(* Title: HOL/Word/Bits_Int.thy
Author: Jeremy Dawson and Gerwin Klein, NICTA
*)
section \<open>Bitwise Operations on integers\<close>
theory Bits_Int
imports
"HOL-Library.Bit_Operations"
Traditional_Syntax
begin
subsection \<open>Generic auxiliary\<close>
lemma int_mod_ge: "a < n \<Longrightarrow> 0 < n \<Longrightarrow> a \<le> a mod n"
for a n :: int
by (metis dual_order.trans le_cases mod_pos_pos_trivial pos_mod_conj)
subsection \<open>Implicit bit representation of \<^typ>\<open>int\<close>\<close>
abbreviation (input) bin_last :: "int \<Rightarrow> bool"
where "bin_last \<equiv> odd"
lemma bin_last_def:
"bin_last w \<longleftrightarrow> w mod 2 = 1"
by (fact odd_iff_mod_2_eq_one)
abbreviation (input) bin_rest :: "int \<Rightarrow> int"
where "bin_rest w \<equiv> w div 2"
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
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
lemma bin_rl_eqI: "\<lbrakk>bin_rest x = bin_rest y; bin_last x = bin_last y\<rbrakk> \<Longrightarrow> x = y"
by (auto elim: oddE)
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
lemma bin_rest_gt_0 [simp]: "bin_rest x > 0 \<longleftrightarrow> x > 1"
by auto
subsection \<open>Bit projection\<close>
abbreviation (input) bin_nth :: \<open>int \<Rightarrow> nat \<Rightarrow> bool\<close>
where \<open>bin_nth \<equiv> bit\<close>
lemma bin_nth_eq_iff: "bin_nth x = bin_nth y \<longleftrightarrow> x = y"
by (simp add: bit_eq_iff fun_eq_iff)
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)"
by (fact bit_eq_iff)
lemma bin_nth_zero [simp]: "\<not> bin_nth 0 n"
by simp
lemma bin_nth_1 [simp]: "bin_nth 1 n \<longleftrightarrow> n = 0"
by (cases n) (simp_all add: bit_Suc)
lemma bin_nth_minus1 [simp]: "bin_nth (- 1) n"
by (induction n) (simp_all add: bit_Suc)
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 bit_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 =
bit_0 bit_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>
by (auto simp add: bit_exp_iff)
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: bit_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 (cases n; simp)+
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"
by (simp_all add: bin_sign_def)
lemma bin_sign_rest [simp]: "bin_sign (bin_rest w) = bin_sign w"
by (simp add: bin_sign_def)
abbreviation (input) bintrunc :: \<open>nat \<Rightarrow> int \<Rightarrow> int\<close>
where \<open>bintrunc \<equiv> take_bit\<close>
lemma bintrunc_mod2p: "bintrunc n w = w mod 2 ^ n"
by (fact take_bit_eq_mod)
abbreviation (input) sbintrunc :: \<open>nat \<Rightarrow> int \<Rightarrow> int\<close>
where \<open>sbintrunc \<equiv> signed_take_bit\<close>
abbreviation (input) norm_sint :: \<open>nat \<Rightarrow> int \<Rightarrow> int\<close>
where \<open>norm_sint n \<equiv> signed_take_bit (n - 1)\<close>
lemma sbintrunc_mod2p: "sbintrunc n w = (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n"
by (simp add: bintrunc_mod2p signed_take_bit_eq_take_bit_shift)
lemma sbintrunc_eq_take_bit:
\<open>sbintrunc n k = take_bit (Suc n) (k + 2 ^ n) - 2 ^ n\<close>
by (fact signed_take_bit_eq_take_bit_shift)
lemma sign_bintr: "bin_sign (bintrunc n w) = 0"
by (simp add: bin_sign_def)
lemma bintrunc_n_0: "bintrunc n 0 = 0"
by (fact take_bit_of_0)
lemma sbintrunc_n_0: "sbintrunc n 0 = 0"
by (fact signed_take_bit_of_0)
lemma sbintrunc_n_minus1: "sbintrunc n (- 1) = -1"
by (fact signed_take_bit_of_minus_1)
lemma bintrunc_Suc_numeral:
"bintrunc (Suc n) 1 = 1"
"bintrunc (Suc n) (- 1) = 1 + 2 * bintrunc n (- 1)"
"bintrunc (Suc n) (numeral (Num.Bit0 w)) = 2 * bintrunc n (numeral w)"
"bintrunc (Suc n) (numeral (Num.Bit1 w)) = 1 + 2 * bintrunc n (numeral w)"
"bintrunc (Suc n) (- numeral (Num.Bit0 w)) = 2 * bintrunc n (- numeral w)"
"bintrunc (Suc n) (- numeral (Num.Bit1 w)) = 1 + 2 * bintrunc n (- numeral (w + Num.One))"
by (simp_all add: take_bit_Suc)
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)) = 2 * sbintrunc n (numeral w)"
"sbintrunc (Suc n) (numeral (Num.Bit1 w)) = 1 + 2 * sbintrunc n (numeral w)"
"sbintrunc (Suc n) (- numeral (Num.Bit0 w)) = 2 * sbintrunc n (- numeral w)"
"sbintrunc (Suc n) (- numeral (Num.Bit1 w)) = 1 + 2 * sbintrunc n (- numeral (w + Num.One))"
by (simp_all add: signed_take_bit_Suc)
lemma bin_sign_lem: "(bin_sign (sbintrunc n bin) = -1) = bit bin n"
by (simp add: bin_sign_def)
lemma nth_bintr: "bin_nth (bintrunc m w) n \<longleftrightarrow> n < m \<and> bin_nth w n"
by (fact bit_take_bit_iff)
lemma nth_sbintr: "bin_nth (sbintrunc m w) n = (if n < m then bin_nth w n else bin_nth w m)"
by (simp add: bit_signed_take_bit_iff min_def)
lemma bin_nth_Bit0:
"bin_nth (numeral (Num.Bit0 w)) n \<longleftrightarrow>
(\<exists>m. n = Suc m \<and> bin_nth (numeral w) m)"
using bit_double_iff [of \<open>numeral w :: int\<close> n]
by (auto intro: exI [of _ \<open>n - 1\<close>])
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 even_bit_succ_iff [of \<open>2 * numeral w :: int\<close> n]
bit_double_iff [of \<open>numeral w :: int\<close> n]
by auto
lemma bintrunc_bintrunc_l: "n \<le> m \<Longrightarrow> bintrunc m (bintrunc n w) = bintrunc n w"
by (simp add: min.absorb2)
lemma sbintrunc_sbintrunc_l: "n \<le> m \<Longrightarrow> sbintrunc m (sbintrunc n w) = sbintrunc n w"
by (simp add: min_def)
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 sbintrunc_Suc_Pls =
signed_take_bit_Suc [where k="0", simplified bin_last_numeral_simps bin_rest_numeral_simps]
lemmas sbintrunc_Suc_Min =
signed_take_bit_Suc [where k="-1", simplified bin_last_numeral_simps bin_rest_numeral_simps]
lemmas sbintrunc_Sucs = sbintrunc_Suc_Pls sbintrunc_Suc_Min
sbintrunc_Suc_numeral
lemmas sbintrunc_Pls =
