author | huffman |
Mon, 20 Aug 2007 23:35:51 +0200 | |
changeset 24368 | 4c2e80f30aeb |
parent 24367 | 3e29eafabe16 |
child 24397 | eaf37b780683 |
permissions | -rw-r--r-- |
24333 | 1 |
(* |
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ID: $Id$ |
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Author: Jeremy Dawson and Gerwin Klein, NICTA |
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contains theorems to do with bit-wise (logical) operations on words |
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*) |
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24350 | 7 |
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header {* Bitwise Operations on Words *} |
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theory WordBitwise imports WordArith begin |
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lemmas bin_log_bintrs = bin_trunc_not bin_trunc_xor bin_trunc_and bin_trunc_or |
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(* following definitions require both arithmetic and bit-wise word operations *) |
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(* to get word_no_log_defs from word_log_defs, using bin_log_bintrs *) |
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lemmas wils1 = bin_log_bintrs [THEN word_ubin.norm_eq_iff [THEN iffD1], |
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folded word_ubin.eq_norm, THEN eq_reflection, standard] |
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(* the binary operations only *) |
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lemmas word_log_binary_defs = |
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word_and_def word_or_def word_xor_def |
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lemmas word_no_log_defs [simp] = |
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word_not_def [where a="number_of ?a", |
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unfolded word_no_wi wils1, folded word_no_wi] |
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word_log_binary_defs [where a="number_of ?a" and b="number_of ?b", |
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unfolded word_no_wi wils1, folded word_no_wi] |
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lemmas word_wi_log_defs = word_no_log_defs [unfolded word_no_wi] |
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lemma uint_or: "uint (x OR y) = (uint x) OR (uint y)" |
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by (simp add: word_or_def word_no_wi [symmetric] number_of_is_id |
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bin_trunc_ao(2) [symmetric]) |
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lemma uint_and: "uint (x AND y) = (uint x) AND (uint y)" |
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by (simp add: word_and_def number_of_is_id word_no_wi [symmetric] |
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bin_trunc_ao(1) [symmetric]) |
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lemma word_ops_nth_size: |
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"n < size (x::'a::len0 word) ==> |
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(x OR y) !! n = (x !! n | y !! n) & |
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(x AND y) !! n = (x !! n & y !! n) & |
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(x XOR y) !! n = (x !! n ~= y !! n) & |
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(NOT x) !! n = (~ x !! n)" |
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unfolding word_size word_no_wi word_test_bit_def word_log_defs |
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by (clarsimp simp add : word_ubin.eq_norm nth_bintr bin_nth_ops) |
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lemma word_ao_nth: |
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fixes x :: "'a::len0 word" |
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shows "(x OR y) !! n = (x !! n | y !! n) & |
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(x AND y) !! n = (x !! n & y !! n)" |
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apply (cases "n < size x") |
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apply (drule_tac y = "y" in word_ops_nth_size) |
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apply simp |
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apply (simp add : test_bit_bin word_size) |
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done |
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(* get from commutativity, associativity etc of int_and etc |
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to same for word_and etc *) |
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lemmas bwsimps = |
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word_of_int_homs(2) |
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word_0_wi_Pls |
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word_m1_wi_Min |
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word_wi_log_defs |
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lemma word_bw_assocs: |
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fixes x :: "'a::len0 word" |
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shows |
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"(x AND y) AND z = x AND y AND z" |
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"(x OR y) OR z = x OR y OR z" |
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"(x XOR y) XOR z = x XOR y XOR z" |
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using word_of_int_Ex [where x=x] |
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word_of_int_Ex [where x=y] |
