
   because @{thm uint32_divmod_code} computes both with division only.
 \<close>
 
 code_printing
   constant uint32_div \<rightharpoonup>
   (SML) "Word32.div ((_), (_))" and
   (Haskell) "Prelude.div"
 | constant uint32_mod \<rightharpoonup>
   (SML) "Word32.mod ((_), (_))" and
   (Haskell) "Prelude.mod"
 | constant uint32_divmod \<rightharpoonup>
   (Haskell) "divmod"
 | constant uint32_sdiv \<rightharpoonup>
   (OCaml) "Int32.div" and
   (Scala) "_ '/ _"
 
 definition uint32_test_bit :: "uint32 \<Rightarrow> integer \<Rightarrow> bool"
 where [code del]:
   "uint32_test_bit x n =
   (if n < 0 \<or> 31 < n then undefined (bit :: uint32 \<Rightarrow> _) x n
    else bit x (nat_of_integer n))"
 
 lemma test_bit_uint32_code [code]:
   "bit x n \<longleftrightarrow> n < 32 \<and> uint32_test_bit x (integer_of_nat n)"
   including undefined_transfer integer.lifting unfolding uint32_test_bit_def
   by (transfer, simp, transfer, simp)
 
 lemma uint32_test_bit_code [code]:
   "uint32_test_bit w n =
   (if n < 0 \<or> 31 < n then undefined (bit :: uint32 \<Rightarrow> _) w n else bit (Rep_uint32 w) (nat_of_integer n))"
   unfolding uint32_test_bit_def by(simp add: bit_uint32.rep_eq)
 
 code_printing constant uint32_test_bit \<rightharpoonup>
   (SML) "Uint32.test'_bit" and
   (Haskell) "Data'_Bits.testBitBounded" and
   (OCaml) "Uint32.test'_bit" and
   (Scala) "Uint32.test'_bit" and
   (Eval) "(fn w => fn n => if n < 0 orelse 32 <= n then raise (Fail \"argument to uint32'_test'_bit out of bounds\") else Uint32.test'_bit w n)"
 
 definition uint32_set_bit :: "uint32 \<Rightarrow> integer \<Rightarrow> bool \<Rightarrow> uint32"
 where [code del]:
   "uint32_set_bit x n b =
   (if n < 0 \<or> 31 < n then undefined (set_bit :: uint32 \<Rightarrow> _) x n b
    else set_bit x (nat_of_integer n) b)"
 
 lemma set_bit_uint32_code [code]:
   "set_bit x n b = (if n < 32 then uint32_set_bit x (integer_of_nat n) b else x)"
 including undefined_transfer integer.lifting unfolding uint32_set_bit_def
 by(transfer)(auto cong: conj_cong simp add: not_less set_bit_beyond word_size)
 
 lemma uint32_set_bit_code [code]:
   "Rep_uint32 (uint32_set_bit w n b) = 
   (if n < 0 \<or> 31 < n then Rep_uint32 (undefined (set_bit :: uint32 \<Rightarrow> _) w n b)
    else set_bit (Rep_uint32 w) (nat_of_integer n) b)"
 including undefined_transfer unfolding uint32_set_bit_def by transfer simp
 
 code_printing constant uint32_set_bit \<rightharpoonup>
   (SML) "Uint32.set'_bit" and
   (Haskell) "Data'_Bits.setBitBounded" and
   (OCaml) "Uint32.set'_bit" and
   (Scala) "Uint32.set'_bit" and
   (Eval) "(fn w => fn n => fn b => if n < 0 orelse 32 <= n then raise (Fail \"argument to uint32'_set'_bit out of bounds\") else Uint32.set'_bit x n b)"
 
 definition uint32_shiftl :: "uint32 \<Rightarrow> integer \<Rightarrow> uint32"
 where [code del]:
   "uint32_shiftl x n = (if n < 0 \<or> 32 \<le> n then undefined (push_bit :: nat \<Rightarrow> uint32 \<Rightarrow> _) x n else push_bit (nat_of_integer n) x)"
 
 lemma shiftl_uint32_code [code]: "push_bit n x = (if n < 32 then uint32_shiftl x (integer_of_nat n) else 0)"
   including undefined_transfer integer.lifting unfolding uint32_shiftl_def
   by transfer simp
 
 lemma uint32_shiftl_code [code]:
   "Rep_uint32 (uint32_shiftl w n) =
   (if n < 0 \<or> 32 \<le> n then Rep_uint32 (undefined (push_bit :: nat \<Rightarrow> uint32 \<Rightarrow> _) w n) else push_bit (nat_of_integer n) (Rep_uint32 w))"
   including undefined_transfer unfolding uint32_shiftl_def
   by transfer simp
 
 code_printing constant uint32_shiftl \<rightharpoonup>
   (SML) "Uint32.shiftl" and
   (Haskell) "Data'_Bits.shiftlBounded" and
   (OCaml) "Uint32.shiftl" and
   (Scala) "Uint32.shiftl" and
   (Eval) "(fn x => fn i => if i < 0 orelse i >= 32 then raise Fail \"argument to uint32'_shiftl out of bounds\" else Uint32.shiftl x i)"
 
 definition uint32_shiftr :: "uint32 \<Rightarrow> integer \<Rightarrow> uint32"
 where [code del]:
   "uint32_shiftr x n = (if n < 0 \<or> 32 \<le> n then undefined (drop_bit :: nat \<Rightarrow> uint32 \<Rightarrow> _) x n else drop_bit (nat_of_integer n) x)"
 
 lemma shiftr_uint32_code [code]: "drop_bit n x = (if n < 32 then uint32_shiftr x (integer_of_nat n) else 0)"
   including undefined_transfer integer.lifting unfolding uint32_shiftr_def
   by transfer simp
 
 lemma uint32_shiftr_code [code]:
   "Rep_uint32 (uint32_shiftr w n) =
   (if n < 0 \<or> 32 \<le> n then Rep_uint32 (undefined (drop_bit :: nat \<Rightarrow> uint32 \<Rightarrow> _) w n) else drop_bit (nat_of_integer n) (Rep_uint32 w))"
   including undefined_transfer unfolding uint32_shiftr_def by transfer simp
 
 code_printing constant uint32_shiftr \<rightharpoonup>
   (SML) "Uint32.shiftr" and
   (Haskell) "Data'_Bits.shiftrBounded" and
   (OCaml) "Uint32.shiftr" and
   (Scala) "Uint32.shiftr" and
   (Eval) "(fn x => fn i => if i < 0 orelse i >= 32 then raise Fail \"argument to uint32'_shiftr out of bounds\" else Uint32.shiftr x i)"
 
 definition uint32_sshiftr :: "uint32 \<Rightarrow> integer \<Rightarrow> uint32"
 where [code del]:
   "uint32_sshiftr x n =
   (if n < 0 \<or> 32 \<le> n then undefined signed_drop_bit_uint32 n x else signed_drop_bit_uint32 (nat_of_integer n) x)"
 
 lemma sshiftr_uint32_code [code]:
   "signed_drop_bit_uint32 n x = 
   (if n < 32 then uint32_sshiftr x (integer_of_nat n) else if bit x 31 then - 1 else 0)"
   including undefined_transfer integer.lifting unfolding uint32_sshiftr_def
   by transfer (simp add: not_less signed_drop_bit_beyond)
 
 lemma uint32_sshiftr_code [code]:
   "Rep_uint32 (uint32_sshiftr w n) =
   (if n < 0 \<or> 32 \<le> n then Rep_uint32 (undefined signed_drop_bit_uint32 n w) else signed_drop_bit (nat_of_integer n) (Rep_uint32 w))"
 including undefined_transfer unfolding uint32_sshiftr_def by transfer simp
 
 code_printing constant uint32_sshiftr \<rightharpoonup>
   (SML) "Uint32.shiftr'_signed" and
   (Haskell) 
     "(Prelude.fromInteger (Prelude.toInteger (Data'_Bits.shiftrBounded (Prelude.fromInteger (Prelude.toInteger _) :: Uint32.Int32) _)) :: Uint32.Word32)" and
   (OCaml) "Uint32.shiftr'_signed" and
   (Scala) "Uint32.shiftr'_signed" and
   (Eval) "(fn x => fn i => if i < 0 orelse i >= 32 then raise Fail \"argument to uint32'_shiftr'_signed out of bounds\" else Uint32.shiftr'_signed x i)"
 
 context
   includes bit_operations_syntax
 begin
 
 lemma uint32_msb_test_bit: "msb x \<longleftrightarrow> bit (x :: uint32) 31"
   by transfer (simp add: msb_word_iff_bit)
 
 lemma msb_uint32_code [code]: "msb x \<longleftrightarrow> uint32_test_bit x 31"
   by (simp add: uint32_test_bit_def uint32_msb_test_bit)
 
 lemma uint32_of_int_code [code]:
   "uint32_of_int i = Uint32 (integer_of_int i)"
   including integer.lifting by transfer simp
 
 lemma int_of_uint32_code [code]:
   "int_of_uint32 x = int_of_integer (integer_of_uint32 x)"
   including integer.lifting by transfer simp
 
 lemma uint32_of_nat_code [code]:
   "uint32_of_nat = uint32_of_int \<circ> int"
   by transfer (simp add: fun_eq_iff)
 
 lemma nat_of_uint32_code [code]:
   "nat_of_uint32 x = nat_of_integer (integer_of_uint32 x)"
   unfolding integer_of_uint32_def including integer.lifting by transfer simp
 
 definition integer_of_uint32_signed :: "uint32 \<Rightarrow> integer"
 where
   "integer_of_uint32_signed n = (if bit n 31 then undefined integer_of_uint32 n else integer_of_uint32 n)"
 
 lemma integer_of_uint32_signed_code [code]:
   "integer_of_uint32_signed n =
   (if bit n 31 then undefined integer_of_uint32 n else integer_of_int (uint (Rep_uint32' n)))"
   by (simp add: integer_of_uint32_signed_def integer_of_uint32_def)
 
 lemma integer_of_uint32_code [code]:
   "integer_of_uint32 n =
   (if bit n 31 then integer_of_uint32_signed (n AND 0x7FFFFFFF) OR 0x80000000 else integer_of_uint32_signed n)"
 proof -
   have \<open>integer_of_uint32_signed (n AND 0x7FFFFFFF) OR 0x80000000 = Bit_Operations.set_bit 31 (integer_of_uint32_signed (take_bit 31 n))\<close>
     by (simp add: take_bit_eq_mask set_bit_eq_or push_bit_eq_mult mask_eq_exp_minus_1)
   moreover have \<open>integer_of_uint32 n = Bit_Operations.set_bit 31 (integer_of_uint32 (take_bit 31 n))\<close> if \<open>bit n 31\<close>
   proof (rule bit_eqI)
     fix m
     from that show \<open>bit (integer_of_uint32 n) m = bit (Bit_Operations.set_bit 31 (integer_of_uint32 (take_bit 31 n))) m\<close> for m
       including integer.lifting by transfer (auto simp add: bit_simps dest: bit_imp_le_length)
   qed
   ultimately show ?thesis
     by simp (simp add: integer_of_uint32_signed_def bit_simps)
 qed
 
 end
 
 code_printing
   constant "integer_of_uint32" \<rightharpoonup>
   (SML) "IntInf.fromLarge (Word32.toLargeInt _) : IntInf.int" and
   (Haskell) "Prelude.toInteger"
 | constant "integer_of_uint32_signed" \<rightharpoonup>
   (OCaml) "Z.of'_int32" and
   (Scala) "BigInt"
 
 section \<open>Quickcheck setup\<close>
 
 definition uint32_of_natural :: "natural \<Rightarrow> uint32"
 where "uint32_of_natural x \<equiv> Uint32 (integer_of_natural x)"
 
 instantiation uint32 :: "{random, exhaustive, full_exhaustive}" begin
 definition "random_uint32 \<equiv> qc_random_cnv uint32_of_natural"
 definition "exhaustive_uint32 \<equiv> qc_exhaustive_cnv uint32_of_natural"
 definition "full_exhaustive_uint32 \<equiv> qc_full_exhaustive_cnv uint32_of_natural"
 instance ..
 end
 
 instantiation uint32 :: narrowing begin
 
 interpretation quickcheck_narrowing_samples
   "\<lambda>i. let x = Uint32 i in (x, 0xFFFFFFFF - x)" "0"
   "Typerep.Typerep (STR ''Uint32.uint32'') []" .
 
 definition "narrowing_uint32 d = qc_narrowing_drawn_from (narrowing_samples d) d"
 declare [[code drop: "partial_term_of :: uint32 itself \<Rightarrow> _"]]
 lemmas partial_term_of_uint32 [code] = partial_term_of_code
 
 instance ..
 end
 
 end
diff --git a/thys/Native_Word/Uint64.thy b/thys/Native_Word/Uint64.thy
--- a/thys/Native_Word/Uint64.thy
+++ b/thys/Native_Word/Uint64.thy
@@ -1,888 +1,889 @@
 (*  Title:      Uint64.thy
     Author:     Andreas Lochbihler, ETH Zurich
 *)
 
 chapter \<open>Unsigned words of 64 bits\<close>
 
 theory Uint64 imports
-  Code_Target_Word_Base Word_Type_Copies
+  Word_Type_Copies
+  Code_Target_Integer_Bit
 begin
 
 text \<open>
   PolyML (in version 5.7) provides a Word64 structure only when run in 64-bit mode.
   Therefore, we by default provide an implementation of 64-bit words using \verb$IntInf.int$ and
   masking. The code target \texttt{SML\_word} replaces this implementation and maps the operations
   directly to the \verb$Word64$ structure provided by the Standard ML implementations.
 
