section ‹Bit operations in $\cal Z_2$›
theory Bits_Bit
imports Bits "HOL-Library.Bit"
begin
instantiation bit :: bit
begin
primrec bitNOT_bit
where
"NOT 0 = (1::bit)"
| "NOT 1 = (0::bit)"
primrec bitAND_bit
where
"0 AND y = (0::bit)"
| "1 AND y = (y::bit)"
primrec bitOR_bit
where
"0 OR y = (y::bit)"
| "1 OR y = (1::bit)"
primrec bitXOR_bit
where
"0 XOR y = (y::bit)"
| "1 XOR y = (NOT y :: bit)"
instance ..
end
lemmas bit_simps =
bitNOT_bit.simps bitAND_bit.simps bitOR_bit.simps bitXOR_bit.simps
lemma bit_extra_simps [simp]:
"x AND 0 = 0"
"x AND 1 = x"
"x OR 1 = 1"
"x OR 0 = x"
"x XOR 1 = NOT x"
"x XOR 0 = x"
for x :: bit
by (cases x, auto)+
lemma bit_ops_comm:
"x AND y = y AND x"
"x OR y = y OR x"
"x XOR y = y XOR x"
for x :: bit
by (cases y, auto)+
lemma bit_ops_same [simp]:
"x AND x = x"
"x OR x = x"
"x XOR x = 0"
for x :: bit
by (cases x, auto)+
lemma bit_not_not [simp]: "NOT (NOT x) = x"
for x :: bit
by (cases x) auto
lemma bit_or_def: "b OR c = NOT (NOT b AND NOT c)"
for b c :: bit
by (induct b) simp_all
lemma bit_xor_def: "b XOR c = (b AND NOT c) OR (NOT b AND c)"
for b c :: bit
by (induct b) simp_all
lemma bit_NOT_eq_1_iff [simp]: "NOT b = 1 ⟷ b = 0"
for b :: bit
by (induct b) simp_all
lemma bit_AND_eq_1_iff [simp]: "a AND b = 1 ⟷ a = 1 ∧ b = 1"
for a b :: bit
by (induct a) simp_all
end