author | huffman |
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permissions | -rw-r--r-- |
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(* |
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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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header {* Bitwise Operations on Binary Integers *} |
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theory Bit_Int |
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imports Bit_Representation Bit_Operations |
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begin |
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subsection {* Logical operations *} |
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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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definition int_not_def: |
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"bitNOT = (\<lambda>x::int. - x - 1)" |
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function bitAND_int where |
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"bitAND_int x y = |
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(if x = 0 then 0 else if x = -1 then y else |
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(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 o abs o 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: |
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"bitOR = (\<lambda>x y::int. NOT (NOT x AND NOT y))" |
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definition int_xor_def: |
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"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 {* Basic simplification rules *} |
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lemma int_not_BIT [simp]: |
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"NOT (w BIT b) = (NOT w) BIT (NOT b)" |
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unfolding int_not_def Bit_def by (cases b, simp_all) |
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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) = neg_numeral (w + Num.One)" |
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"NOT (neg_numeral (Num.Bit0 w)::int) = numeral (Num.BitM w)" |
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"NOT (neg_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::int)) = x" |
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unfolding int_not_def by simp |
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lemma int_and_0 [simp]: "(0::int) AND x = 0" |
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by (simp add: bitAND_int.simps) |
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lemma int_and_m1 [simp]: "(-1::int) AND x = x" |
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by (simp add: bitAND_int.simps) |
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lemma Bit_eq_0_iff: "w BIT b = 0 \<longleftrightarrow> w = 0 \<and> b = 0" |
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by (subst BIT_eq_iff [symmetric], simp) |
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lemma Bit_eq_m1_iff: "w BIT b = -1 \<longleftrightarrow> w = -1 \<and> b = 1" |
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by (subst BIT_eq_iff [symmetric], simp) |
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lemma int_and_Bits [simp]: |
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"(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::int) OR x = x" |
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unfolding int_or_def by simp |
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lemma int_or_minus1 [simp]: "(-1::int) OR x = -1" |
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unfolding int_or_def by simp |
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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) (* TODO: move *) |
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lemma int_or_Bits [simp]: |
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"(x BIT b) OR (y BIT c) = (x OR y) BIT (b OR c)" |
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unfolding int_or_def bit_or_def by simp |
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lemma int_xor_zero [simp]: "(0::int) XOR x = x" |
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unfolding int_xor_def by simp |
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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) (* TODO: move *) |
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lemma int_xor_Bits [simp]: |
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"(x BIT b) XOR (y BIT c) = (x XOR y) BIT (b XOR c)" |
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unfolding int_xor_def bit_xor_def by simp |
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subsubsection {* Binary destructors *} |
