1(* Title: HOL/Word/Word_Examples.thy 2 Authors: Gerwin Klein and Thomas Sewell, NICTA 3 4Examples demonstrating and testing various word operations. 5*) 6 7section "Examples of word operations" 8 9theory Word_Examples 10 imports Word_Bitwise 11begin 12 13type_synonym word32 = "32 word" 14type_synonym word8 = "8 word" 15type_synonym byte = word8 16 17text "modulus" 18 19lemma "(27 :: 4 word) = -5" by simp 20 21lemma "(27 :: 4 word) = 11" by simp 22 23lemma "27 \<noteq> (11 :: 6 word)" by simp 24 25text "signed" 26 27lemma "(127 :: 6 word) = -1" by simp 28 29text "number ring simps" 30 31lemma 32 "27 + 11 = (38::'a::len word)" 33 "27 + 11 = (6::5 word)" 34 "7 * 3 = (21::'a::len word)" 35 "11 - 27 = (-16::'a::len word)" 36 "- (- 11) = (11::'a::len word)" 37 "-40 + 1 = (-39::'a::len word)" 38 by simp_all 39 40lemma "word_pred 2 = 1" by simp 41 42lemma "word_succ (- 3) = -2" by simp 43 44lemma "23 < (27::8 word)" by simp 45lemma "23 \<le> (27::8 word)" by simp 46lemma "\<not> 23 < (27::2 word)" by simp 47lemma "0 < (4::3 word)" by simp 48lemma "1 < (4::3 word)" by simp 49lemma "0 < (1::3 word)" by simp 50 51text "ring operations" 52 53lemma "a + 2 * b + c - b = (b + c) + (a :: 32 word)" by simp 54 55text "casting" 56 57lemma "uint (234567 :: 10 word) = 71" by simp 58lemma "uint (-234567 :: 10 word) = 953" by simp 59lemma "sint (234567 :: 10 word) = 71" by simp 60lemma "sint (-234567 :: 10 word) = -71" by simp 61lemma "uint (1 :: 10 word) = 1" by simp 62 63lemma "unat (-234567 :: 10 word) = 953" by simp 64lemma "unat (1 :: 10 word) = 1" by simp 65 66lemma "ucast (0b1010 :: 4 word) = (0b10 :: 2 word)" by simp 67lemma "ucast (0b1010 :: 4 word) = (0b1010 :: 10 word)" by simp 68lemma "scast (0b1010 :: 4 word) = (0b111010 :: 6 word)" by simp 69lemma "ucast (1 :: 4 word) = (1 :: 2 word)" by simp 70 71text "reducing goals to nat or int and arith:" 72lemma "i < x \<Longrightarrow> i < i + 1" for i x :: "'a::len word" 73 by unat_arith 74lemma "i < x \<Longrightarrow> i < i + 1" for i x :: "'a::len word" 75 by unat_arith 76 77text "bool lists" 78 79lemma "of_bl [True, False, True, True] = (0b1011::'a::len word)" by simp 80 81lemma "to_bl (0b110::4 word) = [False, True, True, False]" by simp 82 83text "this is not exactly fast, but bearable" 84lemma "of_bl (replicate 32 True) = (0xFFFFFFFF::32 word)" by simp 85 86text "this works only for replicate n True" 87lemma "of_bl (replicate 32 True) = (0xFFFFFFFF::32 word)" 88 by (unfold mask_bl [symmetric]) (simp add: mask_def) 89 90 91text "bit operations" 92 93lemma "0b110 AND 0b101 = (0b100 :: 32 word)" by simp 94lemma "0b110 OR 0b011 = (0b111 :: 8 word)" by simp 95lemma "0xF0 XOR 0xFF = (0x0F :: byte)" by simp 96lemma "NOT (0xF0 :: 16 word) = 0xFF0F" by simp 97lemma "0 AND 5 = (0 :: byte)" by simp 98lemma "1 AND 1 = (1 :: byte)" by simp 99lemma "1 AND 0 = (0 :: byte)" by simp 100lemma "1 AND 5 = (1 :: byte)" by simp 101lemma "1 OR 6 = (7 :: byte)" by simp 102lemma "1 OR 1 = (1 :: byte)" by simp 103lemma "1 XOR 7 = (6 :: byte)" by simp 104lemma "1 XOR 1 = (0 :: byte)" by simp 105lemma "NOT 1 = (254 :: byte)" by simp 106lemma "NOT 0 = (255 :: byte)" apply simp oops 107(* FIXME: "NOT 0" rewrites to "max_word" instead of "-1" *) 108 109lemma "(-1 :: 32 word) = 0xFFFFFFFF" by simp 110 111lemma "(0b0010 :: 4 word) !! 1" by simp 112lemma "\<not> (0b0010 :: 4 word) !! 