--- a/src/HOL/Word/Bits_Int.thy Sat Apr 20 18:02:22 2019 +0000
+++ b/src/HOL/Word/Bits_Int.thy Mon Apr 22 06:28:17 2019 +0000
@@ -6,12 +6,1183 @@
and converting them to and from lists of bools.
*)
-section \<open>Bitwise Operations on Binary Integers\<close>
+section \<open>Bitwise Operations on integers\<close>
theory Bits_Int
- imports Bits Bit_Representation Bool_List_Representation
+ imports Bits Misc_Auxiliary
begin
+subsection \<open>Implicit bit representation of \<^typ>\<open>int\<close>\<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 (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
+
+lemma uminus_Bit_eq:
+ "- k BIT b = (- k - of_bool b) BIT b"
+ by (cases b) (simp_all add: Bit_def)
+
+lemma power_BIT: "2 ^ Suc n - 1 = (2 ^ n - 1) BIT True"
+ by (simp add: Bit_B1)
+
+definition bin_last :: "int \<Rightarrow> bool"
+ 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"
+ unfolding bin_rest_def bin_last_def Bit_def
+ 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
+
+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"
+ 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
+ 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"
+ by (simp_all add: Bit_def)
+
+lemma Bit_eq_0_iff: "w BIT b = 0 \<longleftrightarrow> w = 0 \<and> \<not> b"
+ by (auto simp: Bit_def) arith
+
+lemma Bit_eq_m1_iff: "w BIT b = -1 \<longleftrightarrow> w = -1 \<and> b"
+ by (auto simp: Bit_def) arith
+
+lemma BitM_inc: "Num.BitM (Num.inc w) = Num.Bit1 w"
+ 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"
+ 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_last Numeral1"
+ "\<not> bin_last (numeral (Num.Bit0 w))"
+ "bin_last (numeral (Num.Bit1 w))"
+ "\<not> bin_last (- numeral (Num.Bit0 w))"
+ "bin_last (- numeral (Num.Bit1 w))"
+ by (simp_all add: bin_last_def zmod_zminus1_eq_if) (auto simp add: divmod_def)
+
+lemma bin_rest_numeral_simps [simp]:
+ "bin_rest 0 = 0"
+ "bin_rest 1 = 0"
+ "bin_rest (- 1) = - 1"
+ "bin_rest Numeral1 = 0"
+ "bin_rest (numeral (Num.Bit0 w)) = numeral w"
+ "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"
+ 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)
+
+lemma succ_BIT_simps [simp]:
+ "x BIT False + 1 = x BIT True"
+ "x BIT True + 1 = (x + 1) BIT False"
+ by (simp_all add: Bit_B0_2t Bit_B1_2t)
+
+lemma add_BIT_simps [simp]:
+ "x BIT False + y BIT False = (x + y) BIT False"
+ "x BIT False + y BIT True = (x + y) BIT True"
+ "x BIT True + y BIT False = (x + y) BIT True"
+ "x BIT True + y BIT True = (x + y + 1) BIT False"
+ by (simp_all add: Bit_B0_2t Bit_B1_2t)
+
+lemma mult_BIT_simps [simp]:
+ "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 \<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: "(\<And>x b. bin = x BIT b \<Longrightarrow> Q) \<Longrightarrow> Q"
+by (metis bin_ex_rl)
+
+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: "\<And>bin bit. P bin \<Longrightarrow> P (bin BIT bit)"
+ shows "P bin"
+ 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_rl_eqI: "\<lbrakk>bin_rest x = bin_rest y; bin_last x = bin_last y\<rbrakk> \<Longrightarrow> x = y"
+ by (metis (mono_tags) BIT_eq_iff bin_ex_rl bin_last_BIT bin_rest_BIT)
+
+lemma twice_conv_BIT: "2 * x = x BIT False"
+ by (rule bin_rl_eqI) (simp_all, simp_all add: bin_rest_def bin_last_def)
+
+lemma BIT_lt0 [simp]: "x BIT b < 0 \<longleftrightarrow> x < 0"
+by(cases b)(auto simp add: Bit_def)
+
+lemma BIT_ge0 [simp]: "x BIT b \<ge> 0 \<longleftrightarrow> x \<ge> 0"
+by(cases b)(auto simp add: Bit_def)
+
+lemma [simp]:
+ shows bin_rest_lt0: "bin_rest i < 0 \<longleftrightarrow> i < 0"
+ and bin_rest_ge_0: "bin_rest i \<ge> 0 \<longleftrightarrow> i \<ge> 0"
+by(auto simp add: bin_rest_def)
+
+lemma bin_rest_gt_0 [simp]: "bin_rest x > 0 \<longleftrightarrow> x > 1"
+by(simp add: bin_rest_def add1_zle_eq pos_imp_zdiv_pos_iff) (metis add1_zle_eq one_add_one)
+
+
+subsection \<open>Explicit bit representation of \<^typ>\<open>int\<close>\<close>
+
+primrec bl_to_bin_aux :: "bool list \<Rightarrow> int \<Rightarrow> int"
+ where
+ Nil: "bl_to_bin_aux [] w = w"
+ | Cons: "bl_to_bin_aux (b # bs) w = bl_to_bin_aux bs (w BIT b)"
+
+definition bl_to_bin :: "bool list \<Rightarrow> int"
+ where "bl_to_bin bs = bl_to_bin_aux bs 0"
+
+primrec bin_to_bl_aux :: "nat \<Rightarrow> int \<Rightarrow> bool list \<Rightarrow> bool list"
+ where
+ Z: "bin_to_bl_aux 0 w bl = bl"
+ | Suc: "bin_to_bl_aux (Suc n) w bl = bin_to_bl_aux n (bin_rest w) ((bin_last w) # bl)"
+
+definition bin_to_bl :: "nat \<Rightarrow> int \<Rightarrow> bool list"
+ where "bin_to_bl n w = bin_to_bl_aux n w []"
+
+lemma bin_to_bl_aux_zero_minus_simp [simp]:
+ "0 < n \<Longrightarrow> bin_to_bl_aux n 0 bl = bin_to_bl_aux (n - 1) 0 (False # bl)"
+ by (cases n) auto
+
+lemma bin_to_bl_aux_minus1_minus_simp [simp]:
+ "0 < n \<Longrightarrow> bin_to_bl_aux n (- 1) bl = bin_to_bl_aux (n - 1) (- 1) (True # bl)"
+ by (cases n) auto
+
+lemma bin_to_bl_aux_one_minus_simp [simp]:
+ "0 < n \<Longrightarrow> bin_to_bl_aux n 1 bl = bin_to_bl_aux (n - 1) 0 (True # bl)"
+ by (cases n) auto
+
+lemma bin_to_bl_aux_Bit_minus_simp [simp]:
+ "0 < n \<Longrightarrow> bin_to_bl_aux n (w BIT b) bl = bin_to_bl_aux (n - 1) w (b # bl)"
+ by (cases n) auto
+
+lemma bin_to_bl_aux_Bit0_minus_simp [simp]:
+ "0 < n \<Longrightarrow>
+ bin_to_bl_aux n (numeral (Num.Bit0 w)) bl = bin_to_bl_aux (n - 1) (numeral w) (False # bl)"
+ by (cases n) auto
+
+lemma bin_to_bl_aux_Bit1_minus_simp [simp]:
+ "0 < n \<Longrightarrow>
+ bin_to_bl_aux n (numeral (Num.Bit1 w)) bl = bin_to_bl_aux (n - 1) (numeral w) (True # bl)"
+ by (cases n) auto
+
+lemma bl_to_bin_aux_append: "bl_to_bin_aux (bs @ cs) w = bl_to_bin_aux cs (bl_to_bin_aux bs w)"
+ by (induct bs arbitrary: w) auto
+
+lemma bin_to_bl_aux_append: "bin_to_bl_aux n w bs @ cs = bin_to_bl_aux n w (bs @ cs)"
+ by (induct n arbitrary: w bs) auto
+
+lemma bl_to_bin_append: "bl_to_bin (bs @ cs) = bl_to_bin_aux cs (bl_to_bin bs)"
+ unfolding bl_to_bin_def by (rule bl_to_bin_aux_append)
+
+lemma bin_to_bl_aux_alt: "bin_to_bl_aux n w bs = bin_to_bl n w @ bs"
+ by (simp add: bin_to_bl_def bin_to_bl_aux_append)
+
+lemma bin_to_bl_0 [simp]: "bin_to_bl 0 bs = []"
+ by (auto simp: bin_to_bl_def)
+
+lemma size_bin_to_bl_aux: "length (bin_to_bl_aux n w bs) = n + length bs"
+ by (induct n arbitrary: w bs) auto
+
+lemma size_bin_to_bl [simp]: "length (bin_to_bl n w) = n"
+ by (simp add: bin_to_bl_def size_bin_to_bl_aux)
+
+lemma bl_bin_bl': "bin_to_bl (n + length bs) (bl_to_bin_aux bs w) = bin_to_bl_aux n w bs"
+ apply (induct bs arbitrary: w n)
+ apply auto
+ apply (simp_all only: add_Suc [symmetric])
+ apply (auto simp add: bin_to_bl_def)
+ done
+
+lemma bl_bin_bl [simp]: "bin_to_bl (length bs) (bl_to_bin bs) = bs"
+ unfolding bl_to_bin_def
+ apply (rule box_equals)
+ apply (rule bl_bin_bl')
+ prefer 2
+ apply (rule bin_to_bl_aux.Z)
+ apply simp
+ done
+
+lemma bl_to_bin_inj: "bl_to_bin bs = bl_to_bin cs \<Longrightarrow> length bs = length cs \<Longrightarrow> bs = cs"
+ apply (rule_tac box_equals)
+ defer
+ apply (rule bl_bin_bl)
+ apply (rule bl_bin_bl)
+ apply simp
+ done
+
+lemma bl_to_bin_False [simp]: "bl_to_bin (False # bl) = bl_to_bin bl"
+ by (auto simp: bl_to_bin_def)
+
+lemma bl_to_bin_Nil [simp]: "bl_to_bin [] = 0"
+ by (auto simp: bl_to_bin_def)
+
+lemma bin_to_bl_zero_aux: "bin_to_bl_aux n 0 bl = replicate n False @ bl"
+ by (induct n arbitrary: bl) (auto simp: replicate_app_Cons_same)
+
+lemma bin_to_bl_zero: "bin_to_bl n 0 = replicate n False"
+ by (simp add: bin_to_bl_def bin_to_bl_zero_aux)
+
+lemma bin_to_bl_minus1_aux: "bin_to_bl_aux n (- 1) bl = replicate n True @ bl"
+ by (induct n arbitrary: bl) (auto simp: replicate_app_Cons_same)
+
+lemma bin_to_bl_minus1: "bin_to_bl n (- 1) = replicate n True"
+ by (simp add: bin_to_bl_def bin_to_bl_minus1_aux)
+
+lemma bl_to_bin_BIT:
+ "bl_to_bin bs BIT b = bl_to_bin (bs @ [b])"
+ by (simp add: bl_to_bin_append)
+
+
