src/HOL/Word/Bit_Representation.thy

-(* 
-  Author: Jeremy Dawson, NICTA
-*) 
+(*  Title:      HOL/Word/Bit_Representation.thy
+    Author:     Jeremy Dawson, NICTA
+*)
 
 section \<open>Integers as implict bit strings\<close>
 
 theory Bit_Representation
-imports Misc_Numeric
+  imports Misc_Numeric
 begin
 
 subsection \<open>Constructors and destructors for binary integers\<close>
 
-definition Bit :: "int \<Rightarrow> bool \<Rightarrow> int" (infixl "BIT" 90)
-where
-  "k BIT b = (if b then 1 else 0) + k + k"
-
-lemma Bit_B0:
-  "k BIT False = k + k"
-   by (unfold Bit_def) simp
-
-lemma Bit_B1:
-  "k BIT True = k + k + 1"
-   by (unfold Bit_def) simp
-  
+definition Bit :: "int \<Rightarrow> bool \<Rightarrow> int"  (infixl "BIT" 90)
+  where "k BIT b = (if b then 1 else 0) + k + k"
+
+lemma Bit_B0: "k BIT False = k + k"
+   by (simp add: Bit_def)
+
+lemma Bit_B1: "k BIT True = k + k + 1"
+   by (simp add: Bit_def)
+
 lemma Bit_B0_2t: "k BIT False = 2 * k"
   by (rule trans, rule Bit_B0) simp
 
 lemma Bit_B1_2t: "k BIT True = 2 * k + 1"
   by (rule trans, rule Bit_B1) simp
 
 definition bin_last :: "int \<Rightarrow> bool"
-where
-  "bin_last w \<longleftrightarrow> w mod 2 = 1"
-
-lemma bin_last_odd:
-  "bin_last = odd"
+  where "bin_last w \<longleftrightarrow> w mod 2 = 1"
+
+lemma bin_last_odd: "bin_last = odd"
   by (rule ext) (simp add: bin_last_def even_iff_mod_2_eq_zero)
 
 definition bin_rest :: "int \<Rightarrow> int"
-where
-  "bin_rest w = w div 2"
-
-lemma bin_rl_simp [simp]:
-  "bin_rest w BIT bin_last w = w"
+  where "bin_rest w = w div 2"
+
+lemma bin_rl_simp [simp]: "bin_rest w BIT bin_last w = w"
   unfolding bin_rest_def bin_last_def Bit_def
-  using div_mult_mod_eq [of w 2]
-  by (cases "w mod 2 = 0", simp_all)
+  by (cases "w mod 2 = 0") (use div_mult_mod_eq [of w 2] in simp_all)
 
 lemma bin_rest_BIT [simp]: "bin_rest (x BIT b) = x"
   unfolding bin_rest_def Bit_def
-  by (cases b, simp_all)
+  by (cases b) simp_all
 
 lemma bin_last_BIT [simp]: "bin_last (x BIT b) = b"
   unfolding bin_last_def Bit_def
   by (cases b) simp_all
 
 lemma BIT_eq_iff [iff]: "u BIT b = v BIT c \<longleftrightarrow> u = v \<and> b = c"
-  apply (auto simp add: Bit_def)
-  apply arith
-  apply arith
-  done
+  by (auto simp: Bit_def) arith+
 
 lemma BIT_bin_simps [simp]:
   "numeral k BIT False = numeral (Num.Bit0 k)"
   "numeral k BIT True = numeral (Num.Bit1 k)"
   "(- numeral k) BIT False = - numeral (Num.Bit0 k)"
   "(- numeral k) BIT True = - numeral (Num.BitM k)"
   unfolding numeral.simps numeral_BitM
-  unfolding Bit_def
-  by (simp_all del: arith_simps add_numeral_special diff_numeral_special)
+  by (simp_all add: Bit_def del: arith_simps add_numeral_special diff_numeral_special)
 
 lemma BIT_special_simps [simp]:
-  shows "0 BIT False = 0" and "0 BIT True = 1"
-  and "1 BIT False = 2" and "1 BIT True = 3"
-  and "(- 1) BIT False = - 2" and "(- 1) BIT True = - 1"
-  unfolding Bit_def by simp_all
+  shows "0 BIT False = 0"
+    and "0 BIT True = 1"
+    and "1 BIT False = 2"
+    and "1 BIT True = 3"
+    and "(- 1) BIT False = - 2"
+    and "(- 1) BIT True = - 1"
+  by (simp_all add: Bit_def)
 
 lemma Bit_eq_0_iff: "w BIT b = 0 \<longleftrightarrow> w = 0 \<and> \<not> b"
-  apply (auto simp add: Bit_def)
-  apply arith
-  done
+  by (auto simp: Bit_def) arith
 
 lemma Bit_eq_m1_iff: "w BIT b = -1 \<longleftrightarrow> w = -1 \<and> b"
-  apply (auto simp add: Bit_def)
-  apply arith
-  done
+  by (auto simp: Bit_def) arith
 
 lemma BitM_inc: "Num.BitM (Num.inc w) = Num.Bit1 w"
-  by (induct w, simp_all)
+  by (induct w) simp_all
 
 lemma expand_BIT:
   "numeral (Num.Bit0 w) = numeral w BIT False"
   "numeral (Num.Bit1 w) = numeral w BIT True"
   "- numeral (Num.Bit0 w) = (- numeral w) BIT False"
   "- numeral (Num.Bit1 w) = (- numeral (w + Num.One)) BIT True"
-  unfolding add_One by (simp_all add: BitM_inc)
+  by (simp_all add: add_One BitM_inc)
 
 lemma bin_last_numeral_simps [simp]:
   "\<not> bin_last 0"
   "bin_last 1"
   "bin_last (- 1)"

   "bin_rest (numeral (Num.Bit1 w)) = numeral w"
   "bin_rest (- numeral (Num.Bit0 w)) = - numeral w"
   "bin_rest (- numeral (Num.Bit1 w)) = - numeral (w + Num.One)"
   by (simp_all add: bin_rest_def zdiv_zminus1_eq_if) (auto simp add: divmod_def)
 
