author | haftmann |
Sat, 11 Jul 2020 06:21:02 +0000 | |
changeset 72009 | febdd4eead56 |
parent 71991 | 8bff286878bf |
child 72010 | a851ce626b78 |
permissions | -rw-r--r-- |
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(* Author: Florian Haftmann, TUM |
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*) |
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section \<open>Bit operations in suitable algebraic structures\<close> |
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theory Bit_Operations |
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imports |
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"HOL-Library.Boolean_Algebra" |
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Main |
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begin |
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subsection \<open>Bit operations\<close> |
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|
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class semiring_bit_operations = semiring_bit_shifts + |
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fixes "and" :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close> (infixr \<open>AND\<close> 64) |
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and or :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close> (infixr \<open>OR\<close> 59) |
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and xor :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close> (infixr \<open>XOR\<close> 59) |
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assumes bit_and_iff: \<open>\<And>n. bit (a AND b) n \<longleftrightarrow> bit a n \<and> bit b n\<close> |
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and bit_or_iff: \<open>\<And>n. bit (a OR b) n \<longleftrightarrow> bit a n \<or> bit b n\<close> |
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and bit_xor_iff: \<open>\<And>n. bit (a XOR b) n \<longleftrightarrow> bit a n \<noteq> bit b n\<close> |
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begin |
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text \<open> |
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We want the bitwise operations to bind slightly weaker |
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than \<open>+\<close> and \<open>-\<close>. |
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For the sake of code generation |
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the operations \<^const>\<open>and\<close>, \<^const>\<open>or\<close> and \<^const>\<open>xor\<close> |
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are specified as definitional class operations. |
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\<close> |
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|
71418 | 31 |
sublocale "and": semilattice \<open>(AND)\<close> |
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by standard (auto simp add: bit_eq_iff bit_and_iff) |
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sublocale or: semilattice_neutr \<open>(OR)\<close> 0 |
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by standard (auto simp add: bit_eq_iff bit_or_iff) |
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sublocale xor: comm_monoid \<open>(XOR)\<close> 0 |
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by standard (auto simp add: bit_eq_iff bit_xor_iff) |
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lemma even_and_iff: |
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\<open>even (a AND b) \<longleftrightarrow> even a \<or> even b\<close> |
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using bit_and_iff [of a b 0] by auto |
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lemma even_or_iff: |
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\<open>even (a OR b) \<longleftrightarrow> even a \<and> even b\<close> |
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using bit_or_iff [of a b 0] by auto |
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lemma even_xor_iff: |
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\<open>even (a XOR b) \<longleftrightarrow> (even a \<longleftrightarrow> even b)\<close> |
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using bit_xor_iff [of a b 0] by auto |
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lemma zero_and_eq [simp]: |
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"0 AND a = 0" |
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by (simp add: bit_eq_iff bit_and_iff) |
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lemma and_zero_eq [simp]: |
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"a AND 0 = 0" |
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by (simp add: bit_eq_iff bit_and_iff) |
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lemma one_and_eq: |
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"1 AND a = a mod 2" |
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by (simp add: bit_eq_iff bit_and_iff) (auto simp add: bit_1_iff) |
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lemma and_one_eq: |
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"a AND 1 = a mod 2" |
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using one_and_eq [of a] by (simp add: ac_simps) |
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lemma one_or_eq: |
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"1 OR a = a + of_bool (even a)" |
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by (simp add: bit_eq_iff bit_or_iff add.commute [of _ 1] even_bit_succ_iff) (auto simp add: bit_1_iff) |
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lemma or_one_eq: |
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"a OR 1 = a + of_bool (even a)" |
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using one_or_eq [of a] by (simp add: ac_simps) |
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lemma one_xor_eq: |
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"1 XOR a = a + of_bool (even a) - of_bool (odd a)" |
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by (simp add: bit_eq_iff bit_xor_iff add.commute [of _ 1] even_bit_succ_iff) (auto simp add: bit_1_iff odd_bit_iff_bit_pred elim: oddE) |
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lemma xor_one_eq: |
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"a XOR 1 = a + of_bool (even a) - of_bool (odd a)" |
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using one_xor_eq [of a] by (simp add: ac_simps) |
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lemma take_bit_and [simp]: |
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\<open>take_bit n (a AND b) = take_bit n a AND take_bit n b\<close> |
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by (auto simp add: bit_eq_iff bit_take_bit_iff bit_and_iff) |
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lemma take_bit_or [simp]: |
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\<open>take_bit n (a OR b) = take_bit n a OR take_bit n b\<close> |
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by (auto simp add: bit_eq_iff bit_take_bit_iff bit_or_iff) |
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lemma take_bit_xor [simp]: |
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\<open>take_bit n (a XOR b) = take_bit n a XOR take_bit n b\<close> |
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by (auto simp add: bit_eq_iff bit_take_bit_iff bit_xor_iff) |
