author | wenzelm |
Mon, 03 Apr 2017 23:12:16 +0200 | |
changeset 65363 | 5eb619751b14 |
parent 61799 | 4cf66f21b764 |
child 66453 | cc19f7ca2ed6 |
permissions | -rw-r--r-- |
55210 | 1 |
(* Title: HOL/Word/Bits_Bit.thy |
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Author: Author: Brian Huffman, PSU and Gerwin Klein, NICTA |
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*) |
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section \<open>Bit operations in $\cal Z_2$\<close> |
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prefer "Bits" as theory name for abstract bit operations, similar to "Orderings", "Lattices", "Groups" etc.
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theory Bits_Bit |
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imports Bits "~~/src/HOL/Library/Bit" |
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begin |
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instantiation bit :: bit |
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begin |
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primrec bitNOT_bit |
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where |
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"NOT 0 = (1::bit)" |
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| "NOT 1 = (0::bit)" |
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primrec bitAND_bit |
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where |
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"0 AND y = (0::bit)" |
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| "1 AND y = (y::bit)" |
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primrec bitOR_bit |
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where |
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"0 OR y = (y::bit)" |
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| "1 OR y = (1::bit)" |
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primrec bitXOR_bit |
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where |
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"0 XOR y = (y::bit)" |
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| "1 XOR y = (NOT y :: bit)" |
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instance .. |
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end |
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lemmas bit_simps = |
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bitNOT_bit.simps bitAND_bit.simps bitOR_bit.simps bitXOR_bit.simps |
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lemma bit_extra_simps [simp]: |
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"x AND 0 = 0" |
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"x AND 1 = x" |
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"x OR 1 = 1" |
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"x OR 0 = x" |
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"x XOR 1 = NOT x" |
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"x XOR 0 = x" |
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for x :: bit |
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by (cases x, auto)+ |
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lemma bit_ops_comm: |
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"x AND y = y AND x" |
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"x OR y = y OR x" |
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"x XOR y = y XOR x" |
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for x :: bit |
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by (cases y, auto)+ |
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lemma bit_ops_same [simp]: |
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"x AND x = x" |
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"x OR x = x" |
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"x XOR x = 0" |
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for x :: bit |
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by (cases x, auto)+ |
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lemma bit_not_not [simp]: "NOT (NOT x) = x" |
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for x :: bit |
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by (cases x) auto |
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lemma bit_or_def: "b OR c = NOT (NOT b AND NOT c)" |
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for b c :: bit |
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by (induct b) simp_all |
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lemma bit_xor_def: "b XOR c = (b AND NOT c) OR (NOT b AND c)" |
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for b c :: bit |
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by (induct b) simp_all |
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lemma bit_NOT_eq_1_iff [simp]: "NOT b = 1 \<longleftrightarrow> b = 0" |
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for b :: bit |
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by (induct b) simp_all |
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lemma bit_AND_eq_1_iff [simp]: "a AND b = 1 \<longleftrightarrow> a = 1 \<and> b = 1" |
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for a b :: bit |
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by (induct a) simp_all |
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end |