author | haftmann |
Tue, 16 Apr 2019 19:50:03 +0000 | |
changeset 70169 | 8bb835f10a39 |
parent 67408 | 4a4c14b24800 |
child 70170 | 56727602d0a5 |
permissions | -rw-r--r-- |
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(* Title: HOL/Word/Bits_Int.thy |
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Author: Jeremy Dawson and Gerwin Klein, NICTA |
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Definitions and basic theorems for bit-wise logical operations |
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for integers expressed using Pls, Min, BIT, |
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and converting them to and from lists of bools. |
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*) |
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section \<open>Bitwise Operations on Binary Integers\<close> |
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theory Bits_Int |
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imports Bits Bit_Representation Bool_List_Representation |
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begin |
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||
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subsection \<open>Logical operations\<close> |
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text "bit-wise logical operations on the int type" |
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instantiation int :: bit |
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begin |
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||
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definition int_not_def: "bitNOT = (\<lambda>x::int. - x - 1)" |
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function bitAND_int |
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where "bitAND_int x y = |
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(if x = 0 then 0 else if x = -1 then y |
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else (bin_rest x AND bin_rest y) BIT (bin_last x \<and> bin_last y))" |
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by pat_completeness simp |
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termination |
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by (relation "measure (nat \<circ> abs \<circ> fst)", simp_all add: bin_rest_def) |
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declare bitAND_int.simps [simp del] |
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definition int_or_def: "bitOR = (\<lambda>x y::int. NOT (NOT x AND NOT y))" |
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definition int_xor_def: "bitXOR = (\<lambda>x y::int. (x AND NOT y) OR (NOT x AND y))" |
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instance .. |
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end |
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subsubsection \<open>Basic simplification rules\<close> |
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lemma int_not_BIT [simp]: "NOT (w BIT b) = (NOT w) BIT (\<not> b)" |
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by (cases b) (simp_all add: int_not_def Bit_def) |
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lemma int_not_simps [simp]: |
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"NOT (0::int) = -1" |
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"NOT (1::int) = -2" |
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"NOT (- 1::int) = 0" |
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"NOT (numeral w::int) = - numeral (w + Num.One)" |
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"NOT (- numeral (Num.Bit0 w)::int) = numeral (Num.BitM w)" |
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"NOT (- numeral (Num.Bit1 w)::int) = numeral (Num.Bit0 w)" |
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unfolding int_not_def by simp_all |
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lemma int_not_not [simp]: "NOT (NOT x) = x" |
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for x :: int |
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unfolding int_not_def by simp |
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lemma int_and_0 [simp]: "0 AND x = 0" |
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for x :: int |
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by (simp add: bitAND_int.simps) |
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lemma int_and_m1 [simp]: "-1 AND x = x" |
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for x :: int |
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by (simp add: bitAND_int.simps) |
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lemma int_and_Bits [simp]: "(x BIT b) AND (y BIT c) = (x AND y) BIT (b \<and> c)" |
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by (subst bitAND_int.simps) (simp add: Bit_eq_0_iff Bit_eq_m1_iff) |
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lemma int_or_zero [simp]: "0 OR x = x" |
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for x :: int |