+subsection \<open>Bit projection\<close>
+
+primrec bin_nth :: "int \<Rightarrow> nat \<Rightarrow> bool"
+ 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_nth_eq_mod:
+ "bin_nth w n \<longleftrightarrow> odd (w div 2 ^ n)"
+ by (induction n arbitrary: w) (simp_all add: bin_last_def bin_rest_def odd_iff_mod_2_eq_one zdiv_zmult2_eq)
+
+lemma bin_nth_eq_iff: "bin_nth x = bin_nth y \<longleftrightarrow> x = y"
+proof -
+ 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 (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 (case_tac y rule: bin_exhaust)
+ apply clarify
+ apply (erule allE)
+ apply (erule impE)
+ prefer 2
+ apply (erule conjI)
+ 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)
+qed
+
+lemma bin_eqI:
+ "x = y" if "\<And>n. bin_nth x n \<longleftrightarrow> bin_nth y n"
+ using that bin_nth_eq_iff [of x y] by (simp add: fun_eq_iff)
+
+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
+
+lemma bin_nth_1 [simp]: "bin_nth 1 n \<longleftrightarrow> n = 0"
+ by (cases n) simp_all
+
+lemma bin_nth_minus1 [simp]: "bin_nth (- 1) n"
+ by (induct n) auto
+
+lemma bin_nth_0_BIT: "bin_nth (w BIT b) 0 \<longleftrightarrow> b"
+ 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 \<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)"
+ 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 =
+ bin_nth.Z bin_nth.Suc bin_nth_zero bin_nth_minus1
+ bin_nth_numeral_simps
+
+lemma nth_2p_bin: "bin_nth (2 ^ n) m = (m = n)" \<comment> \<open>for use when simplifying with \<open>bin_nth_Bit\<close>\<close>
+ apply (induct n arbitrary: m)
+ apply clarsimp
+ apply safe
+ apply (case_tac m)
+ apply (auto simp: Bit_B0_2t [symmetric])
+ done
+
+lemma nth_rest_power_bin: "bin_nth ((bin_rest ^^ k) w) n = bin_nth w (n + k)"
+ apply (induct k arbitrary: n)
+ apply clarsimp
+ apply clarsimp
+ apply (simp only: bin_nth.Suc [symmetric] add_Suc)
+ done
+
+lemma bin_nth_numeral_unfold:
+ "bin_nth (numeral (num.Bit0 x)) n \<longleftrightarrow> n > 0 \<and> bin_nth (numeral x) (n - 1)"
+ "bin_nth (numeral (num.Bit1 x)) n \<longleftrightarrow> (n > 0 \<longrightarrow> bin_nth (numeral x) (n - 1))"
+by(case_tac [!] n) simp_all
+
+
+subsection \<open>Truncating\<close>
+
+definition bin_sign :: "int \<Rightarrow> int"
+ 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"
+ 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 \<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 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"
+proof (induction n arbitrary: w)
+ case 0
+ then show ?case
+ by (auto simp add: bin_last_odd odd_iff_mod_2_eq_one)
+next
+ case (Suc n)
+ moreover have "((bin_rest w + 2 ^ n) mod (2 * 2 ^ n) - 2 ^ n) BIT bin_last w =
+ (w + 2 * 2 ^ n) mod (4 * 2 ^ n) - 2 * 2 ^ n"
+ proof (cases w rule: parity_cases)
+ case even
+ then show ?thesis
+ by (simp add: bin_last_odd bin_rest_def Bit_B0_2t mult_mod_right)
+ next
+ case odd
+ then have "2 * (w div 2) = w - 1"
+ using minus_mod_eq_mult_div [of w 2] by simp
+ moreover have "(2 * 2 ^ n + w - 1) mod (2 * 2 * 2 ^ n) + 1 = (2 * 2 ^ n + w) mod (2 * 2 * 2 ^ n)"
+ using odd emep1 [of "2 * 2 ^ n + w - 1" "2 * 2 * 2 ^ n"] by simp
+ ultimately show ?thesis
+ using odd by (simp add: bin_last_odd bin_rest_def Bit_B1_2t mult_mod_right) (simp add: algebra_simps)
+ qed
+ ultimately show ?case
+ by simp
+qed
+
+lemma sign_bintr: "bin_sign (bintrunc n w) = 0"
+ by (simp add: bintrunc_mod2p bin_sign_def)
+
+lemma bintrunc_n_0 [simp]: "bintrunc n 0 = 0"
+ by (simp add: bintrunc_mod2p)
+
+lemma sbintrunc_n_0 [simp]: "sbintrunc n 0 = 0"
+ by (simp add: sbintrunc_mod2p)
+
+lemma sbintrunc_n_minus1 [simp]: "sbintrunc n (- 1) = -1"
+ by (induct n) auto
+
+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"
+ 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"
+ "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"
+ 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 \<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)"
+ apply (induct n arbitrary: w m)
+ 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 \<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)"
+ using bin_nth_Bit [where w="numeral w" and b="False"] by simp
+
+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 \<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 \<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"
+ 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] =
+ bintrunc.Suc [where bin="-1", simplified bin_last_numeral_simps bin_rest_numeral_simps]
+
+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 =
+ sbintrunc.Suc [where bin="0", simplified bin_last_numeral_simps bin_rest_numeral_simps]
+
+lemmas sbintrunc_Suc_Min =
+ sbintrunc.Suc [where bin="-1", simplified bin_last_numeral_simps bin_rest_numeral_simps]
+
+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_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 \<Longrightarrow> bintrunc (Suc (n - 1)) w = bintrunc n w"
+ by auto
+
+lemma sbintrunc_minus: "0 < n \<Longrightarrow> sbintrunc (Suc (n - 1)) w = sbintrunc n w"
+ by auto
+
+lemmas bintrunc_minus_simps =
+ bintrunc_Sucs [THEN [2] bintrunc_minus [symmetric, THEN trans]]
+lemmas sbintrunc_minus_simps =
+ sbintrunc_Sucs [THEN [2] sbintrunc_minus [symmetric, THEN trans]]
+
+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 \<Longrightarrow> m = Suc n \<Longrightarrow> bintrunc m x = y"
+ by auto
+
+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 =
+ sbintrunc_Sucs(1-3) [THEN thobini1 [THEN [2] trans]]
+
+lemmas sbintrunc_Suc_minus_Is =
+ sbintrunc_minus_simps(1-3) [THEN thobini1 [THEN [2] trans]]
+
+lemma sbintrunc_Suc_lem: "sbintrunc (Suc n) x = y \<Longrightarrow> m = Suc n \<Longrightarrow> sbintrunc m x = y"
+ by auto
+
+lemmas sbintrunc_Suc_Ialts =
+ sbintrunc_Suc_Is [THEN sbintrunc_Suc_lem]
+
+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 \<le> Suc n \<Longrightarrow> bintrunc m (sbintrunc n w) = bintrunc m w"
+ apply (rule bin_eqI)
+ using le_Suc_eq less_Suc_eq_le apply (auto simp: nth_sbintr nth_bintr)
+ 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 \<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"
+ 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]
+
+lemma bintrunc_numeral:
+ "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"
+ 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.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 + 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 \<le> i \<and> i < 2 ^ n}"
+ apply (unfold no_bintr_alt1)
+ apply (auto simp add: image_iff)
+ apply (rule exI)
+ apply (rule sym)
+ using int_mod_lem [symmetric, of "2 ^ n"]
+ apply auto
+ done
+
+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) \<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 + 2^k < 0 \<Longrightarrow> a + 2^k + 2^(Suc k) \<le> (a + 2^k) mod 2^(Suc k)"
+ for a :: int
+ using int_mod_ge' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k"]
+ by simp
+
+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) \<Longrightarrow> x + 2^(Suc n) \<le> sbintrunc n x"
+ unfolding no_sbintr_alt2 by (drule sb_inc_lem') simp
+
+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 ^ 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 \<ge> (2 ^ n) \<Longrightarrow> x - 2 ^ (Suc n) >= sbintrunc n x"
+ unfolding no_sbintr_alt2 by (drule sb_dec_lem') simp
+
+lemma bintr_ge0: "0 \<le> bintrunc n w"
+ by (simp add: bintrunc_mod2p)
+
+lemma bintr_lt2p: "bintrunc n w < 2 ^ n"
+ by (simp add: bintrunc_mod2p)
+
+lemma bintr_Min: "bintrunc n (- 1) = 2 ^ n - 1"
+ by (simp add: bintrunc_mod2p m1mod2k)
+
+lemma sbintr_ge: "- (2 ^ n) \<le> sbintrunc 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 \<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)"
+ 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_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)
+ 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)
+ apply simp
+ apply (case_tac bin rule: bin_exhaust)