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word_of_int_Ex [where x=z] |
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24367
3e29eafabe16
AC rules for bitwise logical operators no longer declared simp
huffman
parents:
24353
diff
changeset
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by (auto simp: bwsimps bbw_assocs) |
24333 | 78 |
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lemma word_bw_comms: |
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fixes x :: "'a::len0 word" |
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shows |
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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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using word_of_int_Ex [where x=x] |
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word_of_int_Ex [where x=y] |
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24367
3e29eafabe16
AC rules for bitwise logical operators no longer declared simp
huffman
parents:
24353
diff
changeset
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by (auto simp: bwsimps bin_ops_comm) |
24333 | 88 |
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lemma word_bw_lcs: |
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fixes x :: "'a::len0 word" |
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shows |
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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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using word_of_int_Ex [where x=x] |
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word_of_int_Ex [where x=y] |
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word_of_int_Ex [where x=z] |
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by (auto simp: bwsimps) |
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lemma word_log_esimps [simp]: |
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fixes x :: "'a::len0 word" |
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shows |
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"x AND 0 = 0" |
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"x AND -1 = x" |
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"x OR 0 = x" |
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"x OR -1 = -1" |
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"x XOR 0 = x" |
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"x XOR -1 = NOT x" |
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"0 AND x = 0" |
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"-1 AND x = x" |
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"0 OR x = x" |
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"-1 OR x = -1" |
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"0 XOR x = x" |
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"-1 XOR x = NOT x" |
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using word_of_int_Ex [where x=x] |
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by (auto simp: bwsimps) |
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lemma word_not_dist: |
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fixes x :: "'a::len0 word" |
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shows |
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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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using word_of_int_Ex [where x=x] |
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word_of_int_Ex [where x=y] |
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by (auto simp: bwsimps bbw_not_dist) |
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lemma word_bw_same: |
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fixes x :: "'a::len0 word" |
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shows |
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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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using word_of_int_Ex [where x=x] |
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by (auto simp: bwsimps) |
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lemma word_ao_absorbs [simp]: |
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fixes x :: "'a::len0 word" |
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shows |
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"x AND (y OR x) = x" |
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"x OR y AND x = x" |
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"x AND (x OR y) = x" |
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"y AND x OR x = x" |
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"(y OR x) AND x = x" |
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"x OR x AND y = x" |
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"(x OR y) AND x = x" |
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"x AND y OR x = x" |
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using word_of_int_Ex [where x=x] |
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word_of_int_Ex [where x=y] |
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by (auto simp: bwsimps) |
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lemma word_not_not [simp]: |
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"NOT NOT (x::'a::len0 word) = x" |
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using word_of_int_Ex [where x=x] |
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by (auto simp: bwsimps) |
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lemma word_ao_dist: |
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fixes x :: "'a::len0 word" |