-lemma less_Bits: 
-  "v BIT b < w BIT c \<longleftrightarrow> v < w \<or> v \<le> w \<and> \<not> b \<and> c"
-  unfolding Bit_def by auto
-
-lemma le_Bits: 
-  "v BIT b \<le> w BIT c \<longleftrightarrow> v < w \<or> v \<le> w \<and> (\<not> b \<or> c)" 
-  unfolding Bit_def by auto
+lemma less_Bits: "v BIT b < w BIT c \<longleftrightarrow> v < w \<or> v \<le> w \<and> \<not> b \<and> c"
+  by (auto simp: Bit_def)
+
+lemma le_Bits: "v BIT b \<le> w BIT c \<longleftrightarrow> v < w \<or> v \<le> w \<and> (\<not> b \<or> c)"
+  by (auto simp: Bit_def)
 
 lemma pred_BIT_simps [simp]:
   "x BIT False - 1 = (x - 1) BIT True"
   "x BIT True - 1 = x BIT False"
   by (simp_all add: Bit_B0_2t Bit_B1_2t)

   "x BIT False * y = (x * y) BIT False"
   "x * y BIT False = (x * y) BIT False"
   "x BIT True * y = (x * y) BIT False + y"
   by (simp_all add: Bit_B0_2t Bit_B1_2t algebra_simps)
 
-lemma B_mod_2': 
-  "X = 2 ==> (w BIT True) mod X = 1 & (w BIT False) mod X = 0"
-  apply (simp (no_asm) only: Bit_B0 Bit_B1)
-  apply simp
-  done
-
-lemma bin_ex_rl: "EX w b. w BIT b = bin"
+lemma B_mod_2': "X = 2 \<Longrightarrow> (w BIT True) mod X = 1 \<and> (w BIT False) mod X = 0"
+  by (simp add: Bit_B0 Bit_B1)
+
+lemma bin_ex_rl: "\<exists>w b. w BIT b = bin"
   by (metis bin_rl_simp)
 
 lemma bin_exhaust:
-  assumes Q: "\<And>x b. bin = x BIT b \<Longrightarrow> Q"
+  assumes that: "\<And>x b. bin = x BIT b \<Longrightarrow> Q"
   shows "Q"
-  apply (insert bin_ex_rl [of bin])  
+  apply (insert bin_ex_rl [of bin])
   apply (erule exE)+
-  apply (rule Q)
+  apply (rule that)
   apply force
   done
 
-primrec bin_nth where
-  Z: "bin_nth w 0 \<longleftrightarrow> bin_last w"
+primrec bin_nth
+  where
+    Z: "bin_nth w 0 \<longleftrightarrow> bin_last w"
   | Suc: "bin_nth w (Suc n) \<longleftrightarrow> bin_nth (bin_rest w) n"
 
-lemma bin_abs_lem:
-  "bin = (w BIT b) ==> bin ~= -1 --> bin ~= 0 -->
-    nat \<bar>w\<bar> < nat \<bar>bin\<bar>"
+lemma bin_abs_lem: "bin = (w BIT b) \<Longrightarrow> bin \<noteq> -1 \<longrightarrow> bin \<noteq> 0 \<longrightarrow> nat \<bar>w\<bar> < nat \<bar>bin\<bar>"
   apply clarsimp
   apply (unfold Bit_def)
   apply (cases b)
    apply (clarsimp, arith)
   apply (clarsimp, arith)
   done
 
 lemma bin_induct:
   assumes PPls: "P 0"
     and PMin: "P (- 1)"
-    and PBit: "!!bin bit. P bin ==> P (bin BIT bit)"
+    and PBit: "\<And>bin bit. P bin \<Longrightarrow> P (bin BIT bit)"
   shows "P bin"
-  apply (rule_tac P=P and a=bin and f1="nat o abs" 
-                  in wf_measure [THEN wf_induct])
+  apply (rule_tac P=P and a=bin and f1="nat \<circ> abs" in wf_measure [THEN wf_induct])
   apply (simp add: measure_def inv_image_def)
   apply (case_tac x rule: bin_exhaust)
   apply (frule bin_abs_lem)
   apply (auto simp add : PPls PMin PBit)
   done
 
 lemma Bit_div2 [simp]: "(w BIT b) div 2 = w"
   unfolding bin_rest_def [symmetric] by (rule bin_rest_BIT)
 
-lemma bin_nth_eq_iff:
-  "bin_nth x = bin_nth y \<longleftrightarrow> x = y"
+lemma bin_nth_eq_iff: "bin_nth x = bin_nth y \<longleftrightarrow> x = y"
 proof -
-  have bin_nth_lem [rule_format]: "ALL y. bin_nth x = bin_nth y --> x = y"
+  have bin_nth_lem [rule_format]: "\<forall>y. bin_nth x = bin_nth y \<longrightarrow> x = y"
     apply (induct x rule: bin_induct)
       apply safe
       apply (erule rev_mp)
       apply (induct_tac y rule: bin_induct)
         apply safe
         apply (drule_tac x=0 in fun_cong, force)
-       apply (erule notE, rule ext, 
-            drule_tac x="Suc x" in fun_cong, force)
+       apply (erule notE, rule ext, drule_tac x="Suc x" in fun_cong, force)
       apply (drule_tac x=0 in fun_cong, force)
      apply (erule rev_mp)
      apply (induct_tac y rule: bin_induct)
        apply safe
        apply (drule_tac x=0 in fun_cong, force)
-      apply (erule notE, rule ext, 
-           drule_tac x="Suc x" in fun_cong, force)
-      apply (metis Bit_eq_m1_iff Z bin_last_BIT)
+      apply (erule notE, rule ext, drule_tac x="Suc x" in fun_cong, force)
+     apply (metis Bit_eq_m1_iff Z bin_last_BIT)
     apply (case_tac y rule: bin_exhaust)
     apply clarify
     apply (erule allE)
     apply (erule impE)
      prefer 2

      apply (drule_tac x=0 in fun_cong, force)
     apply (rule ext)
     apply (drule_tac x="Suc x" for x in fun_cong, force)
     done
   show ?thesis
-  by (auto elim: bin_nth_lem)
+    by (auto elim: bin_nth_lem)
 qed
 
 lemmas bin_eqI = ext [THEN bin_nth_eq_iff [THEN iffD1]]
 