signed_take_bit_0 [where k="0", simplified bin_last_numeral_simps bin_rest_numeral_simps]
lemmas sbintrunc_Min =
signed_take_bit_0 [where k="-1", simplified bin_last_numeral_simps bin_rest_numeral_simps]
lemmas sbintrunc_0_simps =
sbintrunc_Pls sbintrunc_Min
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 sbintrunc_minus_simps =
sbintrunc_Sucs [THEN [2] sbintrunc_minus [symmetric, THEN trans]]
lemma sbintrunc_BIT_I:
\<open>0 < n \<Longrightarrow>
sbintrunc (n - 1) 0 = y \<Longrightarrow>
sbintrunc n 0 = 2 * y\<close>
by simp
lemma sbintrunc_Suc_Is:
\<open>sbintrunc n (- 1) = y \<Longrightarrow>
sbintrunc (Suc n) (- 1) = 1 + 2 * y\<close>
by auto
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) simp_all
lemma sbintrunc_bintrunc' [simp]: "0 < n \<Longrightarrow> sbintrunc (n - 1) (bintrunc n w) = sbintrunc (n - 1) w"
by (cases n) simp_all
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)
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 = of_bool (odd x) + 2 * bintrunc (pred_numeral k) (x div 2)"
by (simp add: numeral_eq_Suc take_bit_Suc mod_2_eq_odd)
lemma sbintrunc_numeral:
"sbintrunc (numeral k) x = of_bool (odd x) + 2 * sbintrunc (pred_numeral k) (x div 2)"
by (simp add: numeral_eq_Suc signed_take_bit_Suc mod2_eq_if)
lemma bintrunc_numeral_simps [simp]:
"bintrunc (numeral k) (numeral (Num.Bit0 w)) =
2 * bintrunc (pred_numeral k) (numeral w)"
"bintrunc (numeral k) (numeral (Num.Bit1 w)) =
1 + 2 * bintrunc (pred_numeral k) (numeral w)"
"bintrunc (numeral k) (- numeral (Num.Bit0 w)) =
2 * bintrunc (pred_numeral k) (- numeral w)"
"bintrunc (numeral k) (- numeral (Num.Bit1 w)) =
1 + 2 * bintrunc (pred_numeral k) (- numeral (w + Num.One))"
"bintrunc (numeral k) 1 = 1"
by (simp_all add: bintrunc_numeral)
lemma sbintrunc_numeral_simps [simp]:
"sbintrunc (numeral k) (numeral (Num.Bit0 w)) =
2 * sbintrunc (pred_numeral k) (numeral w)"
"sbintrunc (numeral k) (numeral (Num.Bit1 w)) =
1 + 2 * sbintrunc (pred_numeral k) (numeral w)"
"sbintrunc (numeral k) (- numeral (Num.Bit0 w)) =
2 * sbintrunc (pred_numeral k) (- numeral w)"
"sbintrunc (numeral k) (- numeral (Num.Bit1 w)) =
1 + 2 * sbintrunc (pred_numeral k) (- numeral (w + Num.One))"
"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}"
by (auto simp add: take_bit_eq_mod image_iff) (metis mod_pos_pos_trivial)
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}"
proof -
have \<open>surj (\<lambda>k::int. k + 2 ^ n)\<close>
by (rule surjI [of _ \<open>(\<lambda>k. k - 2 ^ n)\<close>]) simp
moreover have \<open>sbintrunc n = ((\<lambda>k. k - 2 ^ n) \<circ> take_bit (Suc n) \<circ> (\<lambda>k. k + 2 ^ n))\<close>
by (simp add: sbintrunc_eq_take_bit fun_eq_iff)
ultimately show ?thesis
apply (simp only: fun.set_map range_bintrunc)
apply (auto simp add: image_iff)
apply presburger
done
qed
lemma sbintrunc_inc:
\<open>k + 2 ^ Suc n \<le> sbintrunc n k\<close> if \<open>k < - (2 ^ n)\<close>
using that by (fact signed_take_bit_greater_eq)
lemma sbintrunc_dec:
\<open>sbintrunc n k \<le> k - 2 ^ (Suc n)\<close> if \<open>k \<ge> 2 ^ n\<close>
using that by (fact signed_take_bit_less_eq)
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: stable_imp_take_bit_eq)
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 (simp add: take_bit_rec [of n bin])
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 simp add: take_bit_Suc)
lemma bin_rest_strunc: "bin_rest (sbintrunc (Suc n) bin) = sbintrunc n (bin_rest bin)"
by (simp add: signed_take_bit_Suc)
lemma bintrunc_rest [simp]: "bintrunc n (bin_rest (bintrunc n bin)) = bin_rest (bintrunc n bin)"
by (induct n arbitrary: bin) (simp_all add: take_bit_Suc)
lemma sbintrunc_rest [simp]: "sbintrunc n (bin_rest (sbintrunc n bin)) = bin_rest (sbintrunc n bin)"
by (induct n arbitrary: bin) (simp_all add: signed_take_bit_Suc mod2_eq_if)
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>
definition bin_split :: \<open>nat \<Rightarrow> int \<Rightarrow> int \<times> int\<close>
where [simp]: \<open>bin_split n k = (drop_bit n k, take_bit n k)\<close>
lemma [code]:
"bin_split (Suc n) w = (let (w1, w2) = bin_split n (w div 2) in (w1, of_bool (odd w) + 2 * w2))"
"bin_split 0 w = (w, 0)"
by (simp_all add: drop_bit_Suc take_bit_Suc mod_2_eq_odd)
abbreviation (input) bin_cat :: \<open>int \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int\<close>
where \<open>bin_cat k n l \<equiv> concat_bit n l k\<close>
lemma bin_cat_eq_push_bit_add_take_bit:
\<open>bin_cat k n l = push_bit n k + take_bit n l\<close>
by (simp add: concat_bit_eq)
lemma bin_sign_cat: "bin_sign (bin_cat x n y) = bin_sign x"
proof -
have \<open>0 \<le> x\<close> if \<open>0 \<le> x * 2 ^ n + y mod 2 ^ n\<close>
proof -
have \<open>y mod 2 ^ n < 2 ^ n\<close>
using pos_mod_bound [of \<open>2 ^ n\<close> y] by simp
then have \<open>\<not> y mod 2 ^ n \<ge> 2 ^ n\<close>
by (simp add: less_le)
with that have \<open>x \<noteq> - 1\<close>
by auto
have *: \<open>- 1 \<le> (- (y mod 2 ^ n)) div 2 ^ n\<close>
by (simp add: zdiv_zminus1_eq_if)
from that have \<open>- (y mod 2 ^ n) \<le> x * 2 ^ n\<close>
by simp
then have \<open>(- (y mod 2 ^ n)) div 2 ^ n \<le> (x * 2 ^ n) div 2 ^ n\<close>
using zdiv_mono1 zero_less_numeral zero_less_power by blast
with * have \<open>- 1 \<le> x * 2 ^ n div 2 ^ n\<close> by simp
with \<open>x \<noteq> - 1\<close> show ?thesis
by simp
qed
then show ?thesis
by (simp add: bin_sign_def not_le not_less bin_cat_eq_push_bit_add_take_bit push_bit_eq_mult take_bit_eq_mod)
qed
lemma bin_cat_assoc: "bin_cat (bin_cat x m y) n z = bin_cat x (m + n) (bin_cat y n z)"
by (fact concat_bit_assoc)
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"
by (fact concat_bit_assoc_sym)