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shows "(x OR y) AND z = x AND z OR y AND z" |
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using word_of_int_Ex [where x=x] |
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word_of_int_Ex [where x=y] |
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word_of_int_Ex [where x=z] |
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by (auto simp: bwsimps bbw_ao_dist simp del: bin_ops_comm) |
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lemma word_oa_dist: |
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fixes x :: "'a::len0 word" |
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shows "x AND y OR z = (x OR z) AND (y OR z)" |
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using word_of_int_Ex [where x=x] |
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word_of_int_Ex [where x=y] |
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word_of_int_Ex [where x=z] |
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by (auto simp: bwsimps bbw_oa_dist simp del: bin_ops_comm) |
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lemma word_add_not [simp]: |
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fixes x :: "'a::len0 word" |
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shows "x + NOT x = -1" |
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using word_of_int_Ex [where x=x] |
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by (auto simp: bwsimps bin_add_not) |
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lemma word_plus_and_or [simp]: |
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fixes x :: "'a::len0 word" |
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shows "(x AND y) + (x OR y) = x + y" |
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using word_of_int_Ex [where x=x] |
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word_of_int_Ex [where x=y] |
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by (auto simp: bwsimps plus_and_or) |
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lemma leoa: |
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fixes x :: "'a::len0 word" |
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shows "(w = (x OR y)) ==> (y = (w AND y))" by auto |
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lemma leao: |
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fixes x' :: "'a::len0 word" |
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shows "(w' = (x' AND y')) ==> (x' = (x' OR w'))" by auto |
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lemmas word_ao_equiv = leao [COMP leoa [COMP iffI]] |
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lemma le_word_or2: "x <= x OR (y::'a::len0 word)" |
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unfolding word_le_def uint_or |
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by (auto intro: le_int_or) |
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lemmas le_word_or1 = xtr3 [OF word_bw_comms (2) le_word_or2, standard] |
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lemmas word_and_le1 = |
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xtr3 [OF word_ao_absorbs (4) [symmetric] le_word_or2, standard] |
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lemmas word_and_le2 = |
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xtr3 [OF word_ao_absorbs (8) [symmetric] le_word_or2, standard] |
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lemma bl_word_not: "to_bl (NOT w) = map Not (to_bl w)" |
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unfolding to_bl_def word_log_defs |
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24368 | 206 |
by (simp add: bl_not_bin number_of_is_id word_no_wi [symmetric]) |
24333 | 207 |
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lemma bl_word_xor: "to_bl (v XOR w) = app2 op ~= (to_bl v) (to_bl w)" |
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unfolding to_bl_def word_log_defs bl_xor_bin |
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24368 | 210 |
by (simp add: number_of_is_id word_no_wi [symmetric]) |
24333 | 211 |
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lemma bl_word_or: "to_bl (v OR w) = app2 op | (to_bl v) (to_bl w)" |
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unfolding to_bl_def word_log_defs |
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by (simp add: bl_or_bin number_of_is_id word_no_wi [symmetric]) |
24333 | 215 |
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lemma bl_word_and: "to_bl (v AND w) = app2 op & (to_bl v) (to_bl w)" |
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unfolding to_bl_def word_log_defs |
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24368 | 218 |
by (simp add: bl_and_bin number_of_is_id word_no_wi [symmetric]) |
24333 | 219 |
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lemma word_lsb_alt: "lsb (w::'a::len0 word) = test_bit w 0" |
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by (auto simp: word_test_bit_def word_lsb_def) |
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lemma word_lsb_1_0: "lsb (1::'a::len word) & ~ lsb (0::'b::len0 word)" |
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unfolding word_lsb_def word_1_no word_0_no by auto |
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lemma word_lsb_last: "lsb (w::'a::len word) = last (to_bl w)" |
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apply (unfold word_lsb_def uint_bl bin_to_bl_def) |
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apply (rule_tac bin="uint w" in bin_exhaust) |
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apply (cases "size w") |
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apply auto |