-lemma bin_eq_iff:
-  "x = y \<longleftrightarrow> (\<forall>n. bin_nth x n = bin_nth y n)"
+lemma bin_eq_iff: "x = y \<longleftrightarrow> (\<forall>n. bin_nth x n = bin_nth y n)"
   using bin_nth_eq_iff by auto
 
 lemma bin_nth_zero [simp]: "\<not> bin_nth 0 n"
   by (induct n) auto
 

   by auto
 
 lemma bin_nth_Suc_BIT: "bin_nth (w BIT b) (Suc n) = bin_nth w n"
   by auto
 
-lemma bin_nth_minus [simp]: "0 < n ==> bin_nth (w BIT b) n = bin_nth w (n - 1)"
+lemma bin_nth_minus [simp]: "0 < n \<Longrightarrow> bin_nth (w BIT b) n = bin_nth w (n - 1)"
   by (cases n) auto
 
-lemma bin_nth_numeral:
-  "bin_rest x = y \<Longrightarrow> bin_nth x (numeral n) = bin_nth y (pred_numeral n)"
+lemma bin_nth_numeral: "bin_rest x = y \<Longrightarrow> bin_nth x (numeral n) = bin_nth y (pred_numeral n)"
   by (simp add: numeral_eq_Suc)
 
 lemmas bin_nth_numeral_simps [simp] =
   bin_nth_numeral [OF bin_rest_numeral_simps(2)]
   bin_nth_numeral [OF bin_rest_numeral_simps(5)]
   bin_nth_numeral [OF bin_rest_numeral_simps(6)]
   bin_nth_numeral [OF bin_rest_numeral_simps(7)]
   bin_nth_numeral [OF bin_rest_numeral_simps(8)]
 
-lemmas bin_nth_simps = 
+lemmas bin_nth_simps =
   bin_nth.Z bin_nth.Suc bin_nth_zero bin_nth_minus1
   bin_nth_numeral_simps
 
 
 subsection \<open>Truncating binary integers\<close>
 
 definition bin_sign :: "int \<Rightarrow> int"
-where
-  bin_sign_def: "bin_sign k = (if k \<ge> 0 then 0 else - 1)"
+  where "bin_sign k = (if k \<ge> 0 then 0 else - 1)"
 
 lemma bin_sign_simps [simp]:
   "bin_sign 0 = 0"
   "bin_sign 1 = 0"
   "bin_sign (- 1) = - 1"
   "bin_sign (numeral k) = 0"
   "bin_sign (- numeral k) = -1"
   "bin_sign (w BIT b) = bin_sign w"
-  unfolding bin_sign_def Bit_def
-  by simp_all
-
-lemma bin_sign_rest [simp]: 
-  "bin_sign (bin_rest w) = bin_sign w"
+  by (simp_all add: bin_sign_def Bit_def)
+
+lemma bin_sign_rest [simp]: "bin_sign (bin_rest w) = bin_sign w"
   by (cases w rule: bin_exhaust) auto
 
-primrec bintrunc :: "nat \<Rightarrow> int \<Rightarrow> int" where
-  Z : "bintrunc 0 bin = 0"
-| Suc : "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT (bin_last bin)"
-
-primrec sbintrunc :: "nat => int => int" where
-  Z : "sbintrunc 0 bin = (if bin_last bin then -1 else 0)"
-| Suc : "sbintrunc (Suc n) bin = sbintrunc n (bin_rest bin) BIT (bin_last bin)"
+primrec bintrunc :: "nat \<Rightarrow> int \<Rightarrow> int"
+  where
+    Z : "bintrunc 0 bin = 0"
+  | Suc : "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT (bin_last bin)"
+
+primrec sbintrunc :: "nat \<Rightarrow> int \<Rightarrow> int"
+  where
+    Z : "sbintrunc 0 bin = (if bin_last bin then -1 else 0)"
+  | Suc : "sbintrunc (Suc n) bin = sbintrunc n (bin_rest bin) BIT (bin_last bin)"
 
 lemma sign_bintr: "bin_sign (bintrunc n w) = 0"
   by (induct n arbitrary: w) auto
 
-lemma bintrunc_mod2p: "bintrunc n w = (w mod 2 ^ n)"
-  apply (induct n arbitrary: w, clarsimp)
-  apply (simp add: bin_last_def bin_rest_def Bit_def zmod_zmult2_eq)
-  done
+lemma bintrunc_mod2p: "bintrunc n w = w mod 2 ^ n"
+  by (induct n arbitrary: w) (auto simp add: bin_last_def bin_rest_def Bit_def zmod_zmult2_eq)
 
 lemma sbintrunc_mod2p: "sbintrunc n w = (w + 2 ^ n) mod 2 ^ (Suc n) - 2 ^ n"
   apply (induct n arbitrary: w)
    apply (auto simp add: bin_last_odd bin_rest_def Bit_def elim!: evenE oddE)
    apply (smt pos_zmod_mult_2 zle2p)

 lemma bintrunc_Suc_numeral:
   "bintrunc (Suc n) 1 = 1"
   "bintrunc (Suc n) (- 1) = bintrunc n (- 1) BIT True"
   "bintrunc (Suc n) (numeral (Num.Bit0 w)) = bintrunc n (numeral w) BIT False"
   "bintrunc (Suc n) (numeral (Num.Bit1 w)) = bintrunc n (numeral w) BIT True"
-  "bintrunc (Suc n) (- numeral (Num.Bit0 w)) =
-    bintrunc n (- numeral w) BIT False"
-  "bintrunc (Suc n) (- numeral (Num.Bit1 w)) =
-    bintrunc n (- numeral (w + Num.One)) BIT True"
+  "bintrunc (Suc n) (- numeral (Num.Bit0 w)) = bintrunc n (- numeral w) BIT False"
+  "bintrunc (Suc n) (- numeral (Num.Bit1 w)) = bintrunc n (- numeral (w + Num.One)) BIT True"
   by simp_all
 
 lemma sbintrunc_0_numeral [simp]:
   "sbintrunc 0 1 = -1"
   "sbintrunc 0 (numeral (Num.Bit0 w)) = 0"

