author  haftmann 
Tue, 04 Aug 2020 09:33:05 +0000  
changeset 72082  41393ecb57ac 
parent 71988  ace45a11a45e 
permissions  rwrr 
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(* Title: HOL/Library/Z2.thy 
41959  2 
Author: Brian Huffman 
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*) 
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section \<open>The Field of Integers mod 2\<close> 
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theory Z2 
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imports Main "HOLLibrary.Bit_Operations" 
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begin 
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text \<open> 
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Note that in most cases \<^typ>\<open>bool\<close> is appropriate hen a binary type is needed; the 
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type provided here, for historical reasons named \<^text>\<open>bit\<close>, is only needed if proper 
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field operations are required. 
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\<close> 
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typedef bit = \<open>UNIV :: bool set\<close> .. 
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instantiation bit :: zero_neq_one 
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begin 
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definition zero_bit :: bit 
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where \<open>0 = Abs_bit False\<close> 

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definition one_bit :: bit 
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where \<open>1 = Abs_bit True\<close> 

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instance 
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by standard (simp add: zero_bit_def one_bit_def Abs_bit_inject) 

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end 
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free_constructors case_bit for \<open>0::bit\<close>  \<open>1::bit\<close> 
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proof  
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fix P :: bool 
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fix a :: bit 

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assume \<open>a = 0 \<Longrightarrow> P\<close> and \<open>a = 1 \<Longrightarrow> P\<close> 

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then show P 

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by (cases a) (auto simp add: zero_bit_def one_bit_def Abs_bit_inject) 

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qed simp 

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lemma bit_not_zero_iff [simp]: 

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\<open>a \<noteq> 0 \<longleftrightarrow> a = 1\<close> for a :: bit 

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by (cases a) simp_all 

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lemma bit_not_one_imp [simp]: 

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\<open>a \<noteq> 1 \<longleftrightarrow> a = 0\<close> for a :: bit 

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by (cases a) simp_all 

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instantiation bit :: semidom_modulo 

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begin 

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definition plus_bit :: \<open>bit \<Rightarrow> bit \<Rightarrow> bit\<close> 

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where \<open>a + b = Abs_bit (Rep_bit a \<noteq> Rep_bit b)\<close> 

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definition minus_bit :: \<open>bit \<Rightarrow> bit \<Rightarrow> bit\<close> 

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where [simp]: \<open>minus_bit = plus\<close> 

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definition times_bit :: \<open>bit \<Rightarrow> bit \<Rightarrow> bit\<close> 

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where \<open>a * b = Abs_bit (Rep_bit a \<and> Rep_bit b)\<close> 

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definition divide_bit :: \<open>bit \<Rightarrow> bit \<Rightarrow> bit\<close> 

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where [simp]: \<open>divide_bit = times\<close> 

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definition modulo_bit :: \<open>bit \<Rightarrow> bit \<Rightarrow> bit\<close> 

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where \<open>a mod b = Abs_bit (Rep_bit a \<and> \<not> Rep_bit b)\<close> 

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instance 

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by standard 

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(auto simp flip: Rep_bit_inject 

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simp add: zero_bit_def one_bit_def plus_bit_def times_bit_def modulo_bit_def Abs_bit_inverse Rep_bit_inverse) 

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end 

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lemma bit_2_eq_0 [simp]: 

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\<open>2 = (0::bit)\<close> 

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by (simp flip: one_add_one add: zero_bit_def plus_bit_def) 

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instance bit :: semiring_parity 

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apply standard 

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apply (auto simp flip: Rep_bit_inject simp add: modulo_bit_def Abs_bit_inverse Rep_bit_inverse) 

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apply (auto simp add: zero_bit_def one_bit_def Abs_bit_inverse Rep_bit_inverse) 

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done 

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lemma Abs_bit_eq_of_bool [code_abbrev]: 

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\<open>Abs_bit = of_bool\<close> 

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by (simp add: fun_eq_iff zero_bit_def one_bit_def) 

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lemma Rep_bit_eq_odd: 

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\<open>Rep_bit = odd\<close> 

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proof  

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have \<open>\<not> Rep_bit 0\<close> 

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by (simp only: zero_bit_def) (subst Abs_bit_inverse, auto) 

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then show ?thesis 

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by (auto simp flip: Rep_bit_inject simp add: fun_eq_iff) 

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qed 
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lemma Rep_bit_iff_odd [code_abbrev]: 
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\<open>Rep_bit b \<longleftrightarrow> odd b\<close> 

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by (simp add: Rep_bit_eq_odd) 

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lemma Not_Rep_bit_iff_even [code_abbrev]: 

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\<open>\<not> Rep_bit b \<longleftrightarrow> even b\<close> 

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by (simp add: Rep_bit_eq_odd) 

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lemma Not_Not_Rep_bit [code_unfold]: 

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\<open>\<not> \<not> Rep_bit b \<longleftrightarrow> Rep_bit b\<close> 

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by simp 

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code_datatype \<open>0::bit\<close> \<open>1::bit\<close> 