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by (simp add: int_or_def) |
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lemma int_or_minus1 [simp]: "-1 OR x = -1" |
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for x :: int |
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by (simp add: int_or_def) |
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lemma int_or_Bits [simp]: "(x BIT b) OR (y BIT c) = (x OR y) BIT (b \<or> c)" |
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by (simp add: int_or_def) |
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lemma int_xor_zero [simp]: "0 XOR x = x" |
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for x :: int |
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by (simp add: int_xor_def) |
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lemma int_xor_Bits [simp]: "(x BIT b) XOR (y BIT c) = (x XOR y) BIT ((b \<or> c) \<and> \<not> (b \<and> c))" |
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unfolding int_xor_def by auto |
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subsubsection \<open>Binary destructors\<close> |
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lemma bin_rest_NOT [simp]: "bin_rest (NOT x) = NOT (bin_rest x)" |
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by (cases x rule: bin_exhaust) simp |
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lemma bin_last_NOT [simp]: "bin_last (NOT x) \<longleftrightarrow> \<not> bin_last x" |
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by (cases x rule: bin_exhaust) simp |
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lemma bin_rest_AND [simp]: "bin_rest (x AND y) = bin_rest x AND bin_rest y" |
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by (cases x rule: bin_exhaust, cases y rule: bin_exhaust) simp |
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lemma bin_last_AND [simp]: "bin_last (x AND y) \<longleftrightarrow> bin_last x \<and> bin_last y" |
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by (cases x rule: bin_exhaust, cases y rule: bin_exhaust) simp |
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lemma bin_rest_OR [simp]: "bin_rest (x OR y) = bin_rest x OR bin_rest y" |
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by (cases x rule: bin_exhaust, cases y rule: bin_exhaust) simp |
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lemma bin_last_OR [simp]: "bin_last (x OR y) \<longleftrightarrow> bin_last x \<or> bin_last y" |
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by (cases x rule: bin_exhaust, cases y rule: bin_exhaust) simp |
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lemma bin_rest_XOR [simp]: "bin_rest (x XOR y) = bin_rest x XOR bin_rest y" |
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by (cases x rule: bin_exhaust, cases y rule: bin_exhaust) simp |
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lemma bin_last_XOR [simp]: |
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"bin_last (x XOR y) \<longleftrightarrow> (bin_last x \<or> bin_last y) \<and> \<not> (bin_last x \<and> bin_last y)" |
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by (cases x rule: bin_exhaust, cases y rule: bin_exhaust) simp |
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lemma bin_nth_ops: |
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"\<And>x y. bin_nth (x AND y) n \<longleftrightarrow> bin_nth x n \<and> bin_nth y n" |
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"\<And>x y. bin_nth (x OR y) n \<longleftrightarrow> bin_nth x n \<or> bin_nth y n" |
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"\<And>x y. bin_nth (x XOR y) n \<longleftrightarrow> bin_nth x n \<noteq> bin_nth y n" |
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"\<And>x. bin_nth (NOT x) n \<longleftrightarrow> \<not> bin_nth x n" |
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by (induct n) auto |
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subsubsection \<open>Derived properties\<close> |
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lemma int_xor_minus1 [simp]: "-1 XOR x = NOT x" |
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for x :: int |
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by (auto simp add: bin_eq_iff bin_nth_ops) |
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lemma int_xor_extra_simps [simp]: |
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"w XOR 0 = w" |
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"w XOR -1 = NOT w" |
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for w :: int |
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by (auto simp add: bin_eq_iff bin_nth_ops) |
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lemma int_or_extra_simps [simp]: |
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"w OR 0 = w" |
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"w OR -1 = -1" |
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for w :: int |
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by (auto simp add: bin_eq_iff bin_nth_ops) |