   "sbintrunc 0 (- numeral (Num.Bit1 w)) = -1"
   by simp_all
 
 lemma sbintrunc_Suc_numeral:
   "sbintrunc (Suc n) 1 = 1"
-  "sbintrunc (Suc n) (numeral (Num.Bit0 w)) =
-    sbintrunc n (numeral w) BIT False"
-  "sbintrunc (Suc n) (numeral (Num.Bit1 w)) =
-    sbintrunc n (numeral w) BIT True"
-  "sbintrunc (Suc n) (- numeral (Num.Bit0 w)) =
-    sbintrunc n (- numeral w) BIT False"
-  "sbintrunc (Suc n) (- numeral (Num.Bit1 w)) =
-    sbintrunc n (- numeral (w + Num.One)) BIT True"
+  "sbintrunc (Suc n) (numeral (Num.Bit0 w)) = sbintrunc n (numeral w) BIT False"
+  "sbintrunc (Suc n) (numeral (Num.Bit1 w)) = sbintrunc n (numeral w) BIT True"
+  "sbintrunc (Suc n) (- numeral (Num.Bit0 w)) = sbintrunc n (- numeral w) BIT False"
+  "sbintrunc (Suc n) (- numeral (Num.Bit1 w)) = sbintrunc n (- numeral (w + Num.One)) BIT True"
   by simp_all
 
 lemma bin_sign_lem: "(bin_sign (sbintrunc n bin) = -1) = bin_nth bin n"
   apply (induct n arbitrary: bin)
   apply (case_tac bin rule: bin_exhaust, case_tac b, auto)
   done
 
-lemma nth_bintr: "bin_nth (bintrunc m w) n = (n < m & bin_nth w n)"
+lemma nth_bintr: "bin_nth (bintrunc m w) n \<longleftrightarrow> n < m \<and> bin_nth w n"
   apply (induct n arbitrary: w m)
    apply (case_tac m, auto)[1]
   apply (case_tac m, auto)[1]
   done
 
-lemma nth_sbintr:
-  "bin_nth (sbintrunc m w) n = 
-          (if n < m then bin_nth w n else bin_nth w m)"
+lemma nth_sbintr: "bin_nth (sbintrunc m w) n = (if n < m then bin_nth w n else bin_nth w m)"
   apply (induct n arbitrary: w m)
+   apply (case_tac m)
+    apply simp_all
   apply (case_tac m)
-  apply simp_all
-  apply (case_tac m)
-  apply simp_all
-  done
-
-lemma bin_nth_Bit:
-  "bin_nth (w BIT b) n = (n = 0 & b | (EX m. n = Suc m & bin_nth w m))"
+   apply simp_all
+  done
+
+lemma bin_nth_Bit: "bin_nth (w BIT b) n \<longleftrightarrow> n = 0 \<and> b \<or> (\<exists>m. n = Suc m \<and> bin_nth w m)"
   by (cases n) auto
 
 lemma bin_nth_Bit0:
   "bin_nth (numeral (Num.Bit0 w)) n \<longleftrightarrow>
     (\<exists>m. n = Suc m \<and> bin_nth (numeral w) m)"

 lemma bin_nth_Bit1:
   "bin_nth (numeral (Num.Bit1 w)) n \<longleftrightarrow>
     n = 0 \<or> (\<exists>m. n = Suc m \<and> bin_nth (numeral w) m)"
   using bin_nth_Bit [where w="numeral w" and b="True"] by simp
 
-lemma bintrunc_bintrunc_l:
-  "n <= m ==> (bintrunc m (bintrunc n w) = bintrunc n w)"
-  by (rule bin_eqI) (auto simp add : nth_bintr)
-
-lemma sbintrunc_sbintrunc_l:
-  "n <= m ==> (sbintrunc m (sbintrunc n w) = sbintrunc n w)"
+lemma bintrunc_bintrunc_l: "n \<le> m \<Longrightarrow> bintrunc m (bintrunc n w) = bintrunc n w"
+  by (rule bin_eqI) (auto simp: nth_bintr)
+
+lemma sbintrunc_sbintrunc_l: "n \<le> m \<Longrightarrow> sbintrunc m (sbintrunc n w) = sbintrunc n w"
   by (rule bin_eqI) (auto simp: nth_sbintr)
 
-lemma bintrunc_bintrunc_ge:
-  "n <= m ==> (bintrunc n (bintrunc m w) = bintrunc n w)"
+lemma bintrunc_bintrunc_ge: "n \<le> m \<Longrightarrow> bintrunc n (bintrunc m w) = bintrunc n w"
   by (rule bin_eqI) (auto simp: nth_bintr)
 
-lemma bintrunc_bintrunc_min [simp]:
-  "bintrunc m (bintrunc n w) = bintrunc (min m n) w"
-  apply (rule bin_eqI)
-  apply (auto simp: nth_bintr)
-  done
-
-lemma sbintrunc_sbintrunc_min [simp]:
-  "sbintrunc m (sbintrunc n w) = sbintrunc (min m n) w"
-  apply (rule bin_eqI)
-  apply (auto simp: nth_sbintr min.absorb1 min.absorb2)
-  done
-
-lemmas bintrunc_Pls = 
+lemma bintrunc_bintrunc_min [simp]: "bintrunc m (bintrunc n w) = bintrunc (min m n) w"
+  by (rule bin_eqI) (auto simp: nth_bintr)
+
+lemma sbintrunc_sbintrunc_min [simp]: "sbintrunc m (sbintrunc n w) = sbintrunc (min m n) w"
+  by (rule bin_eqI) (auto simp: nth_sbintr min.absorb1 min.absorb2)
+
+lemmas bintrunc_Pls =
   bintrunc.Suc [where bin="0", simplified bin_last_numeral_simps bin_rest_numeral_simps]
 