   The \verb$Eval$ target used by @{command value} and @{method eval} dynamically tests at
   runtime for the version of PolyML and uses PolyML's Word64 structure if it detects a 64-bit 
   version which does not suffer from a division bug found in PolyML 5.6.
 \<close>
 
 section \<open>Type definition and primitive operations\<close>
 
 typedef uint64 = \<open>UNIV :: 64 word set\<close> ..
 
 global_interpretation uint64: word_type_copy Abs_uint64 Rep_uint64
   using type_definition_uint64 by (rule word_type_copy.intro)
 
 setup_lifting type_definition_uint64
 
 declare uint64.of_word_of [code abstype]
 
 declare Quotient_uint64 [transfer_rule]
 
 instantiation uint64 :: \<open>{comm_ring_1, semiring_modulo, equal, linorder}\<close>
 begin
 
 lift_definition zero_uint64 :: uint64 is 0 .
 lift_definition one_uint64 :: uint64 is 1 .
 lift_definition plus_uint64 :: \<open>uint64 \<Rightarrow> uint64 \<Rightarrow> uint64\<close> is \<open>(+)\<close> .
 lift_definition uminus_uint64 :: \<open>uint64 \<Rightarrow> uint64\<close> is uminus .
 lift_definition minus_uint64 :: \<open>uint64 \<Rightarrow> uint64 \<Rightarrow> uint64\<close> is \<open>(-)\<close> .
 lift_definition times_uint64 :: \<open>uint64 \<Rightarrow> uint64 \<Rightarrow> uint64\<close> is \<open>(*)\<close> .
 lift_definition divide_uint64 :: \<open>uint64 \<Rightarrow> uint64 \<Rightarrow> uint64\<close> is \<open>(div)\<close> .
 lift_definition modulo_uint64 :: \<open>uint64 \<Rightarrow> uint64 \<Rightarrow> uint64\<close> is \<open>(mod)\<close> .
 lift_definition equal_uint64 :: \<open>uint64 \<Rightarrow> uint64 \<Rightarrow> bool\<close> is \<open>HOL.equal\<close> .
 lift_definition less_eq_uint64 :: \<open>uint64 \<Rightarrow> uint64 \<Rightarrow> bool\<close> is \<open>(\<le>)\<close> .
 lift_definition less_uint64 :: \<open>uint64 \<Rightarrow> uint64 \<Rightarrow> bool\<close> is \<open>(<)\<close> .
 
 global_interpretation uint64: word_type_copy_ring Abs_uint64 Rep_uint64
   by standard (fact zero_uint64.rep_eq one_uint64.rep_eq
     plus_uint64.rep_eq uminus_uint64.rep_eq minus_uint64.rep_eq
     times_uint64.rep_eq divide_uint64.rep_eq modulo_uint64.rep_eq
     equal_uint64.rep_eq less_eq_uint64.rep_eq less_uint64.rep_eq)+
 
 instance proof -
   show \<open>OFCLASS(uint64, comm_ring_1_class)\<close>
     by (rule uint64.of_class_comm_ring_1)
   show \<open>OFCLASS(uint64, semiring_modulo_class)\<close>
     by (fact uint64.of_class_semiring_modulo)
   show \<open>OFCLASS(uint64, equal_class)\<close>
     by (fact uint64.of_class_equal)
   show \<open>OFCLASS(uint64, linorder_class)\<close>
     by (fact uint64.of_class_linorder)
 qed
 
 end
 
 instantiation uint64 :: ring_bit_operations
 begin
 
 lift_definition bit_uint64 :: \<open>uint64 \<Rightarrow> nat \<Rightarrow> bool\<close> is bit .
 lift_definition not_uint64 :: \<open>uint64 \<Rightarrow> uint64\<close> is \<open>Bit_Operations.not\<close> .
 lift_definition and_uint64 :: \<open>uint64 \<Rightarrow> uint64 \<Rightarrow> uint64\<close> is \<open>Bit_Operations.and\<close> .
 lift_definition or_uint64 :: \<open>uint64 \<Rightarrow> uint64 \<Rightarrow> uint64\<close> is \<open>Bit_Operations.or\<close> .
 lift_definition xor_uint64 :: \<open>uint64 \<Rightarrow> uint64 \<Rightarrow> uint64\<close> is \<open>Bit_Operations.xor\<close> .
 lift_definition mask_uint64 :: \<open>nat \<Rightarrow> uint64\<close> is mask .
 lift_definition push_bit_uint64 :: \<open>nat \<Rightarrow> uint64 \<Rightarrow> uint64\<close> is push_bit .
 lift_definition drop_bit_uint64 :: \<open>nat \<Rightarrow> uint64 \<Rightarrow> uint64\<close> is drop_bit .
 lift_definition signed_drop_bit_uint64 :: \<open>nat \<Rightarrow> uint64 \<Rightarrow> uint64\<close> is signed_drop_bit .
 lift_definition take_bit_uint64 :: \<open>nat \<Rightarrow> uint64 \<Rightarrow> uint64\<close> is take_bit .
 lift_definition set_bit_uint64 :: \<open>nat \<Rightarrow> uint64 \<Rightarrow> uint64\<close> is Bit_Operations.set_bit .
 lift_definition unset_bit_uint64 :: \<open>nat \<Rightarrow> uint64 \<Rightarrow> uint64\<close> is unset_bit .
 lift_definition flip_bit_uint64 :: \<open>nat \<Rightarrow> uint64 \<Rightarrow> uint64\<close> is flip_bit .
 
 global_interpretation uint64: word_type_copy_bits Abs_uint64 Rep_uint64 signed_drop_bit_uint64
   by standard (fact bit_uint64.rep_eq not_uint64.rep_eq and_uint64.rep_eq or_uint64.rep_eq xor_uint64.rep_eq
     mask_uint64.rep_eq push_bit_uint64.rep_eq drop_bit_uint64.rep_eq signed_drop_bit_uint64.rep_eq take_bit_uint64.rep_eq
     set_bit_uint64.rep_eq unset_bit_uint64.rep_eq flip_bit_uint64.rep_eq)+
 
 instance
   by (fact uint64.of_class_ring_bit_operations)
 
 end
 
 lift_definition uint64_of_nat :: \<open>nat \<Rightarrow> uint64\<close>
   is word_of_nat .
 
 lift_definition nat_of_uint64 :: \<open>uint64 \<Rightarrow> nat\<close>
   is unat .
 
 lift_definition uint64_of_int :: \<open>int \<Rightarrow> uint64\<close>
   is word_of_int .
 
 lift_definition int_of_uint64 :: \<open>uint64 \<Rightarrow> int\<close>
   is uint .
 
 context
   includes integer.lifting
 begin
 
 lift_definition Uint64 :: \<open>integer \<Rightarrow> uint64\<close>
   is word_of_int .
 
 lift_definition integer_of_uint64 :: \<open>uint64 \<Rightarrow> integer\<close>
   is uint .
 
 end
 
 global_interpretation uint64: word_type_copy_more Abs_uint64 Rep_uint64 signed_drop_bit_uint64
   uint64_of_nat nat_of_uint64 uint64_of_int int_of_uint64 Uint64 integer_of_uint64
   apply standard
        apply (simp_all add: uint64_of_nat.rep_eq nat_of_uint64.rep_eq
          uint64_of_int.rep_eq int_of_uint64.rep_eq
          Uint64.rep_eq integer_of_uint64.rep_eq integer_eq_iff)
   done
 
 instantiation uint64 :: "{size, msb, lsb, set_bit, bit_comprehension}"
 begin
 
 lift_definition size_uint64 :: \<open>uint64 \<Rightarrow> nat\<close> is size .
 
 lift_definition msb_uint64 :: \<open>uint64 \<Rightarrow> bool\<close> is msb .
 lift_definition lsb_uint64 :: \<open>uint64 \<Rightarrow> bool\<close> is lsb .
 
 text \<open>Workaround: avoid name space clash by spelling out \<^text>\<open>lift_definition\<close> explicitly.\<close>
 
 definition set_bit_uint64 :: \<open>uint64 \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> uint64\<close>
   where set_bit_uint64_eq: \<open>set_bit_uint64 a n b = (if b then Bit_Operations.set_bit else unset_bit) n a\<close>
 
 context
   includes lifting_syntax
 begin
 
 lemma set_bit_uint64_transfer [transfer_rule]:
   \<open>(cr_uint64 ===> (=) ===> (\<longleftrightarrow>) ===> cr_uint64) Generic_set_bit.set_bit Generic_set_bit.set_bit\<close>
   by (simp only: set_bit_eq [abs_def] set_bit_uint64_eq [abs_def]) transfer_prover
 
 end
 
 lift_definition set_bits_uint64 :: \<open>(nat \<Rightarrow> bool) \<Rightarrow> uint64\<close> is set_bits .
 lift_definition set_bits_aux_uint64 :: \<open>(nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> uint64 \<Rightarrow> uint64\<close> is set_bits_aux .
 
 global_interpretation uint64: word_type_copy_misc Abs_uint64 Rep_uint64 signed_drop_bit_uint64
   uint64_of_nat nat_of_uint64 uint64_of_int int_of_uint64 Uint64 integer_of_uint64 64 set_bits_aux_uint64
   by (standard; transfer) simp_all
 
 instance using uint64.of_class_bit_comprehension
   uint64.of_class_set_bit uint64.of_class_lsb
   by simp_all standard
 
 end
 
 section \<open>Code setup\<close>
 
 text \<open> For SML, we generate an implementation of unsigned 64-bit words using \verb$IntInf.int$.
   If @{ML "LargeWord.wordSize > 63"} of the Isabelle/ML runtime environment holds, then we assume
   that there is also a \<open>Word64\<close> structure available and accordingly replace the implementation
   for the target \verb$Eval$.
 \<close>
 code_printing code_module "Uint64" \<rightharpoonup> (SML) \<open>(* Test that words can handle numbers between 0 and 63 *)
 val _ = if 6 <= Word.wordSize then () else raise (Fail ("wordSize less than 6"));
 
 structure Uint64 : sig
   eqtype uint64;
   val zero : uint64;
   val one : uint64;
   val fromInt : IntInf.int -> uint64;
   val toInt : uint64 -> IntInf.int;
   val toLarge : uint64 -> LargeWord.word;
   val fromLarge : LargeWord.word -> uint64
   val plus : uint64 -> uint64 -> uint64;
   val minus : uint64 -> uint64 -> uint64;
   val times : uint64 -> uint64 -> uint64;
   val divide : uint64 -> uint64 -> uint64;
   val modulus : uint64 -> uint64 -> uint64;
   val negate : uint64 -> uint64;
   val less_eq : uint64 -> uint64 -> bool;
   val less : uint64 -> uint64 -> bool;
   val notb : uint64 -> uint64;
   val andb : uint64 -> uint64 -> uint64;
   val orb : uint64 -> uint64 -> uint64;
   val xorb : uint64 -> uint64 -> uint64;
   val shiftl : uint64 -> IntInf.int -> uint64;
   val shiftr : uint64 -> IntInf.int -> uint64;
   val shiftr_signed : uint64 -> IntInf.int -> uint64;
   val set_bit : uint64 -> IntInf.int -> bool -> uint64;
   val test_bit : uint64 -> IntInf.int -> bool;
 end = struct
 
 type uint64 = IntInf.int;
 
 val mask = 0xFFFFFFFFFFFFFFFF : IntInf.int;
 
 val zero = 0 : IntInf.int;
 
 val one = 1 : IntInf.int;
 
 fun fromInt x = IntInf.andb(x, mask);
 
 fun toInt x = x
 
 fun toLarge x = LargeWord.fromLargeInt (IntInf.toLarge x);
 
 fun fromLarge x = IntInf.fromLarge (LargeWord.toLargeInt x);
 
 fun plus x y = IntInf.andb(IntInf.+(x, y), mask);
 
 fun minus x y = IntInf.andb(IntInf.-(x, y), mask);
 
 fun negate x = IntInf.andb(IntInf.~(x), mask);
 
 fun times x y = IntInf.andb(IntInf.*(x, y), mask);
 
 fun divide x y = IntInf.div(x, y);
 
 fun modulus x y = IntInf.mod(x, y);
 
 fun less_eq x y = IntInf.<=(x, y);
 
 fun less x y = IntInf.<(x, y);
 
 fun notb x = IntInf.andb(IntInf.notb(x), mask);
 
 fun orb x y = IntInf.orb(x, y);
 
 fun andb x y = IntInf.andb(x, y);
 
 fun xorb x y = IntInf.xorb(x, y);
 
 val maxWord = IntInf.pow (2, Word.wordSize);
 
 fun shiftl x n = 
   if n < maxWord then IntInf.andb(IntInf.<< (x, Word.fromLargeInt (IntInf.toLarge n)), mask)
   else 0;
 
 fun shiftr x n =
   if n < maxWord then IntInf.~>> (x, Word.fromLargeInt (IntInf.toLarge n))
   else 0;
 
 val msb_mask = 0x8000000000000000 : IntInf.int;
 
 fun shiftr_signed x i =
   if IntInf.andb(x, msb_mask) = 0 then shiftr x i
   else if i >= 64 then 0xFFFFFFFFFFFFFFFF
   else let
     val x' = shiftr x i
     val m' = IntInf.andb(IntInf.<<(mask, Word.max(0w64 - Word.fromLargeInt (IntInf.toLarge i), 0w0)), mask)
   in IntInf.orb(x', m') end;
 
 fun test_bit x n =
   if n < maxWord then IntInf.andb (x, IntInf.<< (1, Word.fromLargeInt (IntInf.toLarge n))) <> 0
   else false;
 
 fun set_bit x n b =
   if n < 64 then
     if b then IntInf.orb (x, IntInf.<< (1, Word.fromLargeInt (IntInf.toLarge n)))
     else IntInf.andb (x, IntInf.notb (IntInf.<< (1, Word.fromLargeInt (IntInf.toLarge n))))
   else x;
 
 end
 \<close>
 code_reserved SML Uint64
 
 setup \<open>
 let
   val polyml64 = LargeWord.wordSize > 63;
   (* PolyML 5.6 has bugs in its Word64 implementation. We test for one such bug and refrain
      from using Word64 in that case. Testing is done with dynamic code evaluation such that
      the compiler does not choke on the Word64 structure, which need not be present in a 32bit
      environment. *)
   val error_msg = "Buggy Word64 structure";
   val test_code = 
    "val _ = if Word64.div (0w18446744073709551611 : Word64.word, 0w3) = 0w6148914691236517203 then ()\n" ^
    "else raise (Fail \"" ^ error_msg ^ "\");";
   val f = Exn.interruptible_capture (fn () => ML_Compiler.eval ML_Compiler.flags Position.none (ML_Lex.tokenize test_code))
   val use_Word64 = polyml64 andalso
     (case f () of 
        Exn.Res _ => true
      | Exn.Exn (e as ERROR m) => if String.isSuffix error_msg m then false else Exn.reraise e
      | Exn.Exn e => Exn.reraise e)
     ;
 
