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
rep_datatype "0::bit" "1::bit"
proof -
fix P and x :: bit
assume "P (0::bit)" and "P (1::bit)"
then have "∀b. P (Bit b)"
unfolding zero_bit_def one_bit_def
by (simp add: all_bool_eq)
then show "P x"
by (induct x) simp
next
show "(0::bit) ≠ (1::bit)"
unfolding zero_bit_def one_bit_def
by (simp add: Bit_inject)
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 <-> (set x <-> set y)"
by (auto simp add: set_inject)
lemma Bit_inject [simp]:
"Bit P = Bit Q <-> (P <-> Q)"
by (auto simp add: Bit_inject)
lemma set [iff]:
"¬ set 0"
"set 1"
by (simp_all add: zero_bit_def one_bit_def Bit_inverse)
lemma [code]:
"set 0 <-> False"
"set 1 <-> True"
by simp_all
lemma set_iff:
"set b <-> b = 1"
by (cases b) simp_all
lemma bit_eq_iff_set:
"b = 0 <-> ¬ set b"
"b = 1 <-> set b"
by (simp_all add: bit_eq_iff)
lemma Bit [simp, code]:
"Bit False = 0"
"Bit True = 1"
by (simp_all add: zero_bit_def one_bit_def)
lemma bit_not_0_iff [iff]:
"(x::bit) ≠ 0 <-> x = 1"
by (simp add: bit_eq_iff)
lemma bit_not_1_iff [iff]:
"(x::bit) ≠ 1 <-> x = 0"
by (simp add: bit_eq_iff)
lemma [code]:
"HOL.equal 0 b <-> ¬ set b"
"HOL.equal 1 b <-> set b"
by (simp_all add: equal set_iff)
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) <-> x = 1 ∧ 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) <-> x = y"
unfolding plus_bit_def by (simp split: bit.split)
lemma bit_add_eq_1_iff: "x + y = (1 :: bit) <-> x ≠ 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)"
by (simp only: numeral_Bit0 bit_add_self)
lemma bit_numeral_odd [simp]: "numeral (Num.Bit1 w) = (1 :: bit)"
by (simp only: numeral_Bit1 bit_add_self add_0_left)
subsection {* Conversion from @{typ bit} *}
context zero_neq_one
begin
definition of_bit :: "bit => 'a"
where
"of_bit b = bit_case 0 1 b"
lemma of_bit_eq [simp, code]:
"of_bit 0 = 0"
"of_bit 1 = 1"
by (simp_all add: of_bit_def)
lemma of_bit_eq_iff:
"of_bit x = of_bit y <-> 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