-lemmas bintrunc_Min [simp] = 
+lemmas bintrunc_Min [simp] =
   bintrunc.Suc [where bin="-1", simplified bin_last_numeral_simps bin_rest_numeral_simps]
 
-lemmas bintrunc_BIT  [simp] = 
+lemmas bintrunc_BIT  [simp] =
   bintrunc.Suc [where bin="w BIT b", simplified bin_last_BIT bin_rest_BIT] for w b
 
 lemmas bintrunc_Sucs = bintrunc_Pls bintrunc_Min bintrunc_BIT
   bintrunc_Suc_numeral
 
-lemmas sbintrunc_Suc_Pls = 
+lemmas sbintrunc_Suc_Pls =
   sbintrunc.Suc [where bin="0", simplified bin_last_numeral_simps bin_rest_numeral_simps]
 
-lemmas sbintrunc_Suc_Min = 
+lemmas sbintrunc_Suc_Min =
   sbintrunc.Suc [where bin="-1", simplified bin_last_numeral_simps bin_rest_numeral_simps]
 
-lemmas sbintrunc_Suc_BIT [simp] = 
+lemmas sbintrunc_Suc_BIT [simp] =
   sbintrunc.Suc [where bin="w BIT b", simplified bin_last_BIT bin_rest_BIT] for w b
 
 lemmas sbintrunc_Sucs = sbintrunc_Suc_Pls sbintrunc_Suc_Min sbintrunc_Suc_BIT
   sbintrunc_Suc_numeral
 
-lemmas sbintrunc_Pls = 
-  sbintrunc.Z [where bin="0", 
-               simplified bin_last_numeral_simps bin_rest_numeral_simps]
-
-lemmas sbintrunc_Min = 
-  sbintrunc.Z [where bin="-1",
-               simplified bin_last_numeral_simps bin_rest_numeral_simps]
-
-lemmas sbintrunc_0_BIT_B0 [simp] = 
-  sbintrunc.Z [where bin="w BIT False", 
-               simplified bin_last_numeral_simps bin_rest_numeral_simps] for w
-
-lemmas sbintrunc_0_BIT_B1 [simp] = 
-  sbintrunc.Z [where bin="w BIT True", 
-               simplified bin_last_BIT bin_rest_numeral_simps] for w
+lemmas sbintrunc_Pls =
+  sbintrunc.Z [where bin="0", simplified bin_last_numeral_simps bin_rest_numeral_simps]
+
+lemmas sbintrunc_Min =
+  sbintrunc.Z [where bin="-1", simplified bin_last_numeral_simps bin_rest_numeral_simps]
+
+lemmas sbintrunc_0_BIT_B0 [simp] =
+  sbintrunc.Z [where bin="w BIT False", simplified bin_last_numeral_simps bin_rest_numeral_simps]
+  for w
+
+lemmas sbintrunc_0_BIT_B1 [simp] =
+  sbintrunc.Z [where bin="w BIT True", simplified bin_last_BIT bin_rest_numeral_simps]
+  for w
 
 lemmas sbintrunc_0_simps =
   sbintrunc_Pls sbintrunc_Min sbintrunc_0_BIT_B0 sbintrunc_0_BIT_B1
 
 lemmas bintrunc_simps = bintrunc.Z bintrunc_Sucs
 lemmas sbintrunc_simps = sbintrunc_0_simps sbintrunc_Sucs
 
-lemma bintrunc_minus:
-  "0 < n ==> bintrunc (Suc (n - 1)) w = bintrunc n w"
+lemma bintrunc_minus: "0 < n \<Longrightarrow> bintrunc (Suc (n - 1)) w = bintrunc n w"
   by auto
 
-lemma sbintrunc_minus:
-  "0 < n ==> sbintrunc (Suc (n - 1)) w = sbintrunc n w"
+lemma sbintrunc_minus: "0 < n \<Longrightarrow> sbintrunc (Suc (n - 1)) w = sbintrunc n w"
   by auto
 
-lemmas bintrunc_minus_simps = 
+lemmas bintrunc_minus_simps =
   bintrunc_Sucs [THEN [2] bintrunc_minus [symmetric, THEN trans]]
-lemmas sbintrunc_minus_simps = 
+lemmas sbintrunc_minus_simps =
   sbintrunc_Sucs [THEN [2] sbintrunc_minus [symmetric, THEN trans]]
 
-lemmas thobini1 = arg_cong [where f = "%w. w BIT b"] for b
+lemmas thobini1 = arg_cong [where f = "\<lambda>w. w BIT b"] for b
 
 lemmas bintrunc_BIT_I = trans [OF bintrunc_BIT thobini1]
 lemmas bintrunc_Min_I = trans [OF bintrunc_Min thobini1]
 
 lemmas bmsts = bintrunc_minus_simps(1-3) [THEN thobini1 [THEN [2] trans]]
 lemmas bintrunc_Pls_minus_I = bmsts(1)
 lemmas bintrunc_Min_minus_I = bmsts(2)
 lemmas bintrunc_BIT_minus_I = bmsts(3)
 
-lemma bintrunc_Suc_lem:
-  "bintrunc (Suc n) x = y ==> m = Suc n ==> bintrunc m x = y"
+lemma bintrunc_Suc_lem: "bintrunc (Suc n) x = y \<Longrightarrow> m = Suc n \<Longrightarrow> bintrunc m x = y"
   by auto
 
-lemmas bintrunc_Suc_Ialts = 
+lemmas bintrunc_Suc_Ialts =
   bintrunc_Min_I [THEN bintrunc_Suc_lem]
   bintrunc_BIT_I [THEN bintrunc_Suc_lem]
 
 lemmas sbintrunc_BIT_I = trans [OF sbintrunc_Suc_BIT thobini1]
 
-lemmas sbintrunc_Suc_Is = 
+lemmas sbintrunc_Suc_Is =
   sbintrunc_Sucs(1-3) [THEN thobini1 [THEN [2] trans]]
 
-lemmas sbintrunc_Suc_minus_Is = 
+lemmas sbintrunc_Suc_minus_Is =
   sbintrunc_minus_simps(1-3) [THEN thobini1 [THEN [2] trans]]
 
