diff --git a/src/HOL/Word/Conversions.thy b/src/HOL/Word/Conversions.thy
--- a/src/HOL/Word/Conversions.thy
+++ b/src/HOL/Word/Conversions.thy
@@ -1,478 +1,496 @@
 (*  Author:  Florian Haftmann, TUM
 *)
 
 section \<open>Proof of concept for conversions for algebraically founded bit word types\<close>
 
 theory Conversions
   imports
     Main
     "HOL-Library.Type_Length"
     "HOL-Library.Bit_Operations"
     "HOL-Word.Word"
 begin
 
 hide_const (open) unat uint sint word_of_int ucast scast
 
 
 subsection \<open>Conversions to word\<close>
 
 abbreviation word_of_nat :: \<open>nat \<Rightarrow> 'a::len word\<close>
   where \<open>word_of_nat \<equiv> of_nat\<close>
 
 abbreviation word_of_int :: \<open>int \<Rightarrow> 'a::len word\<close>
   where \<open>word_of_int \<equiv> of_int\<close>
 
 lemma Word_eq_word_of_int [simp]:
   \<open>Word.Word = word_of_int\<close>
   by (rule ext; transfer) simp
 
 lemma word_of_nat_eq_iff:
   \<open>word_of_nat m = (word_of_nat n :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) m = take_bit LENGTH('a) n\<close>
   by transfer (simp add: take_bit_of_nat)
 
 lemma word_of_int_eq_iff:
   \<open>word_of_int k = (word_of_int l :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) k = take_bit LENGTH('a) l\<close>
   by transfer rule
 
 lemma word_of_nat_less_eq_iff:
   \<open>word_of_nat m \<le> (word_of_nat n :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) m \<le> take_bit LENGTH('a) n\<close>
   by transfer (simp add: take_bit_of_nat)
 
 lemma word_of_int_less_eq_iff:
   \<open>word_of_int k \<le> (word_of_int l :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) k \<le> take_bit LENGTH('a) l\<close>
   by transfer rule
 
 lemma word_of_nat_less_iff:
   \<open>word_of_nat m < (word_of_nat n :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) m < take_bit LENGTH('a) n\<close>
   by transfer (simp add: take_bit_of_nat)
 
 lemma word_of_int_less_iff:
   \<open>word_of_int k < (word_of_int l :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) k < take_bit LENGTH('a) l\<close>
   by transfer rule
 
 lemma word_of_nat_eq_0_iff [simp]:
   \<open>word_of_nat n = (0 :: 'a::len word) \<longleftrightarrow> 2 ^ LENGTH('a) dvd n\<close>
   using of_nat_word_eq_iff [where ?'a = 'a, of n 0] by (simp add: take_bit_eq_0_iff)
 
 lemma word_of_int_eq_0_iff [simp]:
   \<open>word_of_int k = (0 :: 'a::len word) \<longleftrightarrow> 2 ^ LENGTH('a) dvd k\<close>
   using of_int_word_eq_iff [where ?'a = 'a, of k 0] by (simp add: take_bit_eq_0_iff)
 
 
 subsection \<open>Conversion from word\<close>
 
 subsubsection \<open>Generic unsigned conversion\<close>
 
 context semiring_1
 begin
 
 lift_definition unsigned :: \<open>'b::len word \<Rightarrow> 'a\<close>
   is \<open>of_nat \<circ> nat \<circ> take_bit LENGTH('b)\<close>
   by simp
 
 lemma unsigned_0 [simp]:
   \<open>unsigned 0 = 0\<close>
   by transfer simp
 
 lemma unsigned_1 [simp]:
   \<open>unsigned 1 = 1\<close>
   by transfer simp
 
 end
 
 context semiring_char_0
 begin
 
 lemma unsigned_word_eqI:
   \<open>v = w\<close> if \<open>unsigned v = unsigned w\<close>
   using that by transfer (simp add: eq_nat_nat_iff)
 
 lemma word_eq_iff_unsigned:
   \<open>v = w \<longleftrightarrow> unsigned v = unsigned w\<close>
   by (auto intro: unsigned_word_eqI)
 
 end
 
 context semiring_bits
 begin
 
 lemma bit_unsigned_iff:
   \<open>bit (unsigned w) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> bit w n\<close>
   for w :: \<open>'b::len word\<close>
   by (transfer fixing: bit) (simp add: bit_of_nat_iff bit_nat_iff bit_take_bit_iff)
 
