src/HOL/Library/Bit.thy
 author huffman Sun, 25 Mar 2012 20:15:39 +0200 changeset 47108 2a1953f0d20d parent 45701 615da8b8d758 child 49834 b27bbb021df1 permissions -rw-r--r--
merged fork with new numeral representation (see NEWS)
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(*  Title:      HOL/Library/Bit.thy
Author:     Brian Huffman
*)

header {* The Field of Integers mod 2 *}

theory Bit
imports Main
begin

subsection {* Bits as a datatype *}

typedef (open) bit = "UNIV :: bool set" ..

instantiation bit :: "{zero, one}"
begin

definition zero_bit_def:
"0 = Abs_bit False"

definition one_bit_def:
"1 = Abs_bit True"

instance ..

end

rep_datatype "0::bit" "1::bit"
proof -
fix P and x :: bit
assume "P (0::bit)" and "P (1::bit)"
then have "\<forall>b. P (Abs_bit b)"
unfolding zero_bit_def one_bit_def
then show "P x"
by (induct x) simp
next
show "(0::bit) \<noteq> (1::bit)"
unfolding zero_bit_def one_bit_def
qed

lemma bit_not_0_iff [iff]: "(x::bit) \<noteq> 0 \<longleftrightarrow> x = 1"
by (induct x) simp_all

lemma bit_not_1_iff [iff]: "(x::bit) \<noteq> 1 \<longleftrightarrow> x = 0"
by (induct x) simp_all

subsection {* Type @{typ bit} forms a field *}

instantiation bit :: field_inverse_zero
begin

definition plus_bit_def:
"x + y = bit_case y (bit_case 1 0 y) x"

definition times_bit_def:
"x * y = bit_case 0 y x"

definition uminus_bit_def [simp]:
"- x = (x :: bit)"

definition minus_bit_def [simp]:
"x - y = (x + y :: bit)"

definition inverse_bit_def [simp]:
"inverse x = (x :: bit)"

definition divide_bit_def [simp]:
"x / y = (x * y :: bit)"

lemmas field_bit_defs =
plus_bit_def times_bit_def minus_bit_def uminus_bit_def
divide_bit_def inverse_bit_def

instance proof
qed (unfold field_bit_defs, auto split: bit.split)

end

lemma bit_add_self: "x + x = (0 :: bit)"
unfolding plus_bit_def by (simp split: bit.split)

lemma bit_mult_eq_1_iff [simp]: "x * y = (1 :: bit) \<longleftrightarrow> x = 1 \<and> y = 1"
unfolding times_bit_def by (simp split: bit.split)

text {* Not sure whether the next two should be simp rules. *}

lemma bit_add_eq_0_iff: "x + y = (0 :: bit) \<longleftrightarrow> x = y"
unfolding plus_bit_def by (simp split: bit.split)

lemma bit_add_eq_1_iff: "x + y = (1 :: bit) \<longleftrightarrow> x \<noteq> y"
unfolding plus_bit_def by (simp split: bit.split)

subsection {* Numerals at type @{typ bit} *}

text {* All numerals reduce to either 0 or 1. *}

lemma bit_minus1 [simp]: "-1 = (1 :: bit)"
by (simp only: minus_one [symmetric] uminus_bit_def)

lemma bit_neg_numeral [simp]: "(neg_numeral w :: bit) = numeral w"
by (simp only: neg_numeral_def uminus_bit_def)

lemma bit_numeral_even [simp]: "numeral (Num.Bit0 w) = (0 :: bit)"