+ apply (auto simp: bintrunc_bintrunc_l split: bool.splits)
+ done
+
+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 \<circ> bin_rest \<circ> sbintrunc n = bin_rest \<circ> sbintrunc n"
+ by (rule ext) auto
+
+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'
+ [THEN rco_lem [THEN fun_cong], unfolded o_def]
+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))"
+
+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"
+ | Suc: "bin_cat w (Suc n) v = bin_cat w n (bin_rest v) BIT bin_last v"
+
+lemma bin_sign_cat: "bin_sign (bin_cat x n y) = bin_sign x"
+ by (induct n arbitrary: y) auto
+
+lemma bin_cat_Suc_Bit: "bin_cat w (Suc n) (v BIT b) = bin_cat w n v BIT b"
+ by auto
+
+lemma bin_cat_assoc: "bin_cat (bin_cat x m y) n z = bin_cat x (m + n) (bin_cat y n z)"
+ by (induct n arbitrary: z) auto
+
+lemma bin_cat_assoc_sym: "bin_cat x m (bin_cat y n z) = bin_cat (bin_cat x (m - n) y) (min m n) z"
+ apply (induct n arbitrary: z m)
+ apply clarsimp
+ apply (case_tac m, auto)
+ done
+
+definition bin_rcat :: "nat \<Rightarrow> int list \<Rightarrow> int"
+ where "bin_rcat n = foldl (\<lambda>u v. bin_cat u n v) 0"
+
+fun bin_rsplit_aux :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> int list"
+ where "bin_rsplit_aux n m c bs =
+ (if m = 0 \<or> n = 0 then bs
+ else
+ let (a, b) = bin_split n c
+ in bin_rsplit_aux n (m - n) a (b # bs))"
+
+definition bin_rsplit :: "nat \<Rightarrow> nat \<times> int \<Rightarrow> int list"
+ where "bin_rsplit n w = bin_rsplit_aux n (fst w) (snd w) []"
+
+fun bin_rsplitl_aux :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> int list"
+ where "bin_rsplitl_aux n m c bs =
+ (if m = 0 \<or> n = 0 then bs
+ else
+ let (a, b) = bin_split (min m n) c
+ in bin_rsplitl_aux n (m - n) a (b # bs))"
+
+definition bin_rsplitl :: "nat \<Rightarrow> nat \<times> int \<Rightarrow> int list"
+ where "bin_rsplitl n w = bin_rsplitl_aux n (fst w) (snd w) []"
+
+declare bin_rsplit_aux.simps [simp del]
+declare bin_rsplitl_aux.simps [simp del]
+
+lemma bin_nth_cat:
+ "bin_nth (bin_cat x k y) n =
+ (if n < k then bin_nth y n else bin_nth x (n - k))"
+ apply (induct k arbitrary: n y)
+ apply clarsimp
+ apply (case_tac n, auto)
+ done
+
+lemma bin_nth_split:
+ "bin_split n c = (a, b) \<Longrightarrow>
+ (\<forall>k. bin_nth a k = bin_nth c (n + k)) \<and>
+ (\<forall>k. bin_nth b k = (k < n \<and> bin_nth c k))"
+ apply (induct n arbitrary: b c)
+ apply clarsimp
+ apply (clarsimp simp: Let_def split: prod.split_asm)
+ apply (case_tac k)
+ apply auto
+ done
+
+lemma bin_cat_zero [simp]: "bin_cat 0 n w = bintrunc n w"
+ by (induct n arbitrary: w) auto
+
+lemma bintr_cat1: "bintrunc (k + n) (bin_cat a n b) = bin_cat (bintrunc k a) n b"
+ by (induct n arbitrary: b) auto
+
+lemma bintr_cat: "bintrunc m (bin_cat a n b) =
+ bin_cat (bintrunc (m - n) a) n (bintrunc (min m n) b)"
+ by (rule bin_eqI) (auto simp: bin_nth_cat nth_bintr)
+
+lemma bintr_cat_same [simp]: "bintrunc n (bin_cat a n b) = bintrunc n b"
+ by (auto simp add : bintr_cat)
+
+lemma cat_bintr [simp]: "bin_cat a n (bintrunc n b) = bin_cat a n b"
+ by (induct n arbitrary: b) auto
+
+lemma split_bintrunc: "bin_split n c = (a, b) \<Longrightarrow> b = bintrunc n c"
+ by (induct n arbitrary: b c) (auto simp: Let_def split: prod.split_asm)
+
+lemma bin_cat_split: "bin_split n w = (u, v) \<Longrightarrow> w = bin_cat u n v"
+ by (induct n arbitrary: v w) (auto simp: Let_def split: prod.split_asm)
+
+lemma bin_split_cat: "bin_split n (bin_cat v n w) = (v, bintrunc n w)"
+ by (induct n arbitrary: w) auto
+
+lemma bin_split_zero [simp]: "bin_split n 0 = (0, 0)"
+ by (induct n) auto
+
+lemma bin_split_minus1 [simp]:
+ "bin_split n (- 1) = (- 1, bintrunc n (- 1))"
+ by (induct n) auto
+
+lemma bin_split_trunc:
+ "bin_split (min m n) c = (a, b) \<Longrightarrow>
+ bin_split n (bintrunc m c) = (bintrunc (m - n) a, b)"
+ apply (induct n arbitrary: m b c, clarsimp)
+ apply (simp add: bin_rest_trunc Let_def split: prod.split_asm)
+ apply (case_tac m)
+ apply (auto simp: Let_def split: prod.split_asm)
+ done
+
+lemma bin_split_trunc1:
+ "bin_split n c = (a, b) \<Longrightarrow>
+ bin_split n (bintrunc m c) = (bintrunc (m - n) a, bintrunc m b)"
+ apply (induct n arbitrary: m b c, clarsimp)
+ apply (simp add: bin_rest_trunc Let_def split: prod.split_asm)
+ apply (case_tac m)
+ apply (auto simp: Let_def split: prod.split_asm)
+ done
+
+lemma bin_cat_num: "bin_cat a n b = a * 2 ^ n + bintrunc n b"
+ apply (induct n arbitrary: b)
+ apply clarsimp
+ apply (simp add: Bit_def)
+ done
+
+lemma bin_split_num: "bin_split n b = (b div 2 ^ n, b mod 2 ^ n)"
+ apply (induct n arbitrary: b)
+ apply simp
+ apply (simp add: bin_rest_def zdiv_zmult2_eq)
+ apply (case_tac b rule: bin_exhaust)
+ apply simp
+ apply (simp add: Bit_def mod_mult_mult1 pos_zmod_mult_2 add.commute)
+ done
+
+lemmas bin_rsplit_aux_simps = bin_rsplit_aux.simps bin_rsplitl_aux.simps
+lemmas rsplit_aux_simps = bin_rsplit_aux_simps
+
+lemmas th_if_simp1 = if_split [where P = "(=) l", THEN iffD1, THEN conjunct1, THEN mp] for l
+lemmas th_if_simp2 = if_split [where P = "(=) l", THEN iffD1, THEN conjunct2, THEN mp] for l
+
+lemmas rsplit_aux_simp1s = rsplit_aux_simps [THEN th_if_simp1]
+
+lemmas rsplit_aux_simp2ls = rsplit_aux_simps [THEN th_if_simp2]
+\<comment> \<open>these safe to \<open>[simp add]\<close> as require calculating \<open>m - n\<close>\<close>
+lemmas bin_rsplit_aux_simp2s [simp] = rsplit_aux_simp2ls [unfolded Let_def]
+lemmas rbscl = bin_rsplit_aux_simp2s (2)
+
+lemmas rsplit_aux_0_simps [simp] =
+ rsplit_aux_simp1s [OF disjI1] rsplit_aux_simp1s [OF disjI2]
+
+lemma bin_rsplit_aux_append: "bin_rsplit_aux n m c (bs @ cs) = bin_rsplit_aux n m c bs @ cs"
+ apply (induct n m c bs rule: bin_rsplit_aux.induct)
+ apply (subst bin_rsplit_aux.simps)
+ apply (subst bin_rsplit_aux.simps)
+ apply (clarsimp split: prod.split)
+ done
+
+lemma bin_rsplitl_aux_append: "bin_rsplitl_aux n m c (bs @ cs) = bin_rsplitl_aux n m c bs @ cs"
+ apply (induct n m c bs rule: bin_rsplitl_aux.induct)
+ apply (subst bin_rsplitl_aux.simps)
+ apply (subst bin_rsplitl_aux.simps)
+ apply (clarsimp split: prod.split)
+ done
+
+lemmas rsplit_aux_apps [where bs = "[]"] =
+ bin_rsplit_aux_append bin_rsplitl_aux_append
+
+lemmas rsplit_def_auxs = bin_rsplit_def bin_rsplitl_def
+
+lemmas rsplit_aux_alts = rsplit_aux_apps
+ [unfolded append_Nil rsplit_def_auxs [symmetric]]
+
+lemma bin_split_minus: "0 < n \<Longrightarrow> bin_split (Suc (n - 1)) w = bin_split n w"
+ by auto
+
+lemmas bin_split_minus_simp =
+ bin_split.Suc [THEN [2] bin_split_minus [symmetric, THEN trans]]
+
+lemma bin_split_pred_simp [simp]:
+ "(0::nat) < numeral bin \<Longrightarrow>
+ bin_split (numeral bin) w =
+ (let (w1, w2) = bin_split (numeral bin - 1) (bin_rest w)
+ in (w1, w2 BIT bin_last w))"
+ by (simp only: bin_split_minus_simp)
+
+lemma bin_rsplit_aux_simp_alt:
+ "bin_rsplit_aux n m c bs =
+ (if m = 0 \<or> n = 0 then bs
+ else let (a, b) = bin_split n c in bin_rsplit n (m - n, a) @ b # bs)"
+ apply (simp add: bin_rsplit_aux.simps [of n m c bs])
+ apply (subst rsplit_aux_alts)
+ apply (simp add: bin_rsplit_def)
+ done
+
+lemmas bin_rsplit_simp_alt =
+ trans [OF bin_rsplit_def bin_rsplit_aux_simp_alt]
+
+lemmas bthrs = bin_rsplit_simp_alt [THEN [2] trans]
+
+lemma bin_rsplit_size_sign' [rule_format]:
+ "n > 0 \<Longrightarrow> rev sw = bin_rsplit n (nw, w) \<Longrightarrow> \<forall>v\<in>set sw. bintrunc n v = v"
+ apply (induct sw arbitrary: nw w)
+ apply clarsimp
+ apply clarsimp
+ apply (drule bthrs)
+ apply (simp (no_asm_use) add: Let_def split: prod.split_asm if_split_asm)
+ apply clarify
+ apply (drule split_bintrunc)
+ apply simp
+ done
+
+lemmas bin_rsplit_size_sign = bin_rsplit_size_sign' [OF asm_rl
+ rev_rev_ident [THEN trans] set_rev [THEN equalityD2 [THEN subsetD]]]
+
+lemma bin_nth_rsplit [rule_format] :
+ "n > 0 \<Longrightarrow> m < n \<Longrightarrow>
+ \<forall>w k nw.
