author  haftmann 
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child 36409  d323e7773aa8 
permissions  rwrr 
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(* Title: Bit.thy 
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Author: Brian Huffman 
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*) 
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header {* The Field of Integers mod 2 *} 
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theory Bit 
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imports Main 
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begin 
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subsection {* Bits as a datatype *} 
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typedef (open) bit = "UNIV :: bool set" .. 
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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 = Abs_bit False" 
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definition one_bit_def: 
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"1 = Abs_bit True" 
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instance .. 
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end 
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rep_datatype (bit) "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 (Abs_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: Abs_bit_inject) 
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qed 
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lemma bit_not_0_iff [iff]: "(x::bit) \<noteq> 0 \<longleftrightarrow> x = 1" 
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by (induct x) simp_all 
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lemma bit_not_1_iff [iff]: "(x::bit) \<noteq> 1 \<longleftrightarrow> x = 0" 
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by (induct x) simp_all 
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subsection {* Type @{typ bit} forms a field *} 
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instantiation bit :: "{field, division_ring_inverse_zero}" 
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begin 
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definition plus_bit_def: 
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"x + y = bit_case y (bit_case 1 0 y) x" 
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definition times_bit_def: 
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"x * y = bit_case 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 / 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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unfolding plus_bit_def by (simp split: bit.split) 

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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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unfolding times_bit_def by (simp split: bit.split) 
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text {* Not sure whether the next two should be simp rules. *} 
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lemma bit_add_eq_0_iff: "x + y = (0 :: bit) \<longleftrightarrow> x = y" 
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unfolding plus_bit_def by (simp split: bit.split) 
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lemma bit_add_eq_1_iff: "x + y = (1 :: bit) \<longleftrightarrow> x \<noteq> y" 
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unfolding plus_bit_def by (simp split: bit.split) 
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subsection {* Numerals at type @{typ bit} *} 
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instantiation bit :: number_ring 
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begin 
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definition number_of_bit_def: 
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"(number_of w :: bit) = of_int w" 
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instance proof 
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qed (rule number_of_bit_def) 
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end 
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text {* All numerals reduce to either 0 or 1. *} 
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lemma bit_minus1 [simp]: "1 = (1 :: bit)" 
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by (simp only: number_of_Min uminus_bit_def) 

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lemma bit_number_of_even [simp]: "number_of (Int.Bit0 w) = (0 :: bit)" 
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by (simp only: number_of_Bit0 add_0_left bit_add_self) 
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lemma bit_number_of_odd [simp]: "number_of (Int.Bit1 w) = (1 :: bit)" 
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by (simp only: number_of_Bit1 add_assoc bit_add_self 
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monoid_add_class.add_0_right) 
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end 