 end
 
 context semiring_bit_shifts
 begin
 
 lemma unsigned_push_bit_eq:
   \<open>unsigned (push_bit n w) = take_bit LENGTH('b) (push_bit n (unsigned w))\<close>
   for w :: \<open>'b::len word\<close>
 proof (rule bit_eqI)
   fix m
   assume \<open>2 ^ m \<noteq> 0\<close>
   show \<open>bit (unsigned (push_bit n w)) m = bit (take_bit LENGTH('b) (push_bit n (unsigned w))) m\<close>
   proof (cases \<open>n \<le> m\<close>)
     case True
     with \<open>2 ^ m \<noteq> 0\<close> have \<open>2 ^ (m - n) \<noteq> 0\<close>
       by (metis (full_types) diff_add exp_add_not_zero_imp)
     with True show ?thesis
       by (simp add: bit_unsigned_iff bit_push_bit_iff Parity.bit_push_bit_iff bit_take_bit_iff ac_simps exp_eq_zero_iff not_le)
   next
     case False
     then show ?thesis
       by (simp add: not_le bit_unsigned_iff bit_push_bit_iff Parity.bit_push_bit_iff bit_take_bit_iff)
   qed
 qed
 
 lemma unsigned_take_bit_eq:
   \<open>unsigned (take_bit n w) = take_bit n (unsigned w)\<close>
   for w :: \<open>'b::len word\<close>
   by (rule bit_eqI) (simp add: bit_unsigned_iff bit_take_bit_iff Parity.bit_take_bit_iff)
 
 end
 
 context semiring_bit_operations
 begin
 
 lemma unsigned_and_eq:
   \<open>unsigned (v AND w) = unsigned v AND unsigned w\<close>
   for v w :: \<open>'b::len word\<close>
   by (rule bit_eqI) (simp add: bit_unsigned_iff bit_and_iff Bit_Operations.bit_and_iff)
 
 lemma unsigned_or_eq:
   \<open>unsigned (v OR w) = unsigned v OR unsigned w\<close>
   for v w :: \<open>'b::len word\<close>
   by (rule bit_eqI) (simp add: bit_unsigned_iff bit_or_iff Bit_Operations.bit_or_iff)
 
 lemma unsigned_xor_eq:
   \<open>unsigned (v XOR w) = unsigned v XOR unsigned w\<close>
   for v w :: \<open>'b::len word\<close>
   by (rule bit_eqI) (simp add: bit_unsigned_iff bit_xor_iff Bit_Operations.bit_xor_iff)
 
 end
 
 context ring_bit_operations
 begin
 
 lemma unsigned_not_eq:
   \<open>unsigned (NOT w) = take_bit LENGTH('b) (NOT (unsigned w))\<close>
   for w :: \<open>'b::len word\<close>
   by (rule bit_eqI)
     (simp add: bit_unsigned_iff bit_take_bit_iff bit_not_iff Bit_Operations.bit_not_iff exp_eq_zero_iff not_le)
 
 end
 
 lemma unsigned_of_nat [simp]:
   \<open>unsigned (word_of_nat n :: 'a::len word) = of_nat (take_bit LENGTH('a) n)\<close>
   by transfer (simp add: nat_eq_iff take_bit_of_nat)
 
 lemma unsigned_of_int [simp]:
   \<open>unsigned (word_of_int n :: 'a::len word) = of_int (take_bit LENGTH('a) n)\<close>
   by transfer (simp add: nat_eq_iff take_bit_of_nat)
 
 context unique_euclidean_semiring_numeral
 begin
 
 lemma unsigned_greater_eq:
   \<open>0 \<le> unsigned w\<close> for w :: \<open>'b::len word\<close>
   by (transfer fixing: less_eq) simp
 
 lemma unsigned_less:
   \<open>unsigned w < 2 ^ LENGTH('b)\<close> for w :: \<open>'b::len word\<close>
   by (transfer fixing: less) (simp add: take_bit_int_less_exp)
 
 end
 
 context linordered_semidom
 begin
 
 lemma word_less_eq_iff_unsigned:
   "a \<le> b \<longleftrightarrow> unsigned a \<le> unsigned b"
   by (transfer fixing: less_eq) (simp add: nat_le_eq_zle)
 