-lemma sbintrunc_Suc_lem:
-  "sbintrunc (Suc n) x = y ==> m = Suc n ==> sbintrunc m x = y"
+lemma sbintrunc_Suc_lem: "sbintrunc (Suc n) x = y \<Longrightarrow> m = Suc n \<Longrightarrow> sbintrunc m x = y"
   by auto
 
-lemmas sbintrunc_Suc_Ialts = 
+lemmas sbintrunc_Suc_Ialts =
   sbintrunc_Suc_Is [THEN sbintrunc_Suc_lem]
 
-lemma sbintrunc_bintrunc_lt:
-  "m > n ==> sbintrunc n (bintrunc m w) = sbintrunc n w"
+lemma sbintrunc_bintrunc_lt: "m > n \<Longrightarrow> sbintrunc n (bintrunc m w) = sbintrunc n w"
   by (rule bin_eqI) (auto simp: nth_sbintr nth_bintr)
 
-lemma bintrunc_sbintrunc_le:
-  "m <= Suc n ==> bintrunc m (sbintrunc n w) = bintrunc m w"
+lemma bintrunc_sbintrunc_le: "m \<le> Suc n \<Longrightarrow> bintrunc m (sbintrunc n w) = bintrunc m w"
   apply (rule bin_eqI)
   apply (auto simp: nth_sbintr nth_bintr)
    apply (subgoal_tac "x=n", safe, arith+)[1]
   apply (subgoal_tac "x=n", safe, arith+)[1]
   done
 
 lemmas bintrunc_sbintrunc [simp] = order_refl [THEN bintrunc_sbintrunc_le]
 lemmas sbintrunc_bintrunc [simp] = lessI [THEN sbintrunc_bintrunc_lt]
 lemmas bintrunc_bintrunc [simp] = order_refl [THEN bintrunc_bintrunc_l]
-lemmas sbintrunc_sbintrunc [simp] = order_refl [THEN sbintrunc_sbintrunc_l] 
-
-lemma bintrunc_sbintrunc' [simp]:
-  "0 < n \<Longrightarrow> bintrunc n (sbintrunc (n - 1) w) = bintrunc n w"
+lemmas sbintrunc_sbintrunc [simp] = order_refl [THEN sbintrunc_sbintrunc_l]
+
+lemma bintrunc_sbintrunc' [simp]: "0 < n \<Longrightarrow> bintrunc n (sbintrunc (n - 1) w) = bintrunc n w"
   by (cases n) (auto simp del: bintrunc.Suc)
 
-lemma sbintrunc_bintrunc' [simp]:
-  "0 < n \<Longrightarrow> sbintrunc (n - 1) (bintrunc n w) = sbintrunc (n - 1) w"
+lemma sbintrunc_bintrunc' [simp]: "0 < n \<Longrightarrow> sbintrunc (n - 1) (bintrunc n w) = sbintrunc (n - 1) w"
   by (cases n) (auto simp del: bintrunc.Suc)
 
-lemma bin_sbin_eq_iff: 
-  "bintrunc (Suc n) x = bintrunc (Suc n) y \<longleftrightarrow> 
-   sbintrunc n x = sbintrunc n y"
+lemma bin_sbin_eq_iff: "bintrunc (Suc n) x = bintrunc (Suc n) y \<longleftrightarrow> sbintrunc n x = sbintrunc n y"
   apply (rule iffI)
    apply (rule box_equals [OF _ sbintrunc_bintrunc sbintrunc_bintrunc])
    apply simp
   apply (rule box_equals [OF _ bintrunc_sbintrunc bintrunc_sbintrunc])
   apply simp
   done
 
 lemma bin_sbin_eq_iff':
-  "0 < n \<Longrightarrow> bintrunc n x = bintrunc n y \<longleftrightarrow> 
-            sbintrunc (n - 1) x = sbintrunc (n - 1) y"
+  "0 < n \<Longrightarrow> bintrunc n x = bintrunc n y \<longleftrightarrow> sbintrunc (n - 1) x = sbintrunc (n - 1) y"
   by (cases n) (simp_all add: bin_sbin_eq_iff del: bintrunc.Suc)
 
 lemmas bintrunc_sbintruncS0 [simp] = bintrunc_sbintrunc' [unfolded One_nat_def]
 lemmas sbintrunc_bintruncS0 [simp] = sbintrunc_bintrunc' [unfolded One_nat_def]
 

 
 (* although bintrunc_minus_simps, if added to default simpset,
   tends to get applied where it's not wanted in developing the theories,
   we get a version for when the word length is given literally *)
 
-lemmas nat_non0_gr = 
+lemmas nat_non0_gr =
   trans [OF iszero_def [THEN Not_eq_iff [THEN iffD2]] refl]
 
 lemma bintrunc_numeral:
-  "bintrunc (numeral k) x =
-    bintrunc (pred_numeral k) (bin_rest x) BIT bin_last x"
+  "bintrunc (numeral k) x = bintrunc (pred_numeral k) (bin_rest x) BIT bin_last x"
   by (simp add: numeral_eq_Suc)
 
 lemma sbintrunc_numeral:
-  "sbintrunc (numeral k) x =
-    sbintrunc (pred_numeral k) (bin_rest x) BIT bin_last x"
+  "sbintrunc (numeral k) x = sbintrunc (pred_numeral k) (bin_rest x) BIT bin_last x"
   by (simp add: numeral_eq_Suc)
 
 lemma bintrunc_numeral_simps [simp]:
-  "bintrunc (numeral k) (numeral (Num.Bit0 w)) =
-    bintrunc (pred_numeral k) (numeral w) BIT False"
-  "bintrunc (numeral k) (numeral (Num.Bit1 w)) =
-    bintrunc (pred_numeral k) (numeral w) BIT True"
-  "bintrunc (numeral k) (- numeral (Num.Bit0 w)) =
-    bintrunc (pred_numeral k) (- numeral w) BIT False"
+  "bintrunc (numeral k) (numeral (Num.Bit0 w)) = bintrunc (pred_numeral k) (numeral w) BIT False"
+  "bintrunc (numeral k) (numeral (Num.Bit1 w)) = bintrunc (pred_numeral k) (numeral w) BIT True"
+  "bintrunc (numeral k) (- numeral (Num.Bit0 w)) = bintrunc (pred_numeral k) (- numeral w) BIT False"
   "bintrunc (numeral k) (- numeral (Num.Bit1 w)) =
     bintrunc (pred_numeral k) (- numeral (w + Num.One)) BIT True"
   "bintrunc (numeral k) 1 = 1"
   by (simp_all add: bintrunc_numeral)
 