   val newline = "\n";
   val content = 
     "structure Uint64 : sig" ^ newline ^
     "  eqtype uint64;" ^ newline ^
     "  val zero : uint64;" ^ newline ^
     "  val one : uint64;" ^ newline ^
     "  val fromInt : IntInf.int -> uint64;" ^ newline ^
     "  val toInt : uint64 -> IntInf.int;" ^ newline ^
     "  val toLarge : uint64 -> LargeWord.word;" ^ newline ^
     "  val fromLarge : LargeWord.word -> uint64" ^ newline ^
     "  val plus : uint64 -> uint64 -> uint64;" ^ newline ^
     "  val minus : uint64 -> uint64 -> uint64;" ^ newline ^
     "  val times : uint64 -> uint64 -> uint64;" ^ newline ^
     "  val divide : uint64 -> uint64 -> uint64;" ^ newline ^
     "  val modulus : uint64 -> uint64 -> uint64;" ^ newline ^
     "  val negate : uint64 -> uint64;" ^ newline ^
     "  val less_eq : uint64 -> uint64 -> bool;" ^ newline ^
     "  val less : uint64 -> uint64 -> bool;" ^ newline ^
     "  val notb : uint64 -> uint64;" ^ newline ^
     "  val andb : uint64 -> uint64 -> uint64;" ^ newline ^
     "  val orb : uint64 -> uint64 -> uint64;" ^ newline ^
     "  val xorb : uint64 -> uint64 -> uint64;" ^ newline ^
     "  val shiftl : uint64 -> IntInf.int -> uint64;" ^ newline ^
     "  val shiftr : uint64 -> IntInf.int -> uint64;" ^ newline ^
     "  val shiftr_signed : uint64 -> IntInf.int -> uint64;" ^ newline ^
     "  val set_bit : uint64 -> IntInf.int -> bool -> uint64;" ^ newline ^
     "  val test_bit : uint64 -> IntInf.int -> bool;" ^ newline ^
     "end = struct" ^ newline ^
     "" ^ newline ^
     "type uint64 = Word64.word;" ^ newline ^
     "" ^ newline ^
     "val zero = (0wx0 : uint64);" ^ newline ^
     "" ^ newline ^
     "val one = (0wx1 : uint64);" ^ newline ^
     "" ^ newline ^
     "fun fromInt x = Word64.fromLargeInt (IntInf.toLarge x);" ^ newline ^
     "" ^ newline ^
     "fun toInt x = IntInf.fromLarge (Word64.toLargeInt x);"  ^ newline ^
     "" ^ newline ^
     "fun fromLarge x = Word64.fromLarge x;" ^ newline ^
     "" ^ newline ^
     "fun toLarge x = Word64.toLarge x;"  ^ newline ^
     "" ^ newline ^
     "fun plus x y = Word64.+(x, y);" ^ newline ^
     "" ^ newline ^
     "fun minus x y = Word64.-(x, y);" ^ newline ^
     "" ^ newline ^
     "fun negate x = Word64.~(x);" ^ newline ^
     "" ^ newline ^
     "fun times x y = Word64.*(x, y);" ^ newline ^
     "" ^ newline ^
     "fun divide x y = Word64.div(x, y);" ^ newline ^
     "" ^ newline ^
     "fun modulus x y = Word64.mod(x, y);" ^ newline ^
     "" ^ newline ^
     "fun less_eq x y = Word64.<=(x, y);" ^ newline ^
     "" ^ newline ^
     "fun less x y = Word64.<(x, y);" ^ newline ^
     "" ^ newline ^
     "fun set_bit x n b =" ^ newline ^
     "  let val mask = Word64.<< (0wx1, Word.fromLargeInt (IntInf.toLarge n))" ^ newline ^
     "  in if b then Word64.orb (x, mask)" ^ newline ^
     "     else Word64.andb (x, Word64.notb mask)" ^ newline ^
     "  end" ^ newline ^
     "" ^ newline ^
     "fun shiftl x n =" ^ newline ^
     "  Word64.<< (x, Word.fromLargeInt (IntInf.toLarge n))" ^ newline ^
     "" ^ newline ^
     "fun shiftr x n =" ^ newline ^
     "  Word64.>> (x, Word.fromLargeInt (IntInf.toLarge n))" ^ newline ^
     "" ^ newline ^
     "fun shiftr_signed x n =" ^ newline ^
     "  Word64.~>> (x, Word.fromLargeInt (IntInf.toLarge n))" ^ newline ^
     "" ^ newline ^
     "fun test_bit x n =" ^ newline ^
     "  Word64.andb (x, Word64.<< (0wx1, Word.fromLargeInt (IntInf.toLarge n))) <> Word64.fromInt 0" ^ newline ^
     "" ^ newline ^
     "val notb = Word64.notb" ^ newline ^
     "" ^ newline ^
     "fun andb x y = Word64.andb(x, y);" ^ newline ^
     "" ^ newline ^
     "fun orb x y = Word64.orb(x, y);" ^ newline ^
     "" ^ newline ^
     "fun xorb x y = Word64.xorb(x, y);" ^ newline ^
     "" ^ newline ^
     "end (*struct Uint64*)"
   val target_SML64 = "SML_word";
 in
   (if use_Word64 then Code_Target.set_printings (Code_Symbol.Module ("Uint64", [(Code_Runtime.target, SOME (content, []))])) else I)
   #> Code_Target.set_printings (Code_Symbol.Module ("Uint64", [(target_SML64, SOME (content, []))]))
 end
 \<close>
 
 code_printing code_module Uint64 \<rightharpoonup> (Haskell)
  \<open>module Uint64(Int64, Word64) where
 
   import Data.Int(Int64)
   import Data.Word(Word64)\<close>
 code_reserved Haskell Uint64
 
 text \<open>
   OCaml and Scala provide only signed 64bit numbers, so we use these and 
   implement sign-sensitive operations like comparisons manually.
 \<close>
 code_printing code_module "Uint64" \<rightharpoonup> (OCaml)
 \<open>module Uint64 : sig
   val less : int64 -> int64 -> bool
   val less_eq : int64 -> int64 -> bool
   val set_bit : int64 -> Z.t -> bool -> int64
   val shiftl : int64 -> Z.t -> int64
   val shiftr : int64 -> Z.t -> int64
   val shiftr_signed : int64 -> Z.t -> int64
   val test_bit : int64 -> Z.t -> bool
 end = struct
 
 (* negative numbers have their highest bit set, 
    so they are greater than positive ones *)
 let less x y =
   if Int64.compare x Int64.zero < 0 then
     Int64.compare y Int64.zero < 0 && Int64.compare x y < 0
   else Int64.compare y Int64.zero < 0 || Int64.compare x y < 0;;
 
 let less_eq x y =
   if Int64.compare x Int64.zero < 0 then
     Int64.compare y Int64.zero < 0 && Int64.compare x y <= 0
   else Int64.compare y Int64.zero < 0 || Int64.compare x y <= 0;;
 
 let set_bit x n b =
   let mask = Int64.shift_left Int64.one (Z.to_int n)
   in if b then Int64.logor x mask
      else Int64.logand x (Int64.lognot mask);;
 
 let shiftl x n = Int64.shift_left x (Z.to_int n);;
 
 let shiftr x n = Int64.shift_right_logical x (Z.to_int n);;
 
 let shiftr_signed x n = Int64.shift_right x (Z.to_int n);;
 
 let test_bit x n =
   Int64.compare 
     (Int64.logand x (Int64.shift_left Int64.one (Z.to_int n)))
     Int64.zero
   <> 0;;
 
 end;; (*struct Uint64*)\<close>
 code_reserved OCaml Uint64
 
 code_printing code_module Uint64 \<rightharpoonup> (Scala)
 \<open>object Uint64 {
 
 def less(x: Long, y: Long) : Boolean =
   if (x < 0) y < 0 && x < y
   else y < 0 || x < y
 
 def less_eq(x: Long, y: Long) : Boolean =
   if (x < 0) y < 0 && x <= y
   else y < 0 || x <= y
 
 def set_bit(x: Long, n: BigInt, b: Boolean) : Long =
   if (b)
     x | (1L << n.intValue)
   else
     x & (1L << n.intValue).unary_~
 
 def shiftl(x: Long, n: BigInt) : Long = x << n.intValue
 
 def shiftr(x: Long, n: BigInt) : Long = x >>> n.intValue
 
 def shiftr_signed(x: Long, n: BigInt) : Long = x >> n.intValue
 
 def test_bit(x: Long, n: BigInt) : Boolean =
   (x & (1L << n.intValue)) != 0
 
 } /* object Uint64 */\<close>
 code_reserved Scala Uint64
 
 text \<open>
   OCaml's conversion from Big\_int to int64 demands that the value fits int a signed 64-bit integer.
   The following justifies the implementation.
 \<close>
 
 context
   includes bit_operations_syntax
 begin
 
 definition Uint64_signed :: "integer \<Rightarrow> uint64" 
 where "Uint64_signed i = (if i < -(0x8000000000000000) \<or> i \<ge> 0x8000000000000000 then undefined Uint64 i else Uint64 i)"
 
 lemma Uint64_code [code]:
   "Uint64 i = 
   (let i' = i AND 0xFFFFFFFFFFFFFFFF
    in if bit i' 63 then Uint64_signed (i' - 0x10000000000000000) else Uint64_signed i')"
   including undefined_transfer integer.lifting unfolding Uint64_signed_def
   apply transfer
   apply (subst word_of_int_via_signed)
      apply (auto simp add: push_bit_of_1 mask_eq_exp_minus_1 word_of_int_via_signed cong del: if_cong)
   done
 
 lemma Uint64_signed_code [code]:
   "Rep_uint64 (Uint64_signed i) = 
   (if i < -(0x8000000000000000) \<or> i \<ge> 0x8000000000000000 then Rep_uint64 (undefined Uint64 i) else word_of_int (int_of_integer_symbolic i))"
 unfolding Uint64_signed_def Uint64_def int_of_integer_symbolic_def word_of_integer_def
 by(simp add: Abs_uint64_inverse)
 
 end
 
 text \<open>
   Avoid @{term Abs_uint64} in generated code, use @{term Rep_uint64'} instead. 
   The symbolic implementations for code\_simp use @{term Rep_uint64}.
 
   The new destructor @{term Rep_uint64'} is executable.
   As the simplifier is given the [code abstract] equations literally, 
   we cannot implement @{term Rep_uint64} directly, because that makes code\_simp loop.
 
   If code generation raises Match, some equation probably contains @{term Rep_uint64} 
   ([code abstract] equations for @{typ uint64} may use @{term Rep_uint64} because
   these instances will be folded away.)
 
   To convert @{typ "64 word"} values into @{typ uint64}, use @{term "Abs_uint64'"}.
 \<close>
 
 definition Rep_uint64' where [simp]: "Rep_uint64' = Rep_uint64"
 
 lemma Rep_uint64'_transfer [transfer_rule]:
   "rel_fun cr_uint64 (=) (\<lambda>x. x) Rep_uint64'"
 unfolding Rep_uint64'_def by(rule uint64.rep_transfer)
 
 lemma Rep_uint64'_code [code]: "Rep_uint64' x = (BITS n. bit x n)"
   by transfer (simp add: set_bits_bit_eq)
 
 lift_definition Abs_uint64' :: "64 word \<Rightarrow> uint64" is "\<lambda>x :: 64 word. x" .
 
 lemma Abs_uint64'_code [code]:
   "Abs_uint64' x = Uint64 (integer_of_int (uint x))"
 including integer.lifting by transfer simp
 
 declare [[code drop: "term_of_class.term_of :: uint64 \<Rightarrow> _"]]
 
 lemma term_of_uint64_code [code]:
   defines "TR \<equiv> typerep.Typerep" and "bit0 \<equiv> STR ''Numeral_Type.bit0''" 
   shows
   "term_of_class.term_of x = 
    Code_Evaluation.App (Code_Evaluation.Const (STR ''Uint64.uint64.Abs_uint64'') (TR (STR ''fun'') [TR (STR ''Word.word'') [TR bit0 [TR bit0 [TR bit0 [TR bit0 [TR bit0 [TR bit0 [TR (STR ''Numeral_Type.num1'') []]]]]]]], TR (STR ''Uint64.uint64'') []]))
        (term_of_class.term_of (Rep_uint64' x))"
 by(simp add: term_of_anything)
 
 code_printing
   type_constructor uint64 \<rightharpoonup>
   (SML) "Uint64.uint64" and
   (Haskell) "Uint64.Word64" and
   (OCaml) "int64" and
   (Scala) "Long"
 | constant Uint64 \<rightharpoonup>
   (SML) "Uint64.fromInt" and
   (Haskell) "(Prelude.fromInteger _ :: Uint64.Word64)" and
   (Haskell_Quickcheck) "(Prelude.fromInteger (Prelude.toInteger _) :: Uint64.Word64)" and
   (Scala) "_.longValue"
 | constant Uint64_signed \<rightharpoonup>
   (OCaml) "Z.to'_int64"
 | constant "0 :: uint64" \<rightharpoonup>
   (SML) "Uint64.zero" and
   (Haskell) "(0 :: Uint64.Word64)" and
   (OCaml) "Int64.zero" and
   (Scala) "0"
 | constant "1 :: uint64" \<rightharpoonup>
   (SML) "Uint64.one" and
   (Haskell) "(1 :: Uint64.Word64)" and
   (OCaml) "Int64.one" and
   (Scala) "1"
 | constant "plus :: uint64 \<Rightarrow> _ " \<rightharpoonup>
   (SML) "Uint64.plus" and
   (Haskell) infixl 6 "+" and
   (OCaml) "Int64.add" and
   (Scala) infixl 7 "+"
 | constant "uminus :: uint64 \<Rightarrow> _" \<rightharpoonup>
   (SML) "Uint64.negate" and
   (Haskell) "negate" and
   (OCaml) "Int64.neg" and
   (Scala) "!(- _)"
 | constant "minus :: uint64 \<Rightarrow> _" \<rightharpoonup>
   (SML) "Uint64.minus" and
   (Haskell) infixl 6 "-" and
   (OCaml) "Int64.sub" and
   (Scala) infixl 7 "-"
 | constant "times :: uint64 \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
   (SML) "Uint64.times" and
   (Haskell) infixl 7 "*" and
   (OCaml) "Int64.mul" and
   (Scala) infixl 8 "*"
 | constant "HOL.equal :: uint64 \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
   (SML) "!((_ : Uint64.uint64) = _)" and
   (Haskell) infix 4 "==" and
   (OCaml) "(Int64.compare _ _ = 0)" and
   (Scala) infixl 5 "=="
 | class_instance uint64 :: equal \<rightharpoonup>
   (Haskell) -
 | constant "less_eq :: uint64 \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
   (SML) "Uint64.less'_eq" and
   (Haskell) infix 4 "<=" and
   (OCaml) "Uint64.less'_eq" and
   (Scala) "Uint64.less'_eq"
 | constant "less :: uint64 \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
   (SML) "Uint64.less" and
   (Haskell) infix 4 "<" and
   (OCaml) "Uint64.less" and
   (Scala) "Uint64.less"
 | constant "Bit_Operations.not :: uint64 \<Rightarrow> _" \<rightharpoonup>
   (SML) "Uint64.notb" and
   (Haskell) "Data'_Bits.complement" and
   (OCaml) "Int64.lognot" and
   (Scala) "_.unary'_~"
 | constant "Bit_Operations.and :: uint64 \<Rightarrow> _" \<rightharpoonup>
   (SML) "Uint64.andb" and
   (Haskell) infixl 7 "Data_Bits..&." and
   (OCaml) "Int64.logand" and
   (Scala) infixl 3 "&"
 | constant "Bit_Operations.or :: uint64 \<Rightarrow> _" \<rightharpoonup>
   (SML) "Uint64.orb" and
   (Haskell) infixl 5 "Data_Bits..|." and
   (OCaml) "Int64.logor" and
   (Scala) infixl 1 "|"
 | constant "Bit_Operations.xor :: uint64 \<Rightarrow> _" \<rightharpoonup>
   (SML) "Uint64.xorb" and
   (Haskell) "Data'_Bits.xor" and
   (OCaml) "Int64.logxor" and
   (Scala) infixl 2 "^"
 
