# HG changeset patch
# User haftmann
# Date 1521657563 -3600
# Node ID 02a14c1cb9175f79818571f163c867ab34f38cca
# Parent  9cc32b18c7851f560a87546c0eda6052785bf38d
prefer convention to place operation name before type name

diff -r 9cc32b18c785 -r 02a14c1cb917 src/HOL/Parity.thy
--- a/src/HOL/Parity.thy	Tue Mar 20 09:27:40 2018 +0000
+++ b/src/HOL/Parity.thy	Wed Mar 21 19:39:23 2018 +0100
@@ -681,29 +681,29 @@
 text \<open>The primary purpose of the following operations is
   to avoid ad-hoc simplification of concrete expressions @{term "2 ^ n"}\<close>
 
-definition bit_push :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
-  where bit_push_eq_mult: "bit_push n a = a * 2 ^ n"
+definition push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
+  where push_bit_eq_mult: "push_bit n a = a * 2 ^ n"
  
-definition bit_take :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
-  where bit_take_eq_mod: "bit_take n a = a mod of_nat (2 ^ n)"
+definition take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
+  where take_bit_eq_mod: "take_bit n a = a mod of_nat (2 ^ n)"
 
-definition bit_drop :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
-  where bit_drop_eq_div: "bit_drop n a = a div of_nat (2 ^ n)"
+definition drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
+  where drop_bit_eq_div: "drop_bit n a = a div of_nat (2 ^ n)"
 
 lemma bit_ident:
-  "bit_push n (bit_drop n a) + bit_take n a = a"
-  using div_mult_mod_eq by (simp add: bit_push_eq_mult bit_take_eq_mod bit_drop_eq_div)
+  "push_bit n (drop_bit n a) + take_bit n a = a"
+  using div_mult_mod_eq by (simp add: push_bit_eq_mult take_bit_eq_mod drop_bit_eq_div)
 
-lemma bit_take_bit_take [simp]:
-  "bit_take n (bit_take n a) = bit_take n a"
-  by (simp add: bit_take_eq_mod)
+lemma take_bit_take_bit [simp]:
+  "take_bit n (take_bit n a) = take_bit n a"
+  by (simp add: take_bit_eq_mod)
   
-lemma bit_take_0 [simp]:
-  "bit_take 0 a = 0"
-  by (simp add: bit_take_eq_mod)
+lemma take_bit_0 [simp]:
+  "take_bit 0 a = 0"
+  by (simp add: take_bit_eq_mod)
 
-lemma bit_take_Suc [simp]:
-  "bit_take (Suc n) a = bit_take n (a div 2) * 2 + of_bool (odd a)"
+lemma take_bit_Suc [simp]:
+  "take_bit (Suc n) a = take_bit n (a div 2) * 2 + of_bool (odd a)"
 proof -
   have "1 + 2 * (a div 2) mod (2 * 2 ^ n) = (a div 2 * 2 + a mod 2) mod (2 * 2 ^ n)"
     if "odd a"
@@ -712,99 +712,99 @@
   also have "\<dots> = a mod (2 * 2 ^ n)"
     by (simp only: div_mult_mod_eq)
   finally show ?thesis
-    by (simp add: bit_take_eq_mod algebra_simps mult_mod_right)
+    by (simp add: take_bit_eq_mod algebra_simps mult_mod_right)
 qed
 
-lemma bit_take_of_0 [simp]:
-  "bit_take n 0 = 0"
-  by (simp add: bit_take_eq_mod)
+lemma take_bit_of_0 [simp]:
+  "take_bit n 0 = 0"
+  by (simp add: take_bit_eq_mod)
 
-lemma bit_take_plus:
-  "bit_take n (bit_take n a + bit_take n b) = bit_take n (a + b)"
-  by (simp add: bit_take_eq_mod mod_simps)
+lemma take_bit_plus:
+  "take_bit n (take_bit n a + take_bit n b) = take_bit n (a + b)"
+  by (simp add: take_bit_eq_mod mod_simps)
 
-lemma bit_take_of_1_eq_0_iff [simp]:
-  "bit_take n 1 = 0 \<longleftrightarrow> n = 0"
-  by (simp add: bit_take_eq_mod)
+lemma take_bit_of_1_eq_0_iff [simp]:
+  "take_bit n 1 = 0 \<longleftrightarrow> n = 0"
+  by (simp add: take_bit_eq_mod)
 
