src/HOL/Word/Bit_Representation.thy

+(* 
+  Author: Jeremy Dawson, NICTA
+
+  contains basic definition to do with integers
+  expressed using Pls, Min, BIT and important resulting theorems, 
+  in particular, bin_rec and related work
+*) 
+
+header {* Basic Definitions for Binary Integers *}
+
+theory Bit_Representation
+imports Misc_Numeric Bit
+begin
+
+subsection {* Further properties of numerals *}
+
+definition Bit :: "int \<Rightarrow> bit \<Rightarrow> int" (infixl "BIT" 90) where
+  "k BIT b = bit_case 0 1 b + k + k"
+
+lemma BIT_B0_eq_Bit0 [simp]: "w BIT 0 = Int.Bit0 w"
+  unfolding Bit_def Bit0_def by simp
+
+lemma BIT_B1_eq_Bit1 [simp]: "w BIT 1 = Int.Bit1 w"
+  unfolding Bit_def Bit1_def by simp
+
+lemmas BIT_simps = BIT_B0_eq_Bit0 BIT_B1_eq_Bit1
+
+lemma Min_ne_Pls [iff]:  
+  "Int.Min ~= Int.Pls"
+  unfolding Min_def Pls_def by auto
+
+lemmas Pls_ne_Min [iff] = Min_ne_Pls [symmetric]
+
+lemmas PlsMin_defs [intro!] = 
+  Pls_def Min_def Pls_def [symmetric] Min_def [symmetric]
+
+lemmas PlsMin_simps [simp] = PlsMin_defs [THEN Eq_TrueI]
+
+lemma number_of_False_cong: 
+  "False \<Longrightarrow> number_of x = number_of y"
+  by (rule FalseE)
+
+(** ways in which type Bin resembles a datatype **)
+
+lemma BIT_eq: "u BIT b = v BIT c ==> u = v & b = c"
+  apply (unfold Bit_def)
+  apply (simp (no_asm_use) split: bit.split_asm)
+     apply simp_all
+   apply (drule_tac f=even in arg_cong, clarsimp)+
+  done
+     
+lemmas BIT_eqE [elim!] = BIT_eq [THEN conjE, standard]
+
+lemma BIT_eq_iff [simp]: 
+  "(u BIT b = v BIT c) = (u = v \<and> b = c)"
+  by (rule iffI) auto
+
+lemmas BIT_eqI [intro!] = conjI [THEN BIT_eq_iff [THEN iffD2]]
+
+lemma less_Bits: 
+  "(v BIT b < w BIT c) = (v < w | v <= w & b = (0::bit) & c = (1::bit))"
+  unfolding Bit_def by (auto split: bit.split)
+
+lemma le_Bits: 
+  "(v BIT b <= w BIT c) = (v < w | v <= w & (b ~= (1::bit) | c ~= (0::bit)))" 
+  unfolding Bit_def by (auto split: bit.split)
+
+lemma no_no [simp]: "number_of (number_of i) = i"
+  unfolding number_of_eq by simp
+
+lemma Bit_B0:
+  "k BIT (0::bit) = k + k"
+   by (unfold Bit_def) simp
+
+lemma Bit_B1:
+  "k BIT (1::bit) = k + k + 1"
+   by (unfold Bit_def) simp
+  
+lemma Bit_B0_2t: "k BIT (0::bit) = 2 * k"
+  by (rule trans, rule Bit_B0) simp
+
+lemma Bit_B1_2t: "k BIT (1::bit) = 2 * k + 1"
+  by (rule trans, rule Bit_B1) simp
+
+lemma B_mod_2': 
+  "X = 2 ==> (w BIT (1::bit)) mod X = 1 & (w BIT (0::bit)) mod X = 0"
+  apply (simp (no_asm) only: Bit_B0 Bit_B1)
+  apply (simp add: z1pmod2)
+  done
+
+lemma B1_mod_2 [simp]: "(Int.Bit1 w) mod 2 = 1"
+  unfolding numeral_simps number_of_is_id by (simp add: z1pmod2)
+
+lemma B0_mod_2 [simp]: "(Int.Bit0 w) mod 2 = 0"
+  unfolding numeral_simps number_of_is_id by simp
+
+lemma neB1E [elim!]:
+  assumes ne: "y \<noteq> (1::bit)"
+  assumes y: "y = (0::bit) \<Longrightarrow> P"
+  shows "P"
+  apply (rule y)
+  apply (cases y rule: bit.exhaust, simp)
+  apply (simp add: ne)
+  done
+
+lemma bin_ex_rl: "EX w b. w BIT b = bin"
+  apply (unfold Bit_def)
+  apply (cases "even bin")
+   apply (clarsimp simp: even_equiv_def)
+   apply (auto simp: odd_equiv_def split: bit.split)
+  done
+
+lemma bin_exhaust:
+  assumes Q: "\<And>x b. bin = x BIT b \<Longrightarrow> Q"
+  shows "Q"
+  apply (insert bin_ex_rl [of bin])  