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lemma int_and_extra_simps [simp]: |
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"w AND 0 = 0" |
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"w AND -1 = w" |
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for w :: int |
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by (auto simp add: bin_eq_iff bin_nth_ops) |
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text \<open>Commutativity of the above.\<close> |
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lemma bin_ops_comm: |
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fixes x y :: int |
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shows int_and_comm: "x AND y = y AND x" |
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and int_or_comm: "x OR y = y OR x" |
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and int_xor_comm: "x XOR y = y XOR x" |
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by (auto simp add: bin_eq_iff bin_nth_ops) |
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lemma bin_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 :: int |
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by (auto simp add: bin_eq_iff bin_nth_ops) |
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lemmas bin_log_esimps = |
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int_and_extra_simps int_or_extra_simps int_xor_extra_simps |
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int_and_0 int_and_m1 int_or_zero int_or_minus1 int_xor_zero int_xor_minus1 |
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subsubsection \<open>Basic properties of logical (bit-wise) operations\<close> |
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lemma bbw_ao_absorb: "x AND (y OR x) = x \<and> x OR (y AND x) = x" |
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for x y :: int |
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by (auto simp add: bin_eq_iff bin_nth_ops) |
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lemma bbw_ao_absorbs_other: |
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"x AND (x OR y) = x \<and> (y AND x) OR x = x" |
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"(y OR x) AND x = x \<and> x OR (x AND y) = x" |
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"(x OR y) AND x = x \<and> (x AND y) OR x = x" |
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for x y :: int |
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by (auto simp add: bin_eq_iff bin_nth_ops) |
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lemmas bbw_ao_absorbs [simp] = bbw_ao_absorb bbw_ao_absorbs_other |
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lemma int_xor_not: "(NOT x) XOR y = NOT (x XOR y) \<and> x XOR (NOT y) = NOT (x XOR y)" |
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for x y :: int |
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by (auto simp add: bin_eq_iff bin_nth_ops) |
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lemma int_and_assoc: "(x AND y) AND z = x AND (y AND z)" |
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for x y z :: int |
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by (auto simp add: bin_eq_iff bin_nth_ops) |
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lemma int_or_assoc: "(x OR y) OR z = x OR (y OR z)" |
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for x y z :: int |
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by (auto simp add: bin_eq_iff bin_nth_ops) |
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lemma int_xor_assoc: "(x XOR y) XOR z = x XOR (y XOR z)" |
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for x y z :: int |
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by (auto simp add: bin_eq_iff bin_nth_ops) |
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lemmas bbw_assocs = int_and_assoc int_or_assoc int_xor_assoc |
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(* BH: Why are these declared as simp rules??? *) |
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lemma bbw_lcs [simp]: |
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"y AND (x AND z) = x AND (y AND z)" |
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"y OR (x OR z) = x OR (y OR z)" |
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"y XOR (x XOR z) = x XOR (y XOR z)" |
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for x y :: int |
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by (auto simp add: bin_eq_iff bin_nth_ops) |
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lemma bbw_not_dist: |
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"NOT (x OR y) = (NOT x) AND (NOT y)" |
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"NOT (x AND y) = (NOT x) OR (NOT y)" |
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for x y :: int |
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by (auto simp add: bin_eq_iff bin_nth_ops) |
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lemma bbw_oa_dist: "(x AND y) OR z = (x OR z) AND (y OR z)" |
