diff --git a/src/HOL/ex/Word_Conversions.thy b/src/HOL/ex/Word_Conversions.thy
--- a/src/HOL/ex/Word_Conversions.thy
+++ b/src/HOL/ex/Word_Conversions.thy
@@ -1,204 +1,202 @@
 (*  Author:  Florian Haftmann, TUM
 *)
 
 section \<open>Proof of concept for conversions for algebraically founded bit word types\<close>
 
 theory Word_Conversions
   imports
     Main
     "HOL-Library.Type_Length"
     "HOL-Library.Bit_Operations"
     "HOL-Word.Word"
 begin
 
 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
 
 lemma unat_unsigned:
   \<open>unat = unsigned\<close>
   by transfer simp
 
 lemma uint_unsigned:
   \<open>uint = unsigned\<close>
   by transfer (simp add: fun_eq_iff)
 
 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 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>
   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
 
 lemma sint_signed:
   \<open>sint = signed\<close>
   by transfer simp
 
 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
 
 abbreviation nat_of_word :: \<open>'a::len word \<Rightarrow> nat\<close>
   where \<open>nat_of_word \<equiv> unsigned\<close>
 
 abbreviation unsigned_int :: \<open>'a::len word \<Rightarrow> int\<close>
   where \<open>unsigned_int \<equiv> unsigned\<close>
 
 abbreviation signed_int :: \<open>'a::len word \<Rightarrow> int\<close>
   where \<open>signed_int \<equiv> signed\<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>
 
-text \<open>TODO rework names from here\<close>
-
 lemma unsigned_of_nat [simp]:
-  \<open>unsigned (of_nat n :: 'a::len word) = take_bit LENGTH('a) n\<close>
+  \<open>unsigned (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 of_nat_unsigned [simp]:
-  \<open>of_nat (unsigned w) = w\<close>
+lemma of_nat_of_word [simp]:
+  \<open>of_nat (nat_of_word w) = unsigned w\<close>
   by transfer simp
 
 lemma of_int_unsigned [simp]:
-  \<open>of_int (unsigned w) = w\<close>
+  \<open>of_int (unsigned_int w) = unsigned w\<close>
   by transfer simp
 
 lemma unsigned_int_greater_eq:
-  \<open>(0::int) \<le> unsigned w\<close> for w :: \<open>'a::len word\<close>
+  \<open>0 \<le> unsigned_int w\<close> for w :: \<open>'a::len word\<close>
   by transfer simp
 
-lemma unsigned_nat_less:
-  \<open>unsigned w < (2 ^ LENGTH('a) :: nat)\<close> for w :: \<open>'a::len word\<close>
+lemma nat_of_word_less:
+  \<open>nat_of_word w < 2 ^ LENGTH('a)\<close> for w :: \<open>'a::len word\<close>
   by transfer (simp add: take_bit_eq_mod)
 
 lemma unsigned_int_less:
-  \<open>unsigned w < (2 ^ LENGTH('a) :: int)\<close> for w :: \<open>'a::len word\<close>
+  \<open>unsigned_int w < 2 ^ LENGTH('a)\<close> for w :: \<open>'a::len word\<close>
   by transfer (simp add: take_bit_eq_mod)
 
 lemma signed_of_int [simp]:
-  \<open>signed (of_int k :: 'a::len word) = signed_take_bit (LENGTH('a) - 1) k\<close>
+  \<open>signed (word_of_int k :: 'a::len word) = of_int (signed_take_bit (LENGTH('a) - 1) k)\<close>
   by transfer simp
 
 lemma of_int_signed [simp]:
-  \<open>of_int (signed a) = a\<close>
+  \<open>of_int (signed_int a) = signed a\<close>
   by transfer (simp_all add: take_bit_signed_take_bit)
 
 lemma signed_int_greater_eq:
-  \<open>- ((2::int) ^ (LENGTH('a) - 1)) \<le> signed w\<close> for w :: \<open>'a::len word\<close>
+  \<open>- (2 ^ (LENGTH('a) - 1)) \<le> signed_int w\<close> for w :: \<open>'a::len word\<close>
 proof (cases \<open>bit w (LENGTH('a) - 1)\<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 signed_int_less:
-  \<open>signed w < (2 ^ (LENGTH('a) - 1) :: int)\<close> for w :: \<open>'a::len word\<close>
+  \<open>signed_int w < 2 ^ (LENGTH('a) - 1)\<close> for w :: \<open>'a::len word\<close>
   by (cases \<open>bit w (LENGTH('a) - 1)\<close>; transfer)
     (simp_all add: signed_take_bit_eq signed_take_bit_eq_or take_bit_int_less_exp not_eq_complement mask_eq_exp_minus_1 OR_upper)
 
 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
 
-lemma of_nat_word_eq_iff:
-  \<open>of_nat m = (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 of_nat_word_less_eq_iff:
-  \<open>of_nat m \<le> (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 of_nat_word_less_iff:
-  \<open>of_nat m < (of_nat n :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) m < take_bit LENGTH('a) n\<close>
+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 of_nat_word_eq_0_iff:
-  \<open>of_nat n = (0 :: 'a::len word) \<longleftrightarrow> 2 ^ LENGTH('a) dvd n\<close>
+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_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_nat_eq_0_iff:
+  \<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 of_int_word_eq_iff:
-  \<open>of_int k = (of_int l :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) k = take_bit LENGTH('a) l\<close>
+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 of_int_word_less_eq_iff:
-  \<open>of_int k \<le> (of_int l :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) k \<le> take_bit LENGTH('a) l\<close>
+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 of_int_word_less_iff:
-  \<open>of_int k < (of_int l :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) k < take_bit LENGTH('a) l\<close>
+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 of_int_word_eq_0_iff:
-  \<open>of_int k = (0 :: 'a::len word) \<longleftrightarrow> 2 ^ LENGTH('a) dvd k\<close>
+lemma word_of_int_eq_0_iff:
+  \<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)
 
 end