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definition mask :: \<open>nat \<Rightarrow> 'a\<close> |
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where mask_eq_exp_minus_1: \<open>mask n = 2 ^ n - 1\<close> |
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lemma bit_mask_iff: |
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\<open>bit (mask m) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n < m\<close> |
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by (simp add: mask_eq_exp_minus_1 bit_mask_iff) |
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lemma even_mask_iff: |
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\<open>even (mask n) \<longleftrightarrow> n = 0\<close> |
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using bit_mask_iff [of n 0] by auto |
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lemma mask_0 [simp, code]: |
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\<open>mask 0 = 0\<close> |
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by (simp add: mask_eq_exp_minus_1) |
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lemma mask_Suc_exp [code]: |
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\<open>mask (Suc n) = 2 ^ n OR mask n\<close> |
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by (rule bit_eqI) |
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(auto simp add: bit_or_iff bit_mask_iff bit_exp_iff not_less le_less_Suc_eq) |
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lemma mask_Suc_double: |
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\<open>mask (Suc n) = 2 * mask n OR 1\<close> |
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proof (rule bit_eqI) |
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fix q |
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assume \<open>2 ^ q \<noteq> 0\<close> |
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show \<open>bit (mask (Suc n)) q \<longleftrightarrow> bit (2 * mask n OR 1) q\<close> |
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by (cases q) |
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(simp_all add: even_mask_iff even_or_iff bit_or_iff bit_mask_iff bit_exp_iff bit_double_iff not_less le_less_Suc_eq bit_1_iff, auto simp add: mult_2) |
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qed |
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lemma take_bit_eq_mask: |
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\<open>take_bit n a = a AND mask n\<close> |
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by (rule bit_eqI) |
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(auto simp add: bit_take_bit_iff bit_and_iff bit_mask_iff) |
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end |
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class ring_bit_operations = semiring_bit_operations + ring_parity + |
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fixes not :: \<open>'a \<Rightarrow> 'a\<close> (\<open>NOT\<close>) |
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assumes bit_not_iff: \<open>\<And>n. bit (NOT a) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> \<not> bit a n\<close> |
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assumes minus_eq_not_minus_1: \<open>- a = NOT (a - 1)\<close> |
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begin |
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text \<open> |
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For the sake of code generation \<^const>\<open>not\<close> is specified as |
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definitional class operation. Note that \<^const>\<open>not\<close> has no |
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sensible definition for unlimited but only positive bit strings |
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(type \<^typ>\<open>nat\<close>). |
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\<close> |
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lemma bits_minus_1_mod_2_eq [simp]: |
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\<open>(- 1) mod 2 = 1\<close> |
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by (simp add: mod_2_eq_odd) |
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lemma not_eq_complement: |
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\<open>NOT a = - a - 1\<close> |
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using minus_eq_not_minus_1 [of \<open>a + 1\<close>] by simp |
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lemma minus_eq_not_plus_1: |
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\<open>- a = NOT a + 1\<close> |
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using not_eq_complement [of a] by simp |
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lemma bit_minus_iff: |
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\<open>bit (- a) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> \<not> bit (a - 1) n\<close> |
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by (simp add: minus_eq_not_minus_1 bit_not_iff) |
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lemma even_not_iff [simp]: |
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"even (NOT a) \<longleftrightarrow> odd a" |
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using bit_not_iff [of a 0] by auto |
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lemma bit_not_exp_iff: |
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\<open>bit (NOT (2 ^ m)) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n \<noteq> m\<close> |
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by (auto simp add: bit_not_iff bit_exp_iff) |
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lemma bit_minus_1_iff [simp]: |
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\<open>bit (- 1) n \<longleftrightarrow> 2 ^ n \<noteq> 0\<close> |
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by (simp add: bit_minus_iff) |
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lemma bit_minus_exp_iff: |
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\<open>bit (- (2 ^ m)) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n \<ge> m\<close> |
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oops |
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lemma bit_minus_2_iff [simp]: |
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\<open>bit (- 2) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n > 0\<close> |
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by (simp add: bit_minus_iff bit_1_iff) |
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lemma not_one [simp]: |
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"NOT 1 = - 2" |
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by (simp add: bit_eq_iff bit_not_iff) (simp add: bit_1_iff) |
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sublocale "and": semilattice_neutr \<open>(AND)\<close> \<open>- 1\<close> |
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apply standard |
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apply (simp add: bit_eq_iff bit_and_iff) |
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apply (auto simp add: exp_eq_0_imp_not_bit bit_exp_iff) |
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71418 | 190 |
done |
