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1 (* Title: HOL/Word/Bit_Bit.thy |
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2 Author: Author: Brian Huffman, PSU and Gerwin Klein, NICTA |
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3 *) |
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4 |
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5 header {* Bit operations in $\cal Z_2$ *} |
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6 |
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7 theory Bits_Bit |
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8 imports Bits "~~/src/HOL/Library/Bit" |
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9 begin |
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10 |
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11 instantiation bit :: bit |
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12 begin |
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13 |
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14 primrec bitNOT_bit where |
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15 "NOT 0 = (1::bit)" |
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16 | "NOT 1 = (0::bit)" |
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17 |
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18 primrec bitAND_bit where |
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19 "0 AND y = (0::bit)" |
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20 | "1 AND y = (y::bit)" |
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21 |
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22 primrec bitOR_bit where |
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23 "0 OR y = (y::bit)" |
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24 | "1 OR y = (1::bit)" |
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25 |
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26 primrec bitXOR_bit where |
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27 "0 XOR y = (y::bit)" |
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28 | "1 XOR y = (NOT y :: bit)" |
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29 |
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30 instance .. |
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31 |
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32 end |
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33 |
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34 lemmas bit_simps = |
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35 bitNOT_bit.simps bitAND_bit.simps bitOR_bit.simps bitXOR_bit.simps |
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36 |
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37 lemma bit_extra_simps [simp]: |
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38 "x AND 0 = (0::bit)" |
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39 "x AND 1 = (x::bit)" |
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40 "x OR 1 = (1::bit)" |
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41 "x OR 0 = (x::bit)" |
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42 "x XOR 1 = NOT (x::bit)" |
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43 "x XOR 0 = (x::bit)" |
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44 by (cases x, auto)+ |
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45 |
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46 lemma bit_ops_comm: |
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47 "(x::bit) AND y = y AND x" |
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48 "(x::bit) OR y = y OR x" |
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49 "(x::bit) XOR y = y XOR x" |
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50 by (cases y, auto)+ |
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51 |
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52 lemma bit_ops_same [simp]: |
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53 "(x::bit) AND x = x" |
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54 "(x::bit) OR x = x" |
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55 "(x::bit) XOR x = 0" |
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56 by (cases x, auto)+ |
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57 |
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58 lemma bit_not_not [simp]: "NOT (NOT (x::bit)) = x" |
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59 by (cases x) auto |
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60 |
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61 lemma bit_or_def: "(b::bit) OR c = NOT (NOT b AND NOT c)" |
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62 by (induct b, simp_all) |
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63 |
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64 lemma bit_xor_def: "(b::bit) XOR c = (b AND NOT c) OR (NOT b AND c)" |
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65 by (induct b, simp_all) |
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66 |
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67 lemma bit_NOT_eq_1_iff [simp]: "NOT (b::bit) = 1 \<longleftrightarrow> b = 0" |
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68 by (induct b, simp_all) |
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69 |
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70 lemma bit_AND_eq_1_iff [simp]: "(a::bit) AND b = 1 \<longleftrightarrow> a = 1 \<and> b = 1" |
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71 by (induct a, simp_all) |
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72 |
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73 end |