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for x y z :: int |
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by (auto simp add: bin_eq_iff bin_nth_ops) |
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lemma bbw_ao_dist: "(x OR y) AND z = (x AND z) OR (y AND z)" |
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for x y z :: int |
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by (auto simp add: bin_eq_iff bin_nth_ops) |
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(* |
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Why were these declared simp??? |
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declare bin_ops_comm [simp] bbw_assocs [simp] |
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*) |
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subsubsection \<open>Simplification with numerals\<close> |
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text \<open>Cases for \<open>0\<close> and \<open>-1\<close> are already covered by other simp rules.\<close> |
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lemma bin_rl_eqI: "\<lbrakk>bin_rest x = bin_rest y; bin_last x = bin_last y\<rbrakk> \<Longrightarrow> x = y" |
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by (metis (mono_tags) BIT_eq_iff bin_ex_rl bin_last_BIT bin_rest_BIT) |
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|
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lemma bin_rest_neg_numeral_BitM [simp]: |
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"bin_rest (- numeral (Num.BitM w)) = - numeral w" |
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by (simp only: BIT_bin_simps [symmetric] bin_rest_BIT) |
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241 |
|
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242 |
lemma bin_last_neg_numeral_BitM [simp]: |
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243 |
"bin_last (- numeral (Num.BitM w))" |
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by (simp only: BIT_bin_simps [symmetric] bin_last_BIT) |
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245 |
|
67120 | 246 |
(* FIXME: The rule sets below are very large (24 rules for each |
247 |
operator). Is there a simpler way to do this? *) |
|
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248 |
|
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249 |
lemma int_and_numerals [simp]: |
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250 |
"numeral (Num.Bit0 x) AND numeral (Num.Bit0 y) = (numeral x AND numeral y) BIT False" |
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251 |
"numeral (Num.Bit0 x) AND numeral (Num.Bit1 y) = (numeral x AND numeral y) BIT False" |
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252 |
"numeral (Num.Bit1 x) AND numeral (Num.Bit0 y) = (numeral x AND numeral y) BIT False" |
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253 |
"numeral (Num.Bit1 x) AND numeral (Num.Bit1 y) = (numeral x AND numeral y) BIT True" |
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254 |
"numeral (Num.Bit0 x) AND - numeral (Num.Bit0 y) = (numeral x AND - numeral y) BIT False" |
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255 |
"numeral (Num.Bit0 x) AND - numeral (Num.Bit1 y) = (numeral x AND - numeral (y + Num.One)) BIT False" |
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256 |
"numeral (Num.Bit1 x) AND - numeral (Num.Bit0 y) = (numeral x AND - numeral y) BIT False" |
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257 |
"numeral (Num.Bit1 x) AND - numeral (Num.Bit1 y) = (numeral x AND - numeral (y + Num.One)) BIT True" |
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258 |
"- numeral (Num.Bit0 x) AND numeral (Num.Bit0 y) = (- numeral x AND numeral y) BIT False" |
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259 |
"- numeral (Num.Bit0 x) AND numeral (Num.Bit1 y) = (- numeral x AND numeral y) BIT False" |
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260 |
"- numeral (Num.Bit1 x) AND numeral (Num.Bit0 y) = (- numeral (x + Num.One) AND numeral y) BIT False" |
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261 |
"- numeral (Num.Bit1 x) AND numeral (Num.Bit1 y) = (- numeral (x + Num.One) AND numeral y) BIT True" |
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262 |
"- numeral (Num.Bit0 x) AND - numeral (Num.Bit0 y) = (- numeral x AND - numeral y) BIT False" |
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263 |
"- numeral (Num.Bit0 x) AND - numeral (Num.Bit1 y) = (- numeral x AND - numeral (y + Num.One)) BIT False" |
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264 |
"- numeral (Num.Bit1 x) AND - numeral (Num.Bit0 y) = (- numeral (x + Num.One) AND - numeral y) BIT False" |
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265 |
"- numeral (Num.Bit1 x) AND - numeral (Num.Bit1 y) = (- numeral (x + Num.One) AND - numeral (y + Num.One)) BIT True" |
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"(1::int) AND numeral (Num.Bit0 y) = 0" |