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))"
by (simp add: bit_concat_bit_iff)
lemma bin_nth_drop_bit_iff:
\<open>bin_nth (drop_bit n c) k \<longleftrightarrow> bin_nth c (n + k)\<close>
by (simp add: bit_drop_bit_eq)
lemma bin_nth_take_bit_iff:
\<open>bin_nth (take_bit n c) k \<longleftrightarrow> k < n \<and> bin_nth c k\<close>
by (fact bit_take_bit_iff)
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))"
by (auto simp add: bin_nth_drop_bit_iff bin_nth_take_bit_iff)
lemma bin_cat_zero [simp]: "bin_cat 0 n w = bintrunc n w"
by (simp add: bin_cat_eq_push_bit_add_take_bit)
lemma bintr_cat1: "bintrunc (k + n) (bin_cat a n b) = bin_cat (bintrunc k a) n b"
by (metis bin_cat_assoc bin_cat_zero)
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 (simp add: bin_cat_eq_push_bit_add_take_bit)
lemma split_bintrunc: "bin_split n c = (a, b) \<Longrightarrow> b = bintrunc n c"
by simp
lemma bin_cat_split: "bin_split n w = (u, v) \<Longrightarrow> w = bin_cat u n v"
by (auto simp add: bin_cat_eq_push_bit_add_take_bit bits_ident)
lemma drop_bit_bin_cat_eq:
\<open>drop_bit n (bin_cat v n w) = v\<close>
by (rule bit_eqI) (simp add: bit_drop_bit_eq bit_concat_bit_iff)
lemma take_bit_bin_cat_eq:
\<open>take_bit n (bin_cat v n w) = take_bit n w\<close>
by (rule bit_eqI) (simp add: bit_concat_bit_iff)
lemma bin_split_cat: "bin_split n (bin_cat v n w) = (v, bintrunc n w)"
by (simp add: drop_bit_bin_cat_eq take_bit_bin_cat_eq)
lemma bin_split_zero [simp]: "bin_split n 0 = (0, 0)"
by simp
lemma bin_split_minus1 [simp]:
"bin_split n (- 1) = (- 1, bintrunc n (- 1))"
by simp
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 drop_bit_Suc take_bit_Suc mod_2_eq_odd 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 drop_bit_Suc take_bit_Suc mod_2_eq_odd split: prod.split_asm)
done
lemma bin_cat_num: "bin_cat a n b = a * 2 ^ n + bintrunc n b"
by (simp add: bin_cat_eq_push_bit_add_take_bit push_bit_eq_mult)
lemma bin_split_num: "bin_split n b = (b div 2 ^ n, b mod 2 ^ n)"
by (simp add: drop_bit_eq_div take_bit_eq_mod)
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
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, of_bool (odd w) + 2 * w2))"
by (simp add: take_bit_rec drop_bit_rec mod_2_eq_odd)
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 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 (erule allE, erule impE, erule exI)
apply (case_tac k)
apply clarsimp
prefer 2
apply clarsimp
apply (erule allE)
apply (erule (1) impE)
apply (simp add: bit_drop_bit_eq ac_simps)
apply (simp add: bit_take_bit_iff 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 (simp add: ac_simps)
apply (subst rsplit_aux_alts(1))
apply (subst rsplit_aux_alts(2))
apply clarsimp
unfolding bin_rsplit_def bin_rsplitl_def
apply (simp add: drop_bit_take_bit)
apply (case_tac \<open>x < n\<close>)
apply (simp_all add: not_less min_def)
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)
apply (simp add: drop_bit_bin_cat_eq take_bit_bin_cat_eq)
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" [of \<open>bin_split n w\<close> \<open>drop_bit n w\<close> \<open>take_bit n w\<close>] \<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) (drop_bit n w) (take_bit n w # cs))"
using bin_rsplit_aux_len by fastforce
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>
primrec bin_sc :: "nat \<Rightarrow> bool \<Rightarrow> int \<Rightarrow> int"
where
Z: "bin_sc 0 b w = of_bool b + 2 * bin_rest w"
| Suc: "bin_sc (Suc n) b w = of_bool (odd w) + 2 * bin_sc n b (w div 2)"
lemma bin_nth_sc [simp]: "bit (bin_sc n b w) n \<longleftrightarrow> b"
by (induction n arbitrary: w) (simp_all add: bit_Suc)
lemma bin_sc_sc_same [simp]: "bin_sc n c (bin_sc n b w) = bin_sc n c w"
by (induction n arbitrary: w) (simp_all add: bit_Suc)
lemma bin_sc_sc_diff: "m \<noteq> n \<Longrightarrow> bin_sc m c (bin_sc n b w) = bin_sc n b (bin_sc m c w)"
apply (induct n arbitrary: w m)
apply (case_tac [!] m)
apply auto
done
lemma bin_nth_sc_gen: "bin_nth (bin_sc n b w) m = (if m = n then b else bin_nth w m)"
apply (induct n arbitrary: w m)
apply (case_tac m; simp add: bit_Suc)
apply (case_tac m; simp add: bit_Suc)
done
lemma bin_sc_eq:
\<open>bin_sc n False = unset_bit n\<close>
\<open>bin_sc n True = Bit_Operations.set_bit n\<close>
by (simp_all add: fun_eq_iff bit_eq_iff)
(simp_all add: bin_nth_sc_gen bit_set_bit_iff bit_unset_bit_iff)
lemma bin_sc_nth [simp]: "bin_sc n (bin_nth w n) w = w"
by (rule bit_eqI) (simp add: bin_nth_sc_gen)
lemma bin_sign_sc [simp]: "bin_sign (bin_sc n b w) = bin_sign w"
proof (induction n arbitrary: w)
case 0
then show ?case
by (auto simp add: bin_sign_def) (use bin_rest_ge_0 in fastforce)
next
case (Suc n)
from Suc [of \<open>w div 2\<close>]
show ?case by (auto simp add: bin_sign_def split: if_splits)
qed
lemma bin_sc_bintr [simp]:
"bintrunc m (bin_sc n x (bintrunc m w)) = bintrunc m (bin_sc n x w)"
apply (cases x)
apply (simp_all add: bin_sc_eq bit_eq_iff)
apply (auto simp add: bit_take_bit_iff bit_set_bit_iff bit_unset_bit_iff)
done
lemma bin_clr_le: "bin_sc n False w \<le> w"
by (simp add: bin_sc_eq unset_bit_less_eq)
lemma bin_set_ge: "bin_sc n True w \<ge> w"
by (simp add: bin_sc_eq set_bit_greater_eq)
lemma bintr_bin_clr_le: "bintrunc n (bin_sc m False w) \<le> bintrunc n w"
by (simp add: bin_sc_eq take_bit_unset_bit_eq unset_bit_less_eq)
lemma bintr_bin_set_ge: "bintrunc n (bin_sc m True w) \<ge> bintrunc n w"
by (simp add: bin_sc_eq take_bit_set_bit_eq set_bit_greater_eq)
lemma bin_sc_FP [simp]: "bin_sc n False 0 = 0"
by (induct n) auto
lemma bin_sc_TM [simp]: "bin_sc n True (- 1) = - 1"
by (induct n) auto
lemmas bin_sc_simps = bin_sc.Z bin_sc.Suc bin_sc_TM bin_sc_FP