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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) = 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) = 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) = 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]: "bin_last (x XOR y) = bin_last x XOR bin_last y" |
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by (cases x rule: bin_exhaust, cases y rule: bin_exhaust, simp) |
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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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lemma bin_nth_ops: |
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"!!x y. bin_nth (x AND y) n = (bin_nth x n & bin_nth y n)" |
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"!!x y. bin_nth (x OR y) n = (bin_nth x n | bin_nth y n)" |
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"!!x y. bin_nth (x XOR y) n = (bin_nth x n ~= bin_nth y n)" |
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"!!x. bin_nth (NOT x) n = (~ bin_nth x n)" |
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by (induct n) auto |
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subsubsection {* Derived properties *} |
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lemma int_xor_minus1 [simp]: "(-1::int) XOR x = NOT x" |
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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::int) = w" |
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"w XOR (-1::int) = NOT w" |
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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::int) = w" |
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"w OR (-1::int) = -1" |
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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::int) = 0" |
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"w AND (-1::int) = w" |
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by (auto simp add: bin_eq_iff bin_nth_ops) |
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(* commutativity of the above *) |
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lemma bin_ops_comm: |
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shows |
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int_and_comm: "!!y::int. x AND y = y AND x" and |
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int_or_comm: "!!y::int. x OR y = y OR x" and |
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int_xor_comm: "!!y::int. 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::int) AND x = x" |
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"(x::int) OR x = x" |
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"(x::int) XOR x = 0" |
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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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(* basic properties of logical (bit-wise) operations *) |
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lemma bbw_ao_absorb: |
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"!!y::int. x AND (y OR x) = x & x OR (y AND x) = x" |
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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::int)" |
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"(y OR x) AND x = x \<and> x OR (x AND y) = (x::int)" |
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"(x OR y) AND x = x \<and> (x AND y) OR x = (x::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: |
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"!!y::int. (NOT x) XOR y = NOT (x XOR y) & |
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x XOR (NOT y) = NOT (x XOR y)" |
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by (auto simp add: bin_eq_iff bin_nth_ops) |
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lemma int_and_assoc: |
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"(x AND y) AND (z::int) = x AND (y AND z)" |
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201 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 202 |
|
203 |
lemma int_or_assoc: |
|
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"(x OR y) OR (z::int) = x OR (y OR z)" |
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205 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 206 |
|
207 |
lemma int_xor_assoc: |
|
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"(x XOR y) XOR (z::int) = x XOR (y XOR z)" |
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209 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 210 |
|
211 |
lemmas bbw_assocs = int_and_assoc int_or_assoc int_xor_assoc |
|
212 |
||
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213 |
(* BH: Why are these declared as simp rules??? *) |
24333 | 214 |
lemma bbw_lcs [simp]: |
24353 | 215 |
