author | haftmann |
Sun, 09 Feb 2020 10:46:32 +0000 | |
changeset 71424 | e83fe2c31088 |
parent 71420 | 572ab9e64e18 |
child 71426 | 745e518d3d0b |
permissions | -rw-r--r-- |
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(* Author: Florian Haftmann, TUM |
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*) |
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section \<open>Proof of concept for purely algebraically founded lists of bits\<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 in suitable algebraic structures\<close> |
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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 "AND" 64) |
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and or :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close> (infixr "OR" 59) |
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and xor :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close> (infixr "XOR" 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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definition map_bit :: \<open>nat \<Rightarrow> (bool \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a\<close> |
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where \<open>map_bit n f a = take_bit n a + push_bit n (of_bool (f (bit a n)) + 2 * drop_bit (Suc n) a)\<close> |
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definition set_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close> |
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where \<open>set_bit n = map_bit n top\<close> |
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definition unset_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close> |
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where \<open>unset_bit n = map_bit n bot\<close> |
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definition flip_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close> |
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where \<open>flip_bit n = map_bit n Not\<close> |
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text \<open> |
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Having |
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\<^const>\<open>set_bit\<close>, \<^const>\<open>unset_bit\<close> and \<^const>\<open>flip_bit\<close> as separate |
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operations allows to implement them using bit masks later. |
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\<close> |
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lemma stable_imp_drop_eq: |
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\<open>drop_bit n a = a\<close> if \<open>a div 2 = a\<close> |
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by (induction n) (simp_all add: that) |
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lemma map_bit_0 [simp]: |
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\<open>map_bit 0 f a = of_bool (f (odd a)) + 2 * (a div 2)\<close> |
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by (simp add: map_bit_def) |
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lemma map_bit_Suc [simp]: |
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\<open>map_bit (Suc n) f a = a mod 2 + 2 * map_bit n f (a div 2)\<close> |
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by (auto simp add: map_bit_def algebra_simps mod2_eq_if push_bit_add mult_2 |
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elim: evenE oddE) |
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lemma set_bit_0 [simp]: |
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\<open>set_bit 0 a = 1 + 2 * (a div 2)\<close> |
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by (simp add: set_bit_def) |
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lemma set_bit_Suc [simp]: |
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\<open>set_bit (Suc n) a = a mod 2 + 2 * set_bit n (a div 2)\<close> |
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by (simp add: set_bit_def) |
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lemma unset_bit_0 [simp]: |
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\<open>unset_bit 0 a = 2 * (a div 2)\<close> |
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by (simp add: unset_bit_def) |
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lemma unset_bit_Suc [simp]: |
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\<open>unset_bit (Suc n) a = a mod 2 + 2 * unset_bit n (a div 2)\<close> |
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by (simp add: unset_bit_def) |
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lemma flip_bit_0 [simp]: |
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\<open>flip_bit 0 a = of_bool (even a) + 2 * (a div 2)\<close> |
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by (simp add: flip_bit_def) |
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lemma flip_bit_Suc [simp]: |
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\<open>flip_bit (Suc n) a = a mod 2 + 2 * flip_bit n (a div 2)\<close> |
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by (simp add: flip_bit_def) |
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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 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 [simp]: |
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"1 AND a = of_bool (odd a)" |
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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 [simp]: |
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"a AND 1 = of_bool (odd a)" |
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using one_and_eq [of a] by (simp add: ac_simps) |
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lemma one_or_eq [simp]: |
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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 [simp]: |
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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 [simp]: |
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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 [simp]: |
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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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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 (auto simp add: bit_eq_iff bit_and_iff) |