lemma bin_sc_minus: "0 < n \<Longrightarrow> bin_sc (Suc (n - 1)) b w = bin_sc n b w"
by auto
lemmas bin_sc_Suc_minus =
trans [OF bin_sc_minus [symmetric] bin_sc.Suc]
lemma bin_sc_numeral [simp]:
"bin_sc (numeral k) b w =
of_bool (odd w) + 2 * bin_sc (pred_numeral k) b (w div 2)"
by (simp add: numeral_eq_Suc)
instantiation int :: semiring_bit_syntax
begin
definition [iff]: "i !! n \<longleftrightarrow> bin_nth i n"
definition "shiftl x n = x * 2 ^ n" for x :: int
definition "shiftr x n = x div 2 ^ n" for x :: int
instance by standard
(simp_all add: fun_eq_iff shiftl_int_def shiftr_int_def push_bit_eq_mult drop_bit_eq_div)
end
lemma shiftl_eq_push_bit:
\<open>k << n = push_bit n k\<close> for k :: int
by (fact shiftl_eq_push_bit)
lemma shiftr_eq_drop_bit:
\<open>k >> n = drop_bit n k\<close> for k :: int
by (fact shiftr_eq_drop_bit)
subsubsection \<open>Basic simplification rules\<close>
lemmas int_not_def = not_int_def
lemma int_not_simps [simp]:
"NOT (0::int) = -1"
"NOT (1::int) = -2"
"NOT (- 1::int) = 0"
"NOT (numeral w::int) = - numeral (w + Num.One)"
"NOT (- numeral (Num.Bit0 w)::int) = numeral (Num.BitM w)"
"NOT (- numeral (Num.Bit1 w)::int) = numeral (Num.Bit0 w)"
by (simp_all add: not_int_def)
lemma int_not_not: "NOT (NOT x) = x"
for x :: int
by (fact bit.double_compl)
lemma int_and_0 [simp]: "0 AND x = 0"
for x :: int
by (fact bit.conj_zero_left)
lemma int_and_m1 [simp]: "-1 AND x = x"
for x :: int
by (fact bit.conj_one_left)
lemma int_or_zero [simp]: "0 OR x = x"
for x :: int
by (fact bit.disj_zero_left)
lemma int_or_minus1 [simp]: "-1 OR x = -1"
for x :: int
by (fact bit.disj_one_left)
lemma int_xor_zero [simp]: "0 XOR x = x"
for x :: int
by (fact bit.xor_zero_left)
subsubsection \<open>Binary destructors\<close>
lemma bin_rest_NOT [simp]: "bin_rest (NOT x) = NOT (bin_rest x)"
by (fact not_int_div_2)
lemma bin_last_NOT [simp]: "bin_last (NOT x) \<longleftrightarrow> \<not> bin_last x"
by simp
lemma bin_rest_AND [simp]: "bin_rest (x AND y) = bin_rest x AND bin_rest y"
by (subst and_int_rec) auto
lemma bin_last_AND [simp]: "bin_last (x AND y) \<longleftrightarrow> bin_last x \<and> bin_last y"
by (subst and_int_rec) auto
lemma bin_rest_OR [simp]: "bin_rest (x OR y) = bin_rest x OR bin_rest y"
by (subst or_int_rec) auto
lemma bin_last_OR [simp]: "bin_last (x OR y) \<longleftrightarrow> bin_last x \<or> bin_last y"
by (subst or_int_rec) auto
lemma bin_rest_XOR [simp]: "bin_rest (x XOR y) = bin_rest x XOR bin_rest y"
by (subst xor_int_rec) auto
lemma bin_last_XOR [simp]: "bin_last (x XOR y) \<longleftrightarrow> (bin_last x \<or> bin_last y) \<and> \<not> (bin_last x \<and> bin_last y)"
by (subst xor_int_rec) auto
lemma bin_nth_ops:
"\<And>x y. bin_nth (x AND y) n \<longleftrightarrow> bin_nth x n \<and> bin_nth y n"
"\<And>x y. bin_nth (x OR y) n \<longleftrightarrow> bin_nth x n \<or> bin_nth y n"
"\<And>x y. bin_nth (x XOR y) n \<longleftrightarrow> bin_nth x n \<noteq> bin_nth y n"
"\<And>x. bin_nth (NOT x) n \<longleftrightarrow> \<not> bin_nth x n"
by (simp_all add: bit_and_iff bit_or_iff bit_xor_iff bit_not_iff)
subsubsection \<open>Derived properties\<close>
lemma int_xor_minus1 [simp]: "-1 XOR x = NOT x"
for x :: int
by (fact bit.xor_one_left)
lemma int_xor_extra_simps [simp]:
"w XOR 0 = w"
"w XOR -1 = NOT w"
for w :: int
by simp_all
lemma int_or_extra_simps [simp]:
"w OR 0 = w"
"w OR -1 = -1"
for w :: int
by simp_all
lemma int_and_extra_simps [simp]:
"w AND 0 = 0"
"w AND -1 = w"
for w :: int
by simp_all
text \<open>Commutativity of the above.\<close>
lemma bin_ops_comm:
fixes x y :: int
shows int_and_comm: "x AND y = y AND x"
and int_or_comm: "x OR y = y OR x"
and int_xor_comm: "x XOR y = y XOR x"
by (simp_all add: ac_simps)
lemma bin_ops_same [simp]:
"x AND x = x"
"x OR x = x"
"x XOR x = 0"
for x :: int
by simp_all
lemmas bin_log_esimps =
int_and_extra_simps int_or_extra_simps int_xor_extra_simps
int_and_0 int_and_m1 int_or_zero int_or_minus1 int_xor_zero int_xor_minus1
subsubsection \<open>Basic properties of logical (bit-wise) operations\<close>
lemma bbw_ao_absorb: "x AND (y OR x) = x \<and> x OR (y AND x) = x"
for x y :: int
by (auto simp add: bin_eq_iff bin_nth_ops)
lemma bbw_ao_absorbs_other:
"x AND (x OR y) = x \<and> (y AND x) OR x = x"
"(y OR x) AND x = x \<and> x OR (x AND y) = x"
"(x OR y) AND x = x \<and> (x AND y) OR x = x"
for x y :: int
by (auto simp add: bin_eq_iff bin_nth_ops)
lemmas bbw_ao_absorbs [simp] = bbw_ao_absorb bbw_ao_absorbs_other
lemma int_xor_not: "(NOT x) XOR y = NOT (x XOR y) \<and> x XOR (NOT y) = NOT (x XOR y)"
for x y :: int
by (auto simp add: bin_eq_iff bin_nth_ops)
lemma int_and_assoc: "(x AND y) AND z = x AND (y AND z)"
for x y z :: int
by (auto simp add: bin_eq_iff bin_nth_ops)
lemma int_or_assoc: "(x OR y) OR z = x OR (y OR z)"
for x y z :: int
by (auto simp add: bin_eq_iff bin_nth_ops)
lemma int_xor_assoc: "(x XOR y) XOR z = x XOR (y XOR z)"
for x y z :: int
by (auto simp add: bin_eq_iff bin_nth_ops)
lemmas bbw_assocs = int_and_assoc int_or_assoc int_xor_assoc
(* BH: Why are these declared as simp rules??? *)
lemma bbw_lcs [simp]:
"y AND (x AND z) = x AND (y AND z)"
"y OR (x OR z) = x OR (y OR z)"
"y XOR (x XOR z) = x XOR (y XOR z)"
for x y :: int
by (auto simp add: bin_eq_iff bin_nth_ops)
lemma bbw_not_dist:
"NOT (x OR y) = (NOT x) AND (NOT y)"
"NOT (x AND y) = (NOT x) OR (NOT y)"
for x y :: int
by (auto simp add: bin_eq_iff bin_nth_ops)
lemma bbw_oa_dist: "(x AND y) OR z = (x OR z) AND (y OR z)"
for x y z :: int
by (auto simp add: bin_eq_iff bin_nth_ops)
lemma bbw_ao_dist: "(x OR y) AND z = (x AND z) OR (y AND z)"
for x y z :: int
by (auto simp add: bin_eq_iff bin_nth_ops)
(*
Why were these declared simp???