-lemma bit_push_eq_0_iff [simp]:
-  "bit_push n a = 0 \<longleftrightarrow> a = 0"
-  by (simp add: bit_push_eq_mult)
+lemma push_bit_eq_0_iff [simp]:
+  "push_bit n a = 0 \<longleftrightarrow> a = 0"
+  by (simp add: push_bit_eq_mult)
 
-lemma bit_drop_0 [simp]:
-  "bit_drop 0 = id"
-  by (simp add: fun_eq_iff bit_drop_eq_div)
+lemma drop_bit_0 [simp]:
+  "drop_bit 0 = id"
+  by (simp add: fun_eq_iff drop_bit_eq_div)
 
-lemma bit_drop_of_0 [simp]:
-  "bit_drop n 0 = 0"
-  by (simp add: bit_drop_eq_div)
+lemma drop_bit_of_0 [simp]:
+  "drop_bit n 0 = 0"
+  by (simp add: drop_bit_eq_div)
 
-lemma bit_drop_Suc [simp]:
-  "bit_drop (Suc n) a = bit_drop n (a div 2)"
+lemma drop_bit_Suc [simp]:
+  "drop_bit (Suc n) a = drop_bit n (a div 2)"
 proof (cases "even a")
   case True
   then obtain b where "a = 2 * b" ..
-  moreover have "bit_drop (Suc n) (2 * b) = bit_drop n b"
-    by (simp add: bit_drop_eq_div)
+  moreover have "drop_bit (Suc n) (2 * b) = drop_bit n b"
+    by (simp add: drop_bit_eq_div)
   ultimately show ?thesis
     by simp
 next
   case False
   then obtain b where "a = 2 * b + 1" ..
-  moreover have "bit_drop (Suc n) (2 * b + 1) = bit_drop n b"
+  moreover have "drop_bit (Suc n) (2 * b + 1) = drop_bit n b"
     using div_mult2_eq' [of "1 + b * 2" 2 "2 ^ n"]
-    by (auto simp add: bit_drop_eq_div ac_simps)
+    by (auto simp add: drop_bit_eq_div ac_simps)
   ultimately show ?thesis
     by simp
 qed
 
-lemma bit_drop_half:
-  "bit_drop n (a div 2) = bit_drop n a div 2"
+lemma drop_bit_half:
+  "drop_bit n (a div 2) = drop_bit n a div 2"
   by (induction n arbitrary: a) simp_all
 
-lemma bit_drop_of_bool [simp]:
-  "bit_drop n (of_bool d) = of_bool (n = 0 \<and> d)"
+lemma drop_bit_of_bool [simp]:
+  "drop_bit n (of_bool d) = of_bool (n = 0 \<and> d)"
   by (cases n) simp_all
 
-lemma even_bit_take_eq [simp]:
-  "even (bit_take n a) \<longleftrightarrow> n = 0 \<or> even a"
-  by (cases n) (simp_all add: bit_take_eq_mod dvd_mod_iff)
+lemma even_take_bit_eq [simp]:
+  "even (take_bit n a) \<longleftrightarrow> n = 0 \<or> even a"
+  by (cases n) (simp_all add: take_bit_eq_mod dvd_mod_iff)
 
-lemma bit_push_0_id [simp]:
-  "bit_push 0 = id"
-  by (simp add: fun_eq_iff bit_push_eq_mult)
+lemma push_bit_0_id [simp]:
+  "push_bit 0 = id"
+  by (simp add: fun_eq_iff push_bit_eq_mult)
 
-lemma bit_push_of_0 [simp]:
-  "bit_push n 0 = 0"
-  by (simp add: bit_push_eq_mult)
+lemma push_bit_of_0 [simp]:
+  "push_bit n 0 = 0"
+  by (simp add: push_bit_eq_mult)
 
-lemma bit_push_of_1:
-  "bit_push n 1 = 2 ^ n"
-  by (simp add: bit_push_eq_mult)
+lemma push_bit_of_1:
+  "push_bit n 1 = 2 ^ n"
+  by (simp add: push_bit_eq_mult)
 
-lemma bit_push_Suc [simp]:
-  "bit_push (Suc n) a = bit_push n (a * 2)"
-  by (simp add: bit_push_eq_mult ac_simps)
+lemma push_bit_Suc [simp]:
+  "push_bit (Suc n) a = push_bit n (a * 2)"
+  by (simp add: push_bit_eq_mult ac_simps)
 