"(y::int) AND (x AND z) = x AND (y AND z)" |
216 |
"(y::int) OR (x OR z) = x OR (y OR z)" |
|
217 |
"(y::int) XOR (x XOR z) = x XOR (y XOR z)" |
|
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218 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 219 |
|
220 |
lemma bbw_not_dist: |
|
24353 | 221 |
"!!y::int. NOT (x OR y) = (NOT x) AND (NOT y)" |
222 |
"!!y::int. NOT (x AND y) = (NOT x) OR (NOT y)" |
|
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223 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 224 |
|
225 |
lemma bbw_oa_dist: |
|
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"!!y z::int. (x AND y) OR z = |
227 |
(x OR z) AND (y OR z)" |
|
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228 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 229 |
|
230 |
lemma bbw_ao_dist: |
|
24353 | 231 |
"!!y z::int. (x OR y) AND z = |
232 |
(x AND z) OR (y AND z)" |
|
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233 |
by (auto simp add: bin_eq_iff bin_nth_ops) |
24333 | 234 |
|
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235 |
(* |
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236 |
Why were these declared simp??? |
24333 | 237 |
declare bin_ops_comm [simp] bbw_assocs [simp] |
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238 |
*) |
24333 | 239 |
|
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240 |
subsubsection {* Simplification with numerals *} |
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241 |
|
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242 |
text {* Cases for @{text "0"} and @{text "-1"} are already covered by |
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243 |
other simp rules. *} |
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244 |
|
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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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246 |
by (metis bin_rl_simp) |
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247 |
|
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248 |
lemma bin_rest_neg_numeral_BitM [simp]: |
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"bin_rest (neg_numeral (Num.BitM w)) = neg_numeral w" |
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250 |
by (simp only: BIT_bin_simps [symmetric] bin_rest_BIT) |
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251 |
|
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252 |
lemma bin_last_neg_numeral_BitM [simp]: |
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253 |
"bin_last (neg_numeral (Num.BitM w)) = 1" |
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254 |
by (simp only: BIT_bin_simps [symmetric] bin_last_BIT) |
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255 |
|
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256 |
text {* FIXME: The rule sets below are very large (24 rules for each |
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257 |
operator). Is there a simpler way to do this? *} |
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258 |
|
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259 |
lemma int_and_numerals [simp]: |
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260 |
"numeral (Num.Bit0 x) AND numeral (Num.Bit0 y) = (numeral x AND numeral y) BIT 0" |
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261 |
"numeral (Num.Bit0 x) AND numeral (Num.Bit1 y) = (numeral x AND numeral y) BIT 0" |
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262 |
"numeral (Num.Bit1 x) AND numeral (Num.Bit0 y) = (numeral x AND numeral y) BIT 0" |
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263 |
"numeral (Num.Bit1 x) AND numeral (Num.Bit1 y) = (numeral x AND numeral y) BIT 1" |
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264 |
"numeral (Num.Bit0 x) AND neg_numeral (Num.Bit0 y) = (numeral x AND neg_numeral y) BIT 0" |
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265 |
"numeral (Num.Bit0 x) AND neg_numeral (Num.Bit1 y) = (numeral x AND neg_numeral (y + Num.One)) BIT 0" |
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266 |
"numeral (Num.Bit1 x) AND neg_numeral (Num.Bit0 y) = (numeral x AND neg_numeral y) BIT 0" |
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267 |
"numeral (Num.Bit1 x) AND neg_numeral (Num.Bit1 y) = (numeral x AND neg_numeral (y + Num.One)) BIT 1" |
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268 |
"neg_numeral (Num.Bit0 x) AND numeral (Num.Bit0 y) = (neg_numeral x AND numeral y) BIT 0" |
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269 |
"neg_numeral (Num.Bit0 x) AND numeral (Num.Bit1 y) = (neg_numeral x AND numeral y) BIT 0" |
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270 |
"neg_numeral (Num.Bit1 x) AND numeral (Num.Bit0 y) = (neg_numeral (x + Num.One) AND numeral y) BIT 0" |
