src/HOL/Library/Bit.thy
 author haftmann Tue Feb 19 19:44:10 2013 +0100 (2013-02-19) changeset 51188 9b5bf1a9a710 parent 49834 b27bbb021df1 child 53063 8f7ac535892f permissions -rw-r--r--
dropped spurious left-over from 0a2371e7ced3
1 (*  Title:      HOL/Library/Bit.thy
2     Author:     Brian Huffman
3 *)
5 header {* The Field of Integers mod 2 *}
7 theory Bit
8 imports Main
9 begin
11 subsection {* Bits as a datatype *}
13 typedef bit = "UNIV :: bool set" ..
15 instantiation bit :: "{zero, one}"
16 begin
18 definition zero_bit_def:
19   "0 = Abs_bit False"
21 definition one_bit_def:
22   "1 = Abs_bit True"
24 instance ..
26 end
28 rep_datatype "0::bit" "1::bit"
29 proof -
30   fix P and x :: bit
31   assume "P (0::bit)" and "P (1::bit)"
32   then have "\<forall>b. P (Abs_bit b)"
33     unfolding zero_bit_def one_bit_def
34     by (simp add: all_bool_eq)
35   then show "P x"
36     by (induct x) simp
37 next
38   show "(0::bit) \<noteq> (1::bit)"
39     unfolding zero_bit_def one_bit_def
40     by (simp add: Abs_bit_inject)
41 qed
43 lemma bit_not_0_iff [iff]: "(x::bit) \<noteq> 0 \<longleftrightarrow> x = 1"
44   by (induct x) simp_all
46 lemma bit_not_1_iff [iff]: "(x::bit) \<noteq> 1 \<longleftrightarrow> x = 0"
47   by (induct x) simp_all
50 subsection {* Type @{typ bit} forms a field *}
52 instantiation bit :: field_inverse_zero
53 begin
55 definition plus_bit_def:
56   "x + y = bit_case y (bit_case 1 0 y) x"
58 definition times_bit_def:
59   "x * y = bit_case 0 y x"
61 definition uminus_bit_def [simp]:
62   "- x = (x :: bit)"
64 definition minus_bit_def [simp]:
65   "x - y = (x + y :: bit)"
67 definition inverse_bit_def [simp]:
68   "inverse x = (x :: bit)"
70 definition divide_bit_def [simp]:
71   "x / y = (x * y :: bit)"
73 lemmas field_bit_defs =
74   plus_bit_def times_bit_def minus_bit_def uminus_bit_def
75   divide_bit_def inverse_bit_def
77 instance proof
78 qed (unfold field_bit_defs, auto split: bit.split)
80 end
82 lemma bit_add_self: "x + x = (0 :: bit)"
83   unfolding plus_bit_def by (simp split: bit.split)
85 lemma bit_mult_eq_1_iff [simp]: "x * y = (1 :: bit) \<longleftrightarrow> x = 1 \<and> y = 1"
86   unfolding times_bit_def by (simp split: bit.split)
88 text {* Not sure whether the next two should be simp rules. *}
90 lemma bit_add_eq_0_iff: "x + y = (0 :: bit) \<longleftrightarrow> x = y"
91   unfolding plus_bit_def by (simp split: bit.split)
93 lemma bit_add_eq_1_iff: "x + y = (1 :: bit) \<longleftrightarrow> x \<noteq> y"
94   unfolding plus_bit_def by (simp split: bit.split)
97 subsection {* Numerals at type @{typ bit} *}
99 text {* All numerals reduce to either 0 or 1. *}
101 lemma bit_minus1 [simp]: "-1 = (1 :: bit)"
102   by (simp only: minus_one [symmetric] uminus_bit_def)
104 lemma bit_neg_numeral [simp]: "(neg_numeral w :: bit) = numeral w"
105   by (simp only: neg_numeral_def uminus_bit_def)
107 lemma bit_numeral_even [simp]: "numeral (Num.Bit0 w) = (0 :: bit)"
108   by (simp only: numeral_Bit0 bit_add_self)
110 lemma bit_numeral_odd [simp]: "numeral (Num.Bit1 w) = (1 :: bit)"
111   by (simp only: numeral_Bit1 bit_add_self add_0_left)
113 end