0" by simp 113lemma "\<not> (0b1000 :: 3 word) !! 4" by simp 114lemma "\<not> (1 :: 3 word) !! 2" by simp 115 116lemma "(0b11000 :: 10 word) !! n = (n = 4 \<or> n = 3)" 117 by (auto simp add: bin_nth_Bit0 bin_nth_Bit1) 118 119lemma "set_bit 55 7 True = (183::'a::len0 word)" by simp 120lemma "set_bit 0b0010 7 True = (0b10000010::'a::len0 word)" by simp 121lemma "set_bit 0b0010 1 False = (0::'a::len0 word)" by simp 122lemma "set_bit 1 3 True = (0b1001::'a::len0 word)" by simp 123lemma "set_bit 1 0 False = (0::'a::len0 word)" by simp 124lemma "set_bit 0 3 True = (0b1000::'a::len0 word)" by simp 125lemma "set_bit 0 3 False = (0::'a::len0 word)" by simp 126 127lemma "lsb (0b0101::'a::len word)" by simp 128lemma "\<not> lsb (0b1000::'a::len word)" by simp 129lemma "lsb (1::'a::len word)" by simp 130lemma "\<not> lsb (0::'a::len word)" by simp 131 132lemma "\<not> msb (0b0101::4 word)" by simp 133lemma "msb (0b1000::4 word)" by simp 134lemma "\<not> msb (1::4 word)" by simp 135lemma "\<not> msb (0::4 word)" by simp 136 137lemma "word_cat (27::4 word) (27::8 word) = (2843::'a::len word)" by simp 138lemma "word_cat (0b0011::4 word) (0b1111::6word) = (0b0011001111 :: 10 word)" 139 by simp 140 141lemma "0b1011 << 2 = (0b101100::'a::len0 word)" by simp 142lemma "0b1011 >> 2 = (0b10::8 word)" by simp 143lemma "0b1011 >>> 2 = (0b10::8 word)" by simp 144lemma "1 << 2 = (0b100::'a::len0 word)" apply simp? oops 145 146lemma "slice 3 (0b101111::6 word) = (0b101::3 word)" by simp 147lemma "slice 3 (1::6 word) = (0::3 word)" apply simp? oops 148 149lemma "word_rotr 2 0b0110 = (0b1001::4 word)" by simp 150lemma "word_rotl 1 0b1110 = (0b1101::4 word)" by simp 151lemma "word_roti 2 0b1110 = (0b1011::4 word)" by simp 152lemma "word_roti (- 2) 0b0110 = (0b1001::4 word)" by simp 153lemma "word_rotr 2 0 = (0::4 word)" by simp 154lemma "word_rotr 2 1 = (0b0100::4 word)" apply simp? oops 155lemma "word_rotl 2 1 = (0b0100::4 word)" apply simp? oops 156lemma "word_roti (- 2) 1 = (0b0100::4 word)" apply simp? oops 157 158lemma "(x AND 0xff00) OR (x AND 0x00ff) = (x::16 word)" 159proof - 160 have "(x AND 0xff00) OR (x AND 0x00ff) = x AND (0xff00 OR 0x00ff)" 161 by (simp only: word_ao_dist2) 162 also have "0xff00 OR 0x00ff = (-1::16 word)" 163 by simp 164 also have "x AND -1 = x" 165 by simp 166 finally show ?thesis . 167qed 168 169text "alternative proof using bitwise expansion" 170 171lemma "(x AND 0xff00) OR (x AND 0x00ff) = (x::16 word)" 172 by word_bitwise 173 174text "more proofs using bitwise expansion" 175 176lemma "(x AND NOT 3) >> 4 << 2 = ((x >> 2) AND NOT 3)" 177 for x :: "10 word" 178 by word_bitwise 179 180lemma "((x AND -8) >> 3) AND 7 = (x AND 56) >> 3" 181 for x :: "12 word" 182 by word_bitwise 183 184text "some problems require further reasoning after bit expansion" 185 186lemma "x \<le> 42 \<Longrightarrow> x \<le> 89" 187 for x :: "8 word" 188 apply word_bitwise 189 apply blast 190 done 191 192lemma "(x AND 1023) = 0 \<Longrightarrow> x \<le> -1024" 193 for x :: word32 194 apply word_bitwise 195 apply clarsimp 196 done 197 198text "operations like shifts by non-numerals will expose some internal list 199 representations but may still be easy to solve" 200 201lemma shiftr_overflow: "32 \<le> a \<Longrightarrow> b >> a = 0" 202 for b :: word32 203 apply word_bitwise 204 apply simp 205 done 206 207end 208