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apply (simp add: bit_exp_iff) |
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apply (metis local.bit_def local.bit_exp_iff local.bits_div_by_0) |
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done |
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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 |
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apply (auto simp add: bit_eq_iff bit_and_iff bit_or_iff bit_not_iff) |
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apply (simp add: bit_exp_iff) |
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apply (metis local.bit_def local.bit_exp_iff local.bits_div_by_0) |
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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 (auto simp add: fun_eq_iff bit.xor_def bit_eq_iff bit_and_iff bit_or_iff bit_not_iff bit_xor_iff) |
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apply (simp_all add: bit_exp_iff, simp_all add: bit_def) |
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apply (metis local.bit_exp_iff local.bits_div_by_0) |
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apply (metis local.bit_exp_iff local.bits_div_by_0) |
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done |
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qed |
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lemma push_bit_minus: |
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\<open>push_bit n (- a) = - push_bit n a\<close> |
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by (simp add: push_bit_eq_mult) |
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lemma take_bit_not_take_bit: |
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\<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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lemma take_bit_not_iff: |
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"take_bit n (NOT a) = take_bit n (NOT b) \<longleftrightarrow> take_bit n a = take_bit n b" |
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apply (simp add: bit_eq_iff bit_not_iff bit_take_bit_iff) |
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apply (simp add: bit_exp_iff) |
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apply (use local.exp_eq_0_imp_not_bit in blast) |
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done |
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end |
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subsubsection \<open>Instance \<^typ>\<open>nat\<close>\<close> |
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locale zip_nat = single: abel_semigroup f |
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for f :: "bool \<Rightarrow> bool \<Rightarrow> bool" (infixl \<open>\<^bold>*\<close> 70) + |
243 |
assumes end_of_bits: \<open>\<not> False \<^bold>* False\<close> |
|
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begin |
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|
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function F :: \<open>nat \<Rightarrow> nat \<Rightarrow> nat\<close> (infixl \<open>\<^bold>\<times>\<close> 70) |
247 |
where \<open>m \<^bold>\<times> n = (if m = 0 \<and> n = 0 then 0 |
|
248 |
else of_bool (odd m \<^bold>* odd n) + 2 * ((m div 2) \<^bold>\<times> (n div 2)))\<close> |
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by auto |
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250 |
|
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termination |
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by (relation "measure (case_prod (+))") auto |
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|
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declare F.simps [simp del] |
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255 |
|
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lemma rec: |
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"m \<^bold>\<times> n = of_bool (odd m \<^bold>* odd n) + (m div 2) \<^bold>\<times> (n div 2) * 2" |
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proof (cases \<open>m = 0 \<and> n = 0\<close>) |
259 |
case True |
|
260 |
then have \<open>m \<^bold>\<times> n = 0\<close> |
|
261 |
using True by (simp add: F.simps [of 0 0]) |
|
262 |
moreover have \<open>(m div 2) \<^bold>\<times> (n div 2) = m \<^bold>\<times> n\<close> |
|
263 |
using True by simp |
|
264 |
ultimately show ?thesis |
|
265 |
using True by (simp add: end_of_bits) |
|
266 |
next |
|
267 |
case False |
|
268 |
then show ?thesis |
|
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by (auto simp add: ac_simps F.simps [of m n]) |
|
270 |
qed |
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|
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lemma bit_eq_iff: |
273 |
\<open>bit (m \<^bold>\<times> n) q \<longleftrightarrow> bit m q \<^bold>* bit n q\<close> |
|
274 |
proof (induction q arbitrary: m n) |
|
275 |
case 0 |
|
276 |
then show ?case |
|
277 |
by (simp add: rec [of m n]) |
|
278 |
next |
|
279 |
case (Suc n) |
|
280 |
then show ?case |
|
281 |
by (simp add: rec [of m n]) |
|
282 |
qed |
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sublocale abel_semigroup F |
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by standard (simp_all add: Parity.bit_eq_iff bit_eq_iff ac_simps) |
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286 |
|
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287 |
end |
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|
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instantiation nat :: semiring_bit_operations |
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290 |
begin |
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291 |
|
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global_interpretation and_nat: zip_nat \<open>(\<and>)\<close> |
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defines and_nat = and_nat.F |
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by standard auto |
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|
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global_interpretation and_nat: semilattice \<open>(AND) :: nat \<Rightarrow> nat \<Rightarrow> nat\<close> |
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proof (rule semilattice.intro, fact and_nat.abel_semigroup_axioms, standard) |
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show \<open>n AND n = n\<close> for n :: nat |
