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permissions  rwrr 
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(* Title: HOL/Library/Bit.thy 
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 Bit 
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imports Main 
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begin 
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subsection \<open>Bits as a datatype\<close> 
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typedef bit = "UNIV :: bool set" 
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morphisms set Bit 
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.. 
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instantiation bit :: "{zero, one}" 
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begin 
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definition zero_bit_def: 
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"0 = Bit False" 
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definition one_bit_def: 
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"1 = Bit True" 
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instance .. 
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end 
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old_rep_datatype "0::bit" "1::bit" 
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proof  
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fix P and x :: bit 
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assume "P (0::bit)" and "P (1::bit)" 
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then have "\<forall>b. P (Bit b)" 
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unfolding zero_bit_def one_bit_def 
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by (simp add: all_bool_eq) 
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then show "P x" 
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by (induct x) simp 
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next 
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show "(0::bit) \<noteq> (1::bit)" 
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unfolding zero_bit_def one_bit_def 
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by (simp add: Bit_inject) 
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qed 
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lemma Bit_set_eq [simp]: 
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"Bit (set b) = b" 
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by (fact set_inverse) 
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lemma set_Bit_eq [simp]: 
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"set (Bit P) = P" 
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by (rule Bit_inverse) rule 
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lemma bit_eq_iff: 
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"x = y \<longleftrightarrow> (set x \<longleftrightarrow> set y)" 
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by (auto simp add: set_inject) 
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lemma Bit_inject [simp]: 
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"Bit P = Bit Q \<longleftrightarrow> (P \<longleftrightarrow> Q)" 
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by (auto simp add: Bit_inject) 
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lemma set [iff]: 
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"\<not> set 0" 
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"set 1" 
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by (simp_all add: zero_bit_def one_bit_def Bit_inverse) 
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lemma [code]: 
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"set 0 \<longleftrightarrow> False" 
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"set 1 \<longleftrightarrow> True" 
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by simp_all 
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lemma set_iff: 
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"set b \<longleftrightarrow> b = 1" 
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by (cases b) simp_all 
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lemma bit_eq_iff_set: 
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"b = 0 \<longleftrightarrow> \<not> set b" 
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"b = 1 \<longleftrightarrow> set b" 
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by (simp_all add: bit_eq_iff) 
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lemma Bit [simp, code]: 
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"Bit False = 0" 
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"Bit True = 1" 
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by (simp_all add: zero_bit_def one_bit_def) 
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lemma bit_not_0_iff [iff]: 
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"(x::bit) \<noteq> 0 \<longleftrightarrow> x = 1" 
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by (simp add: bit_eq_iff) 
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lemma bit_not_1_iff [iff]: 
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"(x::bit) \<noteq> 1 \<longleftrightarrow> x = 0" 
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by (simp add: bit_eq_iff) 
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lemma [code]: 
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"HOL.equal 0 b \<longleftrightarrow> \<not> set b" 
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"HOL.equal 1 b \<longleftrightarrow> set b" 
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by (simp_all add: equal set_iff) 
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subsection \<open>Type @{typ bit} forms a field\<close> 
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instantiation bit :: field 
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begin 
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definition plus_bit_def: 
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"x + y = case_bit y (case_bit 1 0 y) x" 
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definition times_bit_def: 
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"x * y = case_bit 0 y x" 
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definition uminus_bit_def [simp]: 
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" x = (x :: bit)" 
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definition minus_bit_def [simp]: 
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"x  y = (x + y :: bit)" 
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definition inverse_bit_def [simp]: 
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"inverse x = (x :: bit)" 
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definition divide_bit_def [simp]: 
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"x div y = (x * y :: bit)" 
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lemmas field_bit_defs = 
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plus_bit_def times_bit_def minus_bit_def uminus_bit_def 
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divide_bit_def inverse_bit_def 
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instance proof 
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qed (unfold field_bit_defs, auto split: bit.split) 
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end 
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lemma bit_add_self: "x + x = (0 :: bit)" 
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lemma bit_mult_eq_1_iff [simp]: "x * y = (1 :: bit) \<longleftrightarrow> x = 1 \<and> y = 1" 
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text \<open>Not sure whether the next two should be simp rules.\<close> 
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lemma bit_add_eq_0_iff: "x + y = (0 :: bit) \<longleftrightarrow> x = y" 
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lemma bit_add_eq_1_iff: "x + y = (1 :: bit) \<longleftrightarrow> x \<noteq> y" 
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subsection \<open>Numerals at type @{typ bit}\<close> 
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text \<open>All numerals reduce to either 0 or 1.\<close> 
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lemma bit_minus1 [simp]: " 1 = (1 :: bit)" 
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by (simp only: uminus_bit_def) 
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lemma bit_neg_numeral [simp]: "( numeral w :: bit) = numeral w" 
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by (simp only: uminus_bit_def) 
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lemma bit_numeral_even [simp]: "numeral (Num.Bit0 w) = (0 :: bit)" 
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by (simp only: numeral_Bit0 bit_add_self) 
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158 

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lemma bit_numeral_odd [simp]: "numeral (Num.Bit1 w) = (1 :: bit)" 
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by (simp only: numeral_Bit1 bit_add_self add_0_left) 
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161 

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60500  163 
subsection \<open>Conversion from @{typ bit}\<close> 
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164 

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context zero_neq_one 
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begin 
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167 

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definition of_bit :: "bit \<Rightarrow> 'a" 
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where 
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"of_bit b = case_bit 0 1 b" 
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171 

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lemma of_bit_eq [simp, code]: 
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"of_bit 0 = 0" 
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"of_bit 1 = 1" 
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by (simp_all add: of_bit_def) 
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176 

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lemma of_bit_eq_iff: 
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"of_bit x = of_bit y \<longleftrightarrow> x = y" 
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179 
by (cases x) (cases y, simp_all)+ 
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180 

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end 
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182 

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context semiring_1 
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184 
begin 
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185 

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lemma of_nat_of_bit_eq: 
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"of_nat (of_bit b) = of_bit b" 
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by (cases b) simp_all 
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end 
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context ring_1 
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begin 
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194 

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lemma of_int_of_bit_eq: 
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"of_int (of_bit b) = of_bit b" 
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197 
by (cases b) simp_all 
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198 

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end 
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200 

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hide_const (open) set 
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202 

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203 
end 