+ rev sw = bin_rsplit n (nw, w) \<longrightarrow>
+ k < size sw \<longrightarrow> bin_nth (sw ! k) m = bin_nth w (k * n + m)"
+ apply (induct sw)
+ apply clarsimp
+ apply clarsimp
+ apply (drule bthrs)
+ apply (simp (no_asm_use) add: Let_def split: prod.split_asm if_split_asm)
+ apply clarify
+ apply (erule allE, erule impE, erule exI)
+ apply (case_tac k)
+ apply clarsimp
+ prefer 2
+ apply clarsimp
+ apply (erule allE)
+ apply (erule (1) impE)
+ apply (drule bin_nth_split, erule conjE, erule allE, erule trans, simp add: ac_simps)+
+ done
+
+lemma bin_rsplit_all: "0 < nw \<Longrightarrow> nw \<le> n \<Longrightarrow> bin_rsplit n (nw, w) = [bintrunc n w]"
+ by (auto simp: bin_rsplit_def rsplit_aux_simp2ls split: prod.split dest!: split_bintrunc)
+
+lemma bin_rsplit_l [rule_format]:
+ "\<forall>bin. bin_rsplitl n (m, bin) = bin_rsplit n (m, bintrunc m bin)"
+ apply (rule_tac a = "m" in wf_less_than [THEN wf_induct])
+ apply (simp (no_asm) add: bin_rsplitl_def bin_rsplit_def)
+ apply (rule allI)
+ apply (subst bin_rsplitl_aux.simps)
+ apply (subst bin_rsplit_aux.simps)
+ apply (clarsimp simp: Let_def split: prod.split)
+ apply (drule bin_split_trunc)
+ apply (drule sym [THEN trans], assumption)
+ apply (subst rsplit_aux_alts(1))
+ apply (subst rsplit_aux_alts(2))
+ apply clarsimp
+ unfolding bin_rsplit_def bin_rsplitl_def
+ apply simp
+ done
+
+lemma bin_rsplit_rcat [rule_format]:
+ "n > 0 \<longrightarrow> bin_rsplit n (n * size ws, bin_rcat n ws) = map (bintrunc n) ws"
+ apply (unfold bin_rsplit_def bin_rcat_def)
+ apply (rule_tac xs = ws in rev_induct)
+ apply clarsimp
+ apply clarsimp
+ apply (subst rsplit_aux_alts)
+ unfolding bin_split_cat
+ apply simp
+ done
+
+lemma bin_rsplit_aux_len_le [rule_format] :
+ "\<forall>ws m. n \<noteq> 0 \<longrightarrow> ws = bin_rsplit_aux n nw w bs \<longrightarrow>
+ length ws \<le> m \<longleftrightarrow> nw + length bs * n \<le> m * n"
+proof -
+ have *: R
+ if d: "i \<le> j \<or> m < j'"
+ and R1: "i * k \<le> j * k \<Longrightarrow> R"
+ and R2: "Suc m * k' \<le> j' * k' \<Longrightarrow> R"
+ for i j j' k k' m :: nat and R
+ using d
+ apply safe
+ apply (rule R1, erule mult_le_mono1)
+ apply (rule R2, erule Suc_le_eq [THEN iffD2 [THEN mult_le_mono1]])
+ done
+ have **: "0 < sc \<Longrightarrow> sc - n + (n + lb * n) \<le> m * n \<longleftrightarrow> sc + lb * n \<le> m * n"
+ for sc m n lb :: nat
+ apply safe
+ apply arith
+ apply (case_tac "sc \<ge> n")
+ apply arith
+ apply (insert linorder_le_less_linear [of m lb])
+ apply (erule_tac k=n and k'=n in *)
+ apply arith
+ apply simp
+ done
+ show ?thesis
+ apply (induct n nw w bs rule: bin_rsplit_aux.induct)
+ apply (subst bin_rsplit_aux.simps)
+ apply (simp add: ** Let_def split: prod.split)
+ done
+qed
+
+lemma bin_rsplit_len_le: "n \<noteq> 0 \<longrightarrow> ws = bin_rsplit n (nw, w) \<longrightarrow> length ws \<le> m \<longleftrightarrow> nw \<le> m * n"
+ by (auto simp: bin_rsplit_def bin_rsplit_aux_len_le)
+
+lemma bin_rsplit_aux_len:
+ "n \<noteq> 0 \<Longrightarrow> length (bin_rsplit_aux n nw w cs) = (nw + n - 1) div n + length cs"
+ apply (induct n nw w cs rule: bin_rsplit_aux.induct)
+ apply (subst bin_rsplit_aux.simps)
+ apply (clarsimp simp: Let_def split: prod.split)
+ apply (erule thin_rl)
+ apply (case_tac m)
+ apply simp
+ apply (case_tac "m \<le> n")
+ apply (auto simp add: div_add_self2)
+ done
+
+lemma bin_rsplit_len: "n \<noteq> 0 \<Longrightarrow> length (bin_rsplit n (nw, w)) = (nw + n - 1) div n"
+ by (auto simp: bin_rsplit_def bin_rsplit_aux_len)
+
+lemma bin_rsplit_aux_len_indep:
+ "n \<noteq> 0 \<Longrightarrow> length bs = length cs \<Longrightarrow>
+ length (bin_rsplit_aux n nw v bs) =
+ length (bin_rsplit_aux n nw w cs)"
+proof (induct n nw w cs arbitrary: v bs rule: bin_rsplit_aux.induct)
+ case (1 n m w cs v bs)
+ show ?case
+ proof (cases "m = 0")
+ case True
+ with \<open>length bs = length cs\<close> show ?thesis by simp
+ next
+ case False
+ from "1.hyps" \<open>m \<noteq> 0\<close> \<open>n \<noteq> 0\<close>
+ have hyp: "\<And>v bs. length bs = Suc (length cs) \<Longrightarrow>
+ length (bin_rsplit_aux n (m - n) v bs) =
+ length (bin_rsplit_aux n (m - n) (fst (bin_split n w)) (snd (bin_split n w) # cs))"
+ by auto
+ from \<open>length bs = length cs\<close> \<open>n \<noteq> 0\<close> show ?thesis
+ by (auto simp add: bin_rsplit_aux_simp_alt Let_def bin_rsplit_len split: prod.split)
+ qed
+qed
+
+lemma bin_rsplit_len_indep:
+ "n \<noteq> 0 \<Longrightarrow> length (bin_rsplit n (nw, v)) = length (bin_rsplit n (nw, w))"
+ apply (unfold bin_rsplit_def)
+ apply (simp (no_asm))
+ apply (erule bin_rsplit_aux_len_indep)
+ apply (rule refl)
+ done
+
+
subsection \<open>Logical operations\<close>
text "bit-wise logical operations on the int type"
@@ -232,9 +1403,6 @@
text \<open>Cases for \<open>0\<close> and \<open>-1\<close> are already covered by other simp rules.\<close>
-lemma bin_rl_eqI: "\<lbrakk>bin_rest x = bin_rest y; bin_last x = bin_last y\<rbrakk> \<Longrightarrow> x = y"
- by (metis (mono_tags) BIT_eq_iff bin_ex_rl bin_last_BIT bin_rest_BIT)
-
lemma bin_rest_neg_numeral_BitM [simp]:
"bin_rest (- numeral (Num.BitM w)) = - numeral w"
by (simp only: BIT_bin_simps [symmetric] bin_rest_BIT)
@@ -423,7 +1591,7 @@
by (cases bit) (simp_all add: Bit_def)
then have "0 \<le> bin AND bin'" by (rule 3)
with 1 show ?thesis
- by simp (simp add: Bit_def)
+ by simp
qed
qed
@@ -443,7 +1611,7 @@
by (cases bit') (simp_all add: Bit_def)
ultimately have "0 \<le> bin OR bin'" by (rule 3)
with 1 show ?thesis
- by simp (simp add: Bit_def)
+ by simp
qed
qed simp_all
@@ -463,7 +1631,7 @@
by (cases bit') (simp_all add: Bit_def)
ultimately have "0 \<le> bin XOR bin'" by (rule 3)
with 1 show ?thesis
- by simp (simp add: Bit_def)
+ by simp
qed
next
case 2
@@ -534,7 +1702,8 @@
show ?thesis
proof (cases n)
case 0
- with 3 have "bin BIT bit = 0" by simp
+ with 3 have "bin BIT bit = 0"
+ by (simp add: Bit_def)
then have "bin = 0" and "\<not> bit"
by (auto simp add: Bit_def split: if_splits) arith
then show ?thesis using 0 1 \<open>y < 2 ^ n\<close>
@@ -573,7 +1742,8 @@
show ?thesis
proof (cases n)
case 0
- with 3 have "bin BIT bit = 0" by simp
+ with 3 have "bin BIT bit = 0"
+ by (simp add: Bit_def)
then have "bin = 0" and "\<not> bit"
by (auto simp add: Bit_def split: if_splits) arith
then show ?thesis using 0 1 \<open>y < 2 ^ n\<close>
@@ -615,58 +1785,92 @@
lemmas bin_trunc_and = bin_trunc_ao(1) [THEN bintr_bintr_i]
lemmas bin_trunc_or = bin_trunc_ao(2) [THEN bintr_bintr_i]
-lemma bl_xor_aux_bin:
- "map2 (\<lambda>x y. x \<noteq> y) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) =
- bin_to_bl_aux n (v XOR w) (map2 (\<lambda>x y. x \<noteq> y) bs cs)"
- apply (induct n arbitrary: v w bs cs)
- apply simp
- apply (case_tac v rule: bin_exhaust)
- apply (case_tac w rule: bin_exhaust)
- apply clarsimp
- apply (case_tac b)
- apply auto
- done
-
-lemma bl_or_aux_bin:
- "map2 (\<or>) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) =
- bin_to_bl_aux n (v OR w) (map2 (\<or>) bs cs)"
- apply (induct n arbitrary: v w bs cs)
- apply simp
- apply (case_tac v rule: bin_exhaust)
- apply (case_tac w rule: bin_exhaust)
- apply clarsimp
- done
-
-lemma bl_and_aux_bin:
- "map2 (\<and>) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) =
