(*  Title:      HOL/SMT_Examples/SMT_Word_Examples.thy

     Author:     Sascha Boehme, TU Muenchen

 *)

 header {* Word examples for for SMT binding *}

 theory SMT_Word_Examples

 imports Word

 begin

 declare [[smt_oracle = true]]

 declare [[smt_certificates = "SMT_Word_Examples.certs"]]

 declare [[smt_read_only_certificates = true]]

 text {*

 Currently, there is no proof reconstruction for words.

 All lemmas are proved using the oracle mechanism.

 *}

 section {* Bitvector numbers *}

 lemma "(27 :: 4 word) = -5" by smt

 lemma "(27 :: 4 word) = 11" by smt

 lemma "23 < (27::8 word)" by smt

 lemma "27 + 11 = (6::5 word)" by smt

 lemma "7 * 3 = (21::8 word)" by smt

 lemma "11 - 27 = (-16::8 word)" by smt

 lemma "- -11 = (11::5 word)" by smt

 lemma "-40 + 1 = (-39::7 word)" by smt

 lemma "a + 2 * b + c - b = (b + c) + (a :: 32 word)" by smt

 lemma "x = (5 :: 4 word) \<Longrightarrow> 4 * x = 4" by smt

 section {* Bit-level logic *}

 lemma "0b110 AND 0b101 = (0b100 :: 32 word)" by smt

 lemma "0b110 OR 0b011 = (0b111 :: 8 word)" by smt

 lemma "0xF0 XOR 0xFF = (0x0F :: 8 word)" by smt

 lemma "NOT (0xF0 :: 16 word) = 0xFF0F" by smt

 lemma "word_cat (27::4 word) (27::8 word) = (2843::12 word)" by smt

 lemma "word_cat (0b0011::4 word) (0b1111::6word) = (0b0011001111 :: 10 word)"

   by smt

 lemma "slice 1 (0b10110 :: 4 word) = (0b11 :: 2 word)" by smt

 lemma "ucast (0b1010 :: 4 word) = (0b1010 :: 10 word)" by smt

 lemma "scast (0b1010 :: 4 word) = (0b111010 :: 6 word)" by smt

 lemma "0b10011 << 2 = (0b1001100::8 word)" by smt

 lemma "0b11001 >> 2 = (0b110::8 word)" by smt

 lemma "0b10011 >>> 2 = (0b100::8 word)" by smt

 lemma "word_rotr 2 0b0110 = (0b1001::4 word)" by smt

 lemma "word_rotl 1 0b1110 = (0b1101::4 word)" by smt

 lemma "(x AND 0xff00) OR (x AND 0x00ff) = (x::16 word)" by smt

 lemma "w < 256 \<Longrightarrow> (w :: 16 word) AND 0x00FF = w" by smt

 section {* Combined integer-bitvector properties *}

 lemma

   assumes "bv2int 0 = 0"

       and "bv2int 1 = 1"

       and "bv2int 2 = 2"

       and "bv2int 3 = 3"

       and "\<forall>x::2 word. bv2int x > 0"

   shows "\<forall>i::int. i < 0 \<longrightarrow> (\<forall>x::2 word. bv2int x > i)"

   using assms

   using [[z3_options="AUTO_CONFIG=false"]]

   by smt

 lemma "P (0 \<le> (a :: 4 word)) = P True" by smt

 end