declare bin_ops_comm [simp] bbw_assocs [simp]
*)
subsubsection \<open>Simplification with numerals\<close>
text \<open>Cases for \<open>0\<close> and \<open>-1\<close> are already covered by other simp rules.\<close>
lemma bin_rest_neg_numeral_BitM [simp]:
"bin_rest (- numeral (Num.BitM w)) = - numeral w"
by simp
lemma bin_last_neg_numeral_BitM [simp]:
"bin_last (- numeral (Num.BitM w))"
by simp
(* FIXME: The rule sets below are very large (24 rules for each
operator). Is there a simpler way to do this? *)
lemma int_and_numerals [simp]:
"numeral (Num.Bit0 x) AND numeral (Num.Bit0 y) = (2 :: int) * (numeral x AND numeral y)"
"numeral (Num.Bit0 x) AND numeral (Num.Bit1 y) = (2 :: int) * (numeral x AND numeral y)"
"numeral (Num.Bit1 x) AND numeral (Num.Bit0 y) = (2 :: int) * (numeral x AND numeral y)"
"numeral (Num.Bit1 x) AND numeral (Num.Bit1 y) = 1 + (2 :: int) * (numeral x AND numeral y)"
"numeral (Num.Bit0 x) AND - numeral (Num.Bit0 y) = (2 :: int) * (numeral x AND - numeral y)"
"numeral (Num.Bit0 x) AND - numeral (Num.Bit1 y) = (2 :: int) * (numeral x AND - numeral (y + Num.One))"
"numeral (Num.Bit1 x) AND - numeral (Num.Bit0 y) = (2 :: int) * (numeral x AND - numeral y)"
"numeral (Num.Bit1 x) AND - numeral (Num.Bit1 y) = 1 + (2 :: int) * (numeral x AND - numeral (y + Num.One))"
"- numeral (Num.Bit0 x) AND numeral (Num.Bit0 y) = (2 :: int) * (- numeral x AND numeral y)"
"- numeral (Num.Bit0 x) AND numeral (Num.Bit1 y) = (2 :: int) * (- numeral x AND numeral y)"
"- numeral (Num.Bit1 x) AND numeral (Num.Bit0 y) = (2 :: int) * (- numeral (x + Num.One) AND numeral y)"
"- numeral (Num.Bit1 x) AND numeral (Num.Bit1 y) = 1 + (2 :: int) * (- numeral (x + Num.One) AND numeral y)"
"- numeral (Num.Bit0 x) AND - numeral (Num.Bit0 y) = (2 :: int) * (- numeral x AND - numeral y)"
"- numeral (Num.Bit0 x) AND - numeral (Num.Bit1 y) = (2 :: int) * (- numeral x AND - numeral (y + Num.One))"
"- numeral (Num.Bit1 x) AND - numeral (Num.Bit0 y) = (2 :: int) * (- numeral (x + Num.One) AND - numeral y)"
"- numeral (Num.Bit1 x) AND - numeral (Num.Bit1 y) = 1 + (2 :: int) * (- numeral (x + Num.One) AND - numeral (y + Num.One))"
"(1::int) AND numeral (Num.Bit0 y) = 0"
"(1::int) AND numeral (Num.Bit1 y) = 1"
"(1::int) AND - numeral (Num.Bit0 y) = 0"
"(1::int) AND - numeral (Num.Bit1 y) = 1"
"numeral (Num.Bit0 x) AND (1::int) = 0"
"numeral (Num.Bit1 x) AND (1::int) = 1"
"- numeral (Num.Bit0 x) AND (1::int) = 0"
"- numeral (Num.Bit1 x) AND (1::int) = 1"
by (rule bin_rl_eqI; simp)+
lemma int_or_numerals [simp]:
"numeral (Num.Bit0 x) OR numeral (Num.Bit0 y) = (2 :: int) * (numeral x OR numeral y)"
"numeral (Num.Bit0 x) OR numeral (Num.Bit1 y) = 1 + (2 :: int) * (numeral x OR numeral y)"
"numeral (Num.Bit1 x) OR numeral (Num.Bit0 y) = 1 + (2 :: int) * (numeral x OR numeral y)"
"numeral (Num.Bit1 x) OR numeral (Num.Bit1 y) = 1 + (2 :: int) * (numeral x OR numeral y)"
"numeral (Num.Bit0 x) OR - numeral (Num.Bit0 y) = (2 :: int) * (numeral x OR - numeral y)"
"numeral (Num.Bit0 x) OR - numeral (Num.Bit1 y) = 1 + (2 :: int) * (numeral x OR - numeral (y + Num.One))"
"numeral (Num.Bit1 x) OR - numeral (Num.Bit0 y) = 1 + (2 :: int) * (numeral x OR - numeral y)"
"numeral (Num.Bit1 x) OR - numeral (Num.Bit1 y) = 1 + (2 :: int) * (numeral x OR - numeral (y + Num.One))"
"- numeral (Num.Bit0 x) OR numeral (Num.Bit0 y) = (2 :: int) * (- numeral x OR numeral y)"
"- numeral (Num.Bit0 x) OR numeral (Num.Bit1 y) = 1 + (2 :: int) * (- numeral x OR numeral y)"
"- numeral (Num.Bit1 x) OR numeral (Num.Bit0 y) = 1 + (2 :: int) * (- numeral (x + Num.One) OR numeral y)"
"- numeral (Num.Bit1 x) OR numeral (Num.Bit1 y) = 1 + (2 :: int) * (- numeral (x + Num.One) OR numeral y)"
"- numeral (Num.Bit0 x) OR - numeral (Num.Bit0 y) = (2 :: int) * (- numeral x OR - numeral y)"
"- numeral (Num.Bit0 x) OR - numeral (Num.Bit1 y) = 1 + (2 :: int) * (- numeral x OR - numeral (y + Num.One))"
"- numeral (Num.Bit1 x) OR - numeral (Num.Bit0 y) = 1 + (2 :: int) * (- numeral (x + Num.One) OR - numeral y)"
"- numeral (Num.Bit1 x) OR - numeral (Num.Bit1 y) = 1 + (2 :: int) * (- numeral (x + Num.One) OR - numeral (y + Num.One))"
"(1::int) OR numeral (Num.Bit0 y) = numeral (Num.Bit1 y)"
"(1::int) OR numeral (Num.Bit1 y) = numeral (Num.Bit1 y)"
"(1::int) OR - numeral (Num.Bit0 y) = - numeral (Num.BitM y)"
"(1::int) OR - numeral (Num.Bit1 y) = - numeral (Num.Bit1 y)"
"numeral (Num.Bit0 x) OR (1::int) = numeral (Num.Bit1 x)"
"numeral (Num.Bit1 x) OR (1::int) = numeral (Num.Bit1 x)"
"- numeral (Num.Bit0 x) OR (1::int) = - numeral (Num.BitM x)"