299 |
by (simp add: bit_eq_iff and_nat.bit_eq_iff) |
|
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qed |
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301 |
|
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global_interpretation or_nat: zip_nat \<open>(\<or>)\<close> |
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defines or_nat = or_nat.F |
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by standard auto |
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|
71420 | 306 |
global_interpretation or_nat: semilattice \<open>(OR) :: nat \<Rightarrow> nat \<Rightarrow> nat\<close> |
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proof (rule semilattice.intro, fact or_nat.abel_semigroup_axioms, standard) |
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show \<open>n OR n = n\<close> for n :: nat |
309 |
by (simp add: bit_eq_iff or_nat.bit_eq_iff) |
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qed |
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|
71420 | 312 |
global_interpretation xor_nat: zip_nat \<open>(\<noteq>)\<close> |
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defines xor_nat = xor_nat.F |
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by standard auto |
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|
71186 | 316 |
instance proof |
317 |
fix m n q :: nat |
|
318 |
show \<open>bit (m AND n) q \<longleftrightarrow> bit m q \<and> bit n q\<close> |
|
71420 | 319 |
by (fact and_nat.bit_eq_iff) |
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show \<open>bit (m OR n) q \<longleftrightarrow> bit m q \<or> bit n q\<close> |
71420 | 321 |
by (fact or_nat.bit_eq_iff) |
71186 | 322 |
show \<open>bit (m XOR n) q \<longleftrightarrow> bit m q \<noteq> bit n q\<close> |
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by (fact xor_nat.bit_eq_iff) |
71186 | 324 |
qed |
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325 |
|
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326 |
end |
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327 |
|
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lemma Suc_0_and_eq [simp]: |
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\<open>Suc 0 AND n = of_bool (odd n)\<close> |
330 |
using one_and_eq [of n] by simp |
|
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331 |
|
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332 |
lemma and_Suc_0_eq [simp]: |
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\<open>n AND Suc 0 = of_bool (odd n)\<close> |
334 |
using and_one_eq [of n] by simp |
|
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335 |
|
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336 |
lemma Suc_0_or_eq [simp]: |
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\<open>Suc 0 OR n = n + of_bool (even n)\<close> |
338 |
using one_or_eq [of n] by simp |
|
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339 |
|
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340 |
lemma or_Suc_0_eq [simp]: |
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\<open>n OR Suc 0 = n + of_bool (even n)\<close> |
342 |
using or_one_eq [of n] by simp |
|
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343 |
|
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344 |
lemma Suc_0_xor_eq [simp]: |
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\<open>Suc 0 XOR n = n + of_bool (even n) - of_bool (odd n)\<close> |
346 |
using one_xor_eq [of n] by simp |
|
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347 |
|
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348 |
lemma xor_Suc_0_eq [simp]: |
71419 | 349 |
\<open>n XOR Suc 0 = n + of_bool (even n) - of_bool (odd n)\<close> |
350 |
using xor_one_eq [of n] by simp |
|
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351 |
|
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352 |
|
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353 |
subsubsection \<open>Instance \<^typ>\<open>int\<close>\<close> |
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354 |
|
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355 |
locale zip_int = single: abel_semigroup f |
71420 | 356 |
for f :: \<open>bool \<Rightarrow> bool \<Rightarrow> bool\<close> (infixl \<open>\<^bold>*\<close> 70) |
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|
357 |
begin |
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|
358 |
|
71420 | 359 |
function F :: \<open>int \<Rightarrow> int \<Rightarrow> int\<close> (infixl \<open>\<^bold>\<times>\<close> 70) |
360 |
where \<open>k \<^bold>\<times> l = (if k \<in> {0, - 1} \<and> l \<in> {0, - 1} |
|
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|
361 |
then - of_bool (odd k \<^bold>* odd l) |
71420 | 362 |
else of_bool (odd k \<^bold>* odd l) + 2 * ((k div 2) \<^bold>\<times> (l div 2)))\<close> |
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|
363 |
by auto |
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|
364 |
|
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|
365 |
termination |
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|
366 |
by (relation "measure (\<lambda>(k, l). nat (\<bar>k\<bar> + \<bar>l\<bar>))") auto |
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|
367 |
|
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|
368 |
declare F.simps [simp del] |
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|
369 |
|
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|
370 |
lemma rec: |
71420 | 371 |
\<open>k \<^bold>\<times> l = of_bool (odd k \<^bold>* odd l) + 2 * ((k div 2) \<^bold>\<times> (l div 2))\<close> |
372 |
proof (cases \<open>k \<in> {0, - 1} \<and> l \<in> {0, - 1}\<close>) |
|
373 |
case True |
|
374 |
then have \<open>(k div 2) \<^bold>\<times> (l div 2) = k \<^bold>\<times> l\<close> |
|
375 |
by auto |
|
376 |
moreover have \<open>of_bool (odd k \<^bold>* odd l) = - (k \<^bold>\<times> l)\<close> |
|
377 |
using True by (simp add: F.simps [of k l]) |
|
378 |
ultimately show ?thesis by simp |
|
379 |
next |
|
380 |
case False |
|
381 |
then show ?thesis |
|
382 |
by (auto simp add: ac_simps F.simps [of k l]) |
|
383 |
qed |
|
384 |
||
385 |
lemma bit_eq_iff: |
|
386 |
\<open>bit (k \<^bold>\<times> l) n \<longleftrightarrow> bit k n \<^bold>* bit l n\<close> |
|
387 |
proof (induction n arbitrary: k l) |
|
388 |
case 0 |
|
389 |
then show ?case |
|
390 |
