src/HOL/Library/Bit.thy
 author blanchet Thu, 11 Sep 2014 18:54:36 +0200 changeset 58306 117ba6cbe414 parent 55416 dd7992d4a61a child 58881 b9556a055632 permissions -rw-r--r--
renamed 'rep_datatype' to 'old_rep_datatype' (HOL)
```
(*  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 bit = "UNIV :: bool set"
morphisms set Bit
..

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

definition zero_bit_def:
"0 = Bit False"

definition one_bit_def:
"1 = Bit True"

instance ..

end

old_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 (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_set_eq [simp]:
"Bit (set b) = b"
by (fact set_inverse)

lemma set_Bit_eq [simp]:
"set (Bit P) = P"
by (rule Bit_inverse) rule

lemma bit_eq_iff:
"x = y \<longleftrightarrow> (set x \<longleftrightarrow> set y)"

lemma Bit_inject [simp]:
"Bit P = Bit Q \<longleftrightarrow> (P \<longleftrightarrow> Q)"

lemma set [iff]:
"\<not> set 0"
"set 1"
by (simp_all add: zero_bit_def one_bit_def Bit_inverse)

lemma [code]:
"set 0 \<longleftrightarrow> False"
"set 1 \<longleftrightarrow> True"
by simp_all

lemma set_iff:
"set b \<longleftrightarrow> b = 1"
by (cases b) simp_all

lemma bit_eq_iff_set:
"b = 0 \<longleftrightarrow> \<not> set b"
"b = 1 \<longleftrightarrow> set b"

lemma Bit [simp, code]:
"Bit False = 0"
"Bit True = 1"

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

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

lemma [code]:
"HOL.equal 0 b \<longleftrightarrow> \<not> set b"
"HOL.equal 1 b \<longleftrightarrow> set b"

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

instantiation bit :: field_inverse_zero
begin

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

definition times_bit_def:
"x * y = case_bit 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: uminus_bit_def)

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

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

lemma bit_numeral_odd [simp]: "numeral (Num.Bit1 w) = (1 :: bit)"

subsection {* Conversion from @{typ bit} *}

context zero_neq_one
begin

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

lemma of_bit_eq [simp, code]:
"of_bit 0 = 0"
"of_bit 1 = 1"

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

end

context semiring_1
begin

lemma of_nat_of_bit_eq:
"of_nat (of_bit b) = of_bit b"
by (cases b) simp_all

end

context ring_1
begin

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

end

hide_const (open) set

end

```