Theory Bit_Operations

theory Bit_Operations
imports Bit
`(*  Title:      HOL/Word/Bit_Operations.thy    Author:     Author: Brian Huffman, PSU and Gerwin Klein, NICTA*)header {* Syntactic classes for bitwise operations *}theory Bit_Operationsimports "~~/src/HOL/Library/Bit"beginsubsection {* Abstract syntactic bit operations *}class bit =  fixes bitNOT :: "'a => 'a"       ("NOT _" [70] 71)    and bitAND :: "'a => 'a => 'a" (infixr "AND" 64)    and bitOR  :: "'a => 'a => 'a" (infixr "OR"  59)    and bitXOR :: "'a => 'a => 'a" (infixr "XOR" 59)text {*  We want the bitwise operations to bind slightly weaker  than @{text "+"} and @{text "-"}, but @{text "~~"} to   bind slightly stronger than @{text "*"}.*}text {*  Testing and shifting operations.*}class bits = bit +  fixes test_bit :: "'a => nat => bool" (infixl "!!" 100)    and lsb      :: "'a => bool"    and set_bit  :: "'a => nat => bool => 'a"    and set_bits :: "(nat => bool) => 'a" (binder "BITS " 10)    and shiftl   :: "'a => nat => 'a" (infixl "<<" 55)    and shiftr   :: "'a => nat => 'a" (infixl ">>" 55)class bitss = bits +  fixes msb      :: "'a => bool"  subsection {* Bitwise operations on @{typ bit} *}instantiation bit :: bitbeginprimrec 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  ..endlemmas bit_simps =  bitNOT_bit.simps bitAND_bit.simps bitOR_bit.simps bitXOR_bit.simpslemma 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) autolemma 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 <-> b = 0"  by (induct b, simp_all)lemma bit_AND_eq_1_iff [simp]: "(a::bit) AND b = 1 <-> a = 1 ∧ b = 1"  by (induct a, simp_all)end`