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"(1::int) AND numeral (Num.Bit1 y) = 1" |
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"(1::int) AND - numeral (Num.Bit0 y) = 0" |
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"(1::int) AND - numeral (Num.Bit1 y) = 1" |
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"numeral (Num.Bit0 x) AND (1::int) = 0" |
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"numeral (Num.Bit1 x) AND (1::int) = 1" |
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"- numeral (Num.Bit0 x) AND (1::int) = 0" |
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"- numeral (Num.Bit1 x) AND (1::int) = 1" |
67120 | 274 |
by (rule bin_rl_eqI; simp)+ |
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275 |
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276 |
lemma int_or_numerals [simp]: |
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"numeral (Num.Bit0 x) OR numeral (Num.Bit0 y) = (numeral x OR numeral y) BIT False" |
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"numeral (Num.Bit0 x) OR numeral (Num.Bit1 y) = (numeral x OR numeral y) BIT True" |
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279 |
"numeral (Num.Bit1 x) OR numeral (Num.Bit0 y) = (numeral x OR numeral y) BIT True" |
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280 |
"numeral (Num.Bit1 x) OR numeral (Num.Bit1 y) = (numeral x OR numeral y) BIT True" |
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281 |
"numeral (Num.Bit0 x) OR - numeral (Num.Bit0 y) = (numeral x OR - numeral y) BIT False" |
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282 |
"numeral (Num.Bit0 x) OR - numeral (Num.Bit1 y) = (numeral x OR - numeral (y + Num.One)) BIT True" |
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283 |
"numeral (Num.Bit1 x) OR - numeral (Num.Bit0 y) = (numeral x OR - numeral y) BIT True" |
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284 |
"numeral (Num.Bit1 x) OR - numeral (Num.Bit1 y) = (numeral x OR - numeral (y + Num.One)) BIT True" |
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285 |
"- numeral (Num.Bit0 x) OR numeral (Num.Bit0 y) = (- numeral x OR numeral y) BIT False" |
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286 |
"- numeral (Num.Bit0 x) OR numeral (Num.Bit1 y) = (- numeral x OR numeral y) BIT True" |
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287 |
"- numeral (Num.Bit1 x) OR numeral (Num.Bit0 y) = (- numeral (x + Num.One) OR numeral y) BIT True" |
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288 |
"- numeral (Num.Bit1 x) OR numeral (Num.Bit1 y) = (- numeral (x + Num.One) OR numeral y) BIT True" |
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289 |
"- numeral (Num.Bit0 x) OR - numeral (Num.Bit0 y) = (- numeral x OR - numeral y) BIT False" |
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290 |
"- numeral (Num.Bit0 x) OR - numeral (Num.Bit1 y) = (- numeral x OR - numeral (y + Num.One)) BIT True" |
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291 |
"- numeral (Num.Bit1 x) OR - numeral (Num.Bit0 y) = (- numeral (x + Num.One) OR - numeral y) BIT True" |
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292 |
"- numeral (Num.Bit1 x) OR - numeral (Num.Bit1 y) = (- numeral (x + Num.One) OR - numeral (y + Num.One)) BIT True" |
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"(1::int) OR numeral (Num.Bit0 y) = numeral (Num.Bit1 y)" |
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"(1::int) OR numeral (Num.Bit1 y) = numeral (Num.Bit1 y)" |
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"(1::int) OR - numeral (Num.Bit0 y) = - numeral (Num.BitM y)" |
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"(1::int) OR - numeral (Num.Bit1 y) = - numeral (Num.Bit1 y)" |
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"numeral (Num.Bit0 x) OR (1::int) = numeral (Num.Bit1 x)" |
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298 |
"numeral (Num.Bit1 x) OR (1::int) = numeral (Num.Bit1 x)" |
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299 |
"- numeral (Num.Bit0 x) OR (1::int) = - numeral (Num.BitM x)" |
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300 |
"- numeral (Num.Bit1 x) OR (1::int) = - numeral (Num.Bit1 x)" |
67120 | 301 |
by (rule bin_rl_eqI; simp)+ |
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302 |
|
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303 |
lemma int_xor_numerals [simp]: |
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304 |
"numeral (Num.Bit0 x) XOR numeral (Num.Bit0 y) = (numeral x XOR numeral y) BIT False" |
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305 |
"numeral (Num.Bit0 x) XOR numeral (Num.Bit1 y) = (numeral x XOR numeral y) BIT True" |
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306 |
"numeral (Num.Bit1 x) XOR numeral (Num.Bit0 y) = (numeral x XOR numeral y) BIT True" |
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307 |
"numeral (Num.Bit1 x) XOR numeral (Num.Bit1 y) = (numeral x XOR numeral y) BIT False" |
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308 |