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||
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sublocale bit: boolean_algebra \<open>(AND)\<close> \<open>(OR)\<close> NOT 0 \<open>- 1\<close> |
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rewrites \<open>bit.xor = (XOR)\<close> |
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proof - |
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interpret bit: boolean_algebra \<open>(AND)\<close> \<open>(OR)\<close> NOT 0 \<open>- 1\<close> |
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apply standard |
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apply (simp_all add: bit_eq_iff) |
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apply (auto simp add: bit_and_iff bit_or_iff bit_not_iff bit_exp_iff exp_eq_0_imp_not_bit) |
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done |
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show \<open>boolean_algebra (AND) (OR) NOT 0 (- 1)\<close> |
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by standard |
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show \<open>boolean_algebra.xor (AND) (OR) NOT = (XOR)\<close> |
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apply (simp add: fun_eq_iff bit_eq_iff bit.xor_def) |
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apply (auto simp add: bit_and_iff bit_or_iff bit_not_iff bit_xor_iff exp_eq_0_imp_not_bit) |
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done |
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qed |
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71802 | 208 |
lemma and_eq_not_not_or: |
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\<open>a AND b = NOT (NOT a OR NOT b)\<close> |
|
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by simp |
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lemma or_eq_not_not_and: |
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\<open>a OR b = NOT (NOT a AND NOT b)\<close> |
|
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by simp |
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72009 | 216 |
lemma not_add_distrib: |
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\<open>NOT (a + b) = NOT a - b\<close> |
|
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by (simp add: not_eq_complement algebra_simps) |
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||
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lemma not_diff_distrib: |
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\<open>NOT (a - b) = NOT a + b\<close> |
|
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using not_add_distrib [of a \<open>- b\<close>] by simp |
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lemma push_bit_minus: |
225 |
\<open>push_bit n (- a) = - push_bit n a\<close> |
|
226 |
by (simp add: push_bit_eq_mult) |
|
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71409 | 228 |
lemma take_bit_not_take_bit: |
229 |
\<open>take_bit n (NOT (take_bit n a)) = take_bit n (NOT a)\<close> |
|
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by (auto simp add: bit_eq_iff bit_take_bit_iff bit_not_iff) |
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71418 | 232 |
lemma take_bit_not_iff: |
233 |
"take_bit n (NOT a) = take_bit n (NOT b) \<longleftrightarrow> take_bit n a = take_bit n b" |
|
234 |
apply (simp add: bit_eq_iff bit_not_iff bit_take_bit_iff) |
|
235 |
apply (simp add: bit_exp_iff) |
|
236 |
apply (use local.exp_eq_0_imp_not_bit in blast) |
|
237 |
done |
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||
71922 | 239 |
lemma take_bit_minus_one_eq_mask: |
240 |
\<open>take_bit n (- 1) = mask n\<close> |
|
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by (simp add: bit_eq_iff bit_mask_iff bit_take_bit_iff conj_commute) |
|
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||
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lemma push_bit_minus_one_eq_not_mask: |
|
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\<open>push_bit n (- 1) = NOT (mask n)\<close> |
|
245 |
proof (rule bit_eqI) |
|
246 |
fix m |
|
247 |
assume \<open>2 ^ m \<noteq> 0\<close> |
|
248 |
show \<open>bit (push_bit n (- 1)) m \<longleftrightarrow> bit (NOT (mask n)) m\<close> |
|
249 |
proof (cases \<open>n \<le> m\<close>) |
|
250 |
case True |
|
251 |
moreover define q where \<open>q = m - n\<close> |
|
252 |
ultimately have \<open>m = n + q\<close> \<open>m - n = q\<close> |
|
253 |
by simp_all |
|
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with \<open>2 ^ m \<noteq> 0\<close> have \<open>2 ^ n * 2 ^ q \<noteq> 0\<close> |
|
255 |
by (simp add: power_add) |
|
256 |
then have \<open>2 ^ q \<noteq> 0\<close> |
|
257 |
using mult_not_zero by blast |
|
258 |
with \<open>m - n = q\<close> show ?thesis |
|
259 |
by (auto simp add: bit_not_iff bit_mask_iff bit_push_bit_iff not_less) |
|
260 |
next |
|
261 |
case False |
|
262 |
then show ?thesis |
|
263 |
by (simp add: bit_not_iff bit_mask_iff bit_push_bit_iff not_le) |
|
264 |
qed |
|
265 |
qed |
|
266 |
||
71426 | 267 |
definition set_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close> |
71991 | 268 |
where \<open>set_bit n a = a OR push_bit n 1\<close> |
71426 | 269 |
|
270 |
definition unset_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close> |
|
71991 | 271 |
where \<open>unset_bit n a = a AND NOT (push_bit n 1)\<close> |
71426 | 272 |
|
273 |
definition flip_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close> |
|
71991 | 274 |
where \<open>flip_bit n a = a XOR push_bit n 1\<close> |
71426 | 275 |
|
276 |
lemma bit_set_bit_iff: |
|
277 |
\<open>bit (set_bit m a) n \<longleftrightarrow> bit a n \<or> (m = n \<and> 2 ^ n \<noteq> 0)\<close> |
|
71991 | 278 |
by (auto simp add: set_bit_def push_bit_of_1 bit_or_iff bit_exp_iff) |
71426 | 279 |
|
280 |
lemma even_set_bit_iff: |
|
281 |
\<open>even (set_bit m a) \<longleftrightarrow> even a \<and> m \<noteq> 0\<close> |
|
282 |
using bit_set_bit_iff [of m a 0] by auto |
|
283 |
||
284 |
lemma bit_unset_bit_iff: |
|
285 |
\<open>bit (unset_bit m a) n \<longleftrightarrow> bit a n \<and> m \<noteq> n\<close> |
|
71991 | 286 |
by (auto simp add: unset_bit_def push_bit_of_1 bit_and_iff bit_not_iff bit_exp_iff exp_eq_0_imp_not_bit) |
71426 | 287 |
|
288 |
lemma even_unset_bit_iff: |
|
289 |
\<open>even (unset_bit m a) \<longleftrightarrow> even a \<or> m = 0\<close> |
|
290 |
using bit_unset_bit_iff [of m a 0] by auto |
|
291 |
||
292 |
lemma bit_flip_bit_iff: |
|
293 |
\<open>bit (flip_bit m a) n \<longleftrightarrow> (m = n \<longleftrightarrow> \<not> bit a n) \<and> 2 ^ n \<noteq> 0\<close> |
|
71991 | 294 |
by (auto simp add: flip_bit_def push_bit_of_1 bit_xor_iff bit_exp_iff exp_eq_0_imp_not_bit) |
71426 | 295 |
|
296 |
lemma even_flip_bit_iff: |
|
297 |
\<open>even (flip_bit m a) \<longleftrightarrow> \<not> (even a \<longleftrightarrow> m = 0)\<close> |
|
298 |
using bit_flip_bit_iff [of m a 0] by auto |
|
299 |
||
300 |
lemma set_bit_0 [simp]: |
|
301 |
\<open>set_bit 0 a = 1 + 2 * (a div 2)\<close> |
|
302 |
proof (rule bit_eqI) |
|
303 |
fix m |
|
304 |
assume *: \<open>2 ^ m \<noteq> 0\<close> |
|
305 |
then show \<open>bit (set_bit 0 a) m = bit (1 + 2 * (a div 2)) m\<close> |
|
306 |
by (simp add: bit_set_bit_iff bit_double_iff even_bit_succ_iff) |
|
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307 |
(cases m, simp_all add: bit_Suc) |
71426 | 308 |
qed |
309 |
||
