src/HOL/Word/Examples/WordExamples.thy
author wenzelm
Sat Apr 23 13:00:19 2011 +0200 (2011-04-23)
changeset 42463 f270e3e18be5
parent 29640 741f26190c96
child 44762 8f9d09241a68
permissions -rw-r--r--
modernized specifications;
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(* 
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  Author: Gerwin Klein, NICTA
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  Examples demonstrating and testing various word operations.
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*)
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header "Examples of word operations"
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theory WordExamples
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imports Word
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begin
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type_synonym word32 = "32 word"
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type_synonym word8 = "8 word"
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type_synonym byte = word8
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-- "modulus"
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lemma "(27 :: 4 word) = -5" by simp
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lemma "(27 :: 4 word) = 11" by simp
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lemma "27 \<noteq> (11 :: 6 word)" by simp
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-- "signed"
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lemma "(127 :: 6 word) = -1" by simp
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-- "number ring simps"
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lemma 
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  "27 + 11 = (38::'a::len word)"
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  "27 + 11 = (6::5 word)"
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  "7 * 3 = (21::'a::len word)"
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  "11 - 27 = (-16::'a::len word)"
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  "- -11 = (11::'a::len word)"
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  "-40 + 1 = (-39::'a::len word)"
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  by simp_all
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lemma "word_pred 2 = 1" by simp
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lemma "word_succ -3 = -2" by simp
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lemma "23 < (27::8 word)" by simp
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lemma "23 \<le> (27::8 word)" by simp
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lemma "\<not> 23 < (27::2 word)" by simp
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lemma "0 < (4::3 word)" by simp
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-- "ring operations"
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lemma "a + 2 * b + c - b = (b + c) + (a :: 32 word)" by simp
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-- "casting"
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lemma "uint (234567 :: 10 word) = 71" by simp
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lemma "uint (-234567 :: 10 word) = 953" by simp
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lemma "sint (234567 :: 10 word) = 71" by simp
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lemma "sint (-234567 :: 10 word) = -71" by simp
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lemma "unat (-234567 :: 10 word) = 953" by simp
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lemma "ucast (0b1010 :: 4 word) = (0b10 :: 2 word)" by simp
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lemma "ucast (0b1010 :: 4 word) = (0b1010 :: 10 word)" by simp
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lemma "scast (0b1010 :: 4 word) = (0b111010 :: 6 word)" by simp
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-- "reducing goals to nat or int and arith:"
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lemma "i < x ==> i < (i + 1 :: 'a :: len word)" by unat_arith
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lemma "i < x ==> i < (i + 1 :: 'a :: len word)" by uint_arith
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-- "bool lists"
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lemma "of_bl [True, False, True, True] = (0b1011::'a::len word)" by simp
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lemma "to_bl (0b110::4 word) = [False, True, True, False]" by simp
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-- "this is not exactly fast, but bearable"
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lemma "of_bl (replicate 32 True) = (0xFFFFFFFF::32 word)" by simp
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-- "this works only for replicate n True"
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lemma "of_bl (replicate 32 True) = (0xFFFFFFFF::32 word)"
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  by (unfold mask_bl [symmetric]) (simp add: mask_def)
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-- "bit operations"
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lemma "0b110 AND 0b101 = (0b100 :: 32 word)" by simp
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lemma "0b110 OR 0b011 = (0b111 :: 8 word)" by simp
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lemma "0xF0 XOR 0xFF = (0x0F :: byte)" by simp
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lemma "NOT (0xF0 :: 16 word) = 0xFF0F" by simp
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lemma "(-1 :: 32 word) = 0xFFFFFFFF" by simp
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lemma "(0b0010 :: 4 word) !! 1" by simp
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lemma "\<not> (0b0010 :: 4 word) !! 0" by simp
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lemma "\<not> (0b1000 :: 3 word) !! 4" by simp
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lemma "(0b11000 :: 10 word) !! n = (n = 4 \<or> n = 3)" 
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  by (auto simp add: bin_nth_Bit0 bin_nth_Bit1)
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lemma "set_bit 55 7 True = (183::'a::len0 word)" by simp
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lemma "set_bit 0b0010 7 True = (0b10000010::'a::len0 word)" by simp
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lemma "set_bit 0b0010 1 False = (0::'a::len0 word)" by simp
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lemma "lsb (0b0101::'a::len word)" by simp
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lemma "\<not> lsb (0b1000::'a::len word)" by simp
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lemma "\<not> msb (0b0101::4 word)" by simp
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lemma   "msb (0b1000::4 word)" by simp
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lemma "word_cat (27::4 word) (27::8 word) = (2843::'a::len word)" by simp
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lemma "word_cat (0b0011::4 word) (0b1111::6word) = (0b0011001111 :: 10 word)" 
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  by simp
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lemma "0b1011 << 2 = (0b101100::'a::len0 word)" by simp
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lemma "0b1011 >> 2 = (0b10::8 word)" by simp
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lemma "0b1011 >>> 2 = (0b10::8 word)" by simp
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lemma "slice 3 (0b101111::6 word) = (0b101::3 word)" by simp
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lemma "word_rotr 2 0b0110 = (0b1001::4 word)" by simp
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lemma "word_rotl 1 0b1110 = (0b1101::4 word)" by simp
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lemma "word_roti 2 0b1110 = (0b1011::4 word)" by simp
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lemma "word_roti -2 0b0110 = (0b1001::4 word)" by simp
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lemma "(x AND 0xff00) OR (x AND 0x00ff) = (x::16 word)"
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proof -
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  have "(x AND 0xff00) OR (x AND 0x00ff) = x AND (0xff00 OR 0x00ff)"
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    by (simp only: word_ao_dist2)
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  also have "0xff00 OR 0x00ff = (-1::16 word)"
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    by simp
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  also have "x AND -1 = x"
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    by simp
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  finally show ?thesis .
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qed
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end