"numeral (Num.Bit0 x) XOR - numeral (Num.Bit0 y) = (numeral x XOR - numeral y) BIT False" |
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309 |
"numeral (Num.Bit0 x) XOR - numeral (Num.Bit1 y) = (numeral x XOR - numeral (y + Num.One)) BIT True" |
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310 |
"numeral (Num.Bit1 x) XOR - numeral (Num.Bit0 y) = (numeral x XOR - numeral y) BIT True" |
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311 |
"numeral (Num.Bit1 x) XOR - numeral (Num.Bit1 y) = (numeral x XOR - numeral (y + Num.One)) BIT False" |
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312 |
"- numeral (Num.Bit0 x) XOR numeral (Num.Bit0 y) = (- numeral x XOR numeral y) BIT False" |
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313 |
"- numeral (Num.Bit0 x) XOR numeral (Num.Bit1 y) = (- numeral x XOR numeral y) BIT True" |
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314 |
"- numeral (Num.Bit1 x) XOR numeral (Num.Bit0 y) = (- numeral (x + Num.One) XOR numeral y) BIT True" |
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315 |
"- numeral (Num.Bit1 x) XOR numeral (Num.Bit1 y) = (- numeral (x + Num.One) XOR numeral y) BIT False" |
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316 |
"- numeral (Num.Bit0 x) XOR - numeral (Num.Bit0 y) = (- numeral x XOR - numeral y) BIT False" |
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317 |
"- numeral (Num.Bit0 x) XOR - numeral (Num.Bit1 y) = (- numeral x XOR - numeral (y + Num.One)) BIT True" |
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318 |
"- numeral (Num.Bit1 x) XOR - numeral (Num.Bit0 y) = (- numeral (x + Num.One) XOR - numeral y) BIT True" |
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319 |
"- numeral (Num.Bit1 x) XOR - numeral (Num.Bit1 y) = (- numeral (x + Num.One) XOR - numeral (y + Num.One)) BIT False" |
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320 |
"(1::int) XOR numeral (Num.Bit0 y) = numeral (Num.Bit1 y)" |
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321 |
"(1::int) XOR numeral (Num.Bit1 y) = numeral (Num.Bit0 y)" |
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322 |
"(1::int) XOR - numeral (Num.Bit0 y) = - numeral (Num.BitM y)" |
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323 |
"(1::int) XOR - numeral (Num.Bit1 y) = - numeral (Num.Bit0 (y + Num.One))" |
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324 |
"numeral (Num.Bit0 x) XOR (1::int) = numeral (Num.Bit1 x)" |
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325 |
"numeral (Num.Bit1 x) XOR (1::int) = numeral (Num.Bit0 x)" |
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326 |
"- numeral (Num.Bit0 x) XOR (1::int) = - numeral (Num.BitM x)" |
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|
327 |
"- numeral (Num.Bit1 x) XOR (1::int) = - numeral (Num.Bit0 (x + Num.One))" |
67120 | 328 |
by (rule bin_rl_eqI; simp)+ |
329 |
||
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330 |
|
61799 | 331 |
subsubsection \<open>Interactions with arithmetic\<close> |
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332 |
|
67120 | 333 |
lemma plus_and_or [rule_format]: "\<forall>y::int. (x AND y) + (x OR y) = x + y" |
24333 | 334 |
apply (induct x rule: bin_induct) |
335 |
apply clarsimp |
|
336 |
apply clarsimp |
|
337 |
apply clarsimp |
|
338 |
apply (case_tac y rule: bin_exhaust) |
|
339 |
apply clarsimp |
|
340 |
apply (unfold Bit_def) |
|
341 |
apply clarsimp |
|
342 |
apply (erule_tac x = "x" in allE) |
|
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343 |
apply simp |
24333 | 344 |
done |
345 |
||
67120 | 346 |
lemma le_int_or: "bin_sign y = 0 \<Longrightarrow> x \<le> x OR y" |
347 |
for x y :: int |
|
37667 | 348 |
apply (induct y arbitrary: x rule: bin_induct) |
24333 | 349 |
apply clarsimp |
350 |
apply clarsimp |
|
351 |
apply (case_tac x rule: bin_exhaust) |
|
352 |
apply (case_tac b) |
|
353 |
apply (case_tac [!] bit) |
|
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354 |
apply (auto simp: le_Bits) |
24333 | 355 |
done |
356 |
||
357 |
lemmas int_and_le = |
|
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358 |
xtrans(3) [OF bbw_ao_absorbs (2) [THEN conjunct2, symmetric] le_int_or] |
24333 | 359 |
|
67120 | 360 |
text \<open>Interaction between bit-wise and arithmetic: good example of \<open>bin_induction\<close>.\<close> |
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361 |
lemma bin_add_not: "x + NOT x = (-1::int)" |
24364 | 362 |
apply (induct x rule: bin_induct) |
363 |
apply clarsimp |
|
364 |
apply clarsimp |
|
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|
365 |
apply (case_tac bit, auto) |
24364 | 366 |
done |
367 |
||
70169 | 368 |
lemma mod_BIT: "bin BIT bit mod 2 ^ Suc n = (bin mod 2 ^ n) BIT bit" |
369 |
proof - |
|
370 |
have "2 * (bin mod 2 ^ n) + 1 = (2 * bin mod 2 ^ Suc n) + 1" |
|
371 |
by (simp add: mod_mult_mult1) |
|
372 |
also have "\<dots> = ((2 * bin mod 2 ^ Suc n) + 1) mod 2 ^ Suc n" |
|
373 |
by (simp add: ac_simps p1mod22k') |
|
374 |