+  apply (erule exE)+
+  apply (rule Q)
+  apply force
+  done
+
+
+subsection {* Destructors for binary integers *}
+
+definition bin_last :: "int \<Rightarrow> bit" where
+  "bin_last w = (if w mod 2 = 0 then (0::bit) else (1::bit))"
+
+definition bin_rest :: "int \<Rightarrow> int" where
+  "bin_rest w = w div 2"
+
+definition bin_rl :: "int \<Rightarrow> int \<times> bit" where 
+  "bin_rl w = (bin_rest w, bin_last w)"
+
+lemma bin_rl_char: "bin_rl w = (r, l) \<longleftrightarrow> r BIT l = w"
+  apply (cases l)
+  apply (auto simp add: bin_rl_def bin_last_def bin_rest_def)
+  unfolding Pls_def Min_def Bit0_def Bit1_def number_of_is_id
+  apply arith+
+  done
+
+primrec bin_nth where
+  Z: "bin_nth w 0 = (bin_last w = (1::bit))"
+  | Suc: "bin_nth w (Suc n) = bin_nth (bin_rest w) n"
+
+lemma bin_rl_simps [simp]:
+  "bin_rl Int.Pls = (Int.Pls, (0::bit))"
+  "bin_rl Int.Min = (Int.Min, (1::bit))"
+  "bin_rl (Int.Bit0 r) = (r, (0::bit))"
+  "bin_rl (Int.Bit1 r) = (r, (1::bit))"
+  "bin_rl (r BIT b) = (r, b)"
+  unfolding bin_rl_char by simp_all
+
+lemma bin_rl_simp [simp]:
+  "bin_rest w BIT bin_last w = w"
+  by (simp add: iffD1 [OF bin_rl_char bin_rl_def])
+
+lemma bin_abs_lem:
+  "bin = (w BIT b) ==> ~ bin = Int.Min --> ~ bin = Int.Pls -->
+    nat (abs w) < nat (abs bin)"
+  apply (clarsimp simp add: bin_rl_char)
+  apply (unfold Pls_def Min_def Bit_def)
+  apply (cases b)
+   apply (clarsimp, arith)
+  apply (clarsimp, arith)
+  done
+
+lemma bin_induct:
+  assumes PPls: "P Int.Pls"
+    and PMin: "P Int.Min"
+    and PBit: "!!bin bit. P bin ==> 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 (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 numeral_induct:
+  assumes Pls: "P Int.Pls"
+  assumes Min: "P Int.Min"
+  assumes Bit0: "\<And>w. \<lbrakk>P w; w \<noteq> Int.Pls\<rbrakk> \<Longrightarrow> P (Int.Bit0 w)"
+  assumes Bit1: "\<And>w. \<lbrakk>P w; w \<noteq> Int.Min\<rbrakk> \<Longrightarrow> P (Int.Bit1 w)"
+  shows "P x"
+  apply (induct x rule: bin_induct)
+    apply (rule Pls)
+   apply (rule Min)
+  apply (case_tac bit)
+   apply (case_tac "bin = Int.Pls")
+    apply simp
+   apply (simp add: Bit0)
+  apply (case_tac "bin = Int.Min")
+   apply simp
+  apply (simp add: Bit1)
+  done
+
+lemma bin_rest_simps [simp]: 
+  "bin_rest Int.Pls = Int.Pls"
+  "bin_rest Int.Min = Int.Min"
+  "bin_rest (Int.Bit0 w) = w"
+  "bin_rest (Int.Bit1 w) = w"
+  "bin_rest (w BIT b) = w"
+  using bin_rl_simps bin_rl_def by auto
+
+lemma bin_last_simps [simp]: 
+  "bin_last Int.Pls = (0::bit)"
+  "bin_last Int.Min = (1::bit)"
+  "bin_last (Int.Bit0 w) = (0::bit)"
+  "bin_last (Int.Bit1 w) = (1::bit)"
+  "bin_last (w BIT b) = b"
+  using bin_rl_simps bin_rl_def by auto
+
+lemma bin_r_l_extras [simp]:
+  "bin_last 0 = (0::bit)"
+  "bin_last (- 1) = (1::bit)"
+  "bin_last -1 = (1::bit)"
+  "bin_last 1 = (1::bit)"
+  "bin_rest 1 = 0"
+  "bin_rest 0 = 0"
+  "bin_rest (- 1) = - 1"
+  "bin_rest -1 = -1"
+  by (simp_all add: bin_last_def bin_rest_def)
+
+lemma bin_last_mod: 
+  "bin_last w = (if w mod 2 = 0 then (0::bit) else (1::bit))"
+  apply (case_tac w rule: bin_exhaust)
+  apply (case_tac b)
+   apply auto
+  done
+
+lemma bin_rest_div: 
+  "bin_rest w = w div 2"
+  apply (case_tac w rule: bin_exhaust)
+  apply (rule trans)
+   apply clarsimp