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271 |
"neg_numeral (Num.Bit1 x) AND numeral (Num.Bit1 y) = (neg_numeral (x + Num.One) AND numeral y) BIT 1" |
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272 |
"neg_numeral (Num.Bit0 x) AND neg_numeral (Num.Bit0 y) = (neg_numeral x AND neg_numeral y) BIT 0" |
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273 |
"neg_numeral (Num.Bit0 x) AND neg_numeral (Num.Bit1 y) = (neg_numeral x AND neg_numeral (y + Num.One)) BIT 0" |
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274 |
"neg_numeral (Num.Bit1 x) AND neg_numeral (Num.Bit0 y) = (neg_numeral (x + Num.One) AND neg_numeral y) BIT 0" |
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275 |
"neg_numeral (Num.Bit1 x) AND neg_numeral (Num.Bit1 y) = (neg_numeral (x + Num.One) AND neg_numeral (y + Num.One)) BIT 1" |
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276 |
"(1::int) AND numeral (Num.Bit0 y) = 0" |
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277 |
"(1::int) AND numeral (Num.Bit1 y) = 1" |
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278 |
"(1::int) AND neg_numeral (Num.Bit0 y) = 0" |
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279 |
"(1::int) AND neg_numeral (Num.Bit1 y) = 1" |
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280 |
"numeral (Num.Bit0 x) AND (1::int) = 0" |
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281 |
"numeral (Num.Bit1 x) AND (1::int) = 1" |
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282 |
"neg_numeral (Num.Bit0 x) AND (1::int) = 0" |
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283 |
"neg_numeral (Num.Bit1 x) AND (1::int) = 1" |
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284 |
by (rule bin_rl_eqI, simp, simp)+ |
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285 |
|
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286 |
lemma int_or_numerals [simp]: |
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287 |
"numeral (Num.Bit0 x) OR numeral (Num.Bit0 y) = (numeral x OR numeral y) BIT 0" |
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288 |
"numeral (Num.Bit0 x) OR numeral (Num.Bit1 y) = (numeral x OR numeral y) BIT 1" |
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289 |
"numeral (Num.Bit1 x) OR numeral (Num.Bit0 y) = (numeral x OR numeral y) BIT 1" |
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290 |
"numeral (Num.Bit1 x) OR numeral (Num.Bit1 y) = (numeral x OR numeral y) BIT 1" |
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291 |
"numeral (Num.Bit0 x) OR neg_numeral (Num.Bit0 y) = (numeral x OR neg_numeral y) BIT 0" |
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292 |
"numeral (Num.Bit0 x) OR neg_numeral (Num.Bit1 y) = (numeral x OR neg_numeral (y + Num.One)) BIT 1" |
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293 |
"numeral (Num.Bit1 x) OR neg_numeral (Num.Bit0 y) = (numeral x OR neg_numeral y) BIT 1" |
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294 |
"numeral (Num.Bit1 x) OR neg_numeral (Num.Bit1 y) = (numeral x OR neg_numeral (y + Num.One)) BIT 1" |
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295 |
"neg_numeral (Num.Bit0 x) OR numeral (Num.Bit0 y) = (neg_numeral x OR numeral y) BIT 0" |
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296 |
"neg_numeral (Num.Bit0 x) OR numeral (Num.Bit1 y) = (neg_numeral x OR numeral y) BIT 1" |
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297 |
"neg_numeral (Num.Bit1 x) OR numeral (Num.Bit0 y) = (neg_numeral (x + Num.One) OR numeral y) BIT 1" |
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298 |
"neg_numeral (Num.Bit1 x) OR numeral (Num.Bit1 y) = (neg_numeral (x + Num.One) OR numeral y) BIT 1" |
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299 |
"neg_numeral (Num.Bit0 x) OR neg_numeral (Num.Bit0 y) = (neg_numeral x OR neg_numeral y) BIT 0" |
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300 |
"neg_numeral (Num.Bit0 x) OR neg_numeral (Num.Bit1 y) = (neg_numeral x OR neg_numeral (y + Num.One)) BIT 1" |
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301 |
"neg_numeral (Num.Bit1 x) OR neg_numeral (Num.Bit0 y) = (neg_numeral (x + Num.One) OR neg_numeral y) BIT 1" |
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302 |
"neg_numeral (Num.Bit1 x) OR neg_numeral (Num.Bit1 y) = (neg_numeral (x + Num.One) OR neg_numeral (y + Num.One)) BIT 1" |
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303 |
"(1::int) OR numeral (Num.Bit0 y) = numeral (Num.Bit1 y)" |
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304 |
"(1::int) OR numeral (Num.Bit1 y) = numeral (Num.Bit1 y)" |
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305 |
"(1::int) OR neg_numeral (Num.Bit0 y) = neg_numeral (Num.BitM y)" |
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306 |
"(1::int) OR neg_numeral (Num.Bit1 y) = neg_numeral (Num.Bit1 y)" |