 lemma word_less_iff_unsigned:
   "a < b \<longleftrightarrow> unsigned a < unsigned b"
   by (transfer fixing: less) (auto dest: preorder_class.le_less_trans [OF take_bit_nonnegative])
 
 end
 
 
 subsubsection \<open>Generic signed conversion\<close>
 
 context ring_1
 begin
 
 lift_definition signed :: \<open>'b::len word \<Rightarrow> 'a\<close>
-  is \<open>of_int \<circ> signed_take_bit (LENGTH('b) - 1)\<close>
+  is \<open>of_int \<circ> signed_take_bit (LENGTH('b) - Suc 0)\<close>
   by (simp flip: signed_take_bit_decr_length_iff)
 
 lemma signed_0 [simp]:
   \<open>signed 0 = 0\<close>
   by transfer simp
 
 lemma signed_1 [simp]:
   \<open>signed (1 :: 'b::len word) = (if LENGTH('b) = 1 then - 1 else 1)\<close>
   by (transfer fixing: uminus)
     (simp_all add: signed_take_bit_eq not_le Suc_lessI)
 
 lemma signed_minus_1 [simp]:
   \<open>signed (- 1 :: 'b::len word) = - 1\<close>
   by (transfer fixing: uminus) simp
 
 end
 
 context ring_char_0
 begin
 
 lemma signed_word_eqI:
   \<open>v = w\<close> if \<open>signed v = signed w\<close>
   using that by transfer (simp flip: signed_take_bit_decr_length_iff)
 
 lemma word_eq_iff_signed:
   \<open>v = w \<longleftrightarrow> signed v = signed w\<close>
   by (auto intro: signed_word_eqI)
 
 end
 
 context ring_bit_operations
 begin
 
 lemma bit_signed_iff:
   \<open>bit (signed w) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> bit w (min (LENGTH('b) - Suc 0) n)\<close>
   for w :: \<open>'b::len word\<close>
   by (transfer fixing: bit) (auto simp add: bit_of_int_iff bit_signed_take_bit_iff min_def)
 
 lemma signed_push_bit_eq:
   \<open>signed (push_bit n w) = take_bit (LENGTH('b) - Suc 0) (push_bit n (signed w))
     OR of_bool (n < LENGTH('b) \<and> bit w (LENGTH('b) - Suc n)) * NOT (mask (LENGTH('b) - Suc 0))\<close>
   for w :: \<open>'b::len word\<close>
 proof (rule bit_eqI)
   fix m
   assume \<open>2 ^ m \<noteq> 0\<close>
   define q where \<open>q = LENGTH('b) - Suc 0\<close>
   then have *: \<open>LENGTH('b) = Suc q\<close>
     by simp
   show \<open>bit (signed (push_bit n w)) m \<longleftrightarrow>
     bit (take_bit (LENGTH('b) - Suc 0) (push_bit n (signed w)) OR
       of_bool (n < LENGTH('b) \<and> bit w (LENGTH('b) - Suc n)) * NOT (mask (LENGTH('b) - Suc 0))) m\<close>
   proof (cases \<open>n \<le> m\<close>)
     case True
     with \<open>2 ^ m \<noteq> 0\<close> have \<open>2 ^ (m - n) \<noteq> 0\<close>
       by (metis (full_types) diff_add exp_add_not_zero_imp)
     with True show ?thesis
       by (auto simp add: * bit_signed_iff bit_push_bit_iff Parity.bit_push_bit_iff bit_or_iff bit_take_bit_iff bit_not_iff bit_mask_iff exp_eq_zero_iff min_def)
   next
     case False
     then show ?thesis
       by (simp add: * bit_signed_iff bit_push_bit_iff Parity.bit_push_bit_iff bit_or_iff bit_take_bit_iff bit_not_iff bit_mask_iff)
   qed
 qed
 
 lemma signed_take_bit_eq:
   \<open>signed (take_bit n w) = (if n < LENGTH('b) then take_bit n (signed w) else signed w)\<close>
   for w :: \<open>'b::len word\<close>
   by (transfer fixing: take_bit; cases \<open>LENGTH('b)\<close>)
     (auto simp add: signed_take_bit_take_bit take_bit_signed_take_bit take_bit_of_int min_def)
 