71821 | 310 |
lemma set_bit_Suc: |
71426 | 311 |
\<open>set_bit (Suc n) a = a mod 2 + 2 * set_bit n (a div 2)\<close> |
312 |
proof (rule bit_eqI) |
|
313 |
fix m |
|
314 |
assume *: \<open>2 ^ m \<noteq> 0\<close> |
|
315 |
show \<open>bit (set_bit (Suc n) a) m \<longleftrightarrow> bit (a mod 2 + 2 * set_bit n (a div 2)) m\<close> |
|
316 |
proof (cases m) |
|
317 |
case 0 |
|
318 |
then show ?thesis |
|
319 |
by (simp add: even_set_bit_iff) |
|
320 |
next |
|
321 |
case (Suc m) |
|
322 |
with * have \<open>2 ^ m \<noteq> 0\<close> |
|
323 |
using mult_2 by auto |
|
324 |
show ?thesis |
|
325 |
by (cases a rule: parity_cases) |
|
326 |
(simp_all add: bit_set_bit_iff bit_double_iff even_bit_succ_iff *, |
|
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327 |
simp_all add: Suc \<open>2 ^ m \<noteq> 0\<close> bit_Suc) |
71426 | 328 |
qed |
329 |
qed |
|
330 |
||
331 |
lemma unset_bit_0 [simp]: |
|
332 |
\<open>unset_bit 0 a = 2 * (a div 2)\<close> |
|
333 |
proof (rule bit_eqI) |
|
334 |
fix m |
|
335 |
assume *: \<open>2 ^ m \<noteq> 0\<close> |
|
336 |
then show \<open>bit (unset_bit 0 a) m = bit (2 * (a div 2)) m\<close> |
|
337 |
by (simp add: bit_unset_bit_iff bit_double_iff) |
|
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|
338 |
(cases m, simp_all add: bit_Suc) |
71426 | 339 |
qed |
340 |
||
71821 | 341 |
lemma unset_bit_Suc: |
71426 | 342 |
\<open>unset_bit (Suc n) a = a mod 2 + 2 * unset_bit n (a div 2)\<close> |
343 |
proof (rule bit_eqI) |
|
344 |
fix m |
|
345 |
assume *: \<open>2 ^ m \<noteq> 0\<close> |
|
346 |
then show \<open>bit (unset_bit (Suc n) a) m \<longleftrightarrow> bit (a mod 2 + 2 * unset_bit n (a div 2)) m\<close> |
|
347 |
proof (cases m) |
|
348 |
case 0 |
|
349 |
then show ?thesis |
|
350 |
by (simp add: even_unset_bit_iff) |
|
351 |
next |
|
352 |
case (Suc m) |
|
353 |
show ?thesis |
|
354 |
by (cases a rule: parity_cases) |
|
355 |
(simp_all add: bit_unset_bit_iff bit_double_iff even_bit_succ_iff *, |
|
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|
356 |
simp_all add: Suc bit_Suc) |
71426 | 357 |
qed |
358 |
qed |
|
359 |
||
360 |
lemma flip_bit_0 [simp]: |
|
361 |
\<open>flip_bit 0 a = of_bool (even a) + 2 * (a div 2)\<close> |
|
362 |
proof (rule bit_eqI) |
|
363 |
fix m |
|
364 |
assume *: \<open>2 ^ m \<noteq> 0\<close> |
|
365 |
then show \<open>bit (flip_bit 0 a) m = bit (of_bool (even a) + 2 * (a div 2)) m\<close> |
|
366 |
by (simp add: bit_flip_bit_iff bit_double_iff even_bit_succ_iff) |
|
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|
367 |
(cases m, simp_all add: bit_Suc) |
71426 | 368 |
qed |
369 |
||
71821 | 370 |
lemma flip_bit_Suc: |
71426 | 371 |
\<open>flip_bit (Suc n) a = a mod 2 + 2 * flip_bit n (a div 2)\<close> |
372 |
proof (rule bit_eqI) |
|
373 |
fix m |
|
374 |
assume *: \<open>2 ^ m \<noteq> 0\<close> |
|
375 |
show \<open>bit (flip_bit (Suc n) a) m \<longleftrightarrow> bit (a mod 2 + 2 * flip_bit n (a div 2)) m\<close> |
|
376 |
proof (cases m) |
|
377 |
case 0 |
|
378 |
then show ?thesis |
|
379 |
by (simp add: even_flip_bit_iff) |
|
380 |
next |
|
381 |
case (Suc m) |
|
382 |
with * have \<open>2 ^ m \<noteq> 0\<close> |
|
383 |
using mult_2 by auto |
|
384 |
show ?thesis |
|
385 |
by (cases a rule: parity_cases) |
|
386 |
(simp_all add: bit_flip_bit_iff bit_double_iff even_bit_succ_iff, |
|
71535
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changeset
|
387 |
simp_all add: Suc \<open>2 ^ m \<noteq> 0\<close> bit_Suc) |
71426 | 388 |
qed |
389 |
qed |
|
390 |
||
72009 | 391 |
lemma flip_bit_eq_if: |
392 |
\<open>flip_bit n a = (if bit a n then unset_bit else set_bit) n a\<close> |
|
393 |
by (rule bit_eqI) (auto simp add: bit_set_bit_iff bit_unset_bit_iff bit_flip_bit_iff) |
|
394 |
||
71986 | 395 |
lemma take_bit_set_bit_eq: |
72009 | 396 |
\<open>take_bit n (set_bit m a) = (if n \<le> m then take_bit n a else set_bit m (take_bit n a))\<close> |
71986 | 397 |
by (rule bit_eqI) (auto simp add: bit_take_bit_iff bit_set_bit_iff) |
398 |
||
399 |
lemma take_bit_unset_bit_eq: |
|
72009 | 400 |
\<open>take_bit n (unset_bit m a) = (if n \<le> m then take_bit n a else unset_bit m (take_bit n a))\<close> |
71986 | 401 |
by (rule bit_eqI) (auto simp add: bit_take_bit_iff bit_unset_bit_iff) |
402 |
||
403 |
lemma take_bit_flip_bit_eq: |
|
72009 | 404 |
\<open>take_bit n (flip_bit m a) = (if n \<le> m then take_bit n a else flip_bit m (take_bit n a))\<close> |
71986 | 405 |
by (rule bit_eqI) (auto simp add: bit_take_bit_iff bit_flip_bit_iff) |
406 |
||
71042
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proof-of-concept theory for bit operations without a constructivistic representation and a minimal common logical foundation
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diff
changeset
|
407 |
end |
400e9512f1d3
proof-of-concept theory for bit operations without a constructivistic representation and a minimal common logical foundation
haftmann
parents:
diff
changeset
|
408 |
|
400e9512f1d3
proof-of-concept theory for bit operations without a constructivistic representation and a minimal common logical foundation
haftmann
parents:
diff
changeset
|
409 |
|
71956 | 410 |
subsection \<open>Instance \<^typ>\<open>int\<close>\<close> |
71042
400e9512f1d3
proof-of-concept theory for bit operations without a constructivistic representation and a minimal common logical foundation
haftmann
parents:
diff
changeset
|
411 |
|
400e9512f1d3
proof-of-concept theory for bit operations without a constructivistic representation and a minimal common logical foundation
haftmann
parents:
diff
changeset
|
412 |
instantiation int :: ring_bit_operations |
400e9512f1d3
proof-of-concept theory for bit operations without a constructivistic representation and a minimal common logical foundation
haftmann
parents:
diff
changeset
|
413 |
begin |
400e9512f1d3
proof-of-concept theory for bit operations without a constructivistic representation and a minimal common logical foundation
haftmann
parents:
diff
changeset
|
414 |
|
71420 | 415 |
definition not_int :: \<open>int \<Rightarrow> int\<close> |
416 |
where \<open>not_int k = - k - 1\<close> |
|
417 |
||
418 |
lemma not_int_rec: |
|
419 |
"NOT k = of_bool (even k) + 2 * NOT (k div 2)" for k :: int |
|
420 |
by (auto simp add: not_int_def elim: oddE) |
|
421 |
||
422 |
lemma even_not_iff_int: |
|
423 |
\<open>even (NOT k) \<longleftrightarrow> odd k\<close> for k :: int |
|
424 |
by (simp add: not_int_def) |
|
425 |
||
426 |
lemma not_int_div_2: |
|
427 |
\<open>NOT k div 2 = NOT (k div 2)\<close> for k :: int |
|
428 |
by (simp add: not_int_def) |
|
71042
400e9512f1d3
proof-of-concept theory for bit operations without a constructivistic representation and a minimal common logical foundation
haftmann
parents:
diff
changeset
|
429 |
|
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
430 |
lemma bit_not_int_iff: |
71186 | 431 |
\<open>bit (NOT k) n \<longleftrightarrow> \<not> bit k n\<close> |
432 |
for k :: int |
|
71535
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haftmann
parents:
71442
diff
changeset
|
433 |
by (induction n arbitrary: k) (simp_all add: not_int_div_2 even_not_iff_int bit_Suc) |
71186 | 434 |
|
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
435 |
function and_int :: \<open>int \<Rightarrow> int \<Rightarrow> int\<close> |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
436 |