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307 |
"numeral (Num.Bit0 x) OR (1::int) = numeral (Num.Bit1 x)" |
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308 |
"numeral (Num.Bit1 x) OR (1::int) = numeral (Num.Bit1 x)" |
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309 |
"neg_numeral (Num.Bit0 x) OR (1::int) = neg_numeral (Num.BitM x)" |
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310 |
"neg_numeral (Num.Bit1 x) OR (1::int) = neg_numeral (Num.Bit1 x)" |
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311 |
by (rule bin_rl_eqI, simp, simp)+ |
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312 |
|
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313 |
lemma int_xor_numerals [simp]: |
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314 |
"numeral (Num.Bit0 x) XOR numeral (Num.Bit0 y) = (numeral x XOR numeral y) BIT 0" |
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315 |
"numeral (Num.Bit0 x) XOR numeral (Num.Bit1 y) = (numeral x XOR numeral y) BIT 1" |
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316 |
"numeral (Num.Bit1 x) XOR numeral (Num.Bit0 y) = (numeral x XOR numeral y) BIT 1" |
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317 |
"numeral (Num.Bit1 x) XOR numeral (Num.Bit1 y) = (numeral x XOR numeral y) BIT 0" |
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318 |
"numeral (Num.Bit0 x) XOR neg_numeral (Num.Bit0 y) = (numeral x XOR neg_numeral y) BIT 0" |
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319 |
"numeral (Num.Bit0 x) XOR neg_numeral (Num.Bit1 y) = (numeral x XOR neg_numeral (y + Num.One)) BIT 1" |
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320 |
"numeral (Num.Bit1 x) XOR neg_numeral (Num.Bit0 y) = (numeral x XOR neg_numeral y) BIT 1" |
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321 |
"numeral (Num.Bit1 x) XOR neg_numeral (Num.Bit1 y) = (numeral x XOR neg_numeral (y + Num.One)) BIT 0" |
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322 |
"neg_numeral (Num.Bit0 x) XOR numeral (Num.Bit0 y) = (neg_numeral x XOR numeral y) BIT 0" |
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323 |
"neg_numeral (Num.Bit0 x) XOR numeral (Num.Bit1 y) = (neg_numeral x XOR numeral y) BIT 1" |
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|
324 |
"neg_numeral (Num.Bit1 x) XOR numeral (Num.Bit0 y) = (neg_numeral (x + Num.One) XOR numeral y) BIT 1" |
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|
325 |
"neg_numeral (Num.Bit1 x) XOR numeral (Num.Bit1 y) = (neg_numeral (x + Num.One) XOR numeral y) BIT 0" |
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|
326 |
"neg_numeral (Num.Bit0 x) XOR neg_numeral (Num.Bit0 y) = (neg_numeral x XOR neg_numeral y) BIT 0" |
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|
327 |
"neg_numeral (Num.Bit0 x) XOR neg_numeral (Num.Bit1 y) = (neg_numeral x XOR neg_numeral (y + Num.One)) BIT 1" |
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|
328 |
"neg_numeral (Num.Bit1 x) XOR neg_numeral (Num.Bit0 y) = (neg_numeral (x + Num.One) XOR neg_numeral y) BIT 1" |
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|
329 |
"neg_numeral (Num.Bit1 x) XOR neg_numeral (Num.Bit1 y) = (neg_numeral (x + Num.One) XOR neg_numeral (y + Num.One)) BIT 0" |
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|
330 |
"(1::int) XOR numeral (Num.Bit0 y) = numeral (Num.Bit1 y)" |
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|
331 |
"(1::int) XOR numeral (Num.Bit1 y) = numeral (Num.Bit0 y)" |
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|
332 |
"(1::int) XOR neg_numeral (Num.Bit0 y) = neg_numeral (Num.BitM y)" |
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|
333 |
"(1::int) XOR neg_numeral (Num.Bit1 y) = neg_numeral (Num.Bit0 (y + Num.One))" |
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|
334 |
"numeral (Num.Bit0 x) XOR (1::int) = numeral (Num.Bit1 x)" |
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|
335 |
"numeral (Num.Bit1 x) XOR (1::int) = numeral (Num.Bit0 x)" |
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|
336 |
"neg_numeral (Num.Bit0 x) XOR (1::int) = neg_numeral (Num.BitM x)" |
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|
337 |
"neg_numeral (Num.Bit1 x) XOR (1::int) = neg_numeral (Num.Bit0 (x + Num.One))" |
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|
338 |
by (rule bin_rl_eqI, simp, simp)+ |
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|
339 |
|
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|
340 |
subsubsection {* Interactions with arithmetic *} |
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341 |
|
24333 | 342 |
lemma plus_and_or [rule_format]: |
24353 | 343 |
"ALL y::int. (x AND y) + (x OR y) = x + y" |
24333 | 344 |
apply (induct x rule: bin_induct) |
345 |
apply clarsimp |
|
346 |
apply clarsimp |
|
347 |
apply clarsimp |
|
348 |
