src/HOL/Word/Bits_Bit.thy
 author paulson Tue Apr 25 16:39:54 2017 +0100 (2017-04-25) changeset 65578 e4997c181cce parent 65363 5eb619751b14 child 66453 cc19f7ca2ed6 permissions -rw-r--r--
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
     1 (*  Title:      HOL/Word/Bits_Bit.thy

     2     Author:     Author: Brian Huffman, PSU and Gerwin Klein, NICTA

     3 *)

     4

     5 section \<open>Bit operations in $\cal Z_2$\<close>

     6

     7 theory Bits_Bit

     8 imports Bits "~~/src/HOL/Library/Bit"

     9 begin

    10

    11 instantiation bit :: bit

    12 begin

    13

    14 primrec bitNOT_bit

    15   where

    16     "NOT 0 = (1::bit)"

    17   | "NOT 1 = (0::bit)"

    18

    19 primrec bitAND_bit

    20   where

    21     "0 AND y = (0::bit)"

    22   | "1 AND y = (y::bit)"

    23

    24 primrec bitOR_bit

    25   where

    26     "0 OR y = (y::bit)"

    27   | "1 OR y = (1::bit)"

    28

    29 primrec bitXOR_bit

    30   where

    31     "0 XOR y = (y::bit)"

    32   | "1 XOR y = (NOT y :: bit)"

    33

    34 instance  ..

    35

    36 end

    37

    38 lemmas bit_simps =

    39   bitNOT_bit.simps bitAND_bit.simps bitOR_bit.simps bitXOR_bit.simps

    40

    41 lemma bit_extra_simps [simp]:

    42   "x AND 0 = 0"

    43   "x AND 1 = x"

    44   "x OR 1 = 1"

    45   "x OR 0 = x"

    46   "x XOR 1 = NOT x"

    47   "x XOR 0 = x"

    48   for x :: bit

    49   by (cases x, auto)+

    50

    51 lemma bit_ops_comm:

    52   "x AND y = y AND x"

    53   "x OR y = y OR x"

    54   "x XOR y = y XOR x"

    55   for x :: bit

    56   by (cases y, auto)+

    57

    58 lemma bit_ops_same [simp]:

    59   "x AND x = x"

    60   "x OR x = x"

    61   "x XOR x = 0"

    62   for x :: bit

    63   by (cases x, auto)+

    64

    65 lemma bit_not_not [simp]: "NOT (NOT x) = x"

    66   for x :: bit

    67   by (cases x) auto

    68

    69 lemma bit_or_def: "b OR c = NOT (NOT b AND NOT c)"

    70   for b c :: bit

    71   by (induct b) simp_all

    72

    73 lemma bit_xor_def: "b XOR c = (b AND NOT c) OR (NOT b AND c)"

    74   for b c :: bit

    75   by (induct b) simp_all

    76

    77 lemma bit_NOT_eq_1_iff [simp]: "NOT b = 1 \<longleftrightarrow> b = 0"

    78   for b :: bit

    79   by (induct b) simp_all

    80

    81 lemma bit_AND_eq_1_iff [simp]: "a AND b = 1 \<longleftrightarrow> a = 1 \<and> b = 1"

    82   for a b :: bit

    83   by (induct a) simp_all

    84

    85 end