# Theory Bits_Bit

theory Bits_Bit
imports Bits Bit
(*  Title:      HOL/Word/Bits_Bit.thy
Author:     Author: Brian Huffman, PSU and Gerwin Klein, NICTA
*)

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