 definition uint64_divmod :: "uint64 \<Rightarrow> uint64 \<Rightarrow> uint64 \<times> uint64" where
   "uint64_divmod x y = 
   (if y = 0 then (undefined ((div) :: uint64 \<Rightarrow> _) x (0 :: uint64), undefined ((mod) :: uint64 \<Rightarrow> _) x (0 :: uint64)) 
   else (x div y, x mod y))"
 
 definition uint64_div :: "uint64 \<Rightarrow> uint64 \<Rightarrow> uint64" 
 where "uint64_div x y = fst (uint64_divmod x y)"
 
 definition uint64_mod :: "uint64 \<Rightarrow> uint64 \<Rightarrow> uint64" 
 where "uint64_mod x y = snd (uint64_divmod x y)"
 
 lemma div_uint64_code [code]: "x div y = (if y = 0 then 0 else uint64_div x y)"
 including undefined_transfer unfolding uint64_divmod_def uint64_div_def
 by transfer (simp add: word_div_def)
 
 lemma mod_uint64_code [code]: "x mod y = (if y = 0 then x else uint64_mod x y)"
 including undefined_transfer unfolding uint64_mod_def uint64_divmod_def
 by transfer (simp add: word_mod_def)
 
 definition uint64_sdiv :: "uint64 \<Rightarrow> uint64 \<Rightarrow> uint64"
 where [code del]:
   "uint64_sdiv x y =
    (if y = 0 then undefined ((div) :: uint64 \<Rightarrow> _) x (0 :: uint64)
     else Abs_uint64 (Rep_uint64 x sdiv Rep_uint64 y))"
 
 definition div0_uint64 :: "uint64 \<Rightarrow> uint64"
 where [code del]: "div0_uint64 x = undefined ((div) :: uint64 \<Rightarrow> _) x (0 :: uint64)"
 declare [[code abort: div0_uint64]]
 
 definition mod0_uint64 :: "uint64 \<Rightarrow> uint64"
 where [code del]: "mod0_uint64 x = undefined ((mod) :: uint64 \<Rightarrow> _) x (0 :: uint64)"
 declare [[code abort: mod0_uint64]]
 
 lemma uint64_divmod_code [code]:
   "uint64_divmod x y =
   (if 0x8000000000000000 \<le> y then if x < y then (0, x) else (1, x - y)
    else if y = 0 then (div0_uint64 x, mod0_uint64 x)
    else let q = push_bit 1 (uint64_sdiv (drop_bit 1 x) y);
             r = x - q * y
         in if r \<ge> y then (q + 1, r - y) else (q, r))"
   including undefined_transfer unfolding uint64_divmod_def uint64_sdiv_def div0_uint64_def mod0_uint64_def
     less_eq_uint64.rep_eq
   apply transfer
   apply (simp add: divmod_via_sdivmod push_bit_eq_mult)
   done
 
 lemma uint64_sdiv_code [code]:
   "Rep_uint64 (uint64_sdiv x y) =
    (if y = 0 then Rep_uint64 (undefined ((div) :: uint64 \<Rightarrow> _) x (0 :: uint64))
     else Rep_uint64 x sdiv Rep_uint64 y)"
 unfolding uint64_sdiv_def by(simp add: Abs_uint64_inverse)
 
 text \<open>
   Note that we only need a translation for signed division, but not for the remainder
   because @{thm uint64_divmod_code} computes both with division only.
 \<close>
 
 code_printing
   constant uint64_div \<rightharpoonup>
   (SML) "Uint64.divide" and
   (Haskell) "Prelude.div"
 | constant uint64_mod \<rightharpoonup>
   (SML) "Uint64.modulus" and
   (Haskell) "Prelude.mod"
 | constant uint64_divmod \<rightharpoonup>
   (Haskell) "divmod"
 | constant uint64_sdiv \<rightharpoonup>
   (OCaml) "Int64.div" and
   (Scala) "_ '/ _"
 
 definition uint64_test_bit :: "uint64 \<Rightarrow> integer \<Rightarrow> bool"
 where [code del]:
   "uint64_test_bit x n =
   (if n < 0 \<or> 63 < n then undefined (bit :: uint64 \<Rightarrow> _) x n
    else bit x (nat_of_integer n))"
 
 lemma bit_uint64_code [code]:
   "bit x n \<longleftrightarrow> n < 64 \<and> uint64_test_bit x (integer_of_nat n)"
   including undefined_transfer integer.lifting unfolding uint64_test_bit_def
   by transfer (auto dest: bit_imp_le_length)
 
 lemma uint64_test_bit_code [code]:
   "uint64_test_bit w n =
   (if n < 0 \<or> 63 < n then undefined (bit :: uint64 \<Rightarrow> _) w n else bit (Rep_uint64 w) (nat_of_integer n))"
   unfolding uint64_test_bit_def by(simp add: bit_uint64.rep_eq)
 
 code_printing constant uint64_test_bit \<rightharpoonup>
   (SML) "Uint64.test'_bit" and
   (Haskell) "Data'_Bits.testBitBounded" and
   (OCaml) "Uint64.test'_bit" and
   (Scala) "Uint64.test'_bit" and
   (Eval) "(fn x => fn i => if i < 0 orelse i >= 64 then raise (Fail \"argument to uint64'_test'_bit out of bounds\") else Uint64.test'_bit x i)"
 
 definition uint64_set_bit :: "uint64 \<Rightarrow> integer \<Rightarrow> bool \<Rightarrow> uint64"
 where [code del]:
   "uint64_set_bit x n b =
   (if n < 0 \<or> 63 < n then undefined (set_bit :: uint64 \<Rightarrow> _) x n b
    else set_bit x (nat_of_integer n) b)"
 
 lemma set_bit_uint64_code [code]:
   "set_bit x n b = (if n < 64 then uint64_set_bit x (integer_of_nat n) b else x)"
 including undefined_transfer integer.lifting unfolding uint64_set_bit_def
 by(transfer)(auto cong: conj_cong simp add: not_less set_bit_beyond word_size)
 
 lemma uint64_set_bit_code [code]:
   "Rep_uint64 (uint64_set_bit w n b) = 
   (if n < 0 \<or> 63 < n then Rep_uint64 (undefined (set_bit :: uint64 \<Rightarrow> _) w n b)
    else set_bit (Rep_uint64 w) (nat_of_integer n) b)"
 including undefined_transfer unfolding uint64_set_bit_def by transfer simp
 
 code_printing constant uint64_set_bit \<rightharpoonup>
   (SML) "Uint64.set'_bit" and
   (Haskell) "Data'_Bits.setBitBounded" and
   (OCaml) "Uint64.set'_bit" and
   (Scala) "Uint64.set'_bit" and
   (Eval) "(fn x => fn i => fn b => if i < 0 orelse i >= 64 then raise (Fail \"argument to uint64'_set'_bit out of bounds\") else Uint64.set'_bit x i b)"
 
 definition uint64_shiftl :: "uint64 \<Rightarrow> integer \<Rightarrow> uint64"
 where [code del]:
   "uint64_shiftl x n = (if n < 0 \<or> 64 \<le> n then undefined (push_bit :: nat \<Rightarrow> uint64 \<Rightarrow> _) x n else push_bit (nat_of_integer n) x)"
 
 lemma shiftl_uint64_code [code]: "push_bit n x = (if n < 64 then uint64_shiftl x (integer_of_nat n) else 0)"
   including undefined_transfer integer.lifting unfolding uint64_shiftl_def
   by transfer simp
 
 lemma uint64_shiftl_code [code]:
   "Rep_uint64 (uint64_shiftl w n) =
   (if n < 0 \<or> 64 \<le> n then Rep_uint64 (undefined (push_bit :: nat \<Rightarrow> uint64 \<Rightarrow> _) w n) else push_bit (nat_of_integer n) (Rep_uint64 w))"
 including undefined_transfer unfolding uint64_shiftl_def by transfer simp
 
 code_printing constant uint64_shiftl \<rightharpoonup>
   (SML) "Uint64.shiftl" and
   (Haskell) "Data'_Bits.shiftlBounded" and
   (OCaml) "Uint64.shiftl" and
   (Scala) "Uint64.shiftl" and
   (Eval) "(fn x => fn i => if i < 0 orelse i >= 64 then raise (Fail \"argument to uint64'_shiftl out of bounds\") else Uint64.shiftl x i)"
 
 definition uint64_shiftr :: "uint64 \<Rightarrow> integer \<Rightarrow> uint64"
 where [code del]:
   "uint64_shiftr x n = (if n < 0 \<or> 64 \<le> n then undefined (drop_bit :: nat \<Rightarrow> uint64 \<Rightarrow> _) x n else drop_bit (nat_of_integer n) x)"
 
 lemma shiftr_uint64_code [code]: "drop_bit n x = (if n < 64 then uint64_shiftr x (integer_of_nat n) else 0)"
   including undefined_transfer integer.lifting unfolding uint64_shiftr_def
   by transfer simp
 
 lemma uint64_shiftr_code [code]:
   "Rep_uint64 (uint64_shiftr w n) =
   (if n < 0 \<or> 64 \<le> n then Rep_uint64 (undefined (drop_bit :: nat \<Rightarrow> uint64 \<Rightarrow> _) w n) else drop_bit (nat_of_integer n) (Rep_uint64 w))"
   including undefined_transfer unfolding uint64_shiftr_def by transfer simp
 
 code_printing constant uint64_shiftr \<rightharpoonup>
   (SML) "Uint64.shiftr" and
   (Haskell) "Data'_Bits.shiftrBounded" and
   (OCaml) "Uint64.shiftr" and
   (Scala) "Uint64.shiftr" and
   (Eval) "(fn x => fn i => if i < 0 orelse i >= 64 then raise (Fail \"argument to uint64'_shiftr out of bounds\") else Uint64.shiftr x i)"
 
 definition uint64_sshiftr :: "uint64 \<Rightarrow> integer \<Rightarrow> uint64"
 where [code del]:
   "uint64_sshiftr x n =
   (if n < 0 \<or> 64 \<le> n then undefined signed_drop_bit_uint64 n x else signed_drop_bit_uint64 (nat_of_integer n) x)"
 
 lemma sshiftr_uint64_code [code]:
   "signed_drop_bit_uint64 n x = 
   (if n < 64 then uint64_sshiftr x (integer_of_nat n) else if bit x 63 then - 1 else 0)"
   including undefined_transfer integer.lifting unfolding uint64_sshiftr_def
   by transfer (simp add: not_less signed_drop_bit_beyond)
 
 lemma uint64_sshiftr_code [code]:
   "Rep_uint64 (uint64_sshiftr w n) =
   (if n < 0 \<or> 64 \<le> n then Rep_uint64 (undefined signed_drop_bit_uint64 n w) else signed_drop_bit (nat_of_integer n) (Rep_uint64 w))"
 including undefined_transfer unfolding uint64_sshiftr_def by transfer simp
 
 code_printing constant uint64_sshiftr \<rightharpoonup>
   (SML) "Uint64.shiftr'_signed" and
   (Haskell) 
     "(Prelude.fromInteger (Prelude.toInteger (Data'_Bits.shiftrBounded (Prelude.fromInteger (Prelude.toInteger _) :: Uint64.Int64) _)) :: Uint64.Word64)" and
   (OCaml) "Uint64.shiftr'_signed" and
   (Scala) "Uint64.shiftr'_signed" and
   (Eval) "(fn x => fn i => if i < 0 orelse i >= 64 then raise (Fail \"argument to uint64'_shiftr'_signed out of bounds\") else Uint64.shiftr'_signed x i)"
 
 context
   includes bit_operations_syntax
 begin
 
 lemma uint64_msb_test_bit: "msb x \<longleftrightarrow> bit (x :: uint64) 63"
   by transfer (simp add: msb_word_iff_bit)
 
 lemma msb_uint64_code [code]: "msb x \<longleftrightarrow> uint64_test_bit x 63"
   by (simp add: uint64_test_bit_def uint64_msb_test_bit)
 
 lemma uint64_of_int_code [code]:
   "uint64_of_int i = Uint64 (integer_of_int i)"
   including integer.lifting by transfer simp
 
 lemma int_of_uint64_code [code]:
   "int_of_uint64 x = int_of_integer (integer_of_uint64 x)"
   including integer.lifting by transfer simp
 
 lemma uint64_of_nat_code [code]:
   "uint64_of_nat = uint64_of_int \<circ> int"
   by transfer (simp add: fun_eq_iff)
 
 lemma nat_of_uint64_code [code]:
   "nat_of_uint64 x = nat_of_integer (integer_of_uint64 x)"
   unfolding integer_of_uint64_def including integer.lifting by transfer simp
 
 definition integer_of_uint64_signed :: "uint64 \<Rightarrow> integer"
 where
   "integer_of_uint64_signed n = (if bit n 63 then undefined integer_of_uint64 n else integer_of_uint64 n)"
 
 lemma integer_of_uint64_signed_code [code]:
   "integer_of_uint64_signed n =
   (if bit n 63 then undefined integer_of_uint64 n else integer_of_int (uint (Rep_uint64' n)))"
   by (simp add: integer_of_uint64_signed_def integer_of_uint64_def)
 
