Shweta (Jul 10 2024 at 14:17): Shweta (Jul 10 2024 at 14:17):Hi. I am working with the More_Word.thy file and am trying to understand the semantics for the bit vector multiplication it uses:
class times =
fixes times :: "'a ⇒ 'a ⇒ 'a" (infixl "*" 70) in Groups.thy
Is there a resource I can use to find out exactly what this function does on a bit level?
From examples:
value "(3::3 word)*(3::3 word)" is 1::3 word
value "(8::3 word)*(3::3 word)" is 0::3 word (so 000*011=000), and the lemma
Word.uint_word_arith_bintrs(3): uint (?a * ?b) = take_bit LENGTH(?'a) (uint ?a * uint ?b)
I understand that it does the multiplication modulo 2^(LENGTH('a)) and ignores the carry overs beyond length n.
Lukas Stevens (Jul 10 2024 at 14:26):If you look at thm times_word_def you see that (*) on words is defined by lifting (*) on ints. This means that the integer multiplication is performed on the the integers that are represented by the words and then the result gets truncated.
Shweta (Jul 10 2024 at 14:47):Thank you! As a follow up, I am trying to prove this lemma but get a counterexample:
lemma bvmul_ref6_h2:
fixes x1 :: "'a::len word" and x2 :: "'a::len word"
shows "(1::int) < int (size (x1::'a::len word))
⟶ (smt_extract 2 0 ((x1*x2)::'a::len word)::'a::len word)
= (((smt_extract 2 0 x1)::'a::len word)
*((smt_extract 2 0 x2)::'a::len word)::'a::len word)"
I get this counterexample:
Auto Quickcheck found a counterexample:
x1 = 3::Enum.finite_3 word
x2 = 3::Enum.finite_3 word
Evaluated terms:
smt_extract (2::nat) (0::nat) ((x1::Enum.finite_3 word) * (x2::Enum.finite_3 word)) =
1::Enum.finite_3 word
smt_extract (2::nat) (0::nat) (x1::Enum.finite_3 word) *
smt_extract (2::nat) (0::nat) (x2::Enum.finite_3 word) =
9::Enum.finite_3 word
However,
value "(smt_extract 2 0 ((3::3 word)*(3::3 word)))::3 word"
value "((smt_extract 2 0 (3::3 word))::3 word)
*((smt_extract 2 0 (3::3 word))::3 word)" both return 1::3 word.
Why is 001 interpreted as 9::3 word, instead of 1::3 word in the counterexample?
Mathias Fleury (Jul 10 2024 at 14:50):quickcheck sometimes does weird things…
Mathias Fleury (Jul 10 2024 at 14:58):If you fix the type from 'a to 3, the counter examples disappears (and nitpick did not find any in the first place)
Shweta (Jul 11 2024 at 09:43):I see. I do need the lemma for arbitrary lengths though. Will I be able to prove it even with a quickcheck counterexample? Does a quickcheck counterexample mean that the prover believes the lemma to be untrue?
Mathias Fleury (Jul 11 2024 at 10:00):I have not tried to prove it, but the lemma looks correct
Shweta (Jul 11 2024 at 10:02):Got it, thanks!
Manuel Eberl (Jul 16 2024 at 18:18):Shweta said:
I see. I do need the lemma for arbitrary lengths though. Will I be able to prove it even with a quickcheck counterexample? Does a quickcheck counterexample mean that the prover believes the lemma to be untrue?
Quickcheck and nitpick are diagnostic tools. There is a lot of extra code in their trusted code base that is not verified in any way. When they give you a counterexample, it's a good indication that your lemma might not be true in general, but it's definitely not a proof or a guarantee. Sometimes they produce spurious "counterexamples". I don't know the technical details.
Last updated: Dec 21 2024 at 16:20 UTC