also have "\<dots> = (2 * bin + 1) mod 2 ^ Suc n" |
|
375 |
by (simp only: mod_simps) |
|
376 |
finally show ?thesis |
|
377 |
by (auto simp add: Bit_def) |
|
378 |
qed |
|
379 |
||
380 |
lemma AND_mod: "x AND 2 ^ n - 1 = x mod 2 ^ n" |
|
381 |
for x :: int |
|
382 |
proof (induct x arbitrary: n rule: bin_induct) |
|
383 |
case 1 |
|
384 |
then show ?case |
|
385 |
by simp |
|
386 |
next |
|
387 |
case 2 |
|
388 |
then show ?case |
|
389 |
by (simp, simp add: m1mod2k) |
|
390 |
next |
|
391 |
case (3 bin bit) |
|
392 |
show ?case |
|
393 |
proof (cases n) |
|
394 |
case 0 |
|
395 |
then show ?thesis by simp |
|
396 |
next |
|
397 |
case (Suc m) |
|
398 |
with 3 show ?thesis |
|
399 |
by (simp only: power_BIT mod_BIT int_and_Bits) simp |
|
400 |
qed |
|
401 |
qed |
|
402 |
||
67120 | 403 |
|
61799 | 404 |
subsubsection \<open>Truncating results of bit-wise operations\<close> |
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|
405 |
|
65363 | 406 |
lemma bin_trunc_ao: |
67120 | 407 |
"bintrunc n x AND bintrunc n y = bintrunc n (x AND y)" |
408 |
"bintrunc n x OR bintrunc n y = bintrunc n (x OR y)" |
|
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|
409 |
by (auto simp add: bin_eq_iff bin_nth_ops nth_bintr) |
24364 | 410 |
|
67120 | 411 |
lemma bin_trunc_xor: "bintrunc n (bintrunc n x XOR bintrunc n y) = bintrunc n (x XOR y)" |
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|
412 |
by (auto simp add: bin_eq_iff bin_nth_ops nth_bintr) |
24364 | 413 |
|
67120 | 414 |
lemma bin_trunc_not: "bintrunc n (NOT (bintrunc n x)) = bintrunc n (NOT x)" |
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|
415 |
by (auto simp add: bin_eq_iff bin_nth_ops nth_bintr) |
24364 | 416 |
|
67120 | 417 |
text \<open>Want theorems of the form of \<open>bin_trunc_xor\<close>.\<close> |
418 |
lemma bintr_bintr_i: "x = bintrunc n y \<Longrightarrow> bintrunc n x = bintrunc n y" |
|
24364 | 419 |
by auto |
420 |
||
421 |
lemmas bin_trunc_and = bin_trunc_ao(1) [THEN bintr_bintr_i] |
|
422 |
lemmas bin_trunc_or = bin_trunc_ao(2) [THEN bintr_bintr_i] |
|
423 |
||
70169 | 424 |
lemma bl_xor_aux_bin: |
425 |
"map2 (\<lambda>x y. x \<noteq> y) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) = |
|
426 |
bin_to_bl_aux n (v XOR w) (map2 (\<lambda>x y. x \<noteq> y) bs cs)" |
|
427 |
apply (induct n arbitrary: v w bs cs) |
|
428 |
apply simp |
|
429 |
apply (case_tac v rule: bin_exhaust) |
|
430 |
apply (case_tac w rule: bin_exhaust) |
|
431 |
apply clarsimp |
|
432 |
apply (case_tac b) |
|
433 |
apply auto |
|
434 |
done |
|
435 |
||
436 |
lemma bl_or_aux_bin: |
|
437 |
"map2 (\<or>) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) = |
|
438 |
bin_to_bl_aux n (v OR w) (map2 (\<or>) bs cs)" |
|
439 |
apply (induct n arbitrary: v w bs cs) |
|
440 |
apply simp |
|
441 |
apply (case_tac v rule: bin_exhaust) |
|
442 |
apply (case_tac w rule: bin_exhaust) |
|
443 |
apply clarsimp |
|
444 |
done |
|
445 |
||
446 |
lemma bl_and_aux_bin: |
|
447 |
"map2 (\<and>) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) = |
|
448 |
bin_to_bl_aux n (v AND w) (map2 (\<and>) bs cs)" |
|
449 |
apply (induct n arbitrary: v w bs cs) |
|
450 |
apply simp |
|
451 |
apply (case_tac v rule: bin_exhaust) |
|
452 |
apply (case_tac w rule: bin_exhaust) |
|
453 |
apply clarsimp |
|
454 |
done |
|
455 |
||
456 |
lemma bl_not_aux_bin: "map Not (bin_to_bl_aux n w cs) = bin_to_bl_aux n (NOT w) (map Not cs)" |
|
457 |
by (induct n arbitrary: w cs) auto |
|
458 |
||
459 |
lemma bl_not_bin: "map Not (bin_to_bl n w) = bin_to_bl n (NOT w)" |
|
460 |
by (simp add: bin_to_bl_def bl_not_aux_bin) |
|
461 |
||
462 |
lemma bl_and_bin: "map2 (\<and>) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v AND w)" |
|
463 |
by (simp add: bin_to_bl_def bl_and_aux_bin) |
|
464 |
||
465 |
lemma bl_or_bin: "map2 (\<or>) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v OR w)" |
|
466 |
by (simp add: bin_to_bl_def bl_or_aux_bin) |
|
467 |
||
468 |
lemma bl_xor_bin: "map2 (\<lambda>x y. x \<noteq> y) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v XOR w)" |
|
469 |
by (simp only: bin_to_bl_def bl_xor_aux_bin map2_Nil) |
|
470 |
||
67120 | 471 |
|
61799 | 472 |
subsection \<open>Setting and clearing bits\<close> |
24364 | 473 |
|
67120 | 474 |
text \<open>nth bit, set/clear\<close> |
54874 | 475 |
|
67120 | 476 |
primrec bin_sc :: "nat \<Rightarrow> bool \<Rightarrow> int \<Rightarrow> int" |
477 |
where |
|
478 |
Z: "bin_sc 0 b w = bin_rest w BIT b" |
|
26558 | 479 |
| Suc: "bin_sc (Suc n) b w = bin_sc n b (bin_rest w) BIT bin_last w" |
24364 | 480 |
|
67120 | 481 |
lemma bin_nth_sc [simp]: "bin_nth (bin_sc n b w) n \<longleftrightarrow> b" |
45955 | 482 |
by (induct n arbitrary: w) auto |
24333 | 483 |
|
67120 | 484 |
lemma bin_sc_sc_same [simp]: "bin_sc n c (bin_sc n b w) = bin_sc n c w" |
45955 | 485 |
by (induct n arbitrary: w) auto |
24333 | 486 |
|
67120 | 487 |
lemma bin_sc_sc_diff: "m \<noteq> n \<Longrightarrow> bin_sc m c (bin_sc n b w) = bin_sc n b (bin_sc m c w)" |
45955 | 488 |
apply (induct n arbitrary: w m) |
24333 | 489 |
