author | blanchet |
Thu, 28 Aug 2014 07:30:16 +0200 | |
changeset 58066 | 96e987003a01 |
parent 55210 | d1e3b708d74b |
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permissions | -rw-r--r-- |
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(* 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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proper latex -- NB: cannot use antiquotation here;
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header {* Bit operations in $\cal Z_2$ *} |
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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 where |
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"NOT 0 = (1::bit)" |
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| "NOT 1 = (0::bit)" |
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primrec bitAND_bit 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 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 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::bit)" |
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"x AND 1 = (x::bit)" |
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"x OR 1 = (1::bit)" |
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"x OR 0 = (x::bit)" |
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"x XOR 1 = NOT (x::bit)" |
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"x XOR 0 = (x::bit)" |
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by (cases x, auto)+ |
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lemma bit_ops_comm: |
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"(x::bit) AND y = y AND x" |
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"(x::bit) OR y = y OR x" |
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"(x::bit) XOR y = y XOR x" |
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by (cases y, auto)+ |
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lemma bit_ops_same [simp]: |
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"(x::bit) AND x = x" |
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"(x::bit) OR x = x" |
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"(x::bit) XOR x = 0" |
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by (cases x, auto)+ |
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lemma bit_not_not [simp]: "NOT (NOT (x::bit)) = x" |
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by (cases x) auto |
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lemma bit_or_def: "(b::bit) OR c = NOT (NOT b AND NOT c)" |
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by (induct b, simp_all) |
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lemma bit_xor_def: "(b::bit) XOR c = (b AND NOT c) OR (NOT b AND c)" |
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by (induct b, simp_all) |
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lemma bit_NOT_eq_1_iff [simp]: "NOT (b::bit) = 1 \<longleftrightarrow> b = 0" |
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by (induct b, simp_all) |
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lemma bit_AND_eq_1_iff [simp]: "(a::bit) AND b = 1 \<longleftrightarrow> a = 1 \<and> b = 1" |
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by (induct a, simp_all) |
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end |