-lemma bit_push_double:
-  "bit_push n (a * 2) = bit_push n a * 2"
-  by (simp add: bit_push_eq_mult ac_simps)
+lemma push_bit_double:
+  "push_bit n (a * 2) = push_bit n a * 2"
+  by (simp add: push_bit_eq_mult ac_simps)
 
-lemma bit_drop_bit_take [simp]:
-  "bit_drop n (bit_take n a) = 0"
-  by (simp add: bit_drop_eq_div bit_take_eq_mod)
+lemma drop_bit_take_bit [simp]:
+  "drop_bit n (take_bit n a) = 0"
+  by (simp add: drop_bit_eq_div take_bit_eq_mod)
 
-lemma bit_take_bit_drop_commute:
-  "bit_drop m (bit_take n a) = bit_take (n - m) (bit_drop m a)"
+lemma take_bit_drop_bit_commute:
+  "drop_bit m (take_bit n a) = take_bit (n - m) (drop_bit m a)"
   for m n :: nat
 proof (cases "n \<ge> m")
   case True
   moreover define q where "q = n - m"
   ultimately have "n - m = q" and "n = m + q"
     by simp_all
-  moreover have "bit_drop m (bit_take (m + q) a) = bit_take q (bit_drop m a)"
+  moreover have "drop_bit m (take_bit (m + q) a) = take_bit q (drop_bit m a)"
     using mod_mult2_eq' [of a "2 ^ m" "2 ^ q"]
-    by (simp add: bit_drop_eq_div bit_take_eq_mod power_add)
+    by (simp add: drop_bit_eq_div take_bit_eq_mod power_add)
   ultimately show ?thesis
     by simp
 next
@@ -812,9 +812,9 @@
   moreover define q where "q = m - n"
   ultimately have "m - n = q" and "m = n + q"
     by simp_all
-  moreover have "bit_drop (n + q) (bit_take n a) = 0"
+  moreover have "drop_bit (n + q) (take_bit n a) = 0"
     using div_mult2_eq' [of "a mod 2 ^ n" "2 ^ n" "2 ^ q"]
-    by (simp add: bit_drop_eq_div bit_take_eq_mod power_add div_mult2_eq)
+    by (simp add: drop_bit_eq_div take_bit_eq_mod power_add div_mult2_eq)
   ultimately show ?thesis
     by simp
 qed
diff -r 9cc32b18c785 -r 02a14c1cb917 src/HOL/Set_Interval.thy
--- a/src/HOL/Set_Interval.thy	Tue Mar 20 09:27:40 2018 +0000
+++ b/src/HOL/Set_Interval.thy	Wed Mar 21 19:39:23 2018 +0100
@@ -1929,8 +1929,8 @@
 lemma sum_diff_distrib: "\<forall>x. Q x \<le> P x  \<Longrightarrow> (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x :: nat)"
   by (subst sum_subtractf_nat) auto
 
-lemma bit_take_sum_nat:
-  "bit_take n m = (\<Sum>k = 0..<n. bit_push k (of_bool (odd (bit_drop k m))))"
+lemma take_bit_sum_nat:
+  "take_bit n m = (\<Sum>k = 0..<n. push_bit k (of_bool (odd (drop_bit k m))))"
   for n m :: nat
 proof (induction n arbitrary: m)
   case 0
@@ -1938,12 +1938,12 @@
     by simp
 next
   case (Suc n)
-  have "(\<Sum>k = 0..<Suc n. bit_push k (of_bool (odd (bit_drop k m)))) = 
-    of_bool (odd m) + (\<Sum>k = Suc 0..<Suc n. bit_push k (of_bool (odd (bit_drop k m))))"
+  have "(\<Sum>k = 0..<Suc n. push_bit k (of_bool (odd (drop_bit k m)))) = 
+    of_bool (odd m) + (\<Sum>k = Suc 0..<Suc n. push_bit k (of_bool (odd (drop_bit k m))))"
     by (simp add: sum_head_upt_Suc ac_simps)
-  also have "(\<Sum>k = Suc 0..<Suc n. bit_push k (of_bool (odd (bit_drop k m))))
-    = (\<Sum>k = 0..<n. bit_push k (of_bool (odd (bit_drop k (m div 2))))) * (2::nat)"
-    by (simp only: sum.atLeast_Suc_lessThan_Suc_shift) (simp add: sum_distrib_right bit_push_double)
+  also have "(\<Sum>k = Suc 0..<Suc n. push_bit k (of_bool (odd (drop_bit k m))))
+    = (\<Sum>k = 0..<n. push_bit k (of_bool (odd (drop_bit k (m div 2))))) * (2::nat)"
+    by (simp only: sum.atLeast_Suc_lessThan_Suc_shift) (simp add: sum_distrib_right push_bit_double)
   finally show ?case
     using Suc [of "m div 2"] by simp
 qed    
diff -r 9cc32b18c785 -r 02a14c1cb917 src/HOL/ex/Word_Type.thy
--- a/src/HOL/ex/Word_Type.thy	Tue Mar 20 09:27:40 2018 +0000
+++ b/src/HOL/ex/Word_Type.thy	Wed Mar 21 19:39:23 2018 +0100
@@ -9,32 +9,32 @@
     "HOL-Library.Type_Length"
 begin
 