apply (case_tac y rule: bin_exhaust) |
|
349 |
apply clarsimp |
|
350 |
apply (unfold Bit_def) |
|
351 |
apply clarsimp |
|
352 |
apply (erule_tac x = "x" in allE) |
|
37667 | 353 |
apply (simp add: bitval_def split: bit.split) |
24333 | 354 |
done |
355 |
||
356 |
lemma le_int_or: |
|
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|
357 |
"bin_sign (y::int) = 0 ==> x <= x OR y" |
37667 | 358 |
apply (induct y arbitrary: x rule: bin_induct) |
24333 | 359 |
apply clarsimp |
360 |
apply clarsimp |
|
361 |
apply (case_tac x rule: bin_exhaust) |
|
362 |
apply (case_tac b) |
|
363 |
apply (case_tac [!] bit) |
|
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|
364 |
apply (auto simp: le_Bits) |
24333 | 365 |
done |
366 |
||
367 |
lemmas int_and_le = |
|
45475 | 368 |
xtr3 [OF bbw_ao_absorbs (2) [THEN conjunct2, symmetric] le_int_or] |
24333 | 369 |
|
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|
370 |
lemma add_BIT_simps [simp]: (* FIXME: move *) |
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|
371 |
"x BIT 0 + y BIT 0 = (x + y) BIT 0" |
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|
372 |
"x BIT 0 + y BIT 1 = (x + y) BIT 1" |
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|
373 |
"x BIT 1 + y BIT 0 = (x + y) BIT 1" |
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|
374 |
"x BIT 1 + y BIT 1 = (x + y + 1) BIT 0" |
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|
375 |
by (simp_all add: Bit_B0_2t Bit_B1_2t) |
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|
376 |
|
24364 | 377 |
(* interaction between bit-wise and arithmetic *) |
378 |
(* good example of bin_induction *) |
|
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|
379 |
lemma bin_add_not: "x + NOT x = (-1::int)" |
24364 | 380 |
apply (induct x rule: bin_induct) |
381 |
apply clarsimp |
|
382 |
apply clarsimp |
|
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|
383 |
apply (case_tac bit, auto) |
24364 | 384 |
done |
385 |
||
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|
386 |
subsubsection {* Truncating results of bit-wise operations *} |
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|
387 |
|
24364 | 388 |
lemma bin_trunc_ao: |
389 |
"!!x y. (bintrunc n x) AND (bintrunc n y) = bintrunc n (x AND y)" |
|
390 |
"!!x y. (bintrunc n x) OR (bintrunc n y) = bintrunc n (x OR y)" |
|
45543
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|
391 |
by (auto simp add: bin_eq_iff bin_nth_ops nth_bintr) |
24364 | 392 |
|
393 |
lemma bin_trunc_xor: |
|
394 |
"!!x y. bintrunc n (bintrunc n x XOR bintrunc n y) = |
|
395 |
bintrunc n (x XOR y)" |
|
45543
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|
396 |
by (auto simp add: bin_eq_iff bin_nth_ops nth_bintr) |
24364 | 397 |
|
398 |
lemma bin_trunc_not: |
|
399 |
"!!x. bintrunc n (NOT (bintrunc n x)) = bintrunc n (NOT x)" |
|
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|
400 |
by (auto simp add: bin_eq_iff bin_nth_ops nth_bintr) |
24364 | 401 |
|
402 |
(* want theorems of the form of bin_trunc_xor *) |
|
403 |
lemma bintr_bintr_i: |
|
404 |
"x = bintrunc n y ==> bintrunc n x = bintrunc n y" |
|
405 |
by auto |
|
406 |
||
407 |
lemmas bin_trunc_and = bin_trunc_ao(1) [THEN bintr_bintr_i] |
|
408 |
lemmas bin_trunc_or = bin_trunc_ao(2) [THEN bintr_bintr_i] |
|
409 |
||
410 |
subsection {* Setting and clearing bits *} |
|
411 |
||
26558 | 412 |
primrec |
24364 | 413 |
bin_sc :: "nat => bit => int => int" |
26558 | 414 |
where |
415 |
Z: "bin_sc 0 b w = bin_rest w BIT b" |
|
416 |
| Suc: "bin_sc (Suc n) b w = bin_sc n b (bin_rest w) BIT bin_last w" |
|
24364 | 417 |
|
24333 | 418 |
(** nth bit, set/clear **) |
419 |
||
420 |
lemma bin_nth_sc [simp]: |
|
45955 | 421 |
"bin_nth (bin_sc n b w) n = (b = 1)" |
422 |
by (induct n arbitrary: w) auto |
|
24333 | 423 |
|
424 |
lemma bin_sc_sc_same [simp]: |
|
45955 | 425 |
"bin_sc n c (bin_sc n b w) = bin_sc n c w" |
426 |
by (induct n arbitrary: w) auto |
|
24333 | 427 |
|
428 |
lemma bin_sc_sc_diff: |
|
45955 | 429 |
"m ~= n ==> |
24333 | 430 |
bin_sc m c (bin_sc n b w) = bin_sc n b (bin_sc m c w)" |
45955 | 431 |
apply (induct n arbitrary: w m) |
24333 | 432 |
apply (case_tac [!] m) |
433 |
apply auto |
|
434 |
done |
|
435 |
||
436 |
lemma bin_nth_sc_gen: |
|
45955 | 437 |
"bin_nth (bin_sc n b w) m = (if m = n then b = 1 else bin_nth w m)" |
438 |
by (induct n arbitrary: w m) (case_tac [!] m, auto) |
|
24333 | 439 |
|
440 |
lemma bin_sc_nth [simp]: |
|
45955 | 441 |
"(bin_sc n (If (bin_nth w n) 1 0) w) = w" |
442 |
by (induct n arbitrary: w) auto |
|
24333 | 443 |
|
444 |
lemma bin_sign_sc [simp]: |
|
45955 | 445 |
"bin_sign (bin_sc n b w) = bin_sign w" |
446 |
by (induct n arbitrary: w) auto |
|
24333 | 447 |
|
448 |
lemma bin_sc_bintr [simp]: |
|
45955 | 449 |
"bintrunc m (bin_sc n x (bintrunc m (w))) = bintrunc m (bin_sc n x w)" |
450 |