- bin_to_bl_aux n (v AND w) (map2 (\<and>) bs cs)"
- apply (induct n arbitrary: v w bs cs)
- apply simp
- apply (case_tac v rule: bin_exhaust)
- apply (case_tac w rule: bin_exhaust)
- apply clarsimp
- done
-
-lemma bl_not_aux_bin: "map Not (bin_to_bl_aux n w cs) = bin_to_bl_aux n (NOT w) (map Not cs)"
- by (induct n arbitrary: w cs) auto
-
-lemma bl_not_bin: "map Not (bin_to_bl n w) = bin_to_bl n (NOT w)"
- by (simp add: bin_to_bl_def bl_not_aux_bin)
-
-lemma bl_and_bin: "map2 (\<and>) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v AND w)"
- by (simp add: bin_to_bl_def bl_and_aux_bin)
-
-lemma bl_or_bin: "map2 (\<or>) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v OR w)"
- by (simp add: bin_to_bl_def bl_or_aux_bin)
-
-lemma bl_xor_bin: "map2 (\<lambda>x y. x \<noteq> y) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v XOR w)"
- by (simp only: bin_to_bl_def bl_xor_aux_bin map2_Nil)
+
+subsubsection \<open>More lemmas\<close>
+
+lemma not_int_cmp_0 [simp]:
+ fixes i :: int shows
+ "0 < NOT i \<longleftrightarrow> i < -1"
+ "0 \<le> NOT i \<longleftrightarrow> i < 0"
+ "NOT i < 0 \<longleftrightarrow> i \<ge> 0"
+ "NOT i \<le> 0 \<longleftrightarrow> i \<ge> -1"
+by(simp_all add: int_not_def) arith+
+
+lemma bbw_ao_dist2: "(x :: int) AND (y OR z) = x AND y OR x AND z"
+by(metis int_and_comm bbw_ao_dist)
+
+lemmas int_and_ac = bbw_lcs(1) int_and_comm int_and_assoc
+
+lemma int_nand_same [simp]: fixes x :: int shows "x AND NOT x = 0"
+by(induct x y\<equiv>"NOT x" rule: bitAND_int.induct)(subst bitAND_int.simps, clarsimp)
+
+lemma int_nand_same_middle: fixes x :: int shows "x AND y AND NOT x = 0"
+by (metis bbw_lcs(1) int_and_0 int_nand_same)
+
+lemma and_xor_dist: fixes x :: int shows
+ "x AND (y XOR z) = (x AND y) XOR (x AND z)"
+by(simp add: int_xor_def bbw_ao_dist2 bbw_not_dist int_and_ac int_nand_same_middle)
+
+lemma int_and_lt0 [simp]: fixes x y :: int shows
+ "x AND y < 0 \<longleftrightarrow> x < 0 \<and> y < 0"
+by(induct x y rule: bitAND_int.induct)(subst bitAND_int.simps, simp)
+
+lemma int_and_ge0 [simp]: fixes x y :: int shows
+ "x AND y \<ge> 0 \<longleftrightarrow> x \<ge> 0 \<or> y \<ge> 0"
+by (metis int_and_lt0 linorder_not_less)
+
+lemma int_and_1: fixes x :: int shows "x AND 1 = x mod 2"
+by(subst bitAND_int.simps)(simp add: Bit_def bin_last_def zmod_minus1)
+
+lemma int_1_and: fixes x :: int shows "1 AND x = x mod 2"
+by(subst int_and_comm)(simp add: int_and_1)
+
+lemma int_or_lt0 [simp]: fixes x y :: int shows
+ "x OR y < 0 \<longleftrightarrow> x < 0 \<or> y < 0"
+by(simp add: int_or_def)
+
+lemma int_xor_lt0 [simp]: fixes x y :: int shows
+ "x XOR y < 0 \<longleftrightarrow> ((x < 0) \<noteq> (y < 0))"
+by(auto simp add: int_xor_def)
+
+lemma int_xor_ge0 [simp]: fixes x y :: int shows
+ "x XOR y \<ge> 0 \<longleftrightarrow> ((x \<ge> 0) \<longleftrightarrow> (y \<ge> 0))"
+by (metis int_xor_lt0 linorder_not_le)
+
+lemma bin_last_conv_AND:
+ "bin_last i \<longleftrightarrow> i AND 1 \<noteq> 0"
+proof -
+ obtain x b where "i = x BIT b" by(cases i rule: bin_exhaust)
+ hence "i AND 1 = 0 BIT b"
+ by(simp add: BIT_special_simps(2)[symmetric] del: BIT_special_simps(2))
+ thus ?thesis using \<open>i = x BIT b\<close> by(cases b) simp_all
+qed
+
+lemma bitval_bin_last:
+ "of_bool (bin_last i) = i AND 1"
+proof -
+ obtain x b where "i = x BIT b" by(cases i rule: bin_exhaust)
+ hence "i AND 1 = 0 BIT b"
+ by(simp add: BIT_special_simps(2)[symmetric] del: BIT_special_simps(2))
+ thus ?thesis by(cases b)(simp_all add: bin_last_conv_AND)
+qed
+
+lemma bin_sign_and:
+ "bin_sign (i AND j) = - (bin_sign i * bin_sign j)"
+by(simp add: bin_sign_def)
+
+lemma minus_BIT_0: fixes x y :: int shows "x BIT b - y BIT False = (x - y) BIT b"
+by(simp add: Bit_def)
+
+lemma int_not_neg_numeral: "NOT (- numeral n) = (Num.sub n num.One :: int)"
+by(simp add: int_not_def)
+
+lemma int_neg_numeral_pOne_conv_not: "- numeral (n + num.One) = (NOT (numeral n) :: int)"
+by(simp add: int_not_def)
subsection \<open>Setting and clearing bits\<close>
-text \<open>nth bit, set/clear\<close>
-
primrec bin_sc :: "nat \<Rightarrow> bool \<Rightarrow> int \<Rightarrow> int"
where
Z: "bin_sc 0 b w = bin_rest w BIT b"
@@ -775,127 +1979,6 @@
end
-
-subsection \<open>More lemmas\<close>
-
-lemma twice_conv_BIT: "2 * x = x BIT False"
- by (rule bin_rl_eqI) (simp_all, simp_all add: bin_rest_def bin_last_def)
-
-lemma not_int_cmp_0 [simp]:
- fixes i :: int shows
- "0 < NOT i \<longleftrightarrow> i < -1"
- "0 \<le> NOT i \<longleftrightarrow> i < 0"
- "NOT i < 0 \<longleftrightarrow> i \<ge> 0"
- "NOT i \<le> 0 \<longleftrightarrow> i \<ge> -1"
-by(simp_all add: int_not_def) arith+
-
-lemma bbw_ao_dist2: "(x :: int) AND (y OR z) = x AND y OR x AND z"
-by(metis int_and_comm bbw_ao_dist)
-
-lemmas int_and_ac = bbw_lcs(1) int_and_comm int_and_assoc
-
-lemma int_nand_same [simp]: fixes x :: int shows "x AND NOT x = 0"
-by(induct x y\<equiv>"NOT x" rule: bitAND_int.induct)(subst bitAND_int.simps, clarsimp)
-
-lemma int_nand_same_middle: fixes x :: int shows "x AND y AND NOT x = 0"
-by (metis bbw_lcs(1) int_and_0 int_nand_same)
-
-lemma and_xor_dist: fixes x :: int shows
- "x AND (y XOR z) = (x AND y) XOR (x AND z)"
-by(simp add: int_xor_def bbw_ao_dist2 bbw_ao_dist bbw_not_dist int_and_ac int_nand_same_middle)
-
-lemma BIT_lt0 [simp]: "x BIT b < 0 \<longleftrightarrow> x < 0"
-by(cases b)(auto simp add: Bit_def)
-
-lemma BIT_ge0 [simp]: "x BIT b \<ge> 0 \<longleftrightarrow> x \<ge> 0"
-by(cases b)(auto simp add: Bit_def)
-
-lemma [simp]:
- shows bin_rest_lt0: "bin_rest i < 0 \<longleftrightarrow> i < 0"
- and bin_rest_ge_0: "bin_rest i \<ge> 0 \<longleftrightarrow> i \<ge> 0"
-by(auto simp add: bin_rest_def)
-
-lemma bin_rest_gt_0 [simp]: "bin_rest x > 0 \<longleftrightarrow> x > 1"
-by(simp add: bin_rest_def add1_zle_eq pos_imp_zdiv_pos_iff) (metis add1_zle_eq one_add_one)
-
-lemma int_and_lt0 [simp]: fixes x y :: int shows
- "x AND y < 0 \<longleftrightarrow> x < 0 \<and> y < 0"
-by(induct x y rule: bitAND_int.induct)(subst bitAND_int.simps, simp)
-
-lemma int_and_ge0 [simp]: fixes x y :: int shows
- "x AND y \<ge> 0 \<longleftrightarrow> x \<ge> 0 \<or> y \<ge> 0"
-by (metis int_and_lt0 linorder_not_less)
-
-lemma int_and_1: fixes x :: int shows "x AND 1 = x mod 2"
-by(subst bitAND_int.simps)(simp add: Bit_def bin_last_def zmod_minus1)
-
-lemma int_1_and: fixes x :: int shows "1 AND x = x mod 2"
-by(subst int_and_comm)(simp add: int_and_1)
-
-lemma int_or_lt0 [simp]: fixes x y :: int shows
- "x OR y < 0 \<longleftrightarrow> x < 0 \<or> y < 0"
-by(simp add: int_or_def)
-
-lemma int_xor_lt0 [simp]: fixes x y :: int shows
- "x XOR y < 0 \<longleftrightarrow> ((x < 0) \<noteq> (y < 0))"
-by(auto simp add: int_xor_def)
-
-lemma int_xor_ge0 [simp]: fixes x y :: int shows
- "x XOR y \<ge> 0 \<longleftrightarrow> ((x \<ge> 0) \<longleftrightarrow> (y \<ge> 0))"
-by (metis int_xor_lt0 linorder_not_le)
-
-lemma bin_last_conv_AND:
- "bin_last i \<longleftrightarrow> i AND 1 \<noteq> 0"
-proof -
- obtain x b where "i = x BIT b" by(cases i rule: bin_exhaust)
- hence "i AND 1 = 0 BIT b"
- by(simp add: BIT_special_simps(2)[symmetric] del: BIT_special_simps(2))
- thus ?thesis using \<open>i = x BIT b\<close> by(cases b) simp_all
-qed
-
-lemma bitval_bin_last:
- "of_bool (bin_last i) = i AND 1"
-proof -
- obtain x b where "i = x BIT b" by(cases i rule: bin_exhaust)
- hence "i AND 1 = 0 BIT b"
- by(simp add: BIT_special_simps(2)[symmetric] del: BIT_special_simps(2))
- thus ?thesis by(cases b)(simp_all add: bin_last_conv_AND)
-qed
-
-lemma bl_to_bin_BIT:
- "bl_to_bin bs BIT b = bl_to_bin (bs @ [b])"
-by(simp add: bl_to_bin_append)
-
-lemma bin_last_bl_to_bin: "bin_last (bl_to_bin bs) \<longleftrightarrow> bs \<noteq> [] \<and> last bs"
-by(cases "bs = []")(auto simp add: bl_to_bin_def last_bin_last'[where w=0])
-
-lemma bin_rest_bl_to_bin: "bin_rest (bl_to_bin bs) = bl_to_bin (butlast bs)"
-by(cases "bs = []")(simp_all add: bl_to_bin_def butlast_rest_bl2bin_aux)
-
-lemma bin_nth_numeral_unfold:
- "bin_nth (numeral (num.Bit0 x)) n \<longleftrightarrow> n > 0 \<and> bin_nth (numeral x) (n - 1)"
- "bin_nth (numeral (num.Bit1 x)) n \<longleftrightarrow> (n > 0 \<longrightarrow> bin_nth (numeral x) (n - 1))"
-by(case_tac [!] n) simp_all
-
-lemma bin_sign_and:
- "bin_sign (i AND j) = - (bin_sign i * bin_sign j)"
-by(simp add: bin_sign_def)
-
-lemma minus_BIT_0: fixes x y :: int shows "x BIT b - y BIT False = (x - y) BIT b"
-by(simp add: Bit_def)
-
-lemma int_not_neg_numeral: "NOT (- numeral n) = (Num.sub n num.One :: int)"
-by(simp add: int_not_def)
-
-lemma sub_inc_One: "Num.sub (Num.inc n) num.One = numeral n"
-by (metis add_diff_cancel diff_minus_eq_add diff_numeral_special(2) diff_numeral_special(6))
-
-lemma inc_BitM: "Num.inc (Num.BitM n) = num.Bit0 n"
-by(simp add: BitM_plus_one[symmetric] add_One)
-
-lemma int_neg_numeral_pOne_conv_not: "- numeral (n + num.One) = (NOT (numeral n) :: int)"
-by(simp add: int_not_def)
-
lemma int_lsb_BIT [simp]: fixes x :: int shows
"lsb (x BIT b) \<longleftrightarrow> b"
by(simp add: lsb_int_def)
@@ -1047,10 +2130,6 @@
lemma int_shiftr_lt_0 [simp]: fixes i :: int shows "i >> n < 0 \<longleftrightarrow> i < 0"
by (metis int_shiftr_ge_0 not_less)
-lemma uminus_Bit_eq:
- "- k BIT b = (- k - of_bool b) BIT b"
- by (cases b) (simp_all add: Bit_def)
-
lemma int_shiftr_numeral [simp]:
"(1 :: int) >> numeral w' = 0"
"(numeral num.One :: int) >> numeral w' = 0"
@@ -1103,10 +2182,10 @@
by(induct n arbitrary: i)(auto intro: bin_rl_eqI)
lemma int_set_bits_K_True [simp]: "(BITS _. True) = (-1 :: int)"
-by(auto simp add: set_bits_int_def bin_last_bl_to_bin)
+ by (auto simp add: set_bits_int_def bl_to_bin_def)
lemma int_set_bits_K_False [simp]: "(BITS _. False) = (0 :: int)"
-by(auto simp add: set_bits_int_def)
+ by (simp add: set_bits_int_def)
lemma msb_conv_bin_sign: "msb x \<longleftrightarrow> bin_sign x = -1"
by(simp add: bin_sign_def not_le msb_int_def)
@@ -1152,4 +2231,551 @@
"msb (- numeral n :: int) = True"
by(simp_all add: msb_int_def)
+
+subsection \<open>Semantic interpretation of \<^typ>\<open>bool list\<close> as \<^typ>\<open>int\<close>\<close>
+
+lemma bin_bl_bin': "bl_to_bin (bin_to_bl_aux n w bs) = bl_to_bin_aux bs (bintrunc n w)"
+ by (induct n arbitrary: w bs) (auto simp: bl_to_bin_def)
+
+lemma bin_bl_bin [simp]: "bl_to_bin (bin_to_bl n w) = bintrunc n w"
+ by (auto simp: bin_to_bl_def bin_bl_bin')
+
+lemma bl_to_bin_rep_F: "bl_to_bin (replicate n False @ bl) = bl_to_bin bl"
+ by (simp add: bin_to_bl_zero_aux [symmetric] bin_bl_bin') (simp add: bl_to_bin_def)
+
+lemma bin_to_bl_trunc [simp]: "n \<le> m \<Longrightarrow> bin_to_bl n (bintrunc m w) = bin_to_bl n w"
+ by (auto intro: bl_to_bin_inj)
+
+lemma bin_to_bl_aux_bintr:
+ "bin_to_bl_aux n (bintrunc m bin) bl =
+ replicate (n - m) False @ bin_to_bl_aux (min n m) bin bl"
+ apply (induct n arbitrary: m bin bl)
+ apply clarsimp
+ apply clarsimp
+ apply (case_tac "m")
+ apply (clarsimp simp: bin_to_bl_zero_aux)
+ apply (erule thin_rl)
+ apply (induct_tac n)
+ apply auto
+ done
+
+lemma bin_to_bl_bintr:
+ "bin_to_bl n (bintrunc m bin) = replicate (n - m) False @ bin_to_bl (min n m) bin"
+ unfolding bin_to_bl_def by (rule bin_to_bl_aux_bintr)
+
+lemma bl_to_bin_rep_False: "bl_to_bin (replicate n False) = 0"
+ by (induct n) auto
+
+lemma len_bin_to_bl_aux: "length (bin_to_bl_aux n w bs) = n + length bs"
+ by (fact size_bin_to_bl_aux)
+
+lemma len_bin_to_bl: "length (bin_to_bl n w) = n"
+ by (fact size_bin_to_bl) (* FIXME: duplicate *)
+
+lemma sign_bl_bin': "bin_sign (bl_to_bin_aux bs w) = bin_sign w"
+ by (induct bs arbitrary: w) auto
+
+lemma sign_bl_bin: "bin_sign (bl_to_bin bs) = 0"
+ by (simp add: bl_to_bin_def sign_bl_bin')
+
+lemma bl_sbin_sign_aux: "hd (bin_to_bl_aux (Suc n) w bs) = (bin_sign (sbintrunc n w) = -1)"
+ apply (induct n arbitrary: w bs)
+ apply clarsimp
+ apply (cases w rule: bin_exhaust)
+ apply simp
+ done
+
+lemma bl_sbin_sign: "hd (bin_to_bl (Suc n) w) = (bin_sign (sbintrunc n w) = -1)"
+ unfolding bin_to_bl_def by (rule bl_sbin_sign_aux)
+
+lemma bin_nth_of_bl_aux:
+ "bin_nth (bl_to_bin_aux bl w) n =
+ (n < size bl \<and> rev bl ! n \<or> n \<ge> length bl \<and> bin_nth w (n - size bl))"
+ apply (induct bl arbitrary: w)
+ apply clarsimp
+ apply clarsimp
+ apply (cut_tac x=n and y="size bl" in linorder_less_linear)
+ apply (erule disjE, simp add: nth_append)+
+ apply auto
+ done
+
+lemma bin_nth_of_bl: "bin_nth (bl_to_bin bl) n = (n < length bl \<and> rev bl ! n)"
+ by (simp add: bl_to_bin_def bin_nth_of_bl_aux)
+
+lemma bin_nth_bl: "n < m \<Longrightarrow> bin_nth w n = nth (rev (bin_to_bl m w)) n"
+ apply (induct n arbitrary: m w)
+ apply clarsimp
+ apply (case_tac m, clarsimp)
+ apply (clarsimp simp: bin_to_bl_def)
+ apply (simp add: bin_to_bl_aux_alt)
+ apply clarsimp
+ apply (case_tac m, clarsimp)
+ apply (clarsimp simp: bin_to_bl_def)
+ apply (simp add: bin_to_bl_aux_alt)
+ done
+
+lemma nth_bin_to_bl_aux:
+ "n < m + length bl \<Longrightarrow> (bin_to_bl_aux m w bl) ! n =
+ (if n < m then bin_nth w (m - 1 - n) else bl ! (n - m))"
+ apply (induct m arbitrary: w n bl)
+ apply clarsimp
+ apply clarsimp
+ apply (case_tac w rule: bin_exhaust)
+ apply simp
+ done
+
+lemma nth_bin_to_bl: "n < m \<Longrightarrow> (bin_to_bl m w) ! n = bin_nth w (m - Suc n)"
+ by (simp add: bin_to_bl_def nth_bin_to_bl_aux)
+
+lemma bl_to_bin_lt2p_aux: "bl_to_bin_aux bs w < (w + 1) * (2 ^ length bs)"
+ apply (induct bs arbitrary: w)
+ apply clarsimp
+ apply clarsimp
+ apply (drule meta_spec, erule xtrans(8) [rotated], simp add: Bit_def)+
+ done
+
+lemma bl_to_bin_lt2p_drop: "bl_to_bin bs < 2 ^ length (dropWhile Not bs)"
+proof (induct bs)
+ case Nil
+ then show ?case by simp
+next
+ case (Cons b bs)
+ with bl_to_bin_lt2p_aux[where w=1] show ?case
+ by (simp add: bl_to_bin_def)
+qed
+
+lemma bl_to_bin_lt2p: "bl_to_bin bs < 2 ^ length bs"
+ by (metis bin_bl_bin bintr_lt2p bl_bin_bl)
+
+lemma bl_to_bin_ge2p_aux: "bl_to_bin_aux bs w \<ge> w * (2 ^ length bs)"
+ apply (induct bs arbitrary: w)
+ apply clarsimp
+ apply clarsimp
+ apply (drule meta_spec, erule order_trans [rotated],
+ simp add: Bit_B0_2t Bit_B1_2t algebra_simps)+
+ apply (simp add: Bit_def)
+ done
+
+lemma bl_to_bin_ge0: "bl_to_bin bs \<ge> 0"
+ apply (unfold bl_to_bin_def)
+ apply (rule xtrans(4))
+ apply (rule bl_to_bin_ge2p_aux)
+ apply simp
+ done
+
+lemma butlast_rest_bin: "butlast (bin_to_bl n w) = bin_to_bl (n - 1) (bin_rest w)"
+ apply (unfold bin_to_bl_def)
+ apply (cases w rule: bin_exhaust)
+ apply (cases n, clarsimp)
+ apply clarsimp
+ apply (auto simp add: bin_to_bl_aux_alt)