-lemma bit_take_uminus:
+lemma take_bit_uminus:
   fixes k :: int
-  shows "bit_take n (- (bit_take n k)) = bit_take n (- k)"
-  by (simp add: bit_take_eq_mod mod_minus_eq)
+  shows "take_bit n (- (take_bit n k)) = take_bit n (- k)"
+  by (simp add: take_bit_eq_mod mod_minus_eq)
 
-lemma bit_take_minus:
+lemma take_bit_minus:
   fixes k l :: int
-  shows "bit_take n (bit_take n k - bit_take n l) = bit_take n (k - l)"
-  by (simp add: bit_take_eq_mod mod_diff_eq)
+  shows "take_bit n (take_bit n k - take_bit n l) = take_bit n (k - l)"
+  by (simp add: take_bit_eq_mod mod_diff_eq)
 
-lemma bit_take_nonnegative [simp]:
+lemma take_bit_nonnegative [simp]:
   fixes k :: int
-  shows "bit_take n k \<ge> 0"
-  by (simp add: bit_take_eq_mod)
+  shows "take_bit n k \<ge> 0"
+  by (simp add: take_bit_eq_mod)
 
-definition signed_bit_take :: "nat \<Rightarrow> int \<Rightarrow> int"
-  where signed_bit_take_eq_bit_take:
-    "signed_bit_take n k = bit_take (Suc n) (k + 2 ^ n) - 2 ^ n"
+definition signed_take_bit :: "nat \<Rightarrow> int \<Rightarrow> int"
+  where signed_take_bit_eq_take_bit:
+    "signed_take_bit n k = take_bit (Suc n) (k + 2 ^ n) - 2 ^ n"
 
-lemma signed_bit_take_eq_bit_take':
+lemma signed_take_bit_eq_take_bit':
   assumes "n > 0"
-  shows "signed_bit_take (n - Suc 0) k = bit_take n (k + 2 ^ (n - 1)) - 2 ^ (n - 1)"
-  using assms by (simp add: signed_bit_take_eq_bit_take)
+  shows "signed_take_bit (n - Suc 0) k = take_bit n (k + 2 ^ (n - 1)) - 2 ^ (n - 1)"
+  using assms by (simp add: signed_take_bit_eq_take_bit)
   
-lemma signed_bit_take_0 [simp]:
-  "signed_bit_take 0 k = - (k mod 2)"
+lemma signed_take_bit_0 [simp]:
+  "signed_take_bit 0 k = - (k mod 2)"
 proof (cases "even k")
   case True
   then have "odd (k + 1)"
@@ -42,54 +42,54 @@
   then have "(k + 1) mod 2 = 1"
     by (simp add: even_iff_mod_2_eq_zero)
   with True show ?thesis
-    by (simp add: signed_bit_take_eq_bit_take)
+    by (simp add: signed_take_bit_eq_take_bit)
 next
   case False
   then show ?thesis
-    by (simp add: signed_bit_take_eq_bit_take odd_iff_mod_2_eq_one)
+    by (simp add: signed_take_bit_eq_take_bit odd_iff_mod_2_eq_one)
 qed
 
-lemma signed_bit_take_Suc [simp]:
-  "signed_bit_take (Suc n) k = signed_bit_take n (k div 2) * 2 + k mod 2"
-  by (simp add: odd_iff_mod_2_eq_one signed_bit_take_eq_bit_take algebra_simps)
+lemma signed_take_bit_Suc [simp]:
+  "signed_take_bit (Suc n) k = signed_take_bit n (k div 2) * 2 + k mod 2"
+  by (simp add: odd_iff_mod_2_eq_one signed_take_bit_eq_take_bit algebra_simps)
 