where \<open>(k::int) AND l = (if k \<in> {0, - 1} \<and> l \<in> {0, - 1} |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
437 |
then - of_bool (odd k \<and> odd l) |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
438 |
else of_bool (odd k \<and> odd l) + 2 * ((k div 2) AND (l div 2)))\<close> |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
439 |
by auto |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
440 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
441 |
termination |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
442 |
by (relation \<open>measure (\<lambda>(k, l). nat (\<bar>k\<bar> + \<bar>l\<bar>))\<close>) auto |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
443 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
444 |
declare and_int.simps [simp del] |
71802 | 445 |
|
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
446 |
lemma and_int_rec: |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
447 |
\<open>k AND l = of_bool (odd k \<and> odd l) + 2 * ((k div 2) AND (l div 2))\<close> |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
448 |
for k l :: int |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
449 |
proof (cases \<open>k \<in> {0, - 1} \<and> l \<in> {0, - 1}\<close>) |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
450 |
case True |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
451 |
then show ?thesis |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
452 |
by auto (simp_all add: and_int.simps) |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
453 |
next |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
454 |
case False |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
455 |
then show ?thesis |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
456 |
by (auto simp add: ac_simps and_int.simps [of k l]) |
71802 | 457 |
qed |
458 |
||
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
459 |
lemma bit_and_int_iff: |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
460 |
\<open>bit (k AND l) n \<longleftrightarrow> bit k n \<and> bit l n\<close> for k l :: int |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
461 |
proof (induction n arbitrary: k l) |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
462 |
case 0 |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
463 |
then show ?case |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
464 |
by (simp add: and_int_rec [of k l]) |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
465 |
next |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
466 |
case (Suc n) |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
467 |
then show ?case |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
468 |
by (simp add: and_int_rec [of k l] bit_Suc) |
71802 | 469 |
qed |
470 |
||
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
471 |
lemma even_and_iff_int: |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
472 |
\<open>even (k AND l) \<longleftrightarrow> even k \<or> even l\<close> for k l :: int |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
473 |
using bit_and_int_iff [of k l 0] by auto |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
474 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
475 |
definition or_int :: \<open>int \<Rightarrow> int \<Rightarrow> int\<close> |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
476 |
where \<open>k OR l = NOT (NOT k AND NOT l)\<close> for k l :: int |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
477 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
478 |
lemma or_int_rec: |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
479 |
\<open>k OR l = of_bool (odd k \<or> odd l) + 2 * ((k div 2) OR (l div 2))\<close> |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
480 |
for k l :: int |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
481 |
using and_int_rec [of \<open>NOT k\<close> \<open>NOT l\<close>] |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
482 |
by (simp add: or_int_def even_not_iff_int not_int_div_2) |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
483 |
(simp add: not_int_def) |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
484 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
485 |
lemma bit_or_int_iff: |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
486 |
\<open>bit (k OR l) n \<longleftrightarrow> bit k n \<or> bit l n\<close> for k l :: int |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
487 |
by (simp add: or_int_def bit_not_int_iff bit_and_int_iff) |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
488 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
489 |
definition xor_int :: \<open>int \<Rightarrow> int \<Rightarrow> int\<close> |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
490 |
where \<open>k XOR l = k AND NOT l OR NOT k AND l\<close> for k l :: int |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
491 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
492 |
lemma xor_int_rec: |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
493 |
\<open>k XOR l = of_bool (odd k \<noteq> odd l) + 2 * ((k div 2) XOR (l div 2))\<close> |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
494 |
for k l :: int |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
495 |
by (simp add: xor_int_def or_int_rec [of \<open>k AND NOT l\<close> \<open>NOT k AND l\<close>] even_and_iff_int even_not_iff_int) |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
496 |
(simp add: and_int_rec [of \<open>NOT k\<close> \<open>l\<close>] and_int_rec [of \<open>k\<close> \<open>NOT l\<close>] not_int_div_2) |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
497 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
498 |
lemma bit_xor_int_iff: |
6fd70ed18199
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haftmann
parents:
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diff
changeset
|
499 |
\<open>bit (k XOR l) n \<longleftrightarrow> bit k n \<noteq> bit l n\<close> for k l :: int |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
500 |
by (auto simp add: xor_int_def bit_or_int_iff bit_and_int_iff bit_not_int_iff) |
71802 | 501 |
|
71042
400e9512f1d3
proof-of-concept theory for bit operations without a constructivistic representation and a minimal common logical foundation
haftmann
parents:
diff
changeset
|
502 |
instance proof |
71186 | 503 |
fix k l :: int and n :: nat |
71409 | 504 |
show \<open>- k = NOT (k - 1)\<close> |
505 |
by (simp add: not_int_def) |
|
71186 | 506 |
show \<open>bit (k AND l) n \<longleftrightarrow> bit k n \<and> bit l n\<close> |
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
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diff
changeset
|
507 |
by (fact bit_and_int_iff) |
71186 | 508 |
show \<open>bit (k OR l) n \<longleftrightarrow> bit k n \<or> bit l n\<close> |
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
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diff
changeset
|
509 |
by (fact bit_or_int_iff) |
71186 | 510 |
show \<open>bit (k XOR l) n \<longleftrightarrow> bit k n \<noteq> bit l n\<close> |
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
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diff
changeset
|
511 |
by (fact bit_xor_int_iff) |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
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|
512 |
qed (simp_all add: bit_not_int_iff) |
71042
400e9512f1d3