 lemma integer_of_uint64_code [code]:
   "integer_of_uint64 n =
   (if bit n 63 then integer_of_uint64_signed (n AND 0x7FFFFFFFFFFFFFFF) OR 0x8000000000000000 else integer_of_uint64_signed n)"
 proof -
   have \<open>integer_of_uint64_signed (n AND 0x7FFFFFFFFFFFFFFF) OR 0x8000000000000000 = Bit_Operations.set_bit 63 (integer_of_uint64_signed (take_bit 63 n))\<close>
     by (simp add: take_bit_eq_mask set_bit_eq_or push_bit_eq_mult mask_eq_exp_minus_1)
   moreover have \<open>integer_of_uint64 n = Bit_Operations.set_bit 63 (integer_of_uint64 (take_bit 63 n))\<close> if \<open>bit n 63\<close>
   proof (rule bit_eqI)
     fix m
     from that show \<open>bit (integer_of_uint64 n) m = bit (Bit_Operations.set_bit 63 (integer_of_uint64 (take_bit 63 n))) m\<close> for m
       including integer.lifting by transfer (auto simp add: bit_simps dest: bit_imp_le_length)
   qed
   ultimately show ?thesis
     by simp (simp add: integer_of_uint64_signed_def bit_simps)
 qed
 
 end
 
 code_printing
   constant "integer_of_uint64" \<rightharpoonup>
   (SML) "Uint64.toInt" and
   (Haskell) "Prelude.toInteger"
 | constant "integer_of_uint64_signed" \<rightharpoonup>
   (OCaml) "Z.of'_int64" and
   (Scala) "BigInt"
 
 section \<open>Quickcheck setup\<close>
 
 definition uint64_of_natural :: "natural \<Rightarrow> uint64"
 where "uint64_of_natural x \<equiv> Uint64 (integer_of_natural x)"
 
 instantiation uint64 :: "{random, exhaustive, full_exhaustive}" begin
 definition "random_uint64 \<equiv> qc_random_cnv uint64_of_natural"
 definition "exhaustive_uint64 \<equiv> qc_exhaustive_cnv uint64_of_natural"
 definition "full_exhaustive_uint64 \<equiv> qc_full_exhaustive_cnv uint64_of_natural"
 instance ..
 end
 
 instantiation uint64 :: narrowing begin
 
 interpretation quickcheck_narrowing_samples
   "\<lambda>i. let x = Uint64 i in (x, 0xFFFFFFFFFFFFFFFF - x)" "0"
   "Typerep.Typerep (STR ''Uint64.uint64'') []" .
 
 definition "narrowing_uint64 d = qc_narrowing_drawn_from (narrowing_samples d) d"
 declare [[code drop: "partial_term_of :: uint64 itself \<Rightarrow> _"]]
 lemmas partial_term_of_uint64 [code] = partial_term_of_code
 
 instance ..
 end
 
 end
diff --git a/thys/Native_Word/Uint8.thy b/thys/Native_Word/Uint8.thy
--- a/thys/Native_Word/Uint8.thy
+++ b/thys/Native_Word/Uint8.thy
@@ -1,610 +1,611 @@
 (*  Title:      Uint8.thy
     Author:     Andreas Lochbihler, ETH Zurich
 *)
 
 chapter \<open>Unsigned words of 8 bits\<close>
 
 theory Uint8 imports
-  Code_Target_Word_Base Word_Type_Copies
+  Word_Type_Copies
+  Code_Target_Integer_Bit
 begin
 
 text \<open>
   Restriction for OCaml code generation:
   OCaml does not provide an int8 type, so no special code generation 
-  for this type is set up. If the theory \<open>Code_Target_Bits_Int\<close>
+  for this type is set up. If the theory \<^text>\<open>Code_Target_Int_Bit\<close>
   is imported, the type \<open>uint8\<close> is emulated via \<^typ>\<open>8 word\<close>.
 \<close>
 
 section \<open>Type definition and primitive operations\<close>
 
 typedef uint8 = \<open>UNIV :: 8 word set\<close> ..
 
 global_interpretation uint8: word_type_copy Abs_uint8 Rep_uint8
   using type_definition_uint8 by (rule word_type_copy.intro)
 
 setup_lifting type_definition_uint8
 
 declare uint8.of_word_of [code abstype]
 
 declare Quotient_uint8 [transfer_rule]
 
 instantiation uint8 :: \<open>{comm_ring_1, semiring_modulo, equal, linorder}\<close>
 begin
 
 lift_definition zero_uint8 :: uint8 is 0 .
 lift_definition one_uint8 :: uint8 is 1 .
 lift_definition plus_uint8 :: \<open>uint8 \<Rightarrow> uint8 \<Rightarrow> uint8\<close> is \<open>(+)\<close> .
 lift_definition uminus_uint8 :: \<open>uint8 \<Rightarrow> uint8\<close> is uminus .
 lift_definition minus_uint8 :: \<open>uint8 \<Rightarrow> uint8 \<Rightarrow> uint8\<close> is \<open>(-)\<close> .
 lift_definition times_uint8 :: \<open>uint8 \<Rightarrow> uint8 \<Rightarrow> uint8\<close> is \<open>(*)\<close> .
 lift_definition divide_uint8 :: \<open>uint8 \<Rightarrow> uint8 \<Rightarrow> uint8\<close> is \<open>(div)\<close> .
 lift_definition modulo_uint8 :: \<open>uint8 \<Rightarrow> uint8 \<Rightarrow> uint8\<close> is \<open>(mod)\<close> .
 lift_definition equal_uint8 :: \<open>uint8 \<Rightarrow> uint8 \<Rightarrow> bool\<close> is \<open>HOL.equal\<close> .
 lift_definition less_eq_uint8 :: \<open>uint8 \<Rightarrow> uint8 \<Rightarrow> bool\<close> is \<open>(\<le>)\<close> .
 lift_definition less_uint8 :: \<open>uint8 \<Rightarrow> uint8 \<Rightarrow> bool\<close> is \<open>(<)\<close> .
 
 global_interpretation uint8: word_type_copy_ring Abs_uint8 Rep_uint8
   by standard (fact zero_uint8.rep_eq one_uint8.rep_eq
     plus_uint8.rep_eq uminus_uint8.rep_eq minus_uint8.rep_eq
     times_uint8.rep_eq divide_uint8.rep_eq modulo_uint8.rep_eq
     equal_uint8.rep_eq less_eq_uint8.rep_eq less_uint8.rep_eq)+
 
 instance proof -
   show \<open>OFCLASS(uint8, comm_ring_1_class)\<close>
     by (rule uint8.of_class_comm_ring_1)
   show \<open>OFCLASS(uint8, semiring_modulo_class)\<close>
     by (fact uint8.of_class_semiring_modulo)
   show \<open>OFCLASS(uint8, equal_class)\<close>
     by (fact uint8.of_class_equal)
   show \<open>OFCLASS(uint8, linorder_class)\<close>
     by (fact uint8.of_class_linorder)
 qed
 
 end
 
 instantiation uint8 :: ring_bit_operations
 begin
 
 lift_definition bit_uint8 :: \<open>uint8 \<Rightarrow> nat \<Rightarrow> bool\<close> is bit .
 lift_definition not_uint8 :: \<open>uint8 \<Rightarrow> uint8\<close> is \<open>Bit_Operations.not\<close> .
 lift_definition and_uint8 :: \<open>uint8 \<Rightarrow> uint8 \<Rightarrow> uint8\<close> is \<open>Bit_Operations.and\<close> .
 lift_definition or_uint8 :: \<open>uint8 \<Rightarrow> uint8 \<Rightarrow> uint8\<close> is \<open>Bit_Operations.or\<close> .
 lift_definition xor_uint8 :: \<open>uint8 \<Rightarrow> uint8 \<Rightarrow> uint8\<close> is \<open>Bit_Operations.xor\<close> .
 lift_definition mask_uint8 :: \<open>nat \<Rightarrow> uint8\<close> is mask .
 lift_definition push_bit_uint8 :: \<open>nat \<Rightarrow> uint8 \<Rightarrow> uint8\<close> is push_bit .
 lift_definition drop_bit_uint8 :: \<open>nat \<Rightarrow> uint8 \<Rightarrow> uint8\<close> is drop_bit .
 lift_definition signed_drop_bit_uint8 :: \<open>nat \<Rightarrow> uint8 \<Rightarrow> uint8\<close> is signed_drop_bit .
 lift_definition take_bit_uint8 :: \<open>nat \<Rightarrow> uint8 \<Rightarrow> uint8\<close> is take_bit .
 lift_definition set_bit_uint8 :: \<open>nat \<Rightarrow> uint8 \<Rightarrow> uint8\<close> is Bit_Operations.set_bit .
 lift_definition unset_bit_uint8 :: \<open>nat \<Rightarrow> uint8 \<Rightarrow> uint8\<close> is unset_bit .
 lift_definition flip_bit_uint8 :: \<open>nat \<Rightarrow> uint8 \<Rightarrow> uint8\<close> is flip_bit .
 
 global_interpretation uint8: word_type_copy_bits Abs_uint8 Rep_uint8 signed_drop_bit_uint8
   by standard (fact bit_uint8.rep_eq not_uint8.rep_eq and_uint8.rep_eq or_uint8.rep_eq xor_uint8.rep_eq
     mask_uint8.rep_eq push_bit_uint8.rep_eq drop_bit_uint8.rep_eq signed_drop_bit_uint8.rep_eq take_bit_uint8.rep_eq
     set_bit_uint8.rep_eq unset_bit_uint8.rep_eq flip_bit_uint8.rep_eq)+
 
 instance
   by (fact uint8.of_class_ring_bit_operations)
 
 end
 
 lift_definition uint8_of_nat :: \<open>nat \<Rightarrow> uint8\<close>
   is word_of_nat .
 
 lift_definition nat_of_uint8 :: \<open>uint8 \<Rightarrow> nat\<close>
   is unat .
 
 lift_definition uint8_of_int :: \<open>int \<Rightarrow> uint8\<close>
   is word_of_int .
 
 lift_definition int_of_uint8 :: \<open>uint8 \<Rightarrow> int\<close>
   is uint .
 
 context
   includes integer.lifting
 begin
 
 lift_definition Uint8 :: \<open>integer \<Rightarrow> uint8\<close>
   is word_of_int .
 
 lift_definition integer_of_uint8 :: \<open>uint8 \<Rightarrow> integer\<close>
   is uint .
 
 end
 
 global_interpretation uint8: word_type_copy_more Abs_uint8 Rep_uint8 signed_drop_bit_uint8
   uint8_of_nat nat_of_uint8 uint8_of_int int_of_uint8 Uint8 integer_of_uint8
   apply standard
        apply (simp_all add: uint8_of_nat.rep_eq nat_of_uint8.rep_eq
          uint8_of_int.rep_eq int_of_uint8.rep_eq
          Uint8.rep_eq integer_of_uint8.rep_eq integer_eq_iff)
   done
 
 instantiation uint8 :: "{size, msb, lsb, set_bit, bit_comprehension}"
 begin
 
 lift_definition size_uint8 :: \<open>uint8 \<Rightarrow> nat\<close> is size .
 
 lift_definition msb_uint8 :: \<open>uint8 \<Rightarrow> bool\<close> is msb .
 lift_definition lsb_uint8 :: \<open>uint8 \<Rightarrow> bool\<close> is lsb .
 
 text \<open>Workaround: avoid name space clash by spelling out \<^text>\<open>lift_definition\<close> explicitly.\<close>
 
 definition set_bit_uint8 :: \<open>uint8 \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> uint8\<close>
   where set_bit_uint8_eq: \<open>set_bit_uint8 a n b = (if b then Bit_Operations.set_bit else unset_bit) n a\<close>
 
 context
   includes lifting_syntax
 begin
 
 lemma set_bit_uint8_transfer [transfer_rule]:
   \<open>(cr_uint8 ===> (=) ===> (\<longleftrightarrow>) ===> cr_uint8) Generic_set_bit.set_bit Generic_set_bit.set_bit\<close>
   by (simp only: set_bit_eq [abs_def] set_bit_uint8_eq [abs_def]) transfer_prover
 
 end
 
 lift_definition set_bits_uint8 :: \<open>(nat \<Rightarrow> bool) \<Rightarrow> uint8\<close> is set_bits .
 lift_definition set_bits_aux_uint8 :: \<open>(nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> uint8 \<Rightarrow> uint8\<close> is set_bits_aux .
 
 global_interpretation uint8: word_type_copy_misc Abs_uint8 Rep_uint8 signed_drop_bit_uint8
   uint8_of_nat nat_of_uint8 uint8_of_int int_of_uint8 Uint8 integer_of_uint8 8 set_bits_aux_uint8
   by (standard; transfer) simp_all
 
 instance using uint8.of_class_bit_comprehension
   uint8.of_class_set_bit uint8.of_class_lsb
   by simp_all standard
 
 end
 
 section \<open>Code setup\<close>
 
 code_printing code_module Uint8 \<rightharpoonup> (SML)
 \<open>(* Test that words can handle numbers between 0 and 3 *)
 val _ = if 3 <= Word.wordSize then () else raise (Fail ("wordSize less than 3"));
 
 structure Uint8 : sig
   val set_bit : Word8.word -> IntInf.int -> bool -> Word8.word
   val shiftl : Word8.word -> IntInf.int -> Word8.word
   val shiftr : Word8.word -> IntInf.int -> Word8.word
   val shiftr_signed : Word8.word -> IntInf.int -> Word8.word
   val test_bit : Word8.word -> IntInf.int -> bool
 end = struct
 
 fun set_bit x n b =
   let val mask = Word8.<< (0wx1, Word.fromLargeInt (IntInf.toLarge n))
   in if b then Word8.orb (x, mask)
      else Word8.andb (x, Word8.notb mask)
   end
 
 fun shiftl x n =
   Word8.<< (x, Word.fromLargeInt (IntInf.toLarge n))
 
 fun shiftr x n =
   Word8.>> (x, Word.fromLargeInt (IntInf.toLarge n))
 
 fun shiftr_signed x n =
   Word8.~>> (x, Word.fromLargeInt (IntInf.toLarge n))
 
 fun test_bit x n =
   Word8.andb (x, Word8.<< (0wx1, Word.fromLargeInt (IntInf.toLarge n))) <> Word8.fromInt 0
 
 end; (* struct Uint8 *)\<close>
 code_reserved SML Uint8
 
 code_printing code_module Uint8 \<rightharpoonup> (Haskell)
  \<open>module Uint8(Int8, Word8) where
 
   import Data.Int(Int8)
   import Data.Word(Word8)\<close>
 code_reserved Haskell Uint8
 
 text \<open>
   Scala provides only signed 8bit numbers, so we use these and 
   implement sign-sensitive operations like comparisons manually.
 \<close>
 
 code_printing code_module Uint8 \<rightharpoonup> (Scala)
 \<open>object Uint8 {
 
 def less(x: Byte, y: Byte) : Boolean =
   if (x < 0) y < 0 && x < y
   else y < 0 || x < y
 
 def less_eq(x: Byte, y: Byte) : Boolean =
   if (x < 0) y < 0 && x <= y
   else y < 0 || x <= y
 
 def set_bit(x: Byte, n: BigInt, b: Boolean) : Byte =
   if (b)
     (x | (1 << n.intValue)).toByte
   else
     (x & (1 << n.intValue).unary_~).toByte
 
 def shiftl(x: Byte, n: BigInt) : Byte = (x << n.intValue).toByte
 
 def shiftr(x: Byte, n: BigInt) : Byte = ((x & 255) >>> n.intValue).toByte
 
 def shiftr_signed(x: Byte, n: BigInt) : Byte = (x >> n.intValue).toByte
 
 def test_bit(x: Byte, n: BigInt) : Boolean =
   (x & (1 << n.intValue)) != 0
 
 } /* object Uint8 */\<close>
 code_reserved Scala Uint8
 
 text \<open>
   Avoid @{term Abs_uint8} in generated code, use @{term Rep_uint8'} instead. 
   The symbolic implementations for code\_simp use @{term Rep_uint8}.
 