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apply (auto simp add: bin_to_bl_aux_alt) |
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done |
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lemma word_lsb_int: "lsb w = (uint w mod 2 = 1)" |
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unfolding word_lsb_def bin_last_mod by auto |
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lemma word_msb_sint: "msb w = (sint w < 0)" |
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unfolding word_msb_def |
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24368 | 239 |
by (simp add : sign_Min_lt_0 number_of_is_id) |
24333 | 240 |
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lemma word_msb_no': |
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"w = number_of bin ==> msb (w::'a::len word) = bin_nth bin (size w - 1)" |
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unfolding word_msb_def word_number_of_def |
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by (clarsimp simp add: word_sbin.eq_norm word_size bin_sign_lem) |
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lemmas word_msb_no = refl [THEN word_msb_no', unfolded word_size] |
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lemma word_msb_nth': "msb (w::'a::len word) = bin_nth (uint w) (size w - 1)" |
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apply (unfold word_size) |
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apply (rule trans [OF _ word_msb_no]) |
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apply (simp add : word_number_of_def) |
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done |
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lemmas word_msb_nth = word_msb_nth' [unfolded word_size] |
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lemma word_msb_alt: "msb (w::'a::len word) = hd (to_bl w)" |
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apply (unfold word_msb_nth uint_bl) |
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apply (subst hd_conv_nth) |
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apply (rule length_greater_0_conv [THEN iffD1]) |
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apply simp |
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apply (simp add : nth_bin_to_bl word_size) |
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done |
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lemma word_set_nth: |
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"set_bit w n (test_bit w n) = (w::'a::len0 word)" |
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unfolding word_test_bit_def word_set_bit_def by auto |
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lemma bin_nth_uint': |
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"bin_nth (uint w) n = (rev (bin_to_bl (size w) (uint w)) ! n & n < size w)" |
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apply (unfold word_size) |
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apply (safe elim!: bin_nth_uint_imp) |
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apply (frule bin_nth_uint_imp) |
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apply (fast dest!: bin_nth_bl)+ |
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done |
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lemmas bin_nth_uint = bin_nth_uint' [unfolded word_size] |
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lemma test_bit_bl: "w !! n = (rev (to_bl w) ! n & n < size w)" |
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unfolding to_bl_def word_test_bit_def word_size |
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by (rule bin_nth_uint) |
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lemma to_bl_nth: "n < size w ==> to_bl w ! n = w !! (size w - Suc n)" |
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apply (unfold test_bit_bl) |
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apply clarsimp |
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apply (rule trans) |
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apply (rule nth_rev_alt) |
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apply (auto simp add: word_size) |
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done |
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lemma test_bit_set: |
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fixes w :: "'a::len0 word" |
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shows "(set_bit w n x) !! n = (n < size w & x)" |
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unfolding word_size word_test_bit_def word_set_bit_def |
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by (clarsimp simp add : word_ubin.eq_norm nth_bintr) |
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295 |
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lemma test_bit_set_gen: |
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fixes w :: "'a::len0 word" |
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shows "test_bit (set_bit w n x) m = |
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(if m = n then n < size w & x else test_bit w m)" |
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apply (unfold word_size word_test_bit_def word_set_bit_def) |
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apply (clarsimp simp add: word_ubin.eq_norm nth_bintr bin_nth_sc_gen) |
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apply (auto elim!: test_bit_size [unfolded word_size] |
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simp add: word_test_bit_def [symmetric]) |
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304 |
