(* Title: HOL/Word/Bits_Bit.thy
Author: Author: Brian Huffman, PSU and Gerwin Klein, NICTA
*)
section \<open>Bit operations in $\cal Z_2$\<close>
theory Bits_Bit
imports Bits "~~/src/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::bit)"
"x AND 1 = (x::bit)"
"x OR 1 = (1::bit)"
"x OR 0 = (x::bit)"
"x XOR 1 = NOT (x::bit)"
"x XOR 0 = (x::bit)"
by (cases x, auto)+
lemma bit_ops_comm:
"(x::bit) AND y = y AND x"
"(x::bit) OR y = y OR x"
"(x::bit) XOR y = y XOR x"
by (cases y, auto)+
lemma bit_ops_same [simp]:
"(x::bit) AND x = x"
"(x::bit) OR x = x"
"(x::bit) XOR x = 0"
by (cases x, auto)+
lemma bit_not_not [simp]: "NOT (NOT (x::bit)) = x"
by (cases x) auto
lemma bit_or_def: "(b::bit) OR c = NOT (NOT b AND NOT c)"
by (induct b, simp_all)
lemma bit_xor_def: "(b::bit) XOR c = (b AND NOT c) OR (NOT b AND c)"
by (induct b, simp_all)
lemma bit_NOT_eq_1_iff [simp]: "NOT (b::bit) = 1 \<longleftrightarrow> b = 0"
by (induct b, simp_all)
lemma bit_AND_eq_1_iff [simp]: "(a::bit) AND b = 1 \<longleftrightarrow> a = 1 \<and> b = 1"
by (induct a, simp_all)
end