apply (induct n arbitrary: w m) |
|
24333 | 451 |
apply (case_tac [!] w rule: bin_exhaust) |
452 |
apply (case_tac [!] m, auto) |
|
453 |
done |
|
454 |
||
455 |
lemma bin_clr_le: |
|
45955 | 456 |
"bin_sc n 0 w <= w" |
457 |
apply (induct n arbitrary: w) |
|
24333 | 458 |
apply (case_tac [!] w rule: bin_exhaust) |
46605 | 459 |
apply (auto simp: le_Bits) |
24333 | 460 |
done |
461 |
||
462 |
lemma bin_set_ge: |
|
45955 | 463 |
"bin_sc n 1 w >= w" |
464 |
apply (induct n arbitrary: w) |
|
24333 | 465 |
apply (case_tac [!] w rule: bin_exhaust) |
46605 | 466 |
apply (auto simp: le_Bits) |
24333 | 467 |
done |
468 |
||
469 |
lemma bintr_bin_clr_le: |
|
45955 | 470 |
"bintrunc n (bin_sc m 0 w) <= bintrunc n w" |
471 |
apply (induct n arbitrary: w m) |
|
24333 | 472 |
apply simp |
473 |
apply (case_tac w rule: bin_exhaust) |
|
474 |
apply (case_tac m) |
|
46605 | 475 |
apply (auto simp: le_Bits) |
24333 | 476 |
done |
477 |
||
478 |
lemma bintr_bin_set_ge: |
|
45955 | 479 |
"bintrunc n (bin_sc m 1 w) >= bintrunc n w" |
480 |
apply (induct n arbitrary: w m) |
|
24333 | 481 |
apply simp |
482 |
apply (case_tac w rule: bin_exhaust) |
|
483 |
apply (case_tac m) |
|
46605 | 484 |
apply (auto simp: le_Bits) |
24333 | 485 |
done |
486 |
||
46608
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|
487 |
lemma bin_sc_FP [simp]: "bin_sc n 0 0 = 0" |
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|
488 |
by (induct n) auto |
24333 | 489 |
|
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|
490 |
lemma bin_sc_TM [simp]: "bin_sc n 1 -1 = -1" |
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|
491 |
by (induct n) auto |
24333 | 492 |
|
493 |
lemmas bin_sc_simps = bin_sc.Z bin_sc.Suc bin_sc_TM bin_sc_FP |
|
494 |
||
495 |
lemma bin_sc_minus: |
|
496 |
"0 < n ==> bin_sc (Suc (n - 1)) b w = bin_sc n b w" |
|
497 |
by auto |
|
498 |
||
499 |
lemmas bin_sc_Suc_minus = |
|
45604 | 500 |
trans [OF bin_sc_minus [symmetric] bin_sc.Suc] |
24333 | 501 |
|
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|
502 |
lemma bin_sc_numeral [simp]: |
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|
503 |
"bin_sc (numeral k) b w = |
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|
504 |
bin_sc (numeral k - 1) b (bin_rest w) BIT bin_last w" |
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|
505 |
by (subst expand_Suc, rule bin_sc.Suc) |
24333 | 506 |
|
24465 | 507 |
|
24364 | 508 |
subsection {* Splitting and concatenation *} |
24333 | 509 |
|
26558 | 510 |
definition bin_rcat :: "nat \<Rightarrow> int list \<Rightarrow> int" where |
37667 | 511 |
"bin_rcat n = foldl (\<lambda>u v. bin_cat u n v) 0" |
512 |
||
28042 | 513 |
fun bin_rsplit_aux :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> int list" where |
26558 | 514 |
"bin_rsplit_aux n m c bs = |
24364 | 515 |
(if m = 0 | n = 0 then bs else |
516 |
let (a, b) = bin_split n c |
|
26558 | 517 |
in bin_rsplit_aux n (m - n) a (b # bs))" |
24364 | 518 |
|
26558 | 519 |
definition bin_rsplit :: "nat \<Rightarrow> nat \<times> int \<Rightarrow> int list" where |
520 |
"bin_rsplit n w = bin_rsplit_aux n (fst w) (snd w) []" |
|
521 |
||
28042 | 522 |
fun bin_rsplitl_aux :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> int list" where |
26558 | 523 |
"bin_rsplitl_aux n m c bs = |
24364 | 524 |
(if m = 0 | n = 0 then bs else |
525 |
let (a, b) = bin_split (min m n) c |
|
26558 | 526 |
in bin_rsplitl_aux n (m - n) a (b # bs))" |
24364 | 527 |
|
26558 | 528 |
definition bin_rsplitl :: "nat \<Rightarrow> nat \<times> int \<Rightarrow> int list" where |
529 |
"bin_rsplitl n w = bin_rsplitl_aux n (fst w) (snd w) []" |
|
530 |
||
24364 | 531 |
declare bin_rsplit_aux.simps [simp del] |
532 |
declare bin_rsplitl_aux.simps [simp del] |
|
533 |
||
534 |
lemma bin_sign_cat: |
|
45955 | 535 |
"bin_sign (bin_cat x n y) = bin_sign x" |
536 |
by (induct n arbitrary: y) auto |
|
24364 | 537 |
|
538 |
lemma bin_cat_Suc_Bit: |
|
539 |
"bin_cat w (Suc n) (v BIT b) = bin_cat w n v BIT b" |
|
540 |
by auto |
|
541 |
||
542 |
lemma bin_nth_cat: |
|
45955 | 543 |
"bin_nth (bin_cat x k y) n = |
24364 | 544 |
(if n < k then bin_nth y n else bin_nth x (n - k))" |
45955 | 545 |
apply (induct k arbitrary: n y) |
24364 | 546 |
apply clarsimp |
547 |
apply (case_tac n, auto) |
|
24333 | 548 |
done |
549 |
||
24364 | 550 |
lemma bin_nth_split: |
45955 | 551 |
"bin_split n c = (a, b) ==> |
24364 | 552 |
(ALL k. bin_nth a k = bin_nth c (n + k)) & |
553 |
(ALL k. bin_nth b k = (k < n & bin_nth c k))" |
|
45955 | 554 |
apply (induct n arbitrary: b c) |
24364 | 555 |
apply clarsimp |
556 |
apply (clarsimp simp: Let_def split: ls_splits) |
|
557 |
apply (case_tac k) |
|
558 |
apply auto |
|
559 |
done |
|
560 |
||
561 |
lemma bin_cat_assoc: |
|
45955 | 562 |
"bin_cat (bin_cat x m y) n z = bin_cat x (m + n) (bin_cat y n z)" |
563 |
by (induct n arbitrary: z) auto |
|
24364 | 564 |