"- numeral (Num.Bit1 x) OR (1::int) = - numeral (Num.Bit1 x)"
by (rule bin_rl_eqI; simp)+
lemma int_xor_numerals [simp]:
"numeral (Num.Bit0 x) XOR numeral (Num.Bit0 y) = (2 :: int) * (numeral x XOR numeral y)"
"numeral (Num.Bit0 x) XOR numeral (Num.Bit1 y) = 1 + (2 :: int) * (numeral x XOR numeral y)"
"numeral (Num.Bit1 x) XOR numeral (Num.Bit0 y) = 1 + (2 :: int) * (numeral x XOR numeral y)"
"numeral (Num.Bit1 x) XOR numeral (Num.Bit1 y) = (2 :: int) * (numeral x XOR numeral y)"
"numeral (Num.Bit0 x) XOR - numeral (Num.Bit0 y) = (2 :: int) * (numeral x XOR - numeral y)"
"numeral (Num.Bit0 x) XOR - numeral (Num.Bit1 y) = 1 + (2 :: int) * (numeral x XOR - numeral (y + Num.One))"
"numeral (Num.Bit1 x) XOR - numeral (Num.Bit0 y) = 1 + (2 :: int) * (numeral x XOR - numeral y)"
"numeral (Num.Bit1 x) XOR - numeral (Num.Bit1 y) = (2 :: int) * (numeral x XOR - numeral (y + Num.One))"
"- numeral (Num.Bit0 x) XOR numeral (Num.Bit0 y) = (2 :: int) * (- numeral x XOR numeral y)"
"- numeral (Num.Bit0 x) XOR numeral (Num.Bit1 y) = 1 + (2 :: int) * (- numeral x XOR numeral y)"
"- numeral (Num.Bit1 x) XOR numeral (Num.Bit0 y) = 1 + (2 :: int) * (- numeral (x + Num.One) XOR numeral y)"
"- numeral (Num.Bit1 x) XOR numeral (Num.Bit1 y) = (2 :: int) * (- numeral (x + Num.One) XOR numeral y)"
"- numeral (Num.Bit0 x) XOR - numeral (Num.Bit0 y) = (2 :: int) * (- numeral x XOR - numeral y)"
"- numeral (Num.Bit0 x) XOR - numeral (Num.Bit1 y) = 1 + (2 :: int) * (- numeral x XOR - numeral (y + Num.One))"
"- numeral (Num.Bit1 x) XOR - numeral (Num.Bit0 y) = 1 + (2 :: int) * (- numeral (x + Num.One) XOR - numeral y)"
"- numeral (Num.Bit1 x) XOR - numeral (Num.Bit1 y) = (2 :: int) * (- numeral (x + Num.One) XOR - numeral (y + Num.One))"
"(1::int) XOR numeral (Num.Bit0 y) = numeral (Num.Bit1 y)"
"(1::int) XOR numeral (Num.Bit1 y) = numeral (Num.Bit0 y)"
"(1::int) XOR - numeral (Num.Bit0 y) = - numeral (Num.BitM y)"
"(1::int) XOR - numeral (Num.Bit1 y) = - numeral (Num.Bit0 (y + Num.One))"
"numeral (Num.Bit0 x) XOR (1::int) = numeral (Num.Bit1 x)"
"numeral (Num.Bit1 x) XOR (1::int) = numeral (Num.Bit0 x)"
"- numeral (Num.Bit0 x) XOR (1::int) = - numeral (Num.BitM x)"
"- numeral (Num.Bit1 x) XOR (1::int) = - numeral (Num.Bit0 (x + Num.One))"
by (rule bin_rl_eqI; simp)+
subsubsection \<open>Interactions with arithmetic\<close>
lemma plus_and_or: "(x AND y) + (x OR y) = x + y" for x y :: int
proof (induction x arbitrary: y rule: int_bit_induct)
case zero
then show ?case
by simp
next
case minus
then show ?case
by simp
next
case (even x)
from even.IH [of \<open>y div 2\<close>]
show ?case
by (auto simp add: and_int_rec [of _ y] or_int_rec [of _ y] elim: oddE)
next
case (odd x)
from odd.IH [of \<open>y div 2\<close>]
show ?case
by (auto simp add: and_int_rec [of _ y] or_int_rec [of _ y] elim: oddE)
qed
lemma le_int_or: "bin_sign y = 0 \<Longrightarrow> x \<le> x OR y"
for x y :: int
by (simp add: bin_sign_def or_greater_eq split: if_splits)
lemmas int_and_le =
xtrans(3) [OF bbw_ao_absorbs (2) [THEN conjunct2, symmetric] le_int_or]
text \<open>Interaction between bit-wise and arithmetic: good example of \<open>bin_induction\<close>.\<close>
lemma bin_add_not: "x + NOT x = (-1::int)"
by (simp add: not_int_def)
lemma AND_mod: "x AND (2 ^ n - 1) = x mod 2 ^ n"
for x :: int
by (simp flip: take_bit_eq_mod add: take_bit_eq_mask mask_eq_exp_minus_1)
subsubsection \<open>Comparison\<close>
lemma AND_lower [simp]: \<^marker>\<open>contributor \<open>Stefan Berghofer\<close>\<close>
fixes x y :: int
assumes "0 \<le> x"
shows "0 \<le> x AND y"
using assms by simp
lemma OR_lower [simp]: \<^marker>\<open>contributor \<open>Stefan Berghofer\<close>\<close>
fixes x y :: int
assumes "0 \<le> x" "0 \<le> y"
shows "0 \<le> x OR y"
using assms by simp
lemma XOR_lower [simp]: \<^marker>\<open>contributor \<open>Stefan Berghofer\<close>\<close>
fixes x y :: int
assumes "0 \<le> x" "0 \<le> y"
shows "0 \<le> x XOR y"
using assms by simp
lemma AND_upper1 [simp]: \<^marker>\<open>contributor \<open>Stefan Berghofer\<close>\<close>
fixes x y :: int
assumes "0 \<le> x"
shows "x AND y \<le> x"
using assms by (induction x arbitrary: y rule: int_bit_induct)
(simp_all add: and_int_rec [of \<open>_ * 2\<close>] and_int_rec [of \<open>1 + _ * 2\<close>] add_increasing)
lemmas AND_upper1' [simp] = order_trans [OF AND_upper1] \<^marker>\<open>contributor \<open>Stefan Berghofer\<close>\<close>
lemmas AND_upper1'' [simp] = order_le_less_trans [OF AND_upper1] \<^marker>\<open>contributor \<open>Stefan Berghofer\<close>\<close>