proof-of-concept theory for bit operations without a constructivistic representation and a minimal common logical foundation
haftmann
parents:
diff
changeset
|
513 |
|
400e9512f1d3
proof-of-concept theory for bit operations without a constructivistic representation and a minimal common logical foundation
haftmann
parents:
diff
changeset
|
514 |
end |
400e9512f1d3
proof-of-concept theory for bit operations without a constructivistic representation and a minimal common logical foundation
haftmann
parents:
diff
changeset
|
515 |
|
72009 | 516 |
lemma disjunctive_add: |
517 |
\<open>k + l = k OR l\<close> if \<open>\<And>n. \<not> bit k n \<or> \<not> bit l n\<close> for k l :: int |
|
518 |
\<comment> \<open>TODO: may integrate (indirectly) into \<^class>\<open>semiring_bits\<close> premises\<close> |
|
519 |
proof (rule bit_eqI) |
|
520 |
fix n |
|
521 |
from that have \<open>bit (k + l) n \<longleftrightarrow> bit k n \<or> bit l n\<close> |
|
522 |
proof (induction n arbitrary: k l) |
|
523 |
case 0 |
|
524 |
from this [of 0] show ?case |
|
525 |
by auto |
|
526 |
next |
|
527 |
case (Suc n) |
|
528 |
have \<open>bit ((k + l) div 2) n \<longleftrightarrow> bit (k div 2 + l div 2) n\<close> |
|
529 |
using Suc.prems [of 0] div_add1_eq [of k l] by auto |
|
530 |
also have \<open>bit (k div 2 + l div 2) n \<longleftrightarrow> bit (k div 2) n \<or> bit (l div 2) n\<close> |
|
531 |
by (rule Suc.IH) (use Suc.prems in \<open>simp flip: bit_Suc\<close>) |
|
532 |
finally show ?case |
|
533 |
by (simp add: bit_Suc) |
|
534 |
qed |
|
535 |
also have \<open>\<dots> \<longleftrightarrow> bit (k OR l) n\<close> |
|
536 |
by (simp add: bit_or_iff) |
|
537 |
finally show \<open>bit (k + l) n \<longleftrightarrow> bit (k OR l) n\<close> . |
|
538 |
qed |
|
539 |
||
540 |
lemma disjunctive_diff: |
|
541 |
\<open>k - l = k AND NOT l\<close> if \<open>\<And>n. bit l n \<Longrightarrow> bit k n\<close> for k l :: int |
|
542 |
proof - |
|
543 |
have \<open>NOT k + l = NOT k OR l\<close> |
|
544 |
by (rule disjunctive_add) (auto simp add: bit_not_iff dest: that) |
|
545 |
then have \<open>NOT (NOT k + l) = NOT (NOT k OR l)\<close> |
|
546 |
by simp |
|
547 |
then show ?thesis |
|
548 |
by (simp add: not_add_distrib) |
|
549 |
qed |
|
550 |
||
71802 | 551 |
lemma not_nonnegative_int_iff [simp]: |
552 |
\<open>NOT k \<ge> 0 \<longleftrightarrow> k < 0\<close> for k :: int |
|
553 |
by (simp add: not_int_def) |
|
554 |
||
555 |
lemma not_negative_int_iff [simp]: |
|
556 |
\<open>NOT k < 0 \<longleftrightarrow> k \<ge> 0\<close> for k :: int |
|
557 |
by (subst Not_eq_iff [symmetric]) (simp add: not_less not_le) |
|
558 |
||
559 |
lemma and_nonnegative_int_iff [simp]: |
|
560 |
\<open>k AND l \<ge> 0 \<longleftrightarrow> k \<ge> 0 \<or> l \<ge> 0\<close> for k l :: int |
|
561 |
proof (induction k arbitrary: l rule: int_bit_induct) |
|
562 |
case zero |
|
563 |
then show ?case |
|
564 |
by simp |
|
565 |
next |
|
566 |
case minus |
|
567 |
then show ?case |
|
568 |
by simp |
|
569 |
next |
|
570 |
case (even k) |
|
571 |
then show ?case |
|
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
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diff
changeset
|
572 |
using and_int_rec [of \<open>k * 2\<close> l] by (simp add: pos_imp_zdiv_nonneg_iff) |
71802 | 573 |
next |
574 |
case (odd k) |
|
575 |
from odd have \<open>0 \<le> k AND l div 2 \<longleftrightarrow> 0 \<le> k \<or> 0 \<le> l div 2\<close> |
|
576 |
by simp |
|
577 |
then have \<open>0 \<le> (1 + k * 2) div 2 AND l div 2 \<longleftrightarrow> 0 \<le> (1 + k * 2) div 2\<or> 0 \<le> l div 2\<close> |
|
578 |
by simp |
|
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
579 |
with and_int_rec [of \<open>1 + k * 2\<close> l] |
71802 | 580 |
show ?case |
581 |
by auto |
|
582 |
qed |
|
583 |
||
584 |
lemma and_negative_int_iff [simp]: |
|
585 |
\<open>k AND l < 0 \<longleftrightarrow> k < 0 \<and> l < 0\<close> for k l :: int |
|
586 |
by (subst Not_eq_iff [symmetric]) (simp add: not_less) |
|
587 |
||
72009 | 588 |
lemma and_less_eq: |
589 |
\<open>k AND l \<le> k\<close> if \<open>l < 0\<close> for k l :: int |
|
590 |
using that proof (induction k arbitrary: l rule: int_bit_induct) |
|
591 |
case zero |
|
592 |
then show ?case |
|
593 |
by simp |
|
594 |
next |
|
595 |
case minus |
|
596 |
then show ?case |
|
597 |
by simp |
|
598 |
next |
|
599 |
case (even k) |
|
600 |
from even.IH [of \<open>l div 2\<close>] even.hyps even.prems |
|
601 |
show ?case |
|
602 |
by (simp add: and_int_rec [of _ l]) |
|
603 |
next |
|
604 |
case (odd k) |
|
605 |
from odd.IH [of \<open>l div 2\<close>] odd.hyps odd.prems |
|
606 |
show ?case |
|
607 |
by (simp add: and_int_rec [of _ l]) |
|
608 |
qed |
|
609 |
||
71802 | 610 |
lemma or_nonnegative_int_iff [simp]: |
611 |
\<open>k OR l \<ge> 0 \<longleftrightarrow> k \<ge> 0 \<and> l \<ge> 0\<close> for k l :: int |
|
612 |
by (simp only: or_eq_not_not_and not_nonnegative_int_iff) simp |
|
613 |
||
614 |
lemma or_negative_int_iff [simp]: |
|
615 |
\<open>k OR l < 0 \<longleftrightarrow> k < 0 \<or> l < 0\<close> for k l :: int |
|
616 |
by (subst Not_eq_iff [symmetric]) (simp add: not_less) |
|
617 |
||
72009 | 618 |
lemma or_greater_eq: |
619 |
\<open>k OR l \<ge> k\<close> if \<open>l \<ge> 0\<close> for k l :: int |
|
620 |
using that proof (induction k arbitrary: l rule: int_bit_induct) |
|
621 |
case zero |
|
622 |
then show ?case |
|
623 |
by simp |
|
624 |
next |
|
625 |
case minus |
|
626 |
then show ?case |
|
627 |
by simp |
|
628 |
next |
|
629 |
case (even k) |
|
630 |
from even.IH [of \<open>l div 2\<close>] even.hyps even.prems |
|
631 |
show ?case |
|
632 |
by (simp add: or_int_rec [of _ l]) |
|
633 |
next |
|
634 |
case (odd k) |
|
635 |
from odd.IH [of \<open>l div 2\<close>] odd.hyps odd.prems |
|
636 |
show ?case |
|
637 |
by (simp add: or_int_rec [of _ l]) |
|
638 |
qed |
|
639 |
||
71802 | 640 |
lemma xor_nonnegative_int_iff [simp]: |
641 |
\<open>k XOR l \<ge> 0 \<longleftrightarrow> (k \<ge> 0 \<longleftrightarrow> l \<ge> 0)\<close> for k l :: int |
|
642 |
by (simp only: bit.xor_def or_nonnegative_int_iff) auto |
|
643 |
||
644 |
lemma xor_negative_int_iff [simp]: |
|
645 |
\<open>k XOR l < 0 \<longleftrightarrow> (k < 0) \<noteq> (l < 0)\<close> for k l :: int |
|
646 |
by (subst Not_eq_iff [symmetric]) (auto simp add: not_less) |
|
647 |
||
648 |
lemma set_bit_nonnegative_int_iff [simp]: |
|
649 |
\<open>set_bit n k \<ge> 0 \<longleftrightarrow> k \<ge> 0\<close> for k :: int |
|
650 |
by (simp add: set_bit_def) |
|
651 |
||
652 |
lemma set_bit_negative_int_iff [simp]: |
|
653 |
\<open>set_bit n k < 0 \<longleftrightarrow> k < 0\<close> for k :: int |
|
654 |
by (simp add: set_bit_def) |
|
655 |
||
656 |
lemma unset_bit_nonnegative_int_iff [simp]: |
|
657 |
\<open>unset_bit n k \<ge> 0 \<longleftrightarrow> k \<ge> 0\<close> for k :: int |
|
658 |
by (simp add: unset_bit_def) |
|
659 |
||
660 |
lemma unset_bit_negative_int_iff [simp]: |
|
661 |
\<open>unset_bit n k < 0 \<longleftrightarrow> k < 0\<close> for k :: int |
|
662 |
by (simp add: unset_bit_def) |
|
663 |
||
664 |
lemma flip_bit_nonnegative_int_iff [simp]: |
|
665 |
\<open>flip_bit n k \<ge> 0 \<longleftrightarrow> k \<ge> 0\<close> for k :: int |
|
666 |
by (simp add: flip_bit_def) |
|
667 |
||
668 |
lemma flip_bit_negative_int_iff [simp]: |
|
669 |
\<open>flip_bit n k < 0 \<longleftrightarrow> k < 0\<close> for k :: int |
|
670 |
by (simp add: flip_bit_def) |