+ done
+
+lemma butlast_bin_rest: "butlast bl = bin_to_bl (length bl - Suc 0) (bin_rest (bl_to_bin bl))"
+ using butlast_rest_bin [where w="bl_to_bin bl" and n="length bl"] by simp
+
+lemma butlast_rest_bl2bin_aux:
+ "bl \<noteq> [] \<Longrightarrow> bl_to_bin_aux (butlast bl) w = bin_rest (bl_to_bin_aux bl w)"
+ by (induct bl arbitrary: w) auto
+
+lemma butlast_rest_bl2bin: "bl_to_bin (butlast bl) = bin_rest (bl_to_bin bl)"
+ by (cases bl) (auto simp: bl_to_bin_def butlast_rest_bl2bin_aux)
+
+lemma trunc_bl2bin_aux:
+ "bintrunc m (bl_to_bin_aux bl w) =
+ bl_to_bin_aux (drop (length bl - m) bl) (bintrunc (m - length bl) w)"
+proof (induct bl arbitrary: w)
+ case Nil
+ show ?case by simp
+next
+ case (Cons b bl)
+ show ?case
+ proof (cases "m - length bl")
+ case 0
+ then have "Suc (length bl) - m = Suc (length bl - m)" by simp
+ with Cons show ?thesis by simp
+ next
+ case (Suc n)
+ then have "m - Suc (length bl) = n" by simp
+ with Cons Suc show ?thesis by simp
+ qed
+qed
+
+lemma trunc_bl2bin: "bintrunc m (bl_to_bin bl) = bl_to_bin (drop (length bl - m) bl)"
+ by (simp add: bl_to_bin_def trunc_bl2bin_aux)
+
+lemma trunc_bl2bin_len [simp]: "bintrunc (length bl) (bl_to_bin bl) = bl_to_bin bl"
+ by (simp add: trunc_bl2bin)
+
+lemma bl2bin_drop: "bl_to_bin (drop k bl) = bintrunc (length bl - k) (bl_to_bin bl)"
+ apply (rule trans)
+ prefer 2
+ apply (rule trunc_bl2bin [symmetric])
+ apply (cases "k \<le> length bl")
+ apply auto
+ done
+
+lemma take_rest_power_bin: "m \<le> n \<Longrightarrow> take m (bin_to_bl n w) = bin_to_bl m ((bin_rest ^^ (n - m)) w)"
+ apply (rule nth_equalityI)
+ apply simp
+ apply (clarsimp simp add: nth_bin_to_bl nth_rest_power_bin)
+ done
+
+lemma last_bin_last': "size xs > 0 \<Longrightarrow> last xs \<longleftrightarrow> bin_last (bl_to_bin_aux xs w)"
+ by (induct xs arbitrary: w) auto
+
+lemma last_bin_last: "size xs > 0 \<Longrightarrow> last xs \<longleftrightarrow> bin_last (bl_to_bin xs)"
+ unfolding bl_to_bin_def by (erule last_bin_last')
+
+lemma bin_last_last: "bin_last w \<longleftrightarrow> last (bin_to_bl (Suc n) w)"
+ by (simp add: bin_to_bl_def) (auto simp: bin_to_bl_aux_alt)
+
+lemma drop_bin2bl_aux:
+ "drop m (bin_to_bl_aux n bin bs) =
+ bin_to_bl_aux (n - m) bin (drop (m - n) bs)"
+ apply (induct n arbitrary: m bin bs, clarsimp)
+ apply clarsimp
+ apply (case_tac bin rule: bin_exhaust)
+ apply (case_tac "m \<le> n", simp)
+ apply (case_tac "m - n", simp)
+ apply simp
+ apply (rule_tac f = "\<lambda>nat. drop nat bs" in arg_cong)
+ apply simp
+ done
+
+lemma drop_bin2bl: "drop m (bin_to_bl n bin) = bin_to_bl (n - m) bin"
+ by (simp add: bin_to_bl_def drop_bin2bl_aux)
+
+lemma take_bin2bl_lem1: "take m (bin_to_bl_aux m w bs) = bin_to_bl m w"
+ apply (induct m arbitrary: w bs)
+ apply clarsimp
+ apply clarsimp
+ apply (simp add: bin_to_bl_aux_alt)
+ apply (simp add: bin_to_bl_def)
+ apply (simp add: bin_to_bl_aux_alt)
+ done
+
+lemma take_bin2bl_lem: "take m (bin_to_bl_aux (m + n) w bs) = take m (bin_to_bl (m + n) w)"
+ by (induct n arbitrary: w bs) (simp_all (no_asm) add: bin_to_bl_def take_bin2bl_lem1, simp)
+
+lemma bin_split_take: "bin_split n c = (a, b) \<Longrightarrow> bin_to_bl m a = take m (bin_to_bl (m + n) c)"
+ apply (induct n arbitrary: b c)
+ apply clarsimp
+ apply (clarsimp simp: Let_def split: prod.split_asm)
+ apply (simp add: bin_to_bl_def)
+ apply (simp add: take_bin2bl_lem)
+ done
+
+lemma bin_split_take1:
+ "k = m + n \<Longrightarrow> bin_split n c = (a, b) \<Longrightarrow> bin_to_bl m a = take m (bin_to_bl k c)"
+ by (auto elim: bin_split_take)
+
+lemma takefill_bintrunc: "takefill False n bl = rev (bin_to_bl n (bl_to_bin (rev bl)))"
+ apply (rule nth_equalityI)
+ apply simp
+ apply (clarsimp simp: nth_takefill nth_rev nth_bin_to_bl bin_nth_of_bl)
+ done
+
+lemma bl_bin_bl_rtf: "bin_to_bl n (bl_to_bin bl) = rev (takefill False n (rev bl))"
+ by (simp add: takefill_bintrunc)
+
+lemma bl_bin_bl_rep_drop:
+ "bin_to_bl n (bl_to_bin bl) =
+ replicate (n - length bl) False @ drop (length bl - n) bl"
+ by (simp add: bl_bin_bl_rtf takefill_alt rev_take)
+
+lemma bl_to_bin_aux_cat:
+ "\<And>nv v. bl_to_bin_aux bs (bin_cat w nv v) =
+ bin_cat w (nv + length bs) (bl_to_bin_aux bs v)"
+ by (induct bs) (simp, simp add: bin_cat_Suc_Bit [symmetric] del: bin_cat.simps)
+
+lemma bin_to_bl_aux_cat:
+ "\<And>w bs. bin_to_bl_aux (nv + nw) (bin_cat v nw w) bs =
+ bin_to_bl_aux nv v (bin_to_bl_aux nw w bs)"
+ by (induct nw) auto
+
+lemma bl_to_bin_aux_alt: "bl_to_bin_aux bs w = bin_cat w (length bs) (bl_to_bin bs)"
+ using bl_to_bin_aux_cat [where nv = "0" and v = "0"]
+ by (simp add: bl_to_bin_def [symmetric])
+
+lemma bin_to_bl_cat:
+ "bin_to_bl (nv + nw) (bin_cat v nw w) =
+ bin_to_bl_aux nv v (bin_to_bl nw w)"
+ by (simp add: bin_to_bl_def bin_to_bl_aux_cat)
+
+lemmas bl_to_bin_aux_app_cat =
+ trans [OF bl_to_bin_aux_append bl_to_bin_aux_alt]
+
+lemmas bin_to_bl_aux_cat_app =
+ trans [OF bin_to_bl_aux_cat bin_to_bl_aux_alt]
+
+lemma bl_to_bin_app_cat:
+ "bl_to_bin (bsa @ bs) = bin_cat (bl_to_bin bsa) (length bs) (bl_to_bin bs)"
+ by (simp only: bl_to_bin_aux_app_cat bl_to_bin_def)
+
+lemma bin_to_bl_cat_app:
+ "bin_to_bl (n + nw) (bin_cat w nw wa) = bin_to_bl n w @ bin_to_bl nw wa"
+ by (simp only: bin_to_bl_def bin_to_bl_aux_cat_app)
+
+text \<open>\<open>bl_to_bin_app_cat_alt\<close> and \<open>bl_to_bin_app_cat\<close> are easily interderivable.\<close>
+lemma bl_to_bin_app_cat_alt: "bin_cat (bl_to_bin cs) n w = bl_to_bin (cs @ bin_to_bl n w)"
+ by (simp add: bl_to_bin_app_cat)
+
+lemma mask_lem: "(bl_to_bin (True # replicate n False)) = bl_to_bin (replicate n True) + 1"
+ apply (unfold bl_to_bin_def)
+ apply (induct n)
+ apply simp
+ apply (simp only: Suc_eq_plus1 replicate_add append_Cons [symmetric] bl_to_bin_aux_append)
+ apply (simp add: Bit_B0_2t Bit_B1_2t)
+ done
+
+primrec rbl_succ :: "bool list \<Rightarrow> bool list"
+ where
+ Nil: "rbl_succ Nil = Nil"
+ | Cons: "rbl_succ (x # xs) = (if x then False # rbl_succ xs else True # xs)"
+
+primrec rbl_pred :: "bool list \<Rightarrow> bool list"
+ where
+ Nil: "rbl_pred Nil = Nil"
+ | Cons: "rbl_pred (x # xs) = (if x then False # xs else True # rbl_pred xs)"
+
+primrec rbl_add :: "bool list \<Rightarrow> bool list \<Rightarrow> bool list"
+ where \<comment> \<open>result is length of first arg, second arg may be longer\<close>
+ Nil: "rbl_add Nil x = Nil"
+ | Cons: "rbl_add (y # ys) x =
+ (let ws = rbl_add ys (tl x)
+ in (y \<noteq> hd x) # (if hd x \<and> y then rbl_succ ws else ws))"
+
+primrec rbl_mult :: "bool list \<Rightarrow> bool list \<Rightarrow> bool list"
+ where \<comment> \<open>result is length of first arg, second arg may be longer\<close>
+ Nil: "rbl_mult Nil x = Nil"
+ | Cons: "rbl_mult (y # ys) x =
+ (let ws = False # rbl_mult ys x
+ in if y then rbl_add ws x else ws)"
+
+lemma size_rbl_pred: "length (rbl_pred bl) = length bl"
+ by (induct bl) auto
+
+lemma size_rbl_succ: "length (rbl_succ bl) = length bl"
+ by (induct bl) auto
+
+lemma size_rbl_add: "length (rbl_add bl cl) = length bl"
+ by (induct bl arbitrary: cl) (auto simp: Let_def size_rbl_succ)
+
+lemma size_rbl_mult: "length (rbl_mult bl cl) = length bl"
+ by (induct bl arbitrary: cl) (auto simp add: Let_def size_rbl_add)
+
+lemmas rbl_sizes [simp] =