lemma AND_upper2 [simp]: \<^marker>\<open>contributor \<open>Stefan Berghofer\<close>\<close>
fixes x y :: int
assumes "0 \<le> y"
shows "x AND y \<le> y"
using assms AND_upper1 [of y x] by (simp add: ac_simps)
lemmas AND_upper2' [simp] = order_trans [OF AND_upper2] \<^marker>\<open>contributor \<open>Stefan Berghofer\<close>\<close>
lemmas AND_upper2'' [simp] = order_le_less_trans [OF AND_upper2] \<^marker>\<open>contributor \<open>Stefan Berghofer\<close>\<close>
lemma OR_upper: \<^marker>\<open>contributor \<open>Stefan Berghofer\<close>\<close>
fixes x y :: int
assumes "0 \<le> x" "x < 2 ^ n" "y < 2 ^ n"
shows "x OR y < 2 ^ n"
using assms proof (induction x arbitrary: y n rule: int_bit_induct)
case zero
then show ?case
by simp
next
case minus
then show ?case
by simp
next
case (even x)
from even.IH [of \<open>n - 1\<close> \<open>y div 2\<close>] even.prems even.hyps
show ?case
by (cases n) (auto simp add: or_int_rec [of \<open>_ * 2\<close>] elim: oddE)
next
case (odd x)
from odd.IH [of \<open>n - 1\<close> \<open>y div 2\<close>] odd.prems odd.hyps
show ?case
by (cases n) (auto simp add: or_int_rec [of \<open>1 + _ * 2\<close>], linarith)
qed
lemma XOR_upper: \<^marker>\<open>contributor \<open>Stefan Berghofer\<close>\<close>
fixes x y :: int
assumes "0 \<le> x" "x < 2 ^ n" "y < 2 ^ n"
shows "x XOR y < 2 ^ n"
using assms proof (induction x arbitrary: y n rule: int_bit_induct)
case zero
then show ?case
by simp
next
case minus
then show ?case
by simp
next
case (even x)
from even.IH [of \<open>n - 1\<close> \<open>y div 2\<close>] even.prems even.hyps
show ?case
by (cases n) (auto simp add: xor_int_rec [of \<open>_ * 2\<close>] elim: oddE)
next
case (odd x)
from odd.IH [of \<open>n - 1\<close> \<open>y div 2\<close>] odd.prems odd.hyps
show ?case
by (cases n) (auto simp add: xor_int_rec [of \<open>1 + _ * 2\<close>])
qed
subsubsection \<open>Truncating results of bit-wise operations\<close>
lemma bin_trunc_ao:
"bintrunc n x AND bintrunc n y = bintrunc n (x AND y)"
"bintrunc n x OR bintrunc n y = bintrunc n (x OR y)"
by (auto simp add: bin_eq_iff bin_nth_ops nth_bintr)
lemma bin_trunc_xor: "bintrunc n (bintrunc n x XOR bintrunc n y) = bintrunc n (x XOR y)"
by (auto simp add: bin_eq_iff bin_nth_ops nth_bintr)
lemma bin_trunc_not: "bintrunc n (NOT (bintrunc n x)) = bintrunc n (NOT x)"
by (auto simp add: bin_eq_iff bin_nth_ops nth_bintr)
text \<open>Want theorems of the form of \<open>bin_trunc_xor\<close>.\<close>
lemma bintr_bintr_i: "x = bintrunc n y \<Longrightarrow> bintrunc n x = bintrunc n y"
by auto
lemmas bin_trunc_and = bin_trunc_ao(1) [THEN bintr_bintr_i]
lemmas bin_trunc_or = bin_trunc_ao(2) [THEN bintr_bintr_i]
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 (fact bit.conj_disj_distrib)
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 simp
lemma int_nand_same_middle: fixes x :: int shows "x AND y AND NOT x = 0"
by (simp add: bit_eq_iff bit_and_iff bit_not_iff)
lemma and_xor_dist: fixes x :: int shows
"x AND (y XOR z) = (x AND y) XOR (x AND z)"
by (fact bit.conj_xor_distrib)
lemma int_and_lt0 [simp]:
\<open>x AND y < 0 \<longleftrightarrow> x < 0 \<and> y < 0\<close> for x y :: int
by (fact and_negative_int_iff)
lemma int_and_ge0 [simp]:
\<open>x AND y \<ge> 0 \<longleftrightarrow> x \<ge> 0 \<or> y \<ge> 0\<close> for x y :: int
by (fact and_nonnegative_int_iff)
lemma int_and_1: fixes x :: int shows "x AND 1 = x mod 2"
by (fact and_one_eq)
lemma int_1_and: fixes x :: int shows "1 AND x = x mod 2"
by (fact one_and_eq)
lemma int_or_lt0 [simp]:
\<open>x OR y < 0 \<longleftrightarrow> x < 0 \<or> y < 0\<close> for x y :: int
by (fact or_negative_int_iff)
lemma int_or_ge0 [simp]:
\<open>x OR y \<ge> 0 \<longleftrightarrow> x \<ge> 0 \<and> y \<ge> 0\<close> for x y :: int
by (fact or_nonnegative_int_iff)
lemma int_xor_lt0 [simp]:
\<open>x XOR y < 0 \<longleftrightarrow> (x < 0) \<noteq> (y < 0)\<close> for x y :: int
by (fact xor_negative_int_iff)
lemma int_xor_ge0 [simp]:
\<open>x XOR y \<ge> 0 \<longleftrightarrow> (x \<ge> 0 \<longleftrightarrow> y \<ge> 0)\<close> for x y :: int
by (fact xor_nonnegative_int_iff)
lemma even_conv_AND:
\<open>even i \<longleftrightarrow> i AND 1 = 0\<close> for i :: int
by (simp add: and_one_eq mod2_eq_if)
lemma bin_last_conv_AND:
"bin_last i \<longleftrightarrow> i AND 1 \<noteq> 0"
by (simp add: and_one_eq mod2_eq_if)
lemma bitval_bin_last:
"of_bool (bin_last i) = i AND 1"
by (simp add: and_one_eq mod2_eq_if)
lemma bin_sign_and:
"bin_sign (i AND j) = - (bin_sign i * bin_sign j)"
by(simp add: bin_sign_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>