|
671 |
||
71986 | 672 |
lemma set_bit_greater_eq: |
673 |
\<open>set_bit n k \<ge> k\<close> for k :: int |
|
674 |
by (simp add: set_bit_def or_greater_eq) |
|
675 |
||
676 |
lemma unset_bit_less_eq: |
|
677 |
\<open>unset_bit n k \<le> k\<close> for k :: int |
|
678 |
by (simp add: unset_bit_def and_less_eq) |
|
679 |
||
72009 | 680 |
lemma set_bit_eq: |
681 |
\<open>set_bit n k = k + of_bool (\<not> bit k n) * 2 ^ n\<close> for k :: int |
|
682 |
proof (rule bit_eqI) |
|
683 |
fix m |
|
684 |
show \<open>bit (set_bit n k) m \<longleftrightarrow> bit (k + of_bool (\<not> bit k n) * 2 ^ n) m\<close> |
|
685 |
proof (cases \<open>m = n\<close>) |
|
686 |
case True |
|
687 |
then show ?thesis |
|
688 |
apply (simp add: bit_set_bit_iff) |
|
689 |
apply (simp add: bit_iff_odd div_plus_div_distrib_dvd_right) |
|
690 |
done |
|
691 |
next |
|
692 |
case False |
|
693 |
then show ?thesis |
|
694 |
apply (clarsimp simp add: bit_set_bit_iff) |
|
695 |
apply (subst disjunctive_add) |
|
696 |
apply (clarsimp simp add: bit_exp_iff) |
|
697 |
apply (clarsimp simp add: bit_or_iff bit_exp_iff) |
|
698 |
done |
|
699 |
qed |
|
700 |
qed |
|
701 |
||
702 |
lemma unset_bit_eq: |
|
703 |
\<open>unset_bit n k = k - of_bool (bit k n) * 2 ^ n\<close> for k :: int |
|
704 |
proof (rule bit_eqI) |
|
705 |
fix m |
|
706 |
show \<open>bit (unset_bit n k) m \<longleftrightarrow> bit (k - of_bool (bit k n) * 2 ^ n) m\<close> |
|
707 |
proof (cases \<open>m = n\<close>) |
|
708 |
case True |
|
709 |
then show ?thesis |
|
710 |
apply (simp add: bit_unset_bit_iff) |
|
711 |
apply (simp add: bit_iff_odd) |
|
712 |
using div_plus_div_distrib_dvd_right [of \<open>2 ^ n\<close> \<open>- (2 ^ n)\<close> k] |
|
713 |
apply (simp add: dvd_neg_div) |
|
714 |
done |
|
715 |
next |
|
716 |
case False |
|
717 |
then show ?thesis |
|
718 |
apply (clarsimp simp add: bit_unset_bit_iff) |
|
719 |
apply (subst disjunctive_diff) |
|
720 |
apply (clarsimp simp add: bit_exp_iff) |
|
721 |
apply (clarsimp simp add: bit_and_iff bit_not_iff bit_exp_iff) |
|
722 |
done |
|
723 |
qed |
|
724 |
qed |
|
725 |
||
71442 | 726 |
|
71956 | 727 |
subsection \<open>Instance \<^typ>\<open>nat\<close>\<close> |
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
728 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
729 |
instantiation nat :: semiring_bit_operations |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
730 |
begin |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
731 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
732 |
definition and_nat :: \<open>nat \<Rightarrow> nat \<Rightarrow> nat\<close> |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
733 |
where \<open>m AND n = nat (int m AND int n)\<close> for m n :: nat |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
734 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
735 |
definition or_nat :: \<open>nat \<Rightarrow> nat \<Rightarrow> nat\<close> |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
736 |
where \<open>m OR n = nat (int m OR int n)\<close> for m n :: nat |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
737 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
738 |
definition xor_nat :: \<open>nat \<Rightarrow> nat \<Rightarrow> nat\<close> |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
739 |
where \<open>m XOR n = nat (int m XOR int n)\<close> for m n :: nat |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
740 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
741 |
instance proof |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
742 |
fix m n q :: nat |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
743 |
show \<open>bit (m AND n) q \<longleftrightarrow> bit m q \<and> bit n q\<close> |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
744 |
by (auto simp add: and_nat_def bit_and_iff less_le bit_eq_iff) |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
745 |
show \<open>bit (m OR n) q \<longleftrightarrow> bit m q \<or> bit n q\<close> |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
746 |
by (auto simp add: or_nat_def bit_or_iff less_le bit_eq_iff) |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
747 |
show \<open>bit (m XOR n) q \<longleftrightarrow> bit m q \<noteq> bit n q\<close> |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
748 |
by (auto simp add: xor_nat_def bit_xor_iff less_le bit_eq_iff) |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
749 |
qed |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
750 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
751 |
end |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
752 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
753 |
lemma and_nat_rec: |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
754 |
\<open>m AND n = of_bool (odd m \<and> odd n) + 2 * ((m div 2) AND (n div 2))\<close> for m n :: nat |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
755 |
by (simp add: and_nat_def and_int_rec [of \<open>int m\<close> \<open>int n\<close>] zdiv_int nat_add_distrib nat_mult_distrib) |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
756 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
757 |
lemma or_nat_rec: |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
758 |
\<open>m OR n = of_bool (odd m \<or> odd n) + 2 * ((m div 2) OR (n div 2))\<close> for m n :: nat |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
759 |
by (simp add: or_nat_def or_int_rec [of \<open>int m\<close> \<open>int n\<close>] zdiv_int nat_add_distrib nat_mult_distrib) |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
760 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
761 |
lemma xor_nat_rec: |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
762 |
\<open>m XOR n = of_bool (odd m \<noteq> odd n) + 2 * ((m div 2) XOR (n div 2))\<close> for m n :: nat |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
763 |
by (simp add: xor_nat_def xor_int_rec [of \<open>int m\<close> \<open>int n\<close>] zdiv_int nat_add_distrib nat_mult_distrib) |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
764 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
765 |
lemma Suc_0_and_eq [simp]: |
71822 | 766 |
\<open>Suc 0 AND n = n mod 2\<close> |
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
767 |
using one_and_eq [of n] by simp |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
768 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
769 |
lemma and_Suc_0_eq [simp]: |
71822 | 770 |
\<open>n AND Suc 0 = n mod 2\<close> |
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
771 |
using and_one_eq [of n] by simp |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
772 |
|
71822 | 773 |
lemma Suc_0_or_eq: |
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
774 |
\<open>Suc 0 OR n = n + of_bool (even n)\<close> |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
775 |
using one_or_eq [of n] by simp |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
776 |
|
71822 | 777 |
lemma or_Suc_0_eq: |
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
778 |
\<open>n OR Suc 0 = n + of_bool (even n)\<close> |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
779 |
using or_one_eq [of n] by simp |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
780 |
|
71822 | 781 |
lemma Suc_0_xor_eq: |
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
782 |