+   apply (rule refl)
+  apply (drule trans)
+   apply (rule Bit_def)
+  apply (simp add: z1pdiv2 split: bit.split)
+  done
+
+lemma Bit_div2 [simp]: "(w BIT b) div 2 = w"
+  unfolding bin_rest_div [symmetric] by auto
+
+lemma Bit0_div2 [simp]: "(Int.Bit0 w) div 2 = w"
+  using Bit_div2 [where b="(0::bit)"] by simp
+
+lemma Bit1_div2 [simp]: "(Int.Bit1 w) div 2 = w"
+  using Bit_div2 [where b="(1::bit)"] by simp
+
+lemma bin_nth_lem [rule_format]:
+  "ALL y. bin_nth x = bin_nth y --> x = y"
+  apply (induct x rule: bin_induct)
+    apply safe
+    apply (erule rev_mp)
+    apply (induct_tac y rule: bin_induct)
+      apply (safe del: subset_antisym)
+      apply (drule_tac x=0 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 del: subset_antisym)
+     apply (drule_tac x=0 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 (case_tac y rule: bin_exhaust)
+  apply clarify
+  apply (erule allE)
+  apply (erule impE)
+   prefer 2
+   apply (erule BIT_eqI)
+   apply (drule_tac x=0 in fun_cong, force)
+  apply (rule ext)
+  apply (drule_tac x="Suc ?x" in fun_cong, force)
+  done
+
+lemma bin_nth_eq_iff: "(bin_nth x = bin_nth y) = (x = y)"
+  by (auto elim: bin_nth_lem)
+
+lemmas bin_eqI = ext [THEN bin_nth_eq_iff [THEN iffD1], standard]
+
+lemma bin_nth_Pls [simp]: "~ bin_nth Int.Pls n"
+  by (induct n) auto
+
+lemma bin_nth_Min [simp]: "bin_nth Int.Min n"
+  by (induct n) auto
+
+lemma bin_nth_0_BIT: "bin_nth (w BIT b) 0 = (b = (1::bit))"
+  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)"
+  by (cases n) auto
+
+lemma bin_nth_minus_Bit0 [simp]:
+  "0 < n ==> bin_nth (Int.Bit0 w) n = bin_nth w (n - 1)"
+  using bin_nth_minus [where b="(0::bit)"] by simp
+
+lemma bin_nth_minus_Bit1 [simp]:
+  "0 < n ==> bin_nth (Int.Bit1 w) n = bin_nth w (n - 1)"
+  using bin_nth_minus [where b="(1::bit)"] by simp
+
+lemmas bin_nth_0 = bin_nth.simps(1)
+lemmas bin_nth_Suc = bin_nth.simps(2)
+
+lemmas bin_nth_simps = 
+  bin_nth_0 bin_nth_Suc bin_nth_Pls bin_nth_Min bin_nth_minus
+  bin_nth_minus_Bit0 bin_nth_minus_Bit1
+
+
+subsection {* Recursion combinator for binary integers *}
+
+lemma brlem: "(bin = Int.Min) = (- bin + Int.pred 0 = 0)"
+  unfolding Min_def pred_def by arith
+
+function
+  bin_rec :: "'a \<Rightarrow> 'a \<Rightarrow> (int \<Rightarrow> bit \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> int \<Rightarrow> 'a"  
+where 
+  "bin_rec f1 f2 f3 bin = (if bin = Int.Pls then f1 
+    else if bin = Int.Min then f2
+    else case bin_rl bin of (w, b) => f3 w b (bin_rec f1 f2 f3 w))"
+  by pat_completeness auto
+
+termination 
+  apply (relation "measure (nat o abs o snd o snd o snd)")
+  apply (auto simp add: bin_rl_def bin_last_def bin_rest_def)
+  unfolding Pls_def Min_def Bit0_def Bit1_def number_of_is_id
+  apply auto
+  done
+
+declare bin_rec.simps [simp del]
+
+lemma bin_rec_PM:
+  "f = bin_rec f1 f2 f3 ==> f Int.Pls = f1 & f Int.Min = f2"
+  by (auto simp add: bin_rec.simps)
+
+lemma bin_rec_Pls: "bin_rec f1 f2 f3 Int.Pls = f1"
+  by (simp add: bin_rec.simps)
+
+lemma bin_rec_Min: "bin_rec f1 f2 f3 Int.Min = f2"
+  by (simp add: bin_rec.simps)
+
+lemma bin_rec_Bit0:
+  "f3 Int.Pls (0::bit) f1 = f1 \<Longrightarrow>
+    bin_rec f1 f2 f3 (Int.Bit0 w) = f3 w (0::bit) (bin_rec f1 f2 f3 w)"