 lemma sbintrunc_numeral_simps [simp]:
-  "sbintrunc (numeral k) (numeral (Num.Bit0 w)) =
-    sbintrunc (pred_numeral k) (numeral w) BIT False"
-  "sbintrunc (numeral k) (numeral (Num.Bit1 w)) =
-    sbintrunc (pred_numeral k) (numeral w) BIT True"
+  "sbintrunc (numeral k) (numeral (Num.Bit0 w)) = sbintrunc (pred_numeral k) (numeral w) BIT False"
+  "sbintrunc (numeral k) (numeral (Num.Bit1 w)) = sbintrunc (pred_numeral k) (numeral w) BIT True"
   "sbintrunc (numeral k) (- numeral (Num.Bit0 w)) =
     sbintrunc (pred_numeral k) (- numeral w) BIT False"
   "sbintrunc (numeral k) (- numeral (Num.Bit1 w)) =
     sbintrunc (pred_numeral k) (- numeral (w + Num.One)) BIT True"
   "sbintrunc (numeral k) 1 = 1"
   by (simp_all add: sbintrunc_numeral)
 
 lemma no_bintr_alt1: "bintrunc n = (\<lambda>w. w mod 2 ^ n :: int)"
   by (rule ext) (rule bintrunc_mod2p)
 
-lemma range_bintrunc: "range (bintrunc n) = {i. 0 <= i & i < 2 ^ n}"
+lemma range_bintrunc: "range (bintrunc n) = {i. 0 \<le> i \<and> i < 2 ^ n}"
   apply (unfold no_bintr_alt1)
   apply (auto simp add: image_iff)
   apply (rule exI)
   apply (auto intro: int_mod_lem [THEN iffD1, symmetric])
   done
 
-lemma no_sbintr_alt2: 
-  "sbintrunc n = (%w. (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)"
+lemma no_sbintr_alt2: "sbintrunc n = (\<lambda>w. (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)"
   by (rule ext) (simp add : sbintrunc_mod2p)
 
-lemma range_sbintrunc: 
-  "range (sbintrunc n) = {i. - (2 ^ n) <= i & i < 2 ^ n}"
+lemma range_sbintrunc: "range (sbintrunc n) = {i. - (2 ^ n) \<le> i \<and> i < 2 ^ n}"
   apply (unfold no_sbintr_alt2)
   apply (auto simp add: image_iff eq_diff_eq)
   apply (rule exI)
   apply (auto intro: int_mod_lem [THEN iffD1, symmetric])
   done
 
-lemma sb_inc_lem:
-  "(a::int) + 2^k < 0 \<Longrightarrow> a + 2^k + 2^(Suc k) <= (a + 2^k) mod 2^(Suc k)"
+lemma sb_inc_lem: "a + 2^k < 0 \<Longrightarrow> a + 2^k + 2^(Suc k) \<le> (a + 2^k) mod 2^(Suc k)"
+  for a :: int
   apply (erule int_mod_ge' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k", simplified zless2p])
   apply (rule TrueI)
   done
 
-lemma sb_inc_lem':
-  "(a::int) < - (2^k) \<Longrightarrow> a + 2^k + 2^(Suc k) <= (a + 2^k) mod 2^(Suc k)"
+lemma sb_inc_lem': "a < - (2^k) \<Longrightarrow> a + 2^k + 2^(Suc k) \<le> (a + 2^k) mod 2^(Suc k)"
+  for a :: int
   by (rule sb_inc_lem) simp
 
-lemma sbintrunc_inc:
-  "x < - (2^n) ==> x + 2^(Suc n) <= sbintrunc n x"
+lemma sbintrunc_inc: "x < - (2^n) \<Longrightarrow> x + 2^(Suc n) \<le> sbintrunc n x"
   unfolding no_sbintr_alt2 by (drule sb_inc_lem') simp
 
-lemma sb_dec_lem:
-  "(0::int) \<le> - (2 ^ k) + a \<Longrightarrow> (a + 2 ^ k) mod (2 * 2 ^ k) \<le> - (2 ^ k) + a"
+lemma sb_dec_lem: "0 \<le> - (2 ^ k) + a \<Longrightarrow> (a + 2 ^ k) mod (2 * 2 ^ k) \<le> - (2 ^ k) + a"
+  for a :: int
   using int_mod_le'[where n = "2 ^ (Suc k)" and b = "a + 2 ^ k"] by simp
 
-lemma sb_dec_lem':
-  "(2::int) ^ k \<le> a \<Longrightarrow> (a + 2 ^ k) mod (2 * 2 ^ k) \<le> - (2 ^ k) + a"
+lemma sb_dec_lem': "2 ^ k \<le> a \<Longrightarrow> (a + 2 ^ k) mod (2 * 2 ^ k) \<le> - (2 ^ k) + a"
+  for a :: int
   by (rule sb_dec_lem) simp
 
-lemma sbintrunc_dec:
-  "x >= (2 ^ n) ==> x - 2 ^ (Suc n) >= sbintrunc n x"
+lemma sbintrunc_dec: "x \<ge> (2 ^ n) \<Longrightarrow> x - 2 ^ (Suc n) >= sbintrunc n x"
   unfolding no_sbintr_alt2 by (drule sb_dec_lem') simp
 
 lemmas m2pths = pos_mod_sign pos_mod_bound [OF zless2p]
 
 lemma bintr_ge0: "0 \<le> bintrunc n w"

   by (simp add: sbintrunc_mod2p)
 
 lemma sbintr_lt: "sbintrunc n w < 2 ^ n"
   by (simp add: sbintrunc_mod2p)
 
