# Theory SMT_Word_Examples

section ‹Word examples for for SMT binding›
theory SMT_Word_Examples
imports "HOL-Library.Word"
begin
declare [[smt_oracle = true]]
declare [[z3_extensions = 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) ⟹ 4 * x = 4" by smt
section ‹Bit-level logic›
context
includes bit_operations_syntax
begin
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 "push_bit 2 0b10011 = (0b1001100::8 word)" by smt
lemma "drop_bit 2 0b11001 = (0b110::8 word)" by smt
lemma "signed_drop_bit 2 0b10011 = (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 ⟹ (w :: 16 word) AND 0x00FF = w" by smt
end
section ‹Combined integer-bitvector properties›
lemma
assumes "bv2int 0 = 0"
and "bv2int 1 = 1"
and "bv2int 2 = 2"
and "bv2int 3 = 3"
and "∀x::2 word. bv2int x > 0"
shows "∀i::int. i < 0 ⟶ (∀x::2 word. bv2int x > i)"
using assms by smt
lemma "P (0 ≤ (a :: 4 word)) = P True" by smt
end