-lemma signed_bit_take_of_0 [simp]:
-  "signed_bit_take n 0 = 0"
-  by (simp add: signed_bit_take_eq_bit_take bit_take_eq_mod)
+lemma signed_take_bit_of_0 [simp]:
+  "signed_take_bit n 0 = 0"
+  by (simp add: signed_take_bit_eq_take_bit take_bit_eq_mod)
 
-lemma signed_bit_take_of_minus_1 [simp]:
-  "signed_bit_take n (- 1) = - 1"
+lemma signed_take_bit_of_minus_1 [simp]:
+  "signed_take_bit n (- 1) = - 1"
   by (induct n) simp_all
 
-lemma signed_bit_take_eq_iff_bit_take_eq:
+lemma signed_take_bit_eq_iff_take_bit_eq:
   assumes "n > 0"
-  shows "signed_bit_take (n - Suc 0) k = signed_bit_take (n - Suc 0) l \<longleftrightarrow> bit_take n k = bit_take n l" (is "?P \<longleftrightarrow> ?Q")
+  shows "signed_take_bit (n - Suc 0) k = signed_take_bit (n - Suc 0) l \<longleftrightarrow> take_bit n k = take_bit n l" (is "?P \<longleftrightarrow> ?Q")
 proof -
   from assms obtain m where m: "n = Suc m"
     by (cases n) auto
   show ?thesis
   proof 
     assume ?Q
-    have "bit_take (Suc m) (k + 2 ^ m) =
-      bit_take (Suc m) (bit_take (Suc m) k + bit_take (Suc m) (2 ^ m))"
-      by (simp only: bit_take_plus)
+    have "take_bit (Suc m) (k + 2 ^ m) =
+      take_bit (Suc m) (take_bit (Suc m) k + take_bit (Suc m) (2 ^ m))"
+      by (simp only: take_bit_plus)
     also have "\<dots> =
-      bit_take (Suc m) (bit_take (Suc m) l + bit_take (Suc m) (2 ^ m))"
+      take_bit (Suc m) (take_bit (Suc m) l + take_bit (Suc m) (2 ^ m))"
       by (simp only: \<open>?Q\<close> m [symmetric])
-    also have "\<dots> = bit_take (Suc m) (l + 2 ^ m)"
-      by (simp only: bit_take_plus)
+    also have "\<dots> = take_bit (Suc m) (l + 2 ^ m)"
+      by (simp only: take_bit_plus)
     finally show ?P
-      by (simp only: signed_bit_take_eq_bit_take m) simp
+      by (simp only: signed_take_bit_eq_take_bit m) simp
   next
     assume ?P
     with assms have "(k + 2 ^ (n - Suc 0)) mod 2 ^ n = (l + 2 ^ (n - Suc 0)) mod 2 ^ n"
-      by (simp add: signed_bit_take_eq_bit_take' bit_take_eq_mod)
+      by (simp add: signed_take_bit_eq_take_bit' take_bit_eq_mod)
     then have "(i + (k + 2 ^ (n - Suc 0))) mod 2 ^ n = (i + (l + 2 ^ (n - Suc 0))) mod 2 ^ n" for i
       by (metis mod_add_eq)
     then have "k mod 2 ^ n = l mod 2 ^ n"
       by (metis add_diff_cancel_right' uminus_add_conv_diff)
     then show ?Q
-      by (simp add: bit_take_eq_mod)
+      by (simp add: take_bit_eq_mod)
   qed
 qed 
 
@@ -98,7 +98,7 @@
 
 subsubsection \<open>Basic properties\<close>
 
-quotient_type (overloaded) 'a word = int / "\<lambda>k l. bit_take LENGTH('a) k = bit_take LENGTH('a::len0) l"
+quotient_type (overloaded) 'a word = int / "\<lambda>k l. take_bit LENGTH('a) k = take_bit LENGTH('a::len0) l"
   by (auto intro!: equivpI reflpI sympI transpI)
 
 instantiation word :: (len0) "{semiring_numeral, comm_semiring_0, comm_ring}"
@@ -114,19 +114,19 @@
 
 lift_definition plus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
   is plus
-  by (subst bit_take_plus [symmetric]) (simp add: bit_take_plus)
+  by (subst take_bit_plus [symmetric]) (simp add: take_bit_plus)
 
 lift_definition uminus_word :: "'a word \<Rightarrow> 'a word"
   is uminus
-  by (subst bit_take_uminus [symmetric]) (simp add: bit_take_uminus)
+  by (subst take_bit_uminus [symmetric]) (simp add: take_bit_uminus)
 