+ size_rbl_pred size_rbl_succ size_rbl_add size_rbl_mult
+
+lemmas rbl_Nils =
+ rbl_pred.Nil rbl_succ.Nil rbl_add.Nil rbl_mult.Nil
+
+lemma rbl_add_app2: "length blb \<ge> length bla \<Longrightarrow> rbl_add bla (blb @ blc) = rbl_add bla blb"
+ apply (induct bla arbitrary: blb)
+ apply simp
+ apply clarsimp
+ apply (case_tac blb, clarsimp)
+ apply (clarsimp simp: Let_def)
+ done
+
+lemma rbl_add_take2:
+ "length blb \<ge> length bla \<Longrightarrow> rbl_add bla (take (length bla) blb) = rbl_add bla blb"
+ apply (induct bla arbitrary: blb)
+ apply simp
+ apply clarsimp
+ apply (case_tac blb, clarsimp)
+ apply (clarsimp simp: Let_def)
+ done
+
+lemma rbl_mult_app2: "length blb \<ge> length bla \<Longrightarrow> rbl_mult bla (blb @ blc) = rbl_mult bla blb"
+ apply (induct bla arbitrary: blb)
+ apply simp
+ apply clarsimp
+ apply (case_tac blb, clarsimp)
+ apply (clarsimp simp: Let_def rbl_add_app2)
+ done
+
+lemma rbl_mult_take2:
+ "length blb \<ge> length bla \<Longrightarrow> rbl_mult bla (take (length bla) blb) = rbl_mult bla blb"
+ apply (rule trans)
+ apply (rule rbl_mult_app2 [symmetric])
+ apply simp
+ apply (rule_tac f = "rbl_mult bla" in arg_cong)
+ apply (rule append_take_drop_id)
+ done
+
+lemma rbl_add_split:
+ "P (rbl_add (y # ys) (x # xs)) =
+ (\<forall>ws. length ws = length ys \<longrightarrow> ws = rbl_add ys xs \<longrightarrow>
+ (y \<longrightarrow> ((x \<longrightarrow> P (False # rbl_succ ws)) \<and> (\<not> x \<longrightarrow> P (True # ws)))) \<and>
+ (\<not> y \<longrightarrow> P (x # ws)))"
+ by (cases y) (auto simp: Let_def)
+
+lemma rbl_mult_split:
+ "P (rbl_mult (y # ys) xs) =
+ (\<forall>ws. length ws = Suc (length ys) \<longrightarrow> ws = False # rbl_mult ys xs \<longrightarrow>
+ (y \<longrightarrow> P (rbl_add ws xs)) \<and> (\<not> y \<longrightarrow> P ws))"
+ by (auto simp: Let_def)
+
+lemma rbl_pred: "rbl_pred (rev (bin_to_bl n bin)) = rev (bin_to_bl n (bin - 1))"
+ apply (unfold bin_to_bl_def)
+ apply (induct n arbitrary: bin)
+ apply simp
+ apply clarsimp
+ apply (case_tac bin rule: bin_exhaust)
+ apply (case_tac b)
+ apply (clarsimp simp: bin_to_bl_aux_alt)+
+ done
+
+lemma rbl_succ: "rbl_succ (rev (bin_to_bl n bin)) = rev (bin_to_bl n (bin + 1))"
+ apply (unfold bin_to_bl_def)
+ apply (induct n arbitrary: bin)
+ apply simp
+ apply clarsimp
+ apply (case_tac bin rule: bin_exhaust)
+ apply (case_tac b)
+ apply (clarsimp simp: bin_to_bl_aux_alt)+
+ done
+
+lemma rbl_add:
+ "\<And>bina binb. rbl_add (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb)) =
+ rev (bin_to_bl n (bina + binb))"
+ apply (unfold bin_to_bl_def)
+ apply (induct n)
+ apply simp
+ apply clarsimp
+ apply (case_tac bina rule: bin_exhaust)
+ apply (case_tac binb rule: bin_exhaust)
+ apply (case_tac b)
+ apply (case_tac [!] "ba")
+ apply (auto simp: rbl_succ bin_to_bl_aux_alt Let_def ac_simps)
+ done
+
+lemma rbl_add_long:
+ "m \<ge> n \<Longrightarrow> rbl_add (rev (bin_to_bl n bina)) (rev (bin_to_bl m binb)) =
+ rev (bin_to_bl n (bina + binb))"
+ apply (rule box_equals [OF _ rbl_add_take2 rbl_add])
+ apply (rule_tac f = "rbl_add (rev (bin_to_bl n bina))" in arg_cong)
+ apply (rule rev_swap [THEN iffD1])
+ apply (simp add: rev_take drop_bin2bl)
+ apply simp
+ done
+
+lemma rbl_mult_gt1:
+ "m \<ge> length bl \<Longrightarrow>
+ rbl_mult bl (rev (bin_to_bl m binb)) =
+ rbl_mult bl (rev (bin_to_bl (length bl) binb))"
+ apply (rule trans)
+ apply (rule rbl_mult_take2 [symmetric])
+ apply simp_all
+ apply (rule_tac f = "rbl_mult bl" in arg_cong)
+ apply (rule rev_swap [THEN iffD1])
+ apply (simp add: rev_take drop_bin2bl)
+ done
+
+lemma rbl_mult_gt:
+ "m > n \<Longrightarrow>
+ rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl m binb)) =
+ rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb))"
+ by (auto intro: trans [OF rbl_mult_gt1])
+
+lemmas rbl_mult_Suc = lessI [THEN rbl_mult_gt]
+
+lemma rbbl_Cons: "b # rev (bin_to_bl n x) = rev (bin_to_bl (Suc n) (x BIT b))"
+ by (simp add: bin_to_bl_def) (simp add: bin_to_bl_aux_alt)
+
+lemma rbl_mult:
+ "rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb)) =
+ rev (bin_to_bl n (bina * binb))"
+ apply (induct n arbitrary: bina binb)
+ apply simp
+ apply (unfold bin_to_bl_def)
+ apply clarsimp
+ apply (case_tac bina rule: bin_exhaust)
+ apply (case_tac binb rule: bin_exhaust)
+ apply (case_tac b)
+ apply (case_tac [!] "ba")
+ apply (auto simp: bin_to_bl_aux_alt Let_def)
+ apply (auto simp: rbbl_Cons rbl_mult_Suc rbl_add)
+ done
+
+lemma sclem: "size (concat (map (bin_to_bl n) xs)) = length xs * n"
+ by (induct xs) auto
+
+lemma bin_cat_foldl_lem:
+ "foldl (\<lambda>u. bin_cat u n) x xs =
+ bin_cat x (size xs * n) (foldl (\<lambda>u. bin_cat u n) y xs)"
+ apply (induct xs arbitrary: x)
+ apply simp
+ apply (simp (no_asm))
+ apply (frule asm_rl)
+ apply (drule meta_spec)
+ apply (erule trans)
+ apply (drule_tac x = "bin_cat y n a" in meta_spec)
+ apply (simp add: bin_cat_assoc_sym min.absorb2)
+ done
+
+lemma bin_rcat_bl: "bin_rcat n wl = bl_to_bin (concat (map (bin_to_bl n) wl))"
+ apply (unfold bin_rcat_def)
+ apply (rule sym)
+ apply (induct wl)
+ apply (auto simp add: bl_to_bin_append)
+ apply (simp add: bl_to_bin_aux_alt sclem)
+ apply (simp add: bin_cat_foldl_lem [symmetric])
+ done
+
+lemma bin_last_bl_to_bin: "bin_last (bl_to_bin bs) \<longleftrightarrow> bs \<noteq> [] \<and> last bs"
+by(cases "bs = []")(auto simp add: bl_to_bin_def last_bin_last'[where w=0])
+
+lemma bin_rest_bl_to_bin: "bin_rest (bl_to_bin bs) = bl_to_bin (butlast bs)"
+by(cases "bs = []")(simp_all add: bl_to_bin_def butlast_rest_bl2bin_aux)
+
+lemma bl_xor_aux_bin:
+ "map2 (\<lambda>x y. x \<noteq> y) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) =
+ bin_to_bl_aux n (v XOR w) (map2 (\<lambda>x y. x \<noteq> y) bs cs)"
+ apply (induct n arbitrary: v w bs cs)
+ apply simp
+ apply (case_tac v rule: bin_exhaust)
+ apply (case_tac w rule: bin_exhaust)
+ apply clarsimp
+ apply (case_tac b)
+ apply auto
+ done
+
+lemma bl_or_aux_bin:
+ "map2 (\<or>) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) =
+ bin_to_bl_aux n (v OR w) (map2 (\<or>) bs cs)"
+ apply (induct n arbitrary: v w bs cs)
+ apply simp
+ apply (case_tac v rule: bin_exhaust)
+ apply (case_tac w rule: bin_exhaust)
+ apply clarsimp
+ done
+
+lemma bl_and_aux_bin:
+ "map2 (\<and>) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) =
+ bin_to_bl_aux n (v AND w) (map2 (\<and>) bs cs)"
+ apply (induct n arbitrary: v w bs cs)
+ apply simp
+ apply (case_tac v rule: bin_exhaust)
+ apply (case_tac w rule: bin_exhaust)
+ apply clarsimp
+ done
+
+lemma bl_not_aux_bin: "map Not (bin_to_bl_aux n w cs) = bin_to_bl_aux n (NOT w) (map Not cs)"
+ by (induct n arbitrary: w cs) auto
+
+lemma bl_not_bin: "map Not (bin_to_bl n w) = bin_to_bl n (NOT w)"
+ by (simp add: bin_to_bl_def bl_not_aux_bin)
+
+lemma bl_and_bin: "map2 (\<and>) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v AND w)"
+ by (simp add: bin_to_bl_def bl_and_aux_bin)
+
+lemma bl_or_bin: "map2 (\<or>) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v OR w)"
+ by (simp add: bin_to_bl_def bl_or_aux_bin)
+
+lemma bl_xor_bin: "map2 (\<lambda>x y. x \<noteq> y) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v XOR w)"
+ by (simp only: bin_to_bl_def bl_xor_aux_bin map2_Nil)
+
end