+  by (simp add: bin_rec_Pls bin_rec.simps [of _ _ _ "Int.Bit0 w"])
+
+lemma bin_rec_Bit1:
+  "f3 Int.Min (1::bit) f2 = f2 \<Longrightarrow>
+    bin_rec f1 f2 f3 (Int.Bit1 w) = f3 w (1::bit) (bin_rec f1 f2 f3 w)"
+  by (simp add: bin_rec_Min bin_rec.simps [of _ _ _ "Int.Bit1 w"])
+  
+lemma bin_rec_Bit:
+  "f = bin_rec f1 f2 f3  ==> f3 Int.Pls (0::bit) f1 = f1 ==> 
+    f3 Int.Min (1::bit) f2 = f2 ==> f (w BIT b) = f3 w b (f w)"
+  by (cases b, simp add: bin_rec_Bit0, simp add: bin_rec_Bit1)
+
+lemmas bin_rec_simps = refl [THEN bin_rec_Bit] bin_rec_Pls bin_rec_Min
+  bin_rec_Bit0 bin_rec_Bit1
+
+
+subsection {* Truncating binary integers *}
+
+definition
+  bin_sign_def [code del] : "bin_sign = bin_rec Int.Pls Int.Min (%w b s. s)"
+
+lemma bin_sign_simps [simp]:
+  "bin_sign Int.Pls = Int.Pls"
+  "bin_sign Int.Min = Int.Min"
+  "bin_sign (Int.Bit0 w) = bin_sign w"
+  "bin_sign (Int.Bit1 w) = bin_sign w"
+  "bin_sign (w BIT b) = bin_sign w"
+  unfolding bin_sign_def by (auto simp: bin_rec_simps)
+
+declare bin_sign_simps(1-4) [code]
+
+lemma bin_sign_rest [simp]: 
+  "bin_sign (bin_rest w) = (bin_sign w)"
+  by (cases w rule: bin_exhaust) auto
+
+consts
+  bintrunc :: "nat => int => int"
+primrec 
+  Z : "bintrunc 0 bin = Int.Pls"
+  Suc : "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT (bin_last bin)"
+
+consts
+  sbintrunc :: "nat => int => int" 
+primrec 
+  Z : "sbintrunc 0 bin = 
+    (case bin_last bin of (1::bit) => Int.Min | (0::bit) => Int.Pls)"
+  Suc : "sbintrunc (Suc n) bin = sbintrunc n (bin_rest bin) BIT (bin_last bin)"
+
+lemma sign_bintr:
+  "!!w. bin_sign (bintrunc n w) = Int.Pls"
+  by (induct n) auto
+
+lemma bintrunc_mod2p:
+  "!!w. bintrunc n w = (w mod 2 ^ n :: int)"
+  apply (induct n, clarsimp)
+  apply (simp add: bin_last_mod bin_rest_div Bit_def zmod_zmult2_eq
+              cong: number_of_False_cong)
+  done
+
+lemma sbintrunc_mod2p:
+  "!!w. sbintrunc n w = ((w + 2 ^ n) mod 2 ^ (Suc n) - 2 ^ n :: int)"
+  apply (induct n)
+   apply clarsimp
+   apply (subst mod_add_left_eq)
+   apply (simp add: bin_last_mod)
+   apply (simp add: number_of_eq)
+  apply clarsimp
+  apply (simp add: bin_last_mod bin_rest_div Bit_def 
+              cong: number_of_False_cong)
+  apply (clarsimp simp: mod_mult_mult1 [symmetric] 
+         zmod_zdiv_equality [THEN diff_eq_eq [THEN iffD2 [THEN sym]]])
+  apply (rule trans [symmetric, OF _ emep1])
+     apply auto
+  apply (auto simp: even_def)
+  done
+
+subsection "Simplifications for (s)bintrunc"
+
+lemma bit_bool:
+  "(b = (b' = (1::bit))) = (b' = (if b then (1::bit) else (0::bit)))"
+  by (cases b') auto
+
+lemmas bit_bool1 [simp] = refl [THEN bit_bool [THEN iffD1], symmetric]
+
+lemma bin_sign_lem:
+  "!!bin. (bin_sign (sbintrunc n bin) = Int.Min) = bin_nth bin n"
+  apply (induct n)
+   apply (case_tac bin rule: bin_exhaust, case_tac b, auto)+
+  done
+
+lemma nth_bintr:
+  "!!w m. bin_nth (bintrunc m w) n = (n < m & bin_nth w n)"
+  apply (induct n)
+   apply (case_tac m, auto)[1]
+  apply (case_tac m, auto)[1]
+  done
+
+lemma nth_sbintr:
+  "!!w m. bin_nth (sbintrunc m w) n = 
+          (if n < m then bin_nth w n else bin_nth w m)"
+  apply (induct n)
+   apply (case_tac m, simp_all split: bit.splits)[1]
+  apply (case_tac m, simp_all split: bit.splits)[1]
+  done
+
+lemma bin_nth_Bit:
+  "bin_nth (w BIT b) n = (n = 0 & b = (1::bit) | (EX m. n = Suc m & bin_nth w m))"
+  by (cases n) auto
+
+lemma bin_nth_Bit0:
+  "bin_nth (Int.Bit0 w) n = (EX m. n = Suc m & bin_nth w m)"
+  using bin_nth_Bit [where b="(0::bit)"] by simp
+
+lemma bin_nth_Bit1:
+  "bin_nth (Int.Bit1 w) n = (n = 0 | (EX m. n = Suc m & bin_nth w m))"
+  using bin_nth_Bit [where b="(1::bit)"] 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)"
+  by (rule bin_eqI) (auto simp: nth_sbintr)
+
+lemma bintrunc_bintrunc_ge:
+  "n <= m ==> (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_max.inf_absorb1 min_max.inf_absorb2)
+  done
+
+lemmas bintrunc_Pls = 
+  bintrunc.Suc [where bin="Int.Pls", simplified bin_last_simps bin_rest_simps, standard]
+
+lemmas bintrunc_Min [simp] = 
+  bintrunc.Suc [where bin="Int.Min", simplified bin_last_simps bin_rest_simps, standard]
+
+lemmas bintrunc_BIT  [simp] = 
+  bintrunc.Suc [where bin="w BIT b", simplified bin_last_simps bin_rest_simps, standard]
+
+lemma bintrunc_Bit0 [simp]:
+  "bintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (bintrunc n w)"
+  using bintrunc_BIT [where b="(0::bit)"] by simp
+
+lemma bintrunc_Bit1 [simp]:
+  "bintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (bintrunc n w)"
+  using bintrunc_BIT [where b="(1::bit)"] by simp
+
+lemmas bintrunc_Sucs = bintrunc_Pls bintrunc_Min bintrunc_BIT
+  bintrunc_Bit0 bintrunc_Bit1
+
+lemmas sbintrunc_Suc_Pls = 
+  sbintrunc.Suc [where bin="Int.Pls", simplified bin_last_simps bin_rest_simps, standard]
+
+lemmas sbintrunc_Suc_Min = 
+  sbintrunc.Suc [where bin="Int.Min", simplified bin_last_simps bin_rest_simps, standard]
+
+lemmas sbintrunc_Suc_BIT [simp] = 
+  sbintrunc.Suc [where bin="w BIT b", simplified bin_last_simps bin_rest_simps, standard]
+
+lemma sbintrunc_Suc_Bit0 [simp]:
+  "sbintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (sbintrunc n w)"
+  using sbintrunc_Suc_BIT [where b="(0::bit)"] by simp
+
+lemma sbintrunc_Suc_Bit1 [simp]:
+  "sbintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (sbintrunc n w)"
+  using sbintrunc_Suc_BIT [where b="(1::bit)"] by simp
+
+lemmas sbintrunc_Sucs = sbintrunc_Suc_Pls sbintrunc_Suc_Min sbintrunc_Suc_BIT
+  sbintrunc_Suc_Bit0 sbintrunc_Suc_Bit1
+
+lemmas sbintrunc_Pls = 
+  sbintrunc.Z [where bin="Int.Pls", 
+               simplified bin_last_simps bin_rest_simps bit.simps, standard]
+
+lemmas sbintrunc_Min = 
+  sbintrunc.Z [where bin="Int.Min", 
+               simplified bin_last_simps bin_rest_simps bit.simps, standard]
+
+lemmas sbintrunc_0_BIT_B0 [simp] = 
+  sbintrunc.Z [where bin="w BIT (0::bit)", 
+               simplified bin_last_simps bin_rest_simps bit.simps, standard]
+
+lemmas sbintrunc_0_BIT_B1 [simp] = 
+  sbintrunc.Z [where bin="w BIT (1::bit)", 
+               simplified bin_last_simps bin_rest_simps bit.simps, standard]
+
+lemma sbintrunc_0_Bit0 [simp]: "sbintrunc 0 (Int.Bit0 w) = Int.Pls"
+  using sbintrunc_0_BIT_B0 by simp
+
+lemma sbintrunc_0_Bit1 [simp]: "sbintrunc 0 (Int.Bit1 w) = Int.Min"
+  using sbintrunc_0_BIT_B1 by simp
+
+lemmas sbintrunc_0_simps =
+  sbintrunc_Pls sbintrunc_Min sbintrunc_0_BIT_B0 sbintrunc_0_BIT_B1
+  sbintrunc_0_Bit0 sbintrunc_0_Bit1
+
+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"
+  by auto
+
+lemma sbintrunc_minus:
+  "0 < n ==> sbintrunc (Suc (n - 1)) w = sbintrunc n w"
+  by auto
+
+lemmas bintrunc_minus_simps = 
+  bintrunc_Sucs [THEN [2] bintrunc_minus [symmetric, THEN trans], standard]
+lemmas sbintrunc_minus_simps = 
+  sbintrunc_Sucs [THEN [2] sbintrunc_minus [symmetric, THEN trans], standard]
+
+lemma bintrunc_n_Pls [simp]:
+  "bintrunc n Int.Pls = Int.Pls"
+  by (induct n) auto
+
+lemma sbintrunc_n_PM [simp]:
+  "sbintrunc n Int.Pls = Int.Pls"
+  "sbintrunc n Int.Min = Int.Min"
+  by (induct n) auto
+
+lemmas thobini1 = arg_cong [where f = "%w. w BIT b", standard]
+
+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], standard]
+lemmas bintrunc_Pls_minus_I = bmsts(1)
+lemmas bintrunc_Min_minus_I = bmsts(2)
+lemmas bintrunc_BIT_minus_I = bmsts(3)
+
+lemma bintrunc_0_Min: "bintrunc 0 Int.Min = Int.Pls"
+  by auto
+lemma bintrunc_0_BIT: "bintrunc 0 (w BIT b) = Int.Pls"
+  by auto
+
+lemma bintrunc_Suc_lem:
+  "bintrunc (Suc n) x = y ==> m = Suc n ==> bintrunc m x = y"
+  by auto
+
+lemmas bintrunc_Suc_Ialts = 
+  bintrunc_Min_I [THEN bintrunc_Suc_lem, standard]
+  bintrunc_BIT_I [THEN bintrunc_Suc_lem, standard]
+
+lemmas sbintrunc_BIT_I = trans [OF sbintrunc_Suc_BIT thobini1]
+
+lemmas sbintrunc_Suc_Is = 
+  sbintrunc_Sucs(1-3) [THEN thobini1 [THEN [2] trans], standard]
+
+lemmas sbintrunc_Suc_minus_Is = 
+  sbintrunc_minus_simps(1-3) [THEN thobini1 [THEN [2] trans], standard]
+
+lemma sbintrunc_Suc_lem:
+  "sbintrunc (Suc n) x = y ==> m = Suc n ==> sbintrunc m x = y"
+  by auto
+
+lemmas sbintrunc_Suc_Ialts = 
+  sbintrunc_Suc_Is [THEN sbintrunc_Suc_lem, standard]
+
+lemma sbintrunc_bintrunc_lt:
+  "m > n ==> 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"
+  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"
+  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"
+  by (cases n) (auto simp del: bintrunc.Suc)
+
+lemma bin_sbin_eq_iff: 
+  "bintrunc (Suc n) x = bintrunc (Suc n) y <-> 
+   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 <-> 
+            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]
+
+lemmas bintrunc_bintrunc_l' = le_add1 [THEN bintrunc_bintrunc_l]
+lemmas sbintrunc_sbintrunc_l' = le_add1 [THEN sbintrunc_sbintrunc_l]
+
+(* 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 = 
+  trans [OF iszero_def [THEN Not_eq_iff [THEN iffD2]] refl, standard]
+
+lemmas bintrunc_pred_simps [simp] = 
+  bintrunc_minus_simps [of "number_of bin", simplified nobm1, standard]
+
+lemmas sbintrunc_pred_simps [simp] = 
+  sbintrunc_minus_simps [of "number_of bin", simplified nobm1, standard]
+
+lemma no_bintr_alt:
+  "number_of (bintrunc n w) = w mod 2 ^ n"
+  by (simp add: number_of_eq bintrunc_mod2p)
+
+lemma no_bintr_alt1: "bintrunc n = (%w. w mod 2 ^ n :: int)"
+  by (rule ext) (rule bintrunc_mod2p)
+
+lemma range_bintrunc: "range (bintrunc n) = {i. 0 <= i & 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_bintr: 
+  "number_of (bintrunc n w) = (number_of w mod 2 ^ n :: int)"
+  by (simp add : bintrunc_mod2p number_of_eq)
+
+lemma no_sbintr_alt2: 
+  "sbintrunc n = (%w. (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)"
+  by (rule ext) (simp add : sbintrunc_mod2p)
+
+lemma no_sbintr: 
+  "number_of (sbintrunc n w) = 
+   ((number_of w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)"
+  by (simp add : no_sbintr_alt2 number_of_eq)