by (simp add: rec [of k l]) |
|
391 |
next |
|
392 |
case (Suc n) |
|
393 |
then show ?case |
|
394 |
by (simp add: rec [of k l]) |
|
395 |
qed |
|
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|
396 |
|
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|
397 |
sublocale abel_semigroup F |
71420 | 398 |
by standard (simp_all add: Parity.bit_eq_iff bit_eq_iff ac_simps) |
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|
399 |
|
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|
400 |
end |
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haftmann
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|
401 |
|
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haftmann
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|
402 |
instantiation int :: ring_bit_operations |
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haftmann
parents:
diff
changeset
|
403 |
begin |
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haftmann
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|
404 |
|
400e9512f1d3
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haftmann
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|
405 |
global_interpretation and_int: zip_int "(\<and>)" |
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haftmann
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|
406 |
defines and_int = and_int.F |
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proof-of-concept theory for bit operations without a constructivistic representation and a minimal common logical foundation
haftmann
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diff
changeset
|
407 |
by standard |
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haftmann
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changeset
|
408 |
|
400e9512f1d3
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haftmann
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|
409 |
global_interpretation and_int: semilattice "(AND) :: int \<Rightarrow> int \<Rightarrow> int" |
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haftmann
parents:
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changeset
|
410 |
proof (rule semilattice.intro, fact and_int.abel_semigroup_axioms, standard) |
400e9512f1d3
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haftmann
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changeset
|
411 |
show "k AND k = k" for k :: int |
71420 | 412 |
by (simp add: bit_eq_iff and_int.bit_eq_iff) |
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|
413 |
qed |
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|
414 |
|
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haftmann
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|
415 |
global_interpretation or_int: zip_int "(\<or>)" |
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haftmann
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|
416 |
defines or_int = or_int.F |
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417 |
by standard |
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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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418 |
|
400e9512f1d3
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419 |
global_interpretation or_int: semilattice "(OR) :: int \<Rightarrow> int \<Rightarrow> int" |
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420 |
proof (rule semilattice.intro, fact or_int.abel_semigroup_axioms, standard) |
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421 |
show "k OR k = k" for k :: int |
71420 | 422 |
by (simp add: bit_eq_iff or_int.bit_eq_iff) |
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|
423 |
qed |
400e9512f1d3
proof-of-concept theory for bit operations without a constructivistic representation and a minimal common logical foundation
haftmann
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|
424 |
|
400e9512f1d3
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425 |
global_interpretation xor_int: zip_int "(\<noteq>)" |
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426 |
defines xor_int = xor_int.F |
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|
427 |
by standard |
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428 |
|
71420 | 429 |
definition not_int :: \<open>int \<Rightarrow> int\<close> |
430 |
where \<open>not_int k = - k - 1\<close> |
|
431 |
||
432 |
lemma not_int_rec: |
|
433 |
"NOT k = of_bool (even k) + 2 * NOT (k div 2)" for k :: int |
|
434 |
by (auto simp add: not_int_def elim: oddE) |
|
435 |
||
436 |
lemma even_not_iff_int: |
|
437 |
\<open>even (NOT k) \<longleftrightarrow> odd k\<close> for k :: int |
|
438 |
by (simp add: not_int_def) |
|
439 |
||
440 |
lemma not_int_div_2: |
|
441 |
\<open>NOT k div 2 = NOT (k div 2)\<close> for k :: int |
|
442 |
by (simp add: not_int_def) |
|
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443 |
|
71186 | 444 |
lemma bit_not_iff_int: |
445 |
\<open>bit (NOT k) n \<longleftrightarrow> \<not> bit k n\<close> |
|
446 |
for k :: int |
|
71420 | 447 |
by (induction n arbitrary: k) (simp_all add: not_int_div_2 even_not_iff_int) |
71186 | 448 |
|
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449 |
instance proof |
71186 | 450 |
fix k l :: int and n :: nat |
71409 | 451 |
show \<open>- k = NOT (k - 1)\<close> |
452 |
by (simp add: not_int_def) |
|
71186 | 453 |
show \<open>bit (k AND l) n \<longleftrightarrow> bit k n \<and> bit l n\<close> |
71420 | 454 |
by (fact and_int.bit_eq_iff) |
71186 | 455 |
show \<open>bit (k OR l) n \<longleftrightarrow> bit k n \<or> bit l n\<close> |
71420 | 456 |
by (fact or_int.bit_eq_iff) |
71186 | 457 |
show \<open>bit (k XOR l) n \<longleftrightarrow> bit k n \<noteq> bit l n\<close> |
71420 | 458 |
by (fact xor_int.bit_eq_iff) |
459 |
qed (simp_all add: bit_not_iff_int) |
|
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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parents:
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|
460 |
|
400e9512f1d3
proof-of-concept theory for bit operations without a constructivistic representation and a minimal common logical foundation
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|
461 |
end |
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|
462 |
|
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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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|
463 |
end |