|
45955 | 565 |
lemma bin_cat_assoc_sym: |
566 |
"bin_cat x m (bin_cat y n z) = bin_cat (bin_cat x (m - n) y) (min m n) z" |
|
567 |
apply (induct n arbitrary: z m, clarsimp) |
|
24364 | 568 |
apply (case_tac m, auto) |
24333 | 569 |
done |
570 |
||
45956 | 571 |
lemma bin_cat_zero [simp]: "bin_cat 0 n w = bintrunc n w" |
46001
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huffman
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diff
changeset
|
572 |
by (induct n arbitrary: w) auto |
45956 | 573 |
|
24364 | 574 |
lemma bintr_cat1: |
45955 | 575 |
"bintrunc (k + n) (bin_cat a n b) = bin_cat (bintrunc k a) n b" |
576 |
by (induct n arbitrary: b) auto |
|
24364 | 577 |
|
578 |
lemma bintr_cat: "bintrunc m (bin_cat a n b) = |
|
579 |
bin_cat (bintrunc (m - n) a) n (bintrunc (min m n) b)" |
|
580 |
by (rule bin_eqI) (auto simp: bin_nth_cat nth_bintr) |
|
581 |
||
582 |
lemma bintr_cat_same [simp]: |
|
583 |
"bintrunc n (bin_cat a n b) = bintrunc n b" |
|
584 |
by (auto simp add : bintr_cat) |
|
585 |
||
586 |
lemma cat_bintr [simp]: |
|
45955 | 587 |
"bin_cat a n (bintrunc n b) = bin_cat a n b" |
588 |
by (induct n arbitrary: b) auto |
|
24364 | 589 |
|
590 |
lemma split_bintrunc: |
|
45955 | 591 |
"bin_split n c = (a, b) ==> b = bintrunc n c" |
592 |
by (induct n arbitrary: b c) (auto simp: Let_def split: ls_splits) |
|
24364 | 593 |
|
594 |
lemma bin_cat_split: |
|
45955 | 595 |
"bin_split n w = (u, v) ==> w = bin_cat u n v" |
596 |
by (induct n arbitrary: v w) (auto simp: Let_def split: ls_splits) |
|
24364 | 597 |
|
598 |
lemma bin_split_cat: |
|
45955 | 599 |
"bin_split n (bin_cat v n w) = (v, bintrunc n w)" |
600 |
by (induct n arbitrary: w) auto |
|
24364 | 601 |
|
45956 | 602 |
lemma bin_split_zero [simp]: "bin_split n 0 = (0, 0)" |
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
45956
diff
changeset
|
603 |
by (induct n) auto |
45956 | 604 |
|
46610
0c3a5e28f425
make uses of bin_split respect int/bin distinction
huffman
parents:
46609
diff
changeset
|
605 |
lemma bin_split_minus1 [simp]: |
0c3a5e28f425
make uses of bin_split respect int/bin distinction
huffman
parents:
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diff
changeset
|
606 |
"bin_split n -1 = (-1, bintrunc n -1)" |
0c3a5e28f425
make uses of bin_split respect int/bin distinction
huffman
parents:
46609
diff
changeset
|
607 |
by (induct n) auto |
24364 | 608 |
|
609 |
lemma bin_split_trunc: |
|
45955 | 610 |
"bin_split (min m n) c = (a, b) ==> |
24364 | 611 |
bin_split n (bintrunc m c) = (bintrunc (m - n) a, b)" |
45955 | 612 |
apply (induct n arbitrary: m b c, clarsimp) |
24364 | 613 |
apply (simp add: bin_rest_trunc Let_def split: ls_splits) |
614 |
apply (case_tac m) |
|
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
45956
diff
changeset
|
615 |
apply (auto simp: Let_def split: ls_splits) |
24333 | 616 |
done |
617 |
||
24364 | 618 |
lemma bin_split_trunc1: |
45955 | 619 |
"bin_split n c = (a, b) ==> |
24364 | 620 |
bin_split n (bintrunc m c) = (bintrunc (m - n) a, bintrunc m b)" |
45955 | 621 |
apply (induct n arbitrary: m b c, clarsimp) |
24364 | 622 |
apply (simp add: bin_rest_trunc Let_def split: ls_splits) |
623 |
apply (case_tac m) |
|
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
45956
diff
changeset
|
624 |
apply (auto simp: Let_def split: ls_splits) |
24364 | 625 |
done |
24333 | 626 |
|
24364 | 627 |
lemma bin_cat_num: |
45955 | 628 |
"bin_cat a n b = a * 2 ^ n + bintrunc n b" |
629 |
apply (induct n arbitrary: b, clarsimp) |
|
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
45956
diff
changeset
|
630 |
apply (simp add: Bit_def) |
24364 | 631 |
done |
632 |
||
633 |
lemma bin_split_num: |
|
45955 | 634 |
"bin_split n b = (b div 2 ^ n, b mod 2 ^ n)" |
46610
0c3a5e28f425
make uses of bin_split respect int/bin distinction
huffman
parents:
46609
diff
changeset
|
635 |
apply (induct n arbitrary: b, simp) |
45529
0e1037d4e049
remove redundant lemmas bin_last_mod and bin_rest_div, use bin_last_def and bin_rest_def instead
huffman
parents:
45475
diff
changeset
|
636 |
apply (simp add: bin_rest_def zdiv_zmult2_eq) |
24364 | 637 |
apply (case_tac b rule: bin_exhaust) |
638 |
apply simp |
|
37667 | 639 |
apply (simp add: Bit_def mod_mult_mult1 p1mod22k bitval_def |
45955 | 640 |
split: bit.split) |
641 |
done |
|
24364 | 642 |
|
643 |
subsection {* Miscellaneous lemmas *} |
|
24333 | 644 |
|
645 |
lemma nth_2p_bin: |
|
45955 | 646 |
"bin_nth (2 ^ n) m = (m = n)" |
647 |
apply (induct n arbitrary: m) |
|
24333 | 648 |
apply clarsimp |
649 |
apply safe |
|
650 |
apply (case_tac m) |
|
651 |
apply (auto simp: Bit_B0_2t [symmetric]) |
|
652 |
done |
|
653 |
||
654 |
(* for use when simplifying with bin_nth_Bit *) |
|
655 |
||
656 |
lemma ex_eq_or: |
|
657 |
"(EX m. n = Suc m & (m = k | P m)) = (n = Suc k | (EX m. n = Suc m & P m))" |
|
658 |
by auto |
|
659 |
||
660 |
end |