 lift_definition minus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
   is minus
-  by (subst bit_take_minus [symmetric]) (simp add: bit_take_minus)
+  by (subst take_bit_minus [symmetric]) (simp add: take_bit_minus)
 
 lift_definition times_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
   is times
-  by (auto simp add: bit_take_eq_mod intro: mod_mult_cong)
+  by (auto simp add: take_bit_eq_mod intro: mod_mult_cong)
 
 instance
   by standard (transfer; simp add: algebra_simps)+
@@ -159,7 +159,7 @@
 begin
 
 lift_definition unsigned :: "'b::len0 word \<Rightarrow> 'a"
-  is "of_nat \<circ> nat \<circ> bit_take LENGTH('b)"
+  is "of_nat \<circ> nat \<circ> take_bit LENGTH('b)"
   by simp
 
 lemma unsigned_0 [simp]:
@@ -181,8 +181,8 @@
 begin
 
 lift_definition signed :: "'b::len word \<Rightarrow> 'a"
-  is "of_int \<circ> signed_bit_take (LENGTH('b) - 1)"
-  by (simp add: signed_bit_take_eq_iff_bit_take_eq [symmetric])
+  is "of_int \<circ> signed_take_bit (LENGTH('b) - 1)"
+  by (simp add: signed_take_bit_eq_iff_take_bit_eq [symmetric])
 
 lemma signed_0 [simp]:
   "signed 0 = 0"
@@ -191,8 +191,8 @@
 end
 
 lemma unsigned_of_nat [simp]:
-  "unsigned (of_nat n :: 'a word) = bit_take LENGTH('a::len) n"
-  by transfer (simp add: nat_eq_iff bit_take_eq_mod zmod_int)
+  "unsigned (of_nat n :: 'a word) = take_bit LENGTH('a::len) n"
+  by transfer (simp add: nat_eq_iff take_bit_eq_mod zmod_int)
 
 lemma of_nat_unsigned [simp]:
   "of_nat (unsigned a) = a"
@@ -207,17 +207,17 @@
 
 lemma word_eq_iff_signed:
   "a = b \<longleftrightarrow> signed a = signed b"
-  by safe (transfer; auto simp add: signed_bit_take_eq_iff_bit_take_eq)
+  by safe (transfer; auto simp add: signed_take_bit_eq_iff_take_bit_eq)
 
 end
 
 lemma signed_of_int [simp]:
-  "signed (of_int k :: 'a word) = signed_bit_take (LENGTH('a::len) - 1) k"
+  "signed (of_int k :: 'a word) = signed_take_bit (LENGTH('a::len) - 1) k"
   by transfer simp
 
 lemma of_int_signed [simp]:
   "of_int (signed a) = a"
-  by transfer (simp add: signed_bit_take_eq_bit_take bit_take_eq_mod mod_simps)
+  by transfer (simp add: signed_take_bit_eq_take_bit take_bit_eq_mod mod_simps)
 
 
 subsubsection \<open>Properties\<close>
@@ -229,11 +229,11 @@
 begin
 
 lift_definition divide_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
-  is "\<lambda>a b. bit_take LENGTH('a) a div bit_take LENGTH('a) b"
+  is "\<lambda>a b. take_bit LENGTH('a) a div take_bit LENGTH('a) b"
   by simp
 
 lift_definition modulo_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
-  is "\<lambda>a b. bit_take LENGTH('a) a mod bit_take LENGTH('a) b"
+  is "\<lambda>a b. take_bit LENGTH('a) a mod take_bit LENGTH('a) b"
   by simp
 
 instance ..
@@ -247,11 +247,11 @@
 begin
 
 lift_definition less_eq_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
-  is "\<lambda>a b. bit_take LENGTH('a) a \<le> bit_take LENGTH('a) b"
+  is "\<lambda>a b. take_bit LENGTH('a) a \<le> take_bit LENGTH('a) b"
   by simp
 
 lift_definition less_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
-  is "\<lambda>a b. bit_take LENGTH('a) a < bit_take LENGTH('a) b"
+  is "\<lambda>a b. take_bit LENGTH('a) a < take_bit LENGTH('a) b"
   by simp
 
 instance
@@ -268,7 +268,7 @@
 
 lemma word_less_iff_unsigned:
   "a < b \<longleftrightarrow> unsigned a < unsigned b"
-  by (transfer fixing: less) (auto dest: preorder_class.le_less_trans [OF bit_take_nonnegative])
+  by (transfer fixing: less) (auto dest: preorder_class.le_less_trans [OF take_bit_nonnegative])
 
 end