+
+lemma range_sbintrunc: 
+  "range (sbintrunc n) = {i. - (2 ^ n) <= i & 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)"
+  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)"
+  by (rule sb_inc_lem) simp
+
+lemma sbintrunc_inc:
+  "x < - (2^n) ==> x + 2^(Suc n) <= sbintrunc n x"
+  unfolding no_sbintr_alt2 by (drule sb_inc_lem') simp
+
+lemma sb_dec_lem:
+  "(0::int) <= - (2^k) + a ==> (a + 2^k) mod (2 * 2 ^ k) <= - (2 ^ k) + a"
+  by (rule int_mod_le' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k",
+    simplified zless2p, OF _ TrueI, simplified])
+
+lemma sb_dec_lem':
+  "(2::int) ^ k <= a ==> (a + 2 ^ k) mod (2 * 2 ^ k) <= - (2 ^ k) + a"
+  by (rule iffD1 [OF diff_le_eq', THEN sb_dec_lem, simplified])
+
+lemma sbintrunc_dec:
+  "x >= (2 ^ n) ==> x - 2 ^ (Suc n) >= sbintrunc n x"
+  unfolding no_sbintr_alt2 by (drule sb_dec_lem') simp
+
+lemmas zmod_uminus' = zmod_uminus [where b="c", standard]
+lemmas zpower_zmod' = zpower_zmod [where m="c" and y="k", standard]
+
+lemmas brdmod1s' [symmetric] = 
+  mod_add_left_eq mod_add_right_eq 
+  zmod_zsub_left_eq zmod_zsub_right_eq 
+  zmod_zmult1_eq zmod_zmult1_eq_rev 
+
+lemmas brdmods' [symmetric] = 
+  zpower_zmod' [symmetric]
+  trans [OF mod_add_left_eq mod_add_right_eq] 
+  trans [OF zmod_zsub_left_eq zmod_zsub_right_eq] 
+  trans [OF zmod_zmult1_eq zmod_zmult1_eq_rev] 
+  zmod_uminus' [symmetric]
+  mod_add_left_eq [where b = "1::int"]
+  zmod_zsub_left_eq [where b = "1"]
+
+lemmas bintr_arith1s =
+  brdmod1s' [where c="2^n::int", folded pred_def succ_def bintrunc_mod2p, standard]
+lemmas bintr_ariths =
+  brdmods' [where c="2^n::int", folded pred_def succ_def bintrunc_mod2p, standard]
+
+lemmas m2pths = pos_mod_sign pos_mod_bound [OF zless2p, standard] 
+
+lemma bintr_ge0: "(0 :: int) <= number_of (bintrunc n w)"
+  by (simp add : no_bintr m2pths)
+
+lemma bintr_lt2p: "number_of (bintrunc n w) < (2 ^ n :: int)"
+  by (simp add : no_bintr m2pths)
+
+lemma bintr_Min: 
+  "number_of (bintrunc n Int.Min) = (2 ^ n :: int) - 1"
+  by (simp add : no_bintr m1mod2k)
+
+lemma sbintr_ge: "(- (2 ^ n) :: int) <= number_of (sbintrunc n w)"
+  by (simp add : no_sbintr m2pths)
+
+lemma sbintr_lt: "number_of (sbintrunc n w) < (2 ^ n :: int)"
+  by (simp add : no_sbintr m2pths)
+
+lemma bintrunc_Suc:
+  "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT bin_last bin"
+  by (case_tac bin rule: bin_exhaust) auto
+
+lemma sign_Pls_ge_0: 
+  "(bin_sign bin = Int.Pls) = (number_of bin >= (0 :: int))"
+  by (induct bin rule: numeral_induct) auto
+
+lemma sign_Min_lt_0: 
+  "(bin_sign bin = Int.Min) = (number_of bin < (0 :: int))"
+  by (induct bin rule: numeral_induct) auto
+
+lemmas sign_Min_neg = trans [OF sign_Min_lt_0 neg_def [symmetric]] 
+
+lemma bin_rest_trunc:
+  "!!bin. (bin_rest (bintrunc n bin)) = bintrunc (n - 1) (bin_rest bin)"
+  by (induct n) auto
+
+lemma bin_rest_power_trunc [rule_format] :
+  "(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)"
+  by auto
+
+lemma bin_rest_strunc:
+  "!!bin. bin_rest (sbintrunc (Suc n) bin) = sbintrunc n (bin_rest bin)"
+  by (induct n) auto
+
+lemma bintrunc_rest [simp]: 
+  "!!bin. bintrunc n (bin_rest (bintrunc n bin)) = bin_rest (bintrunc n bin)"
+  apply (induct n, simp)
+  apply (case_tac bin rule: bin_exhaust)