lemma int_shiftl_BIT: fixes x :: int
shows int_shiftl0 [simp]: "x << 0 = x"
and int_shiftl_Suc [simp]: "x << Suc n = 2 * (x << n)"
by (auto simp add: shiftl_int_def)
lemma int_0_shiftl [simp]: "0 << n = (0 :: int)"
by(induct n) simp_all
lemma bin_last_shiftl: "bin_last (x << n) \<longleftrightarrow> n = 0 \<and> bin_last x"
by(cases n)(simp_all)
lemma bin_rest_shiftl: "bin_rest (x << n) = (if n > 0 then x << (n - 1) else bin_rest x)"
by(cases n)(simp_all)
lemma bin_nth_shiftl [simp]: "bin_nth (x << n) m \<longleftrightarrow> n \<le> m \<and> bin_nth x (m - n)"
by (simp add: bit_push_bit_iff_int shiftl_eq_push_bit)
lemma bin_last_shiftr: "odd (x >> n) \<longleftrightarrow> x !! n" for x :: int
by (simp add: shiftr_eq_drop_bit bit_iff_odd_drop_bit)
lemma bin_rest_shiftr [simp]: "bin_rest (x >> n) = x >> Suc n"
by (simp add: bit_eq_iff shiftr_eq_drop_bit drop_bit_Suc bit_drop_bit_eq drop_bit_half)
lemma bin_nth_shiftr [simp]: "bin_nth (x >> n) m = bin_nth x (n + m)"
by (simp add: shiftr_eq_drop_bit bit_drop_bit_eq)
lemma bin_nth_conv_AND:
fixes x :: int shows
"bin_nth x n \<longleftrightarrow> x AND (1 << n) \<noteq> 0"
by (simp add: bit_eq_iff)
(auto simp add: shiftl_eq_push_bit bit_and_iff bit_push_bit_iff bit_exp_iff)
lemma int_shiftl_numeral [simp]:
"(numeral w :: int) << numeral w' = numeral (num.Bit0 w) << pred_numeral w'"
"(- numeral w :: int) << numeral w' = - numeral (num.Bit0 w) << pred_numeral w'"
by(simp_all add: numeral_eq_Suc shiftl_int_def)
(metis add_One mult_inc semiring_norm(11) semiring_norm(13) semiring_norm(2) semiring_norm(6) semiring_norm(87))+
lemma int_shiftl_One_numeral [simp]:
"(1 :: int) << numeral w = 2 << pred_numeral w"
using int_shiftl_numeral [of Num.One w] by simp
lemma shiftl_ge_0 [simp]: fixes i :: int shows "i << n \<ge> 0 \<longleftrightarrow> i \<ge> 0"
by(induct n) simp_all
lemma shiftl_lt_0 [simp]: fixes i :: int shows "i << n < 0 \<longleftrightarrow> i < 0"
by (metis not_le shiftl_ge_0)
lemma int_shiftl_test_bit: "(n << i :: int) !! m \<longleftrightarrow> m \<ge> i \<and> n !! (m - i)"
by simp
lemma int_0shiftr [simp]: "(0 :: int) >> x = 0"
by(simp add: shiftr_int_def)
lemma int_minus1_shiftr [simp]: "(-1 :: int) >> x = -1"
by(simp add: shiftr_int_def div_eq_minus1)
lemma int_shiftr_ge_0 [simp]: fixes i :: int shows "i >> n \<ge> 0 \<longleftrightarrow> i \<ge> 0"
by (simp add: shiftr_eq_drop_bit)
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 int_shiftr_numeral [simp]:
"(1 :: int) >> numeral w' = 0"
"(numeral num.One :: int) >> numeral w' = 0"
"(numeral (num.Bit0 w) :: int) >> numeral w' = numeral w >> pred_numeral w'"
"(numeral (num.Bit1 w) :: int) >> numeral w' = numeral w >> pred_numeral w'"
"(- numeral (num.Bit0 w) :: int) >> numeral w' = - numeral w >> pred_numeral w'"
"(- numeral (num.Bit1 w) :: int) >> numeral w' = - numeral (Num.inc w) >> pred_numeral w'"
by (simp_all add: shiftr_eq_drop_bit numeral_eq_Suc add_One drop_bit_Suc)
lemma int_shiftr_numeral_Suc0 [simp]:
"(1 :: int) >> Suc 0 = 0"
"(numeral num.One :: int) >> Suc 0 = 0"
"(numeral (num.Bit0 w) :: int) >> Suc 0 = numeral w"
"(numeral (num.Bit1 w) :: int) >> Suc 0 = numeral w"
"(- numeral (num.Bit0 w) :: int) >> Suc 0 = - numeral w"
"(- numeral (num.Bit1 w) :: int) >> Suc 0 = - numeral (Num.inc w)"
by (simp_all add: shiftr_eq_drop_bit drop_bit_Suc add_One)
lemma bin_nth_minus_p2:
assumes sign: "bin_sign x = 0"
and y: "y = 1 << n"
and m: "m < n"
and x: "x < y"
shows "bin_nth (x - y) m = bin_nth x m"
proof -
from sign y x have \<open>x \<ge> 0\<close> and \<open>y = 2 ^ n\<close> and \<open>x < 2 ^ n\<close>
by (simp_all add: bin_sign_def shiftl_eq_push_bit push_bit_eq_mult split: if_splits)
from \<open>0 \<le> x\<close> \<open>x < 2 ^ n\<close> \<open>m < n\<close> have \<open>bit x m \<longleftrightarrow> bit (x - 2 ^ n) m\<close>
proof (induction m arbitrary: x n)
case 0
then show ?case
by simp
next
case (Suc m)
moreover define q where \<open>q = n - 1\<close>
ultimately have n: \<open>n = Suc q\<close>
by simp
have \<open>(x - 2 ^ Suc q) div 2 = x div 2 - 2 ^ q\<close>
by simp
moreover from Suc.IH [of \<open>x div 2\<close> q] Suc.prems
have \<open>bit (x div 2) m \<longleftrightarrow> bit (x div 2 - 2 ^ q) m\<close>
by (simp add: n)
ultimately show ?case
by (simp add: bit_Suc n)
qed
with \<open>y = 2 ^ n\<close> show ?thesis
by simp
qed
lemma bin_clr_conv_NAND:
"bin_sc n False i = i AND NOT (1 << n)"
by (induct n arbitrary: i) (rule bin_rl_eqI; simp)+
lemma bin_set_conv_OR:
"bin_sc n True i = i OR (1 << n)"
by (induct n arbitrary: i) (rule bin_rl_eqI; simp)+
end