apply (case_tac [!] m) |
490 |
apply auto |
|
491 |
done |
|
492 |
||
67120 | 493 |
lemma bin_nth_sc_gen: "bin_nth (bin_sc n b w) m = (if m = n then b else bin_nth w m)" |
45955 | 494 |
by (induct n arbitrary: w m) (case_tac [!] m, auto) |
65363 | 495 |
|
67120 | 496 |
lemma bin_sc_nth [simp]: "bin_sc n (bin_nth w n) w = w" |
45955 | 497 |
by (induct n arbitrary: w) auto |
24333 | 498 |
|
67120 | 499 |
lemma bin_sign_sc [simp]: "bin_sign (bin_sc n b w) = bin_sign w" |
45955 | 500 |
by (induct n arbitrary: w) auto |
65363 | 501 |
|
67120 | 502 |
lemma bin_sc_bintr [simp]: "bintrunc m (bin_sc n x (bintrunc m (w))) = bintrunc m (bin_sc n x w)" |
45955 | 503 |
apply (induct n arbitrary: w m) |
24333 | 504 |
apply (case_tac [!] w rule: bin_exhaust) |
505 |
apply (case_tac [!] m, auto) |
|
506 |
done |
|
507 |
||
67120 | 508 |
lemma bin_clr_le: "bin_sc n False w \<le> w" |
45955 | 509 |
apply (induct n arbitrary: w) |
24333 | 510 |
apply (case_tac [!] w rule: bin_exhaust) |
46605 | 511 |
apply (auto simp: le_Bits) |
24333 | 512 |
done |
513 |
||
67120 | 514 |
lemma bin_set_ge: "bin_sc n True w \<ge> w" |
45955 | 515 |
apply (induct n arbitrary: w) |
24333 | 516 |
apply (case_tac [!] w rule: bin_exhaust) |
46605 | 517 |
apply (auto simp: le_Bits) |
24333 | 518 |
done |
519 |
||
67120 | 520 |
lemma bintr_bin_clr_le: "bintrunc n (bin_sc m False w) \<le> bintrunc n w" |
45955 | 521 |
apply (induct n arbitrary: w m) |
24333 | 522 |
apply simp |
523 |
apply (case_tac w rule: bin_exhaust) |
|
524 |
apply (case_tac m) |
|
46605 | 525 |
apply (auto simp: le_Bits) |
24333 | 526 |
done |
527 |
||
67120 | 528 |
lemma bintr_bin_set_ge: "bintrunc n (bin_sc m True w) \<ge> bintrunc n w" |
45955 | 529 |
apply (induct n arbitrary: w m) |
24333 | 530 |
apply simp |
531 |
apply (case_tac w rule: bin_exhaust) |
|
532 |
apply (case_tac m) |
|
46605 | 533 |
apply (auto simp: le_Bits) |
24333 | 534 |
done |
535 |
||
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diff
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|
536 |
lemma bin_sc_FP [simp]: "bin_sc n False 0 = 0" |
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changeset
|
537 |
by (induct n) auto |
24333 | 538 |
|
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explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
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changeset
|
539 |
lemma bin_sc_TM [simp]: "bin_sc n True (- 1) = - 1" |
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46605
diff
changeset
|
540 |
by (induct n) auto |
65363 | 541 |
|
24333 | 542 |
lemmas bin_sc_simps = bin_sc.Z bin_sc.Suc bin_sc_TM bin_sc_FP |
543 |
||
67120 | 544 |
lemma bin_sc_minus: "0 < n \<Longrightarrow> bin_sc (Suc (n - 1)) b w = bin_sc n b w" |
24333 | 545 |
by auto |
546 |
||
65363 | 547 |
lemmas bin_sc_Suc_minus = |
45604 | 548 |
trans [OF bin_sc_minus [symmetric] bin_sc.Suc] |
24333 | 549 |
|
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changeset
|
550 |
lemma bin_sc_numeral [simp]: |
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46610
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changeset
|
551 |
"bin_sc (numeral k) b w = |
47219
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changeset
|
552 |
bin_sc (pred_numeral k) b (bin_rest w) BIT bin_last w" |
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47108
diff
changeset
|
553 |
by (simp add: numeral_eq_Suc) |
24333 | 554 |
|
70169 | 555 |
instantiation int :: bitss |
556 |
begin |
|
24333 | 557 |
|
70169 | 558 |
definition [iff]: "i !! n \<longleftrightarrow> bin_nth i n" |
24333 | 559 |
|
70169 | 560 |
definition "lsb i = i !! 0" for i :: int |
65363 | 561 |
|
70169 | 562 |
definition "set_bit i n b = bin_sc n b i" |
24364 | 563 |
|
70169 | 564 |
definition |
565 |
"set_bits f = |
|
566 |
(if \<exists>n. \<forall>n'\<ge>n. \<not> f n' then |
|
567 |
let n = LEAST n. \<forall>n'\<ge>n. \<not> f n' |
|
568 |
in bl_to_bin (rev (map f [0..<n])) |
|
569 |
else if \<exists>n. \<forall>n'\<ge>n. f n' then |
|
570 |
let n = LEAST n. \<forall>n'\<ge>n. f n' |
|
571 |
in sbintrunc n (bl_to_bin (True # rev (map f [0..<n]))) |
|
572 |
else 0 :: int)" |
|
24333 | 573 |
|
70169 | 574 |
definition "shiftl x n = x * 2 ^ n" for x :: int |
24333 | 575 |
|
70169 | 576 |
definition "shiftr x n = x div 2 ^ n" for x :: int |
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haftmann
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diff
changeset
|
577 |
|
70169 | 578 |
definition "msb x \<longleftrightarrow> x < 0" for x :: int |
54427
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haftmann
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54224
diff
changeset
|
579 |
|
70169 | 580 |
instance .. |
54427
783861a66a60
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haftmann
parents:
54224
diff
changeset
|
581 |
|
24333 | 582 |
end |
53062
3af1a6020014
some vague grouping of related theorems, with slight tuning of headings and sorting out of dubious lemmas into separate theory
haftmann
parents:
47219
diff
changeset
|
583 |
|
70169 | 584 |
end |