   The new destructor @{term Rep_uint8'} is executable.
   As the simplifier is given the [code abstract] equations literally, 
   we cannot implement @{term Rep_uint8} directly, because that makes code\_simp loop.
 
   If code generation raises Match, some equation probably contains @{term Rep_uint8} 
   ([code abstract] equations for @{typ uint8} may use @{term Rep_uint8} because
   these instances will be folded away.)
 
   To convert @{typ "8 word"} values into @{typ uint8}, use @{term "Abs_uint8'"}.
 \<close>
 
 definition Rep_uint8' where [simp]: "Rep_uint8' = Rep_uint8"
 
 lemma Rep_uint8'_transfer [transfer_rule]:
   "rel_fun cr_uint8 (=) (\<lambda>x. x) Rep_uint8'"
 unfolding Rep_uint8'_def by(rule uint8.rep_transfer)
 
 lemma Rep_uint8'_code [code]: "Rep_uint8' x = (BITS n. bit x n)"
   by transfer (simp add: set_bits_bit_eq)
 
 lift_definition Abs_uint8' :: "8 word \<Rightarrow> uint8" is "\<lambda>x :: 8 word. x" .
 
 lemma Abs_uint8'_code [code]: "Abs_uint8' x = Uint8 (integer_of_int (uint x))"
 including integer.lifting by transfer simp
 
 declare [[code drop: "term_of_class.term_of :: uint8 \<Rightarrow> _"]]
 
 lemma term_of_uint8_code [code]:
   defines "TR \<equiv> typerep.Typerep" and "bit0 \<equiv> STR ''Numeral_Type.bit0''" shows
   "term_of_class.term_of x = 
    Code_Evaluation.App (Code_Evaluation.Const (STR ''Uint8.uint8.Abs_uint8'') (TR (STR ''fun'') [TR (STR ''Word.word'') [TR bit0 [TR bit0 [TR bit0 [TR (STR ''Numeral_Type.num1'') []]]]], TR (STR ''Uint8.uint8'') []]))
        (term_of_class.term_of (Rep_uint8' x))"
 by(simp add: term_of_anything)
 
 lemma Uin8_code [code]: "Rep_uint8 (Uint8 i) = word_of_int (int_of_integer_symbolic i)"
 unfolding Uint8_def int_of_integer_symbolic_def by(simp add: Abs_uint8_inverse)
 
 code_printing type_constructor uint8 \<rightharpoonup>
   (SML) "Word8.word" and
   (Haskell) "Uint8.Word8" and
   (Scala) "Byte"
 | constant Uint8 \<rightharpoonup> 
   (SML) "Word8.fromLargeInt (IntInf.toLarge _)" and
   (Haskell) "(Prelude.fromInteger _ :: Uint8.Word8)" and
   (Haskell_Quickcheck) "(Prelude.fromInteger (Prelude.toInteger _) :: Uint8.Word8)" and
   (Scala) "_.byteValue"
 | constant "0 :: uint8" \<rightharpoonup>
   (SML) "(Word8.fromInt 0)" and
   (Haskell) "(0 :: Uint8.Word8)" and
   (Scala) "0.toByte"
 | constant "1 :: uint8" \<rightharpoonup>
   (SML) "(Word8.fromInt 1)" and
   (Haskell) "(1 :: Uint8.Word8)" and
   (Scala) "1.toByte"
 | constant "plus :: uint8 \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word8.+ ((_), (_))" and
   (Haskell) infixl 6 "+" and
   (Scala) "(_ +/ _).toByte"
 | constant "uminus :: uint8 \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word8.~" and
   (Haskell) "negate" and
   (Scala) "(- _).toByte"
 | constant "minus :: uint8 \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word8.- ((_), (_))" and
   (Haskell) infixl 6 "-" and
   (Scala) "(_ -/ _).toByte"
 | constant "times :: uint8 \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word8.* ((_), (_))" and
   (Haskell) infixl 7 "*" and
   (Scala) "(_ */ _).toByte"
 | constant "HOL.equal :: uint8 \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
   (SML) "!((_ : Word8.word) = _)" and
   (Haskell) infix 4 "==" and
   (Scala) infixl 5 "=="
 | class_instance uint8 :: equal \<rightharpoonup> (Haskell) -
 | constant "less_eq :: uint8 \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
   (SML) "Word8.<= ((_), (_))" and
   (Haskell) infix 4 "<=" and
   (Scala) "Uint8.less'_eq"
 | constant "less :: uint8 \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
   (SML) "Word8.< ((_), (_))" and
   (Haskell) infix 4 "<" and
   (Scala) "Uint8.less"
 | constant "Bit_Operations.not :: uint8 \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word8.notb" and
   (Haskell) "Data'_Bits.complement" and
   (Scala) "_.unary'_~.toByte"
 | constant "Bit_Operations.and :: uint8 \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word8.andb ((_),/ (_))" and
   (Haskell) infixl 7 "Data_Bits..&." and
   (Scala) "(_ & _).toByte"
 | constant "Bit_Operations.or :: uint8 \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word8.orb ((_),/ (_))" and
   (Haskell) infixl 5 "Data_Bits..|." and
   (Scala) "(_ | _).toByte"
 | constant "Bit_Operations.xor :: uint8 \<Rightarrow> _" \<rightharpoonup>
   (SML) "Word8.xorb ((_),/ (_))" and
   (Haskell) "Data'_Bits.xor" and
   (Scala) "(_ ^ _).toByte"
 
 definition uint8_divmod :: "uint8 \<Rightarrow> uint8 \<Rightarrow> uint8 \<times> uint8" where
   "uint8_divmod x y = 
   (if y = 0 then (undefined ((div) :: uint8 \<Rightarrow> _) x (0 :: uint8), undefined ((mod) :: uint8 \<Rightarrow> _) x (0 :: uint8)) 
   else (x div y, x mod y))"
 
 definition uint8_div :: "uint8 \<Rightarrow> uint8 \<Rightarrow> uint8" 
 where "uint8_div x y = fst (uint8_divmod x y)"
 
 definition uint8_mod :: "uint8 \<Rightarrow> uint8 \<Rightarrow> uint8" 
 where "uint8_mod x y = snd (uint8_divmod x y)"
 
 lemma div_uint8_code [code]: "x div y = (if y = 0 then 0 else uint8_div x y)"
   including undefined_transfer unfolding uint8_divmod_def uint8_div_def
 by transfer (simp add: word_div_def)
 
 lemma mod_uint8_code [code]: "x mod y = (if y = 0 then x else uint8_mod x y)"
 including undefined_transfer unfolding uint8_mod_def uint8_divmod_def
 by transfer (simp add: word_mod_def)
 
 definition uint8_sdiv :: "uint8 \<Rightarrow> uint8 \<Rightarrow> uint8"
 where
   "uint8_sdiv x y =
    (if y = 0 then undefined ((div) :: uint8 \<Rightarrow> _) x (0 :: uint8)
     else Abs_uint8 (Rep_uint8 x sdiv Rep_uint8 y))"
 
 definition div0_uint8 :: "uint8 \<Rightarrow> uint8"
 where [code del]: "div0_uint8 x = undefined ((div) :: uint8 \<Rightarrow> _) x (0 :: uint8)"
 declare [[code abort: div0_uint8]]
 
 definition mod0_uint8 :: "uint8 \<Rightarrow> uint8"
 where [code del]: "mod0_uint8 x = undefined ((mod) :: uint8 \<Rightarrow> _) x (0 :: uint8)"
 declare [[code abort: mod0_uint8]]
 
 lemma uint8_divmod_code [code]:
   "uint8_divmod x y =
   (if 0x80 \<le> y then if x < y then (0, x) else (1, x - y)
    else if y = 0 then (div0_uint8 x, mod0_uint8 x)
    else let q = push_bit 1 (uint8_sdiv (drop_bit 1 x) y);
             r = x - q * y
         in if r \<ge> y then (q + 1, r - y) else (q, r))"
   including undefined_transfer unfolding uint8_divmod_def uint8_sdiv_def div0_uint8_def mod0_uint8_def
     less_eq_uint8.rep_eq
   apply transfer
   apply (simp add: divmod_via_sdivmod push_bit_eq_mult)
   done
 
 lemma uint8_sdiv_code [code]:
   "Rep_uint8 (uint8_sdiv x y) =
    (if y = 0 then Rep_uint8 (undefined ((div) :: uint8 \<Rightarrow> _) x (0 :: uint8))
     else Rep_uint8 x sdiv Rep_uint8 y)"
 unfolding uint8_sdiv_def by(simp add: Abs_uint8_inverse)
 
 text \<open>
   Note that we only need a translation for signed division, but not for the remainder
   because @{thm uint8_divmod_code} computes both with division only.
 \<close>
 
 code_printing
   constant uint8_div \<rightharpoonup>
   (SML) "Word8.div ((_), (_))" and
   (Haskell) "Prelude.div"
 | constant uint8_mod \<rightharpoonup>
   (SML) "Word8.mod ((_), (_))" and
   (Haskell) "Prelude.mod"
 | constant uint8_divmod \<rightharpoonup>
   (Haskell) "divmod"
 | constant uint8_sdiv \<rightharpoonup>
   (Scala) "(_ '/ _).toByte"
 
 definition uint8_test_bit :: "uint8 \<Rightarrow> integer \<Rightarrow> bool"
 where [code del]:
   "uint8_test_bit x n =
   (if n < 0 \<or> 7 < n then undefined (bit :: uint8 \<Rightarrow> _) x n
    else bit x (nat_of_integer n))"
 
 lemma bit_uint8_code [code]:
   "bit x n \<longleftrightarrow> n < 8 \<and> uint8_test_bit x (integer_of_nat n)"
   including undefined_transfer integer.lifting unfolding uint8_test_bit_def
   by (transfer, simp, transfer, simp)
 
 lemma uint8_test_bit_code [code]:
   "uint8_test_bit w n =
   (if n < 0 \<or> 7 < n then undefined (bit :: uint8 \<Rightarrow> _) w n else bit (Rep_uint8 w) (nat_of_integer n))"
   unfolding uint8_test_bit_def
   by (simp add: bit_uint8.rep_eq)
 
 code_printing constant uint8_test_bit \<rightharpoonup>
   (SML) "Uint8.test'_bit" and
   (Haskell) "Data'_Bits.testBitBounded" and
   (Scala) "Uint8.test'_bit" and
   (Eval) "(fn x => fn i => if i < 0 orelse i >= 8 then raise (Fail \"argument to uint8'_test'_bit out of bounds\") else Uint8.test'_bit x i)"
 
 
 definition uint8_set_bit :: "uint8 \<Rightarrow> integer \<Rightarrow> bool \<Rightarrow> uint8"
 where [code del]:
   "uint8_set_bit x n b =
   (if n < 0 \<or> 7 < n then undefined (set_bit :: uint8 \<Rightarrow> _) x n b
    else set_bit x (nat_of_integer n) b)"
 
 lemma set_bit_uint8_code [code]:
   "set_bit x n b = (if n < 8 then uint8_set_bit x (integer_of_nat n) b else x)"
 including undefined_transfer integer.lifting unfolding uint8_set_bit_def
 by(transfer)(auto cong: conj_cong simp add: not_less set_bit_beyond word_size)
 
 lemma uint8_set_bit_code [code]:
   "Rep_uint8 (uint8_set_bit w n b) = 
   (if n < 0 \<or> 7 < n then Rep_uint8 (undefined (set_bit :: uint8 \<Rightarrow> _) w n b)
    else set_bit (Rep_uint8 w) (nat_of_integer n) b)"
 including undefined_transfer unfolding uint8_set_bit_def by transfer simp
 
 code_printing constant uint8_set_bit \<rightharpoonup>
   (SML) "Uint8.set'_bit" and
   (Haskell) "Data'_Bits.setBitBounded" and
   (Scala) "Uint8.set'_bit" and
   (Eval) "(fn x => fn i => fn b => if i < 0 orelse i >= 8 then raise (Fail \"argument to uint8'_set'_bit out of bounds\") else Uint8.set'_bit x i b)"
 
 
 definition uint8_shiftl :: "uint8 \<Rightarrow> integer \<Rightarrow> uint8"
 where [code del]:
   "uint8_shiftl x n = (if n < 0 \<or> 8 \<le> n then undefined (push_bit :: nat \<Rightarrow> uint8 \<Rightarrow> _) x n else push_bit (nat_of_integer n) x)"
 
 lemma shiftl_uint8_code [code]:
   "push_bit n x = (if n < 8 then uint8_shiftl x (integer_of_nat n) else 0)"
   including undefined_transfer integer.lifting unfolding uint8_shiftl_def
   by transfer simp
 
 lemma uint8_shiftl_code [code]:
   "Rep_uint8 (uint8_shiftl w n) =
   (if n < 0 \<or> 8 \<le> n then Rep_uint8 (undefined (push_bit :: nat \<Rightarrow> uint8 \<Rightarrow> _) w n)
    else push_bit (nat_of_integer n) (Rep_uint8 w))"
   including undefined_transfer unfolding uint8_shiftl_def
   by transfer simp
 