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lemma Abs_bit_code [code]: 
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\<open>Abs_bit False = 0\<close> 

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\<open>Abs_bit True = 1\<close> 

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by (simp_all add: Abs_bit_eq_of_bool) 

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lemma Rep_bit_code [code]: 

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\<open>Rep_bit 0 \<longleftrightarrow> False\<close> 

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\<open>Rep_bit 1 \<longleftrightarrow> True\<close> 

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by (simp_all add: Rep_bit_eq_odd) 

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context zero_neq_one 

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begin 

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abbreviation of_bit :: \<open>bit \<Rightarrow> 'a\<close> 

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where \<open>of_bit b \<equiv> of_bool (odd b)\<close> 

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end 

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context 
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begin 

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qualified lemma bit_eq_iff: 
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\<open>a = b \<longleftrightarrow> (even a \<longleftrightarrow> even b)\<close> for a b :: bit 

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by (cases a; cases b) simp_all 

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end 
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lemma modulo_bit_unfold [simp, code]: 
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\<open>a mod b = of_bool (odd a \<and> even b)\<close> for a b :: bit 

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by (simp add: modulo_bit_def Abs_bit_eq_of_bool Rep_bit_eq_odd) 

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lemma power_bit_unfold [simp, code]: 
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\<open>a ^ n = of_bool (odd a \<or> n = 0)\<close> for a :: bit 

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by (cases a) simp_all 

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instantiation bit :: semiring_bits 
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begin 
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definition bit_bit :: \<open>bit \<Rightarrow> nat \<Rightarrow> bool\<close> 
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where [simp]: \<open>bit_bit b n \<longleftrightarrow> odd b \<and> n = 0\<close> 
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instance 
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apply standard 
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apply auto 
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apply (metis bit.exhaust of_bool_eq(2)) 

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done 
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end 
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instantiation bit :: semiring_bit_shifts 

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begin 

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definition push_bit_bit :: \<open>nat \<Rightarrow> bit \<Rightarrow> bit\<close> 

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where [simp]: \<open>push_bit n b = of_bool (odd b \<and> n = 0)\<close> for b :: bit 
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definition drop_bit_bit :: \<open>nat \<Rightarrow> bit \<Rightarrow> bit\<close> 

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where [simp]: \<open>drop_bit_bit = push_bit\<close> 
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definition take_bit_bit :: \<open>nat \<Rightarrow> bit \<Rightarrow> bit\<close> 
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where [simp]: \<open>take_bit n b = of_bool (odd b \<and> n > 0)\<close> for b :: bit 
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instance 

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by standard simp_all 

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end 

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instantiation bit :: semiring_bit_operations 

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begin 

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definition and_bit :: \<open>bit \<Rightarrow> bit \<Rightarrow> bit\<close> 

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where [simp]: \<open>b AND c = of_bool (odd b \<and> odd c)\<close> for b c :: bit 

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definition or_bit :: \<open>bit \<Rightarrow> bit \<Rightarrow> bit\<close> 

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where [simp]: \<open>b OR c = of_bool (odd b \<or> odd c)\<close> for b c :: bit 

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definition xor_bit :: \<open>bit \<Rightarrow> bit \<Rightarrow> bit\<close> 

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where [simp]: \<open>b XOR c = of_bool (odd b \<noteq> odd c)\<close> for b c :: bit 

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definition mask_bit :: \<open>nat \<Rightarrow> bit\<close> 
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where [simp]: \<open>mask_bit n = of_bool (n > 0)\<close> 

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instance 
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by standard auto 
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end 

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lemma add_bit_eq_xor [simp, code]: 

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\<open>(+) = ((XOR) :: bit \<Rightarrow> _)\<close> 

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by (auto simp add: fun_eq_iff) 

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lemma mult_bit_eq_and [simp, code]: 

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\<open>(*) = ((AND) :: bit \<Rightarrow> _)\<close> 

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by (simp add: fun_eq_iff) 

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instantiation bit :: field 

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begin 

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definition uminus_bit :: \<open>bit \<Rightarrow> bit\<close> 

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where [simp]: \<open>uminus_bit = id\<close> 

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definition inverse_bit :: \<open>bit \<Rightarrow> bit\<close> 

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where [simp]: \<open>inverse_bit = id\<close> 

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instance 

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by standard simp_all 

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end 

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instantiation bit :: ring_bit_operations 
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begin 
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definition not_bit :: \<open>bit \<Rightarrow> bit\<close> 
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where [simp]: \<open>NOT b = of_bool (even b)\<close> for b :: bit 
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instance 
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by standard simp_all 
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end 
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lemma bit_numeral_even [simp]: "numeral (Num.Bit0 w) = (0 :: bit)" 
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by (simp only: Z2.bit_eq_iff even_numeral) simp 

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lemma bit_numeral_odd [simp]: "numeral (Num.Bit1 w) = (1 :: bit)" 
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by (simp only: Z2.bit_eq_iff odd_numeral) simp 

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end 