-lemma sign_Pls_ge_0: 
-  "(bin_sign bin = 0) = (bin >= (0 :: int))"
-  unfolding bin_sign_def by simp
-
-lemma sign_Min_lt_0: 
-  "(bin_sign bin = -1) = (bin < (0 :: int))"
-  unfolding bin_sign_def by simp
-
-lemma bin_rest_trunc:
-  "(bin_rest (bintrunc n bin)) = bintrunc (n - 1) (bin_rest bin)"
+lemma sign_Pls_ge_0: "bin_sign bin = 0 \<longleftrightarrow> bin \<ge> 0"
+  for bin :: int
+  by (simp add: bin_sign_def)
+
+lemma sign_Min_lt_0: "bin_sign bin = -1 \<longleftrightarrow> bin < 0"
+  for bin :: int
+  by (simp add: bin_sign_def)
+
+lemma bin_rest_trunc: "bin_rest (bintrunc n bin) = bintrunc (n - 1) (bin_rest bin)"
   by (induct n arbitrary: bin) auto
 
 lemma bin_rest_power_trunc:
-  "(bin_rest ^^ k) (bintrunc n bin) = 
-    bintrunc (n - k) ((bin_rest ^^ k) bin)"
+  "(bin_rest ^^ k) (bintrunc n bin) = bintrunc (n - k) ((bin_rest ^^ k) bin)"
   by (induct k) (auto simp: bin_rest_trunc)
 
-lemma bin_rest_trunc_i:
-  "bintrunc n (bin_rest bin) = bin_rest (bintrunc (Suc n) bin)"
+lemma bin_rest_trunc_i: "bintrunc n (bin_rest bin) = bin_rest (bintrunc (Suc n) bin)"
   by auto
 
-lemma bin_rest_strunc:
-  "bin_rest (sbintrunc (Suc n) bin) = sbintrunc n (bin_rest bin)"
+lemma bin_rest_strunc: "bin_rest (sbintrunc (Suc n) bin) = sbintrunc n (bin_rest bin)"
   by (induct n arbitrary: bin) auto
 
-lemma bintrunc_rest [simp]: 
-  "bintrunc n (bin_rest (bintrunc n bin)) = bin_rest (bintrunc n bin)"
-  apply (induct n arbitrary: bin, simp)
+lemma bintrunc_rest [simp]: "bintrunc n (bin_rest (bintrunc n bin)) = bin_rest (bintrunc n bin)"
+  apply (induct n arbitrary: bin)
+   apply simp
   apply (case_tac bin rule: bin_exhaust)
   apply (auto simp: bintrunc_bintrunc_l)
   done
 
-lemma sbintrunc_rest [simp]:
-  "sbintrunc n (bin_rest (sbintrunc n bin)) = bin_rest (sbintrunc n bin)"
-  apply (induct n arbitrary: bin, simp)
+lemma sbintrunc_rest [simp]: "sbintrunc n (bin_rest (sbintrunc n bin)) = bin_rest (sbintrunc n bin)"
+  apply (induct n arbitrary: bin)
+   apply simp
   apply (case_tac bin rule: bin_exhaust)
   apply (auto simp: bintrunc_bintrunc_l split: bool.splits)
   done
 
-lemma bintrunc_rest':
-  "bintrunc n o bin_rest o bintrunc n = bin_rest o bintrunc n"
+lemma bintrunc_rest': "bintrunc n \<circ> bin_rest \<circ> bintrunc n = bin_rest \<circ> bintrunc n"
   by (rule ext) auto
 
-lemma sbintrunc_rest' :
-  "sbintrunc n o bin_rest o sbintrunc n = bin_rest o sbintrunc n"
+lemma sbintrunc_rest': "sbintrunc n \<circ> bin_rest \<circ> sbintrunc n = bin_rest \<circ> sbintrunc n"
   by (rule ext) auto
 
-lemma rco_lem:
-  "f o g o f = g o f ==> f o (g o f) ^^ n = g ^^ n o f"
+lemma rco_lem: "f \<circ> g \<circ> f = g \<circ> f \<Longrightarrow> f \<circ> (g \<circ> f) ^^ n = g ^^ n \<circ> f"
   apply (rule ext)
   apply (induct_tac n)
    apply (simp_all (no_asm))
   apply (drule fun_cong)
   apply (unfold o_def)
   apply (erule trans)
   apply simp
   done
 
-lemmas rco_bintr = bintrunc_rest' 
+lemmas rco_bintr = bintrunc_rest'
   [THEN rco_lem [THEN fun_cong], unfolded o_def]
-lemmas rco_sbintr = sbintrunc_rest' 
+lemmas rco_sbintr = sbintrunc_rest'
   [THEN rco_lem [THEN fun_cong], unfolded o_def]
 
-  
+
 subsection \<open>Splitting and concatenation\<close>
 
-primrec bin_split :: "nat \<Rightarrow> int \<Rightarrow> int \<times> int" where
-  Z: "bin_split 0 w = (w, 0)"
-  | Suc: "bin_split (Suc n) w = (let (w1, w2) = bin_split n (bin_rest w)
-        in (w1, w2 BIT bin_last w))"
+primrec bin_split :: "nat \<Rightarrow> int \<Rightarrow> int \<times> int"
+  where
+    Z: "bin_split 0 w = (w, 0)"
+  | Suc: "bin_split (Suc n) w =
+      (let (w1, w2) = bin_split n (bin_rest w)
+       in (w1, w2 BIT bin_last w))"
 
 lemma [code]:
   "bin_split (Suc n) w = (let (w1, w2) = bin_split n (bin_rest w) in (w1, w2 BIT bin_last w))"
   "bin_split 0 w = (w, 0)"
   by simp_all
 
-primrec bin_cat :: "int \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int" where
-  Z: "bin_cat w 0 v = w"
+primrec bin_cat :: "int \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int"
+  where
+    Z: "bin_cat w 0 v = w"
   | Suc: "bin_cat w (Suc n) v = bin_cat w n (bin_rest v) BIT bin_last v"
 
 end
-