+  apply (auto simp: bintrunc_bintrunc_l)
+  done
+
+lemma sbintrunc_rest [simp]:
+  "!!bin. sbintrunc n (bin_rest (sbintrunc n bin)) = bin_rest (sbintrunc n bin)"
+  apply (induct n, simp)
+  apply (case_tac bin rule: bin_exhaust)
+  apply (auto simp: bintrunc_bintrunc_l split: bit.splits)
+  done
+
+lemma bintrunc_rest':
+  "bintrunc n o bin_rest o bintrunc n = bin_rest o bintrunc n"
+  by (rule ext) auto
+
+lemma sbintrunc_rest' :
+  "sbintrunc n o bin_rest o sbintrunc n = bin_rest o 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"
+  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
+
+lemma rco_alt: "(f o g) ^^ n o f = f o (g o f) ^^ n"
+  apply (rule ext)
+  apply (induct n)
+   apply (simp_all add: o_def)
+  done
+
+lemmas rco_bintr = bintrunc_rest' 
+  [THEN rco_lem [THEN fun_cong], unfolded o_def]
+lemmas rco_sbintr = sbintrunc_rest' 
+  [THEN rco_lem [THEN fun_cong], unfolded o_def]
+
+subsection {* Splitting and concatenation *}
+
+primrec bin_split :: "nat \<Rightarrow> int \<Rightarrow> int \<times> int" where
+  Z: "bin_split 0 w = (w, Int.Pls)"
+  | Suc: "bin_split (Suc n) w = (let (w1, w2) = bin_split n (bin_rest w)
+        in (w1, w2 BIT bin_last 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"
+
+subsection {* Miscellaneous lemmas *}
+
+lemma funpow_minus_simp:
+  "0 < n \<Longrightarrow> f ^^ n = f \<circ> f ^^ (n - 1)"
+  by (cases n) simp_all
+
+lemmas funpow_pred_simp [simp] =
+  funpow_minus_simp [of "number_of bin", simplified nobm1, standard]
+
+lemmas replicate_minus_simp = 
+  trans [OF gen_minus [where f = "%n. replicate n x"] replicate.replicate_Suc,
+         standard]
+
+lemmas replicate_pred_simp [simp] =
+  replicate_minus_simp [of "number_of bin", simplified nobm1, standard]
+
+lemmas power_Suc_no [simp] = power_Suc [of "number_of a", standard]
+
+lemmas power_minus_simp = 
+  trans [OF gen_minus [where f = "power f"] power_Suc, standard]
+
+lemmas power_pred_simp = 
+  power_minus_simp [of "number_of bin", simplified nobm1, standard]
+lemmas power_pred_simp_no [simp] = power_pred_simp [where f= "number_of f", standard]
+
+lemma list_exhaust_size_gt0:
+  assumes y: "\<And>a list. y = a # list \<Longrightarrow> P"
+  shows "0 < length y \<Longrightarrow> P"
+  apply (cases y, simp)
+  apply (rule y)
+  apply fastsimp
+  done
+
+lemma list_exhaust_size_eq0:
+  assumes y: "y = [] \<Longrightarrow> P"
+  shows "length y = 0 \<Longrightarrow> P"
+  apply (cases y)
+   apply (rule y, simp)
+  apply simp
+  done
+
+lemma size_Cons_lem_eq:
+  "y = xa # list ==> size y = Suc k ==> size list = k"
+  by auto
+
+lemma size_Cons_lem_eq_bin:
+  "y = xa # list ==> size y = number_of (Int.succ k) ==> 
+    size list = number_of k"
+  by (auto simp: pred_def succ_def split add : split_if_asm)
+
+lemmas ls_splits = 
+  prod.split split_split prod.split_asm split_split_asm split_if_asm
+
+lemma not_B1_is_B0: "y \<noteq> (1::bit) \<Longrightarrow> y = (0::bit)"
+  by (cases y) auto
+
+lemma B1_ass_B0: 
+  assumes y: "y = (0::bit) \<Longrightarrow> y = (1::bit)"
+  shows "y = (1::bit)"
+  apply (rule classical)
+  apply (drule not_B1_is_B0)
+  apply (erule y)
+  done
+
+-- "simplifications for specific word lengths"
+lemmas n2s_ths [THEN eq_reflection] = add_2_eq_Suc add_2_eq_Suc'
+
+lemmas s2n_ths = n2s_ths [symmetric]
+
+end