 code_printing constant uint8_shiftl \<rightharpoonup>
   (SML) "Uint8.shiftl" and
   (Haskell) "Data'_Bits.shiftlBounded" and
   (Scala) "Uint8.shiftl" and
   (Eval) "(fn x => fn i => if i < 0 orelse i >= 8 then raise (Fail \"argument to uint8'_shiftl out of bounds\") else Uint8.shiftl x i)"
 
 definition uint8_shiftr :: "uint8 \<Rightarrow> integer \<Rightarrow> uint8"
 where [code del]:
   "uint8_shiftr x n = (if n < 0 \<or> 8 \<le> n then undefined (drop_bit :: _ \<Rightarrow> _ \<Rightarrow> uint8) x n else drop_bit (nat_of_integer n) x)"
 
 lemma shiftr_uint8_code [code]:
   "drop_bit n x = (if n < 8 then uint8_shiftr x (integer_of_nat n) else 0)"
   including undefined_transfer integer.lifting unfolding uint8_shiftr_def
   by transfer simp
 
 lemma uint8_shiftr_code [code]:
   "Rep_uint8 (uint8_shiftr w n) =
   (if n < 0 \<or> 8 \<le> n then Rep_uint8 (undefined (drop_bit :: _ \<Rightarrow> _ \<Rightarrow> uint8) w n) 
    else drop_bit (nat_of_integer n) (Rep_uint8 w))"
 including undefined_transfer unfolding uint8_shiftr_def by transfer simp
 
 code_printing constant uint8_shiftr \<rightharpoonup>
   (SML) "Uint8.shiftr" and
   (Haskell) "Data'_Bits.shiftrBounded" and
   (Scala) "Uint8.shiftr" and
   (Eval) "(fn x => fn i => if i < 0 orelse i >= 8 then raise (Fail \"argument to uint8'_shiftr out of bounds\") else Uint8.shiftr x i)"
 
 definition uint8_sshiftr :: "uint8 \<Rightarrow> integer \<Rightarrow> uint8"
 where [code del]:
   "uint8_sshiftr x n =
   (if n < 0 \<or> 8 \<le> n then undefined signed_drop_bit_uint8 n x else signed_drop_bit_uint8 (nat_of_integer n) x)"
 
 lemma sshiftr_uint8_code [code]:
   "signed_drop_bit_uint8 n x = 
   (if n < 8 then uint8_sshiftr x (integer_of_nat n) else if bit x 7 then -1 else 0)"
   including undefined_transfer integer.lifting unfolding uint8_sshiftr_def
   by transfer (simp add: not_less signed_drop_bit_beyond word_size)
 
 lemma uint8_sshiftr_code [code]:
   "Rep_uint8 (uint8_sshiftr w n) =
   (if n < 0 \<or> 8 \<le> n then Rep_uint8 (undefined signed_drop_bit_uint8 n w)
    else signed_drop_bit (nat_of_integer n) (Rep_uint8 w))"
   including undefined_transfer unfolding uint8_sshiftr_def
   by transfer simp
 
 code_printing constant uint8_sshiftr \<rightharpoonup>
   (SML) "Uint8.shiftr'_signed" and
   (Haskell) 
     "(Prelude.fromInteger (Prelude.toInteger (Data'_Bits.shiftrBounded (Prelude.fromInteger (Prelude.toInteger _) :: Uint8.Int8) _)) :: Uint8.Word8)" and
   (Scala) "Uint8.shiftr'_signed" and
   (Eval) "(fn x => fn i => if i < 0 orelse i >= 8 then raise (Fail \"argument to uint8'_sshiftr out of bounds\") else Uint8.shiftr'_signed x i)"
 
 context
   includes bit_operations_syntax
 begin
 
 lemma uint8_msb_test_bit: "msb x \<longleftrightarrow> bit (x :: uint8) 7"
   by transfer (simp add: msb_word_iff_bit)
 
 lemma msb_uint16_code [code]: "msb x \<longleftrightarrow> uint8_test_bit x 7"
   by (simp add: uint8_test_bit_def uint8_msb_test_bit)
 
 lemma uint8_of_int_code [code]:
   "uint8_of_int i = Uint8 (integer_of_int i)"
   including integer.lifting by transfer simp
 
 lemma int_of_uint8_code [code]:
   "int_of_uint8 x = int_of_integer (integer_of_uint8 x)"
   by (simp add: int_of_uint8.rep_eq integer_of_uint8_def)
 
 lemma uint8_of_nat_code [code]:
   "uint8_of_nat = uint8_of_int \<circ> int"
   by transfer (simp add: fun_eq_iff)
 
 lemma nat_of_uint8_code [code]:
   "nat_of_uint8 x = nat_of_integer (integer_of_uint8 x)"
   unfolding integer_of_uint8_def including integer.lifting by transfer simp
 
 definition integer_of_uint8_signed :: "uint8 \<Rightarrow> integer"
 where
   "integer_of_uint8_signed n = (if bit n 7 then undefined integer_of_uint8 n else integer_of_uint8 n)"
 
 lemma integer_of_uint8_signed_code [code]:
   "integer_of_uint8_signed n =
   (if bit n 7 then undefined integer_of_uint8 n else integer_of_int (uint (Rep_uint8' n)))"
   by (simp add: integer_of_uint8_signed_def integer_of_uint8_def)
 
 lemma integer_of_uint8_code [code]:
   "integer_of_uint8 n =
   (if bit n 7 then integer_of_uint8_signed (n AND 0x7F) OR 0x80 else integer_of_uint8_signed n)"
 proof -
   have \<open>integer_of_uint8_signed (n AND 0x7F) OR 0x80 = Bit_Operations.set_bit 7 (integer_of_uint8_signed (take_bit 7 n))\<close>
     by (simp add: take_bit_eq_mask set_bit_eq_or push_bit_eq_mult mask_eq_exp_minus_1)
   moreover have \<open>integer_of_uint8 n = Bit_Operations.set_bit 7 (integer_of_uint8 (take_bit 7 n))\<close> if \<open>bit n 7\<close>
   proof (rule bit_eqI)
     fix m
     from that show \<open>bit (integer_of_uint8 n) m = bit (Bit_Operations.set_bit 7 (integer_of_uint8 (take_bit 7 n))) m\<close> for m
       including integer.lifting by transfer (auto simp add: bit_simps dest: bit_imp_le_length)
   qed
   ultimately show ?thesis
     by simp (simp add: integer_of_uint8_signed_def bit_simps)
 qed
 
 end
 
 code_printing
   constant "integer_of_uint8" \<rightharpoonup>
   (SML) "IntInf.fromLarge (Word8.toLargeInt _)" and
   (Haskell) "Prelude.toInteger"
 | constant "integer_of_uint8_signed" \<rightharpoonup>
   (Scala) "BigInt"
 
 section \<open>Quickcheck setup\<close>
 
 definition uint8_of_natural :: "natural \<Rightarrow> uint8"
 where "uint8_of_natural x \<equiv> Uint8 (integer_of_natural x)"
 
 instantiation uint8 :: "{random, exhaustive, full_exhaustive}" begin
 definition "random_uint8 \<equiv> qc_random_cnv uint8_of_natural"
 definition "exhaustive_uint8 \<equiv> qc_exhaustive_cnv uint8_of_natural"
 definition "full_exhaustive_uint8 \<equiv> qc_full_exhaustive_cnv uint8_of_natural"
 instance ..
 end
 
 instantiation uint8 :: narrowing begin
 
 interpretation quickcheck_narrowing_samples
   "\<lambda>i. let x = Uint8 i in (x, 0xFF - x)" "0"
   "Typerep.Typerep (STR ''Uint8.uint8'') []" .
 
 definition "narrowing_uint8 d = qc_narrowing_drawn_from (narrowing_samples d) d"
 declare [[code drop: "partial_term_of :: uint8 itself \<Rightarrow> _"]]
 lemmas partial_term_of_uint8 [code] = partial_term_of_code
 
 instance ..
 end
 
 end
diff --git a/thys/Native_Word/Uint_Userguide.thy b/thys/Native_Word/Uint_Userguide.thy
--- a/thys/Native_Word/Uint_Userguide.thy
+++ b/thys/Native_Word/Uint_Userguide.thy
@@ -1,343 +1,343 @@
 (*  Title:      Uint_Userguide.thy
     Author:     Andreas Lochbihler, ETH Zurich
 *)
 
 chapter \<open>User guide for native words\<close>
 
 (*<*)
 theory Uint_Userguide imports
   Uint32
   Uint16
-  Code_Target_Bits_Int
+  Code_Target_Int_Bit
 begin
 (*>*)
 
 text \<open>
   This tutorial explains how to best use the types for native
   words like @{typ "uint32"} in your formalisation.
   You can base your formalisation
   \begin{enumerate}
   \item either directly on these types,
   \item or on the generic @{typ "'a word"} and only introduce native
     words a posteriori via code generator refinement.
   \end{enumerate}
 
   The first option causes the least overhead if you have to prove only
   little about the words you use and start a fresh formalisation.
   Just use the native type @{typ uint32} instead of @{typ "32 word"}
   and similarly for \<open>uint64\<close>, \<open>uint16\<close>, and \<open>uint8\<close>.
   As native word types are meant only for code generation, the lemmas
   about @{typ "'a word"}  have not been duplicated, but you can transfer
   theorems between native word types and @{typ "'a word"} using the
   transfer package.
 
   Note, however, that this option restricts your work a bit:
   your own functions cannot be ``polymorphic'' in the word length,
   but you have to define a separate function for every word length you need.
 
   The second option is recommended if you already have a formalisation
   based on @{typ "'a word"} or if your proofs involve words and their
   properties. It separates code generation from modelling and proving,
   i.e., you can work with words as usual. Consequently, you have to
   manually setup the code generator to use the native types wherever
   you want. The following describes how to achieve this with moderate
   effort.
 
   Note, however, that some target languages of the code generator
   (especially OCaml) do not support all the native word types provided.
   Therefore, you should only import those types that you need -- the
   theory file for each type mentions at the top the restrictions for
   code generation. For example, PolyML does not provide the Word16
   structure, and OCaml provides neither Word8 nor Word16.
   You can still use these theories provided that you also import
-  the theory @{theory Native_Word.Code_Target_Bits_Int} (which implements
+  the theory @{theory Native_Word.Code_Target_Int_Bit} (which implements
   @{typ int} by target-language integers), but these words will
   be implemented via Isabelle's \<open>Word\<close> library, i.e.,
   you do not gain anything in terms of efficiency.
 
   \textbf{There is a separate code target \<open>SML_word\<close> for SML.}
   If you use one of the native words that PolyML does not support
   (such as \<open>uint16\<close> and \<open>uint64\<close> in 32-bit mode), but would
   like to map its operations to the Standard Basis Library functions,
   make sure to use the target \<open>SML_word\<close> instead of \<open>SML\<close>;
   if you only use native word sizes that PolyML supports, you can stick
   with \<open>SML\<close>.  This ensures that code generation within Isabelle
   as used by \<open>Quickcheck\<close>, \<open>value\<close> and @\{code\} in ML blocks
   continues to work.
 \<close>
 
 section \<open>Lifting functions from @{typ "'a word"} to native words\<close>
 
 text \<open>
   This section shows how to convert functions from @{typ "'a word"} to native 
   words. For example, the following function \<open>sum_squares\<close> computes 
   the sum of the first @{term n} square numbers in 16 bit arithmetic using
   a tail-recursive function \<open>gen_sum_squares\<close> with accumulator;
   for convenience, \<open>sum_squares_int\<close> takes an integer instead of a word.
 \<close>
 
 function gen_sum_squares :: "16 word \<Rightarrow> 16 word \<Rightarrow> 16 word" where (*<*)[simp del]:(*>*)
 
   "gen_sum_squares accum n =
    (if n = 0 then accum else gen_sum_squares (accum + n * n) (n - 1))"
 (*<*)by pat_completeness simp
 termination by (relation \<open>measure (nat \<circ> uint \<circ> snd)\<close>) (simp_all add: measure_unat)(*>*)
 
 definition sum_squares :: "16 word \<Rightarrow> 16 word" where
    "sum_squares = gen_sum_squares 0"
 
 definition sum_squares_int :: "int \<Rightarrow> 16 word" where
   "sum_squares_int n = sum_squares (word_of_int n)"
 
 text \<open>
   The generated code for @{term sum_squares} and @{term sum_squares_int} 
   emulates words with unbounded integers and explicit modulus as specified 
   in the theory @{theory "HOL-Library.Word"}. But for efficiency, we want that the
   generated code uses machine words and machine arithmetic. Unfortunately,
   as @{typ "'a word"} is polymorphic in the word length, the code generator
   can only do this if we use another type for machine words. The theory
   @{theory Native_Word.Uint16} defines the type @{typ uint16} for machine words of
   16~bits. We just have to follow two steps to use it:
   
   First, we lift all our functions from @{typ "16 word"} to @{typ uint16},
   i.e., @{term sum_squares}, @{term gen_sum_squares}, and 
   @{term sum_squares_int} in our case. The theory @{theory Native_Word.Uint16} sets
   up the lifting package for this and has already taken care of the
   arithmetic and bit-wise operations.
 \<close>
 lift_definition gen_sum_squares_uint :: "uint16 \<Rightarrow> uint16 \<Rightarrow> uint16" 
   is gen_sum_squares .
 lift_definition sum_squares_uint :: "uint16 \<Rightarrow> uint16" is sum_squares .
 lift_definition sum_squares_int_uint :: "int \<Rightarrow> uint16" is sum_squares_int .
 