 lemma signed_not_eq:
   \<open>signed (NOT w) = take_bit LENGTH('b) (NOT (signed w)) OR of_bool (bit (NOT (signed w)) LENGTH('b)) * NOT (mask LENGTH('b))\<close>
   for w :: \<open>'b::len word\<close>
 proof (rule bit_eqI)
   fix n
   assume \<open>2 ^ n \<noteq> 0\<close>
   show \<open>bit (signed (NOT w)) n \<longleftrightarrow>
     bit (take_bit LENGTH('b) (NOT (signed w)) OR
     of_bool (bit (NOT (signed w)) LENGTH('b)) * NOT (mask LENGTH('b))) n\<close>
   proof (cases \<open>LENGTH('b) \<le> n\<close>)
     case False
     then show ?thesis
       by (auto simp add: bit_signed_iff bit_not_iff bit_or_iff bit_take_bit_iff bit_mask_iff Bit_Operations.bit_not_iff exp_eq_zero_iff)
   next
     case True
     moreover define q where \<open>q = n - LENGTH('b)\<close>
     ultimately have \<open>n = LENGTH('b) + q\<close>
       by simp
     with \<open>2 ^ n \<noteq> 0\<close> have \<open>2 ^ q \<noteq> 0\<close> \<open>2 ^ LENGTH('b) \<noteq> 0\<close>
       by (simp_all add: power_add) (use mult_not_zero in blast)+
     then show ?thesis
       by (simp add: bit_signed_iff bit_not_iff bit_or_iff bit_take_bit_iff bit_mask_iff Bit_Operations.bit_not_iff exp_eq_zero_iff min_def not_le not_less le_diff_conv le_Suc_eq)
   qed
 qed
 
 lemma signed_and_eq:
   \<open>signed (v AND w) = signed v AND signed w\<close>
   for v w :: \<open>'b::len word\<close>
   by (rule bit_eqI) (simp add: bit_signed_iff bit_and_iff Bit_Operations.bit_and_iff)
 
 lemma signed_or_eq:
   \<open>signed (v OR w) = signed v OR signed w\<close>
   for v w :: \<open>'b::len word\<close>
   by (rule bit_eqI) (simp add: bit_signed_iff bit_or_iff Bit_Operations.bit_or_iff)
 
 lemma signed_xor_eq:
   \<open>signed (v XOR w) = signed v XOR signed w\<close>
   for v w :: \<open>'b::len word\<close>
   by (rule bit_eqI) (simp add: bit_signed_iff bit_xor_iff Bit_Operations.bit_xor_iff)
 
 end
 
 lemma signed_of_nat [simp]:
   \<open>signed (word_of_nat n :: 'a::len word) = of_int (signed_take_bit (LENGTH('a) - Suc 0) (int n))\<close>
   by transfer simp
 
 lemma signed_of_int [simp]:
   \<open>signed (word_of_int n :: 'a::len word) = of_int (signed_take_bit (LENGTH('a) - Suc 0) n)\<close>
   by transfer simp
 
 
 subsubsection \<open>Important special cases\<close>
 
 abbreviation unat :: \<open>'a::len word \<Rightarrow> nat\<close>
   where \<open>unat \<equiv> unsigned\<close>
 
 abbreviation uint :: \<open>'a::len word \<Rightarrow> int\<close>
   where \<open>uint \<equiv> unsigned\<close>
 
 abbreviation sint :: \<open>'a::len word \<Rightarrow> int\<close>
   where \<open>sint \<equiv> signed\<close>
 
 abbreviation ucast :: \<open>'a::len word \<Rightarrow> 'b::len word\<close>
   where \<open>ucast \<equiv> unsigned\<close>
 
 abbreviation scast :: \<open>'a::len word \<Rightarrow> 'b::len word\<close>
   where \<open>scast \<equiv> signed\<close>
 
 context
   includes lifting_syntax
 begin
 
 lemma [transfer_rule]:
   \<open>(pcr_word ===> (=)) (nat \<circ> take_bit LENGTH('a)) (unat :: 'a::len word \<Rightarrow> nat)\<close>
   using unsigned.transfer [where ?'a = nat] by simp
 
 lemma [transfer_rule]:
   \<open>(pcr_word ===> (=)) (take_bit LENGTH('a)) (uint :: 'a::len word \<Rightarrow> int)\<close>
   using unsigned.transfer [where ?'a = int] by (simp add: comp_def)
 
 lemma [transfer_rule]:
   \<open>(pcr_word ===> (=)) (signed_take_bit (LENGTH('a) - Suc 0)) (sint :: 'a::len word \<Rightarrow> int)\<close>
   using signed.transfer [where ?'a = int] by simp
 