done |
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305 |
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306 |
lemma of_bl_rep_False: "of_bl (replicate n False @ bs) = of_bl bs" |
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unfolding of_bl_def bl_to_bin_rep_F by auto |
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308 |
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309 |
lemma msb_nth': |
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310 |
fixes w :: "'a::len word" |
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311 |
shows "msb w = w !! (size w - 1)" |
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unfolding word_msb_nth' word_test_bit_def by simp |
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313 |
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314 |
lemmas msb_nth = msb_nth' [unfolded word_size] |
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315 |
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316 |
lemmas msb0 = len_gt_0 [THEN diff_Suc_less, THEN |
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317 |
word_ops_nth_size [unfolded word_size], standard] |
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lemmas msb1 = msb0 [where i = 0] |
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lemmas word_ops_msb = msb1 [unfolded msb_nth [symmetric, unfolded One_nat_def]] |
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320 |
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321 |
lemmas lsb0 = len_gt_0 [THEN word_ops_nth_size [unfolded word_size], standard] |
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322 |
lemmas word_ops_lsb = lsb0 [unfolded word_lsb_alt] |
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324 |
lemma td_ext_nth': |
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325 |
"n = size (w::'a::len0 word) ==> ofn = set_bits ==> [w, ofn g] = l ==> |
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326 |
td_ext test_bit ofn {f. ALL i. f i --> i < n} (%h i. h i & i < n)" |
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327 |
apply (unfold word_size td_ext_def') |
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apply safe |
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329 |
apply (rule_tac [3] ext) |
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330 |
apply (rule_tac [4] ext) |
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331 |
apply (unfold word_size of_nth_def test_bit_bl) |
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332 |
apply safe |
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333 |
defer |
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334 |
apply (clarsimp simp: word_bl.Abs_inverse)+ |
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335 |
apply (rule word_bl.Rep_inverse') |
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336 |
apply (rule sym [THEN trans]) |
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337 |
apply (rule bl_of_nth_nth) |
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338 |
apply simp |
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339 |
apply (rule bl_of_nth_inj) |
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340 |
apply (clarsimp simp add : test_bit_bl word_size) |
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done |
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342 |
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lemmas td_ext_nth = td_ext_nth' [OF refl refl refl, unfolded word_size] |
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344 |
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345 |
interpretation test_bit: |
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346 |
td_ext ["op !! :: 'a::len0 word => nat => bool" |
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set_bits |
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348 |
"{f. \<forall>i. f i \<longrightarrow> i < len_of TYPE('a::len0)}" |
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349 |
"(\<lambda>h i. h i \<and> i < len_of TYPE('a::len0))"] |
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350 |
by (rule td_ext_nth) |
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351 |
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352 |
declare test_bit.Rep' [simp del] |
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declare test_bit.Rep' [rule del] |
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354 |
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lemmas td_nth = test_bit.td_thm |
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356 |
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357 |
lemma word_set_set_same: |
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358 |
fixes w :: "'a::len0 word" |
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shows "set_bit (set_bit w n x) n y = set_bit w n y" |
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by (rule word_eqI) (simp add : test_bit_set_gen word_size) |
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lemma word_set_set_diff: |
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fixes w :: "'a::len0 word" |
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assumes "m ~= n" |
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shows "set_bit (set_bit w m x) n y = set_bit (set_bit w n y) m x" |
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by (rule word_eqI) (clarsimp simp add : test_bit_set_gen word_size prems) |
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368 |
lemma test_bit_no': |
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fixes w :: "'a::len0 word" |
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370 |