\<open>Suc 0 XOR n = n + of_bool (even n) - of_bool (odd n)\<close> |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
783 |
using one_xor_eq [of n] by simp |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
784 |
|
71822 | 785 |
lemma xor_Suc_0_eq: |
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
786 |
\<open>n XOR Suc 0 = n + of_bool (even n) - of_bool (odd n)\<close> |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
787 |
using xor_one_eq [of n] by simp |
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
788 |
|
6fd70ed18199
simplified construction of binary bit operations
haftmann
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changeset
|
789 |
|
71956 | 790 |
subsection \<open>Instances for \<^typ>\<open>integer\<close> and \<^typ>\<open>natural\<close>\<close> |
71442 | 791 |
|
792 |
unbundle integer.lifting natural.lifting |
|
793 |
||
794 |
instantiation integer :: ring_bit_operations |
|
795 |
begin |
|
796 |
||
797 |
lift_definition not_integer :: \<open>integer \<Rightarrow> integer\<close> |
|
798 |
is not . |
|
799 |
||
800 |
lift_definition and_integer :: \<open>integer \<Rightarrow> integer \<Rightarrow> integer\<close> |
|
801 |
is \<open>and\<close> . |
|
802 |
||
803 |
lift_definition or_integer :: \<open>integer \<Rightarrow> integer \<Rightarrow> integer\<close> |
|
804 |
is or . |
|
805 |
||
806 |
lift_definition xor_integer :: \<open>integer \<Rightarrow> integer \<Rightarrow> integer\<close> |
|
807 |
is xor . |
|
808 |
||
809 |
instance proof |
|
810 |
fix k l :: \<open>integer\<close> and n :: nat |
|
811 |
show \<open>- k = NOT (k - 1)\<close> |
|
812 |
by transfer (simp add: minus_eq_not_minus_1) |
|
813 |
show \<open>bit (NOT k) n \<longleftrightarrow> (2 :: integer) ^ n \<noteq> 0 \<and> \<not> bit k n\<close> |
|
814 |
by transfer (fact bit_not_iff) |
|
815 |
show \<open>bit (k AND l) n \<longleftrightarrow> bit k n \<and> bit l n\<close> |
|
71804
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changeset
|
816 |
by transfer (fact bit_and_iff) |
71442 | 817 |
show \<open>bit (k OR l) n \<longleftrightarrow> bit k n \<or> bit l n\<close> |
71804
6fd70ed18199
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haftmann
parents:
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diff
changeset
|
818 |
by transfer (fact bit_or_iff) |
71442 | 819 |
show \<open>bit (k XOR l) n \<longleftrightarrow> bit k n \<noteq> bit l n\<close> |
71804
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haftmann
parents:
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diff
changeset
|
820 |
by transfer (fact bit_xor_iff) |
71442 | 821 |
qed |
822 |
||
823 |
end |
|
824 |
||
825 |
instantiation natural :: semiring_bit_operations |
|
826 |
begin |
|
827 |
||
828 |
lift_definition and_natural :: \<open>natural \<Rightarrow> natural \<Rightarrow> natural\<close> |
|
829 |
is \<open>and\<close> . |
|
830 |
||
831 |
lift_definition or_natural :: \<open>natural \<Rightarrow> natural \<Rightarrow> natural\<close> |
|
832 |
is or . |
|
833 |
||
834 |
lift_definition xor_natural :: \<open>natural \<Rightarrow> natural \<Rightarrow> natural\<close> |
|
835 |
is xor . |
|
836 |
||
837 |
instance proof |
|
838 |
fix m n :: \<open>natural\<close> and q :: nat |
|
839 |
show \<open>bit (m AND n) q \<longleftrightarrow> bit m q \<and> bit n q\<close> |
|
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
840 |
by transfer (fact bit_and_iff) |
71442 | 841 |
show \<open>bit (m OR n) q \<longleftrightarrow> bit m q \<or> bit n q\<close> |
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
842 |
by transfer (fact bit_or_iff) |
71442 | 843 |
show \<open>bit (m XOR n) q \<longleftrightarrow> bit m q \<noteq> bit n q\<close> |
71804
6fd70ed18199
simplified construction of binary bit operations
haftmann
parents:
71802
diff
changeset
|
844 |
by transfer (fact bit_xor_iff) |
71442 | 845 |
qed |
846 |
||
847 |
end |
|
848 |
||
849 |
lifting_update integer.lifting |
|
850 |
lifting_forget integer.lifting |
|
851 |
||
852 |
lifting_update natural.lifting |
|
853 |
lifting_forget natural.lifting |
|
854 |
||
71800 | 855 |
|
856 |
subsection \<open>Key ideas of bit operations\<close> |
|
857 |
||
858 |
text \<open> |
|
859 |
When formalizing bit operations, it is tempting to represent |
|
860 |
bit values as explicit lists over a binary type. This however |
|
861 |
is a bad idea, mainly due to the inherent ambiguities in |
|
862 |
representation concerning repeating leading bits. |
|
863 |
||
864 |
Hence this approach avoids such explicit lists altogether |
|
865 |
following an algebraic path: |
|
866 |
||
867 |
\<^item> Bit values are represented by numeric types: idealized |
|
868 |
unbounded bit values can be represented by type \<^typ>\<open>int\<close>, |
|
869 |
bounded bit values by quotient types over \<^typ>\<open>int\<close>. |
|
870 |
||
871 |
\<^item> (A special case are idealized unbounded bit values ending |
|
872 |
in @{term [source] 0} which can be represented by type \<^typ>\<open>nat\<close> but |
|
873 |
only support a restricted set of operations). |
|
874 |
||
875 |
\<^item> From this idea follows that |
|
876 |
||
877 |
\<^item> multiplication by \<^term>\<open>2 :: int\<close> is a bit shift to the left and |
|
878 |
||
879 |
\<^item> division by \<^term>\<open>2 :: int\<close> is a bit shift to the right. |
|
880 |
||
881 |
\<^item> Concerning bounded bit values, iterated shifts to the left |
|
882 |
may result in eliminating all bits by shifting them all |
|
883 |
beyond the boundary. The property \<^prop>\<open>(2 :: int) ^ n \<noteq> 0\<close> |
|
884 |
represents that \<^term>\<open>n\<close> is \<^emph>\<open>not\<close> beyond that boundary. |
|
885 |
||
71965
d45f5d4c41bd
more class operations for the sake of efficient generated code
haftmann
parents:
71956
diff
changeset
|
886 |
\<^item> The projection on a single bit is then @{thm bit_iff_odd [where ?'a = int, no_vars]}. |
71800 | 887 |
|
888 |
\<^item> This leads to the most fundamental properties of bit values: |
|
889 |
||
890 |
\<^item> Equality rule: @{thm bit_eqI [where ?'a = int, no_vars]} |
|
891 |
||
892 |
\<^item> Induction rule: @{thm bits_induct [where ?'a = int, no_vars]} |
|
893 |
||
894 |
\<^item> Typical operations are characterized as follows: |
|
895 |
||
896 |
\<^item> Singleton \<^term>\<open>n\<close>th bit: \<^term>\<open>(2 :: int) ^ n\<close> |
|
897 |
||
71956 | 898 |
\<^item> Bit mask upto bit \<^term>\<open>n\<close>: @{thm mask_eq_exp_minus_1 [where ?'a = int, no_vars]} |
71800 | 899 |
|
900 |
\<^item> Left shift: @{thm push_bit_eq_mult [where ?'a = int, no_vars]} |
|
901 |
||
902 |
\<^item> Right shift: @{thm drop_bit_eq_div [where ?'a = int, no_vars]} |
|
903 |
||
904 |
\<^item> Truncation: @{thm take_bit_eq_mod [where ?'a = int, no_vars]} |
|
905 |
||
906 |
\<^item> Negation: @{thm bit_not_iff [where ?'a = int, no_vars]} |
|
907 |
||
908 |
\<^item> And: @{thm bit_and_iff [where ?'a = int, no_vars]} |
|
909 |
||
910 |
\<^item> Or: @{thm bit_or_iff [where ?'a = int, no_vars]} |
|
911 |
||
912 |
\<^item> Xor: @{thm bit_xor_iff [where ?'a = int, no_vars]} |
|
913 |
||
914 |
\<^item> Set a single bit: @{thm set_bit_def [where ?'a = int, no_vars]} |
|
915 |
||
916 |
\<^item> Unset a single bit: @{thm unset_bit_def [where ?'a = int, no_vars]} |
|
917 |
||
918 |
\<^item> Flip a single bit: @{thm flip_bit_def [where ?'a = int, no_vars]} |
|
919 |
\<close> |
|
920 |
||
71442 | 921 |
end |