src/HOL/Word/Bits_Bit.thy
author paulson <lp15@cam.ac.uk>
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