shows "w = number_of bin ==> test_bit w n = (n < size w & bin_nth bin n)" |
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371 |
unfolding word_test_bit_def word_number_of_def word_size |
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372 |
by (simp add : nth_bintr [symmetric] word_ubin.eq_norm) |
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373 |
||
374 |
lemmas test_bit_no = |
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375 |
refl [THEN test_bit_no', unfolded word_size, THEN eq_reflection, standard] |
|
376 |
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377 |
lemma nth_0: "~ (0::'a::len0 word) !! n" |
|
378 |
unfolding test_bit_no word_0_no by auto |
|
379 |
||
380 |
lemma nth_sint: |
|
381 |
fixes w :: "'a::len word" |
|
382 |
defines "l \<equiv> len_of TYPE ('a)" |
|
383 |
shows "bin_nth (sint w) n = (if n < l - 1 then w !! n else w !! (l - 1))" |
|
384 |
unfolding sint_uint l_def |
|
385 |
by (clarsimp simp add: nth_sbintr word_test_bit_def [symmetric]) |
|
386 |
||
387 |
lemma word_lsb_no: |
|
388 |
"lsb (number_of bin :: 'a :: len word) = (bin_last bin = bit.B1)" |
|
389 |
unfolding word_lsb_alt test_bit_no by auto |
|
390 |
||
391 |
lemma word_set_no: |
|
392 |
"set_bit (number_of bin::'a::len0 word) n b = |
|
393 |
number_of (bin_sc n (if b then bit.B1 else bit.B0) bin)" |
|
394 |
apply (unfold word_set_bit_def word_number_of_def [symmetric]) |
|
395 |
apply (rule word_eqI) |
|
24368 | 396 |
apply (clarsimp simp: word_size bin_nth_sc_gen number_of_is_id |
24333 | 397 |
test_bit_no nth_bintr) |
398 |
done |
|
399 |
||
400 |
lemmas setBit_no = setBit_def [THEN trans [OF meta_eq_to_obj_eq word_set_no], |
|
401 |
simplified if_simps, THEN eq_reflection, standard] |
|
402 |
lemmas clearBit_no = clearBit_def [THEN trans [OF meta_eq_to_obj_eq word_set_no], |
|
403 |
simplified if_simps, THEN eq_reflection, standard] |
|
404 |
||
405 |
lemma to_bl_n1: |
|
406 |
"to_bl (-1::'a::len0 word) = replicate (len_of TYPE ('a)) True" |
|
407 |
apply (rule word_bl.Abs_inverse') |
|
408 |
apply simp |
|
409 |
apply (rule word_eqI) |
|
410 |
apply (clarsimp simp add: word_size test_bit_no) |
|
411 |
apply (auto simp add: word_bl.Abs_inverse test_bit_bl word_size) |
|
412 |
done |
|
413 |
||
414 |
lemma word_msb_n1: "msb (-1::'a::len word)" |
|
415 |
unfolding word_msb_alt word_msb_alt to_bl_n1 by simp |
|
416 |
||
417 |
declare word_set_set_same [simp] word_set_nth [simp] |
|
418 |
test_bit_no [simp] word_set_no [simp] nth_0 [simp] |
|
419 |
setBit_no [simp] clearBit_no [simp] |
|
420 |
word_lsb_no [simp] word_msb_no [simp] word_msb_n1 [simp] word_lsb_1_0 [simp] |
|
421 |
||
422 |
lemma word_set_nth_iff: |
|
423 |
"(set_bit w n b = w) = (w !! n = b | n >= size (w::'a::len0 word))" |
|
424 |
apply (rule iffI) |
|
425 |
apply (rule disjCI) |
|
426 |
apply (drule word_eqD) |
|
427 |
apply (erule sym [THEN trans]) |
|
428 |
apply (simp add: test_bit_set) |
|
429 |
apply (erule disjE) |
|
430 |
apply clarsimp |
|
431 |
apply (rule word_eqI) |
|
432 |
apply (clarsimp simp add : test_bit_set_gen) |
|
433 |
apply (drule test_bit_size) |
|
434 |
apply force |
|
435 |
done |
|
436 |
||
437 |
lemma test_bit_2p': |
|
438 |
"w = word_of_int (2 ^ n) ==> |
|
439 |
w !! m = (m = n & m < size (w :: 'a :: len word))" |
|
440 |
unfolding word_test_bit_def word_size |
|
441 |
by (auto simp add: word_ubin.eq_norm nth_bintr nth_2p_bin) |
|
442 |
||
443 |
lemmas test_bit_2p = refl [THEN test_bit_2p', unfolded word_size] |
|
444 |
||
445 |
lemmas nth_w2p = test_bit_2p [unfolded of_int_number_of_eq |
|
446 |
word_of_int [symmetric] of_int_power] |
|
447 |
||
448 |
lemma uint_2p: |
|
449 |
"(0::'a::len word) < 2 ^ n ==> uint (2 ^ n::'a::len word) = 2 ^ n" |
|
450 |
apply (unfold word_arith_power_alt) |
|
451 |
apply (case_tac "len_of TYPE ('a)") |
|
452 |
apply clarsimp |
|
453 |
apply (case_tac "nat") |
|
454 |
apply clarsimp |
|
455 |
apply (case_tac "n") |
|
456 |
apply (clarsimp simp add : word_1_wi [symmetric]) |
|
457 |
apply (clarsimp simp add : word_0_wi [symmetric]) |
|
458 |
apply (drule word_gt_0 [THEN iffD1]) |
|
459 |
apply (safe intro!: word_eqI bin_nth_lem ext) |
|
460 |
apply (auto simp add: test_bit_2p nth_2p_bin word_test_bit_def [symmetric]) |
|
461 |
done |
|
462 |
||
463 |
lemma word_of_int_2p: "(word_of_int (2 ^ n) :: 'a :: len word) = 2 ^ n" |
|
464 |
apply (unfold word_arith_power_alt) |
|
465 |
apply (case_tac "len_of TYPE ('a)") |
|
466 |
apply clarsimp |
|
467 |
apply (case_tac "nat") |
|
468 |
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) |
|
469 |
apply (rule box_equals) |
|
470 |
apply (rule_tac [2] bintr_ariths (1))+ |
|
24368 | 471 |
apply (clarsimp simp add : number_of_is_id) |
24333 | 472 |
apply simp |
473 |
done |
|
474 |
||
475 |
lemma bang_is_le: "x !! m ==> 2 ^ m <= (x :: 'a :: len word)" |
|
476 |
apply (rule xtr3) |
|
477 |
apply (rule_tac [2] y = "x" in le_word_or2) |
|
478 |
apply (rule word_eqI) |
|
479 |
apply (auto simp add: word_ao_nth nth_w2p word_size) |
|
480 |
done |
|
481 |
||
482 |
lemma word_clr_le: |
|
483 |
fixes w :: "'a::len0 word" |
|
484 |
shows "w >= set_bit w n False" |
|
485 |
apply (unfold word_set_bit_def word_le_def word_ubin.eq_norm) |
|
486 |
apply simp |
|
487 |
apply (rule order_trans) |
|
488 |
apply (rule bintr_bin_clr_le) |
|
489 |
apply simp |
|
490 |
done |
|
491 |
||
492 |
lemma word_set_ge: |
|
493 |
fixes w :: "'a::len word" |
|
494 |
shows "w <= set_bit w n True" |
|
495 |
apply (unfold word_set_bit_def word_le_def word_ubin.eq_norm) |
|
496 |
apply simp |
|
497 |
apply (rule order_trans [OF _ bintr_bin_set_ge]) |
|
498 |
apply simp |
|
499 |
done |
|
500 |
||
501 |
end |
|
502 |