 text \<open>
   Second, we also have to transfer the code equations for our functions.
   The attribute \<open>Transfer.transferred\<close> takes care of that, but it is
   better to check that the transfer succeeded: inspect the theorem to check
   that the new constants are used throughout.
 \<close>
 
 lemmas [Transfer.transferred, code] =
   gen_sum_squares.simps
   sum_squares_def
   sum_squares_int_def
 
 text \<open>
   Finally, we export the code to standard ML.  We use the target
   \<open>SML_word\<close> instead of \<open>SML\<close> to have the operations
   on @{typ uint16} mapped to the Standard Basis Library. As PolyML
   does not provide a Word16 type, the mapping for @{typ uint16} is only
   active in the refined target \<open>SML_word\<close>.
 \<close>
 export_code sum_squares_int_uint in SML_word
 
 text \<open>
   Nevertheless, we can still evaluate terms with @{term "uint16"} within 
   Isabelle, i.e., PolyML, but this will be translated to @{typ "16 word"}
   and therefore less efficient.
 \<close>
 
 value "sum_squares_int_uint 40"
 
 section \<open>Storing native words in datatypes\<close>
 
 text \<open>
   The above lifting is necessary for all functions whose type mentions
   the word type. Fortunately, we do not have to duplicate functions that
   merely operate on datatypes that contain words. Nevertheless, we have
   to tell the code generator that these functions should call the new ones,
   which operate on machine words. This section shows how to achieve this
   with data refinement.
 \<close>
 
 subsection \<open>Example: expressions and two semantics\<close>
 
 text \<open>
   As the running example, we consider a language of expressions (literal values, less-than comparisions and conditional) where values are either booleans or 32-bit words.
   The original specification uses the type @{typ "32 word"}.
 \<close>
 
 datatype val = Bool bool | Word "32 word"
 datatype expr = Lit val | LT expr expr | IF expr expr expr
 
 abbreviation (input) word :: "32 word \<Rightarrow> expr" where "word i \<equiv> Lit (Word i)"
 abbreviation (input) bool :: "bool \<Rightarrow> expr" where "bool i \<equiv> Lit (Bool i)"
 
 \<comment> \<open>Denotational semantics of expressions, @{term None} denotes a type error\<close>
 fun eval :: "expr \<Rightarrow> val option" where
   "eval (Lit v) = Some v"
 | "eval (LT e\<^sub>1 e\<^sub>2) = 
   (case (eval e\<^sub>1, eval e\<^sub>2) 
    of (Some (Word i\<^sub>1), Some (Word i\<^sub>2)) \<Rightarrow> Some (Bool (i\<^sub>1 < i\<^sub>2))
    | _ \<Rightarrow> None)"
 | "eval (IF e\<^sub>1 e\<^sub>2 e\<^sub>3) =
   (case eval e\<^sub>1 of Some (Bool b) \<Rightarrow> if b then eval e\<^sub>2 else eval e\<^sub>3
    | _ \<Rightarrow> None)"
 
 \<comment> \<open>Small-step semantics of expressions, it gets stuck upon type errors.\<close>
 inductive step :: "expr \<Rightarrow> expr \<Rightarrow> bool" ("_ \<rightarrow> _" [50, 50] 60) where
   "e \<rightarrow> e' \<Longrightarrow> LT e e\<^sub>2 \<rightarrow> LT e' e\<^sub>2"
 | "e \<rightarrow> e' \<Longrightarrow> LT (word i) e \<rightarrow> LT (word i) e'"
 | "LT (word i\<^sub>1) (word i\<^sub>2) \<rightarrow> bool (i\<^sub>1 < i\<^sub>2)"
 | "e \<rightarrow> e' \<Longrightarrow> IF e e\<^sub>1 e\<^sub>2 \<rightarrow> IF e' e\<^sub>1 e\<^sub>2"
 | "IF (bool True) e\<^sub>1 e\<^sub>2 \<rightarrow> e\<^sub>1"
 | "IF (bool False) e\<^sub>1 e\<^sub>2 \<rightarrow> e\<^sub>2"
 
 \<comment> \<open>Compile the inductive definition with the predicate compiler\<close>
 code_pred (modes: i \<Rightarrow> o \<Rightarrow> bool as reduce, i \<Rightarrow> i \<Rightarrow> bool as step') step .
 
 subsection \<open>Change the datatype to use machine words\<close>
 
 text \<open>
   Now, we want to use @{typ uint32} instead of @{typ "32 word"}.
   The goal is to make the code generator use the new type without
   duplicating any of the types (@{typ val}, @{typ expr}) or the
   functions (@{term eval}, @{term reduce}) on such types.
 
   The constructor @{term Word} has @{typ "32 word"} in its type, so
   we have to lift it to \<open>Word'\<close>, and the same holds for the
   case combinator @{term case_val}, which @{term case_val'} replaces.%
   \footnote{%
     Note that we should not declare a case translation for the new
     case combinator because this will break parsing case expressions
     with old case combinator.
   }
   Next, we set up the code generator accordingly:
   @{term Bool} and @{term Word'} are the new constructors for @{typ val},
   and @{term case_val'} is the new case combinator with an appropriate 
   case certificate.%
   \footnote{%
     Case certificates tell the code generator to replace the HOL
     case combinator for a datatype with the case combinator of the
     target language.  Without a case certificate, the code generator
     generates a function that re-implements the case combinator; 
     in a strict languages like ML or Scala, this means that the code
     evaluates all possible cases before it decides which one is taken.
 
     Case certificates are described in Haftmann's PhD thesis
     \cite[Def.\ 27]{Haftmann2009PhD}. For a datatype \<open>dt\<close>
     with constructors \<open>C\<^sub>1\<close> to \<open>C\<^sub>n\<close>
     where each constructor \<open>C\<^sub>i\<close> takes \<open>k\<^sub>i\<close> parameters,
     the certificate for the case combinator \<open>case_dt\<close>
     looks as follows:
 
     {
       \isamarkuptrue\isacommand{lemma}\isamarkupfalse\isanewline%
       \ \ \isakeyword{assumes}\ {\isachardoublequoteopen}CASE\ {\isasymequiv}\ dt{\isacharunderscore}case\ c\isactrlsub {\isadigit{1}}\ c\isactrlsub {\isadigit{2}}\ \ldots\ c\isactrlsub{n}{\isachardoublequoteclose}\isanewline
       \ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}CASE\ {\isacharparenleft}C\isactrlsub {\isadigit{1}}\ a\isactrlsub {\isadigit{1}}\isactrlsub {\isadigit{1}}\ a\isactrlsub {\isadigit{1}}\isactrlsub {\isadigit{2}}\ \ldots\ a\isactrlsub {\isadigit{1}}\isactrlsub {k\ensuremath{{}_1}}{\isacharparenright}\ {\isasymequiv}\ c\isactrlsub {\isadigit{1}}\ a\isactrlsub {\isadigit{1}}\isactrlsub {\isadigit{1}}\ a\isactrlsub {\isadigit{1}}\isactrlsub {\isadigit{2}}\ \ldots\ a\isactrlsub {\isadigit{1}}\isactrlsub {k\ensuremath{{}_1}}{\isacharparenright}\isanewline
       \ \ \ \ {\isacharampersand}{\isacharampersand}{\isacharampersand}\ {\isacharparenleft}CASE\ {\isacharparenleft}C\isactrlsub {\isadigit{2}}\ a\isactrlsub {\isadigit{2}}\isactrlsub {\isadigit{1}}\ a\isactrlsub {\isadigit{2}}\isactrlsub {\isadigit{2}}\ \ldots\ a\isactrlsub {\isadigit{2}}\isactrlsub {k\ensuremath{{}_2}}{\isacharparenright}\ {\isasymequiv}\ c\isactrlsub {\isadigit{2}}\ a\isactrlsub {\isadigit{2}}\isactrlsub {\isadigit{1}}\ a\isactrlsub {\isadigit{2}}\isactrlsub {\isadigit{2}}\ \ldots\ a\isactrlsub {\isadigit{2}}\isactrlsub {k\ensuremath{{}_2}}{\isacharparenright}\isanewline
       \ \ \ \ {\isacharampersand}{\isacharampersand}{\isacharampersand}\ \ldots\isanewline
       \ \ \ \ {\isacharampersand}{\isacharampersand}{\isacharampersand}\ {\isacharparenleft}CASE\ {\isacharparenleft}C\isactrlsub {n}\ a\isactrlsub {n}\isactrlsub {\isadigit{1}}\ a\isactrlsub {n}\isactrlsub {\isadigit{2}}\ \ldots\ a\isactrlsub {n}\isactrlsub {k\ensuremath{{}_n}}{\isacharparenright}\ {\isasymequiv}\ c\isactrlsub {n}\ a\isactrlsub {n}\isactrlsub {\isadigit{1}}\ a\isactrlsub {n}\isactrlsub {\isadigit{2}}\ \ldots\ a\isactrlsub {n}\isactrlsub {k\ensuremath{{}_n}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
     }
   }
   We delete the code equations for the old constructor @{term Word}
   and case combinator @{term case_val} such that the code generator
   reports missing adaptations.
 \<close>
 
 lift_definition Word' :: "uint32 \<Rightarrow> val" is Word .
 
 code_datatype Bool Word'
 
 lift_definition case_val' :: "(bool \<Rightarrow> 'a) \<Rightarrow> (uint32 \<Rightarrow> 'a) \<Rightarrow> val \<Rightarrow> 'a" is case_val .
 
 lemmas [code, simp] = val.case [Transfer.transferred]
 
 lemma case_val'_cert:
   fixes bool word' b w
   assumes "CASE \<equiv> case_val' bool word'"
   shows "(CASE (Bool b) \<equiv> bool b) &&& (CASE (Word' w) \<equiv> word' w)"
   by (simp_all add: assms)
 
 setup \<open>Code.declare_case_global @{thm case_val'_cert}\<close>
 
 declare [[code drop: case_val Word]]
 
 
 subsection \<open>Make functions use functions on machine words\<close>
 
 text \<open>
   Finally, we merely have to change the code equations to use the 
   new functions that operate on @{typ uint32}. As before, the
   attribute \<open>Transfer.transferred\<close> does the job. In our example,
   we adapt the equality test on @{typ val} (code equations
   @{thm [source] val.eq.simps}) and the denotational and small-step 
   semantics (code equations @{thm [source] eval.simps} and
   @{thm [source] step.equation}, respectively).
 
   We check that the adaptation has suceeded by exporting the functions.
   As we only use native word sizes that PolyML supports, we can use 
   the usual target \<open>SML\<close> instead of \<open>SML_word\<close>.
 \<close>
 
 lemmas [code] = 
   val.eq.simps[THEN meta_eq_to_obj_eq, Transfer.transferred, THEN eq_reflection]
   eval.simps[Transfer.transferred]
   step.equation[Transfer.transferred]
 
 export_code reduce step' eval checking SML
 
 section \<open>Troubleshooting\<close>
 
 text \<open>
   This section explains some possible problems when using native words.
   If you experience other difficulties, please contact the author.
 \<close>
 
 subsection \<open>\<open>export_code\<close> raises an exception \label{section:export_code:exception}\<close>
 
 text \<open>
   Probably, you have defined and are using a function on a native word type,
   but the code equation refers to emulated words. For example, the following
   defines a function \<open>double\<close> that doubles a word. When we try to export
   code for \<open>double\<close> without any further setup, \<open>export_code\<close> will
   raise an exception or generate code that does not compile.
 \<close>
 
 lift_definition double :: "uint32 \<Rightarrow> uint32" is "\<lambda>x. x + x" .
 
 text \<open>
   We have to prove a code equation that only uses the existing operations on
   @{typ uint32}. Then, \<open>export_code\<close> works again.
 \<close>
 
 lemma double_code [code]: "double n = n + n"
 by transfer simp
 
 subsection \<open>The generated code does not compile\<close>
 
 text \<open>
   Probably, you have been exporting to a target language for which there
   is no setup, or your compiler does not provide the required API. Every
   theory for native words mentions at the start the limitations on code
   generation. Check that your concrete application meets all the
   requirements.
 
   Alternatively, this might be an instance of the problem described 
   in \S\ref{section:export_code:exception}.
 
   For Haskell, you have to enable the extension TypeSynonymInstances with \texttt{-XTypeSynonymInstances}
   if you are using polymorphic bit operations on the native word types.
 \<close>
 
 subsection \<open>The generated code is too slow\<close>
 
 text \<open>
   The generated code will most likely not be as fast as a direct implementation in the target language with manual tuning.
   This is because we want the configuration of the code generation to be sound (as it can be used to prove theorems in Isabelle).
   Therefore, the bit operations sometimes perform range checks before they call the target language API.
   Here are some examples:
   \begin{itemize}
   \item Shift distances and bit indices in target languages are often expected to fit into a bounded integer or word.
     However, the size of these types varies across target languages and platforms.
     Hence, no Isabelle/HOL type can model uniformly all of them.
     Instead, the bit operations use arbitrary-precision integers for such quantities and check at run-time that the values fit into a bounded integer or word, respectively -- if not, they raise an exception.
   
   \item Division and modulo operations explicitly test whether the divisor is $0$ and return the HOL value of division by $0$ in that case.
     This is necessary because some languages leave the behaviour of division by 0 unspecified.
   \end{itemize}
   
   If you have better ideas how to eliminate such checks and speed up the generated code without sacrificing soundness, please contact the author!
 \<close>
 
 (*<*)end(*>*)
diff --git a/thys/WebAssembly/Wasm_Printing/Wasm_Interpreter_Printing_Pure.thy b/thys/WebAssembly/Wasm_Printing/Wasm_Interpreter_Printing_Pure.thy
--- a/thys/WebAssembly/Wasm_Printing/Wasm_Interpreter_Printing_Pure.thy
+++ b/thys/WebAssembly/Wasm_Printing/Wasm_Interpreter_Printing_Pure.thy
@@ -1,36 +1,36 @@
 theory Wasm_Interpreter_Printing_Pure
   imports
     "../Wasm_Interpreter_Properties"
     Wasm_Type_Abs_Printing
    "HOL-Library.Code_Target_Nat"
-   "Native_Word.Code_Target_Bits_Int"
+   "Native_Word.Code_Target_Int_Bit"
 begin
 
 axiomatization where
   mem_grow_impl_is[code]: "mem_grow_impl m n = Some (mem_grow m n)"
 
 definition "run = run_v (2^63) 300"
 
 code_printing
   constant host_apply_impl \<rightharpoonup> (OCaml) "ImplWrapper.host'_apply'_impl"
 
 declare Rep_bytes_inverse[code abstype]
 declare Rep_mem_inverse[code abstype]
 
 declare write_bytes.rep_eq[code abstract]
 and read_bytes.rep_eq[code abstract]
 and mem_append.rep_eq[code abstract]
 
 lemma bytes_takefill_rep_eq[code abstract]:
   "Rep_bytes (bytes_takefill b n bs) = takefill b n (Rep_bytes bs)"
   using bytes_takefill.rep_eq Rep_uint8_inverse
   by simp
 
 lemma bytes_replicate_rep_eq[code abstract]:
   "Rep_bytes (bytes_replicate n b) = replicate n b"
   using bytes_replicate.rep_eq Rep_uint8_inverse
   by simp
 
 export_code open run in OCaml
 
 end