 lemma [transfer_rule]:
   \<open>(pcr_word ===> pcr_word) (take_bit LENGTH('a)) (ucast :: 'a::len word \<Rightarrow> 'b::len word)\<close>
 proof (rule rel_funI)
   fix k :: int and w :: \<open>'a word\<close>
   assume \<open>pcr_word k w\<close>
   then have \<open>w = word_of_int k\<close>
     by (simp add: pcr_word_def cr_word_def relcompp_apply)
   moreover have \<open>pcr_word (take_bit LENGTH('a) k) (ucast (word_of_int k :: 'a word))\<close>
     by transfer (simp add: pcr_word_def cr_word_def relcompp_apply)
   ultimately show \<open>pcr_word (take_bit LENGTH('a) k) (ucast w)\<close>
     by simp
 qed
 
 lemma [transfer_rule]:
-  \<open>(pcr_word ===> pcr_word) (signed_take_bit (LENGTH('a) - 1)) (scast :: 'a::len word \<Rightarrow> 'b::len word)\<close>
+  \<open>(pcr_word ===> pcr_word) (signed_take_bit (LENGTH('a) - Suc 0)) (scast :: 'a::len word \<Rightarrow> 'b::len word)\<close>
 proof (rule rel_funI)
   fix k :: int and w :: \<open>'a word\<close>
   assume \<open>pcr_word k w\<close>
   then have \<open>w = word_of_int k\<close>
     by (simp add: pcr_word_def cr_word_def relcompp_apply)
-  moreover have \<open>pcr_word (signed_take_bit (LENGTH('a) - 1) k) (scast (word_of_int k :: 'a word))\<close>
+  moreover have \<open>pcr_word (signed_take_bit (LENGTH('a) - Suc 0) k) (scast (word_of_int k :: 'a word))\<close>
     by transfer (simp add: pcr_word_def cr_word_def relcompp_apply)
-  ultimately show \<open>pcr_word (signed_take_bit (LENGTH('a) - 1) k) (scast w)\<close>
+  ultimately show \<open>pcr_word (signed_take_bit (LENGTH('a) - Suc 0) k) (scast w)\<close>
     by simp
 qed
 
 end
 
 lemma of_nat_unat [simp]:
   \<open>of_nat (unat w) = unsigned w\<close>
   by transfer simp
 
 lemma of_int_uint [simp]:
   \<open>of_int (uint w) = unsigned w\<close>
   by transfer simp
 
 lemma unat_div_distrib:
   \<open>unat (v div w) = unat v div unat w\<close>
 proof transfer
   fix k l
   have \<open>nat (take_bit LENGTH('a) k) div nat (take_bit LENGTH('a) l) \<le> nat (take_bit LENGTH('a) k)\<close>
     by (rule div_le_dividend)
   also have \<open>nat (take_bit LENGTH('a) k) < 2 ^ LENGTH('a)\<close>
     by (simp add: nat_less_iff take_bit_int_less_exp)
   finally show \<open>(nat \<circ> take_bit LENGTH('a)) (take_bit LENGTH('a) k div take_bit LENGTH('a) l) =
     (nat \<circ> take_bit LENGTH('a)) k div (nat \<circ> take_bit LENGTH('a)) l\<close>
     by (simp add: nat_take_bit_eq div_int_pos_iff nat_div_distrib take_bit_eq_self)
 qed
 
 lemma unat_mod_distrib:
   \<open>unat (v mod w) = unat v mod unat w\<close>
 proof transfer
   fix k l
   have \<open>nat (take_bit LENGTH('a) k) mod nat (take_bit LENGTH('a) l) \<le> nat (take_bit LENGTH('a) k)\<close>
     by (rule mod_less_eq_dividend)
   also have \<open>nat (take_bit LENGTH('a) k) < 2 ^ LENGTH('a)\<close>
     by (simp add: nat_less_iff take_bit_int_less_exp)
   finally show \<open>(nat \<circ> take_bit LENGTH('a)) (take_bit LENGTH('a) k mod take_bit LENGTH('a) l) =
     (nat \<circ> take_bit LENGTH('a)) k mod (nat \<circ> take_bit LENGTH('a)) l\<close>
     by (simp add: nat_take_bit_eq mod_int_pos_iff less_le nat_mod_distrib take_bit_eq_self)
 qed
 
 lemma uint_div_distrib:
   \<open>uint (v div w) = uint v div uint w\<close>
 proof -
   have \<open>int (unat (v div w)) = int (unat v div unat w)\<close>
     by (simp add: unat_div_distrib)
   then show ?thesis
     by (simp add: of_nat_div)
 qed
 
 lemma uint_mod_distrib:
   \<open>uint (v mod w) = uint v mod uint w\<close>
 proof -
   have \<open>int (unat (v mod w)) = int (unat v mod unat w)\<close>
     by (simp add: unat_mod_distrib)
   then show ?thesis
     by (simp add: of_nat_mod)
 qed
 
 lemma of_int_sint [simp]:
   \<open>of_int (sint a) = signed a\<close>
   by transfer (simp_all add: take_bit_signed_take_bit)
 
 lemma sint_not_eq:
   \<open>sint (NOT w) = signed_take_bit LENGTH('a) (NOT (sint w))\<close>
   for w :: \<open>'a::len word\<close>
   by (simp add: signed_not_eq signed_take_bit_unfold)
 
 lemma sint_push_bit_eq:
-  \<open>signed (push_bit n w) = signed_take_bit (LENGTH('a) - 1) (push_bit n (signed w))\<close>
+  \<open>signed (push_bit n w) = signed_take_bit (LENGTH('a) - Suc 0) (push_bit n (signed w))\<close>
   for w :: \<open>'a::len word\<close>
   by (transfer fixing: n; cases \<open>LENGTH('a)\<close>)
      (auto simp add: signed_take_bit_def bit_concat_bit_iff bit_push_bit_iff bit_take_bit_iff bit_or_iff le_diff_conv2,
         auto simp add: take_bit_push_bit not_less concat_bit_eq_iff take_bit_concat_bit_eq le_diff_conv2)
 
 lemma sint_greater_eq:
-  \<open>- (2 ^ (LENGTH('a) - 1)) \<le> sint w\<close> for w :: \<open>'a::len word\<close>
-proof (cases \<open>bit w (LENGTH('a) - 1)\<close>)
+  \<open>- (2 ^ (LENGTH('a) - Suc 0)) \<le> sint w\<close> for w :: \<open>'a::len word\<close>
+proof (cases \<open>bit w (LENGTH('a) - Suc 0)\<close>)
   case True
   then show ?thesis
     by transfer (simp add: signed_take_bit_eq_or minus_exp_eq_not_mask or_greater_eq ac_simps)
 next
   have *: \<open>- (2 ^ (LENGTH('a) - Suc 0)) \<le> (0::int)\<close>
     by simp
   case False
   then show ?thesis
     by transfer (auto simp add: signed_take_bit_eq intro: order_trans *)
 qed
 
 lemma sint_less:
-  \<open>sint w < 2 ^ (LENGTH('a) - 1)\<close> for w :: \<open>'a::len word\<close>
-  by (cases \<open>bit w (LENGTH('a) - 1)\<close>; transfer)
+  \<open>sint w < 2 ^ (LENGTH('a) - Suc 0)\<close> for w :: \<open>'a::len word\<close>
+  by (cases \<open>bit w (LENGTH('a) - Suc 0)\<close>; transfer)
     (simp_all add: signed_take_bit_eq signed_take_bit_unfold take_bit_int_less_exp not_eq_complement mask_eq_exp_minus_1 OR_upper)
 
+context semiring_1
+begin
+
+lemma unsigned_ucase_eq:
+  \<open>unsigned (ucast w :: 'c::len word) = unsigned (take_bit LENGTH('c) w)\<close>
+  for w :: \<open>'b::len word\<close>
+  by transfer (simp add: ac_simps)
+
+lemma signed_ucast_eq:
+  \<open>signed (ucast w :: 'c::len word) =
+    take_bit (min LENGTH('b) LENGTH('c)) w OR of_bool (bit w (LENGTH('c) - Suc 0)) * NOT (mask LENGTH('c))\<close>
+  for w :: \<open>'b::len word\<close>
+  by (transfer, rule bit_eqI)
+    (simp_all add: bit_take_bit_iff bit_signed_take_bit_iff bit_or_iff bit_not_iff bit_mask_iff
+      not_less not_le min_def le_diff_conv less_diff_conv2 le_Suc_eq)
+
 end
+
+end