[isabelle] Issue with smt and linear arithmetic

[Lukas Bulwahn]

Dear Isabelle developers,

in my latest Isabelle formalization, I stumbled upon an
issue with the linear arithmetic method that the smt method
internally uses.

I am especially reporting this as this tool asks me to do
so with this warning:
Linear arithmetic should have refuted the assumptions.
Please inform Tobias Nipkow.

As it is unclear to me as an user, which tool is to `blame`,
I must leave it to you, Jasmin, Sascha and Tobias, to find the
details of tool interaction below.

The following theory fragment shows the issue I encountered
in Isabelle2016 (on Windows 10), but I could not test it
on a recent Isabelle developer version yet:

›

theory Issue imports Transcendental
begin

text ‹
First, here is the original formulation of the lemma as I encountered the issue.
Sledgehammer found the smt proof, but smt then fails.
›
lemma sin_diff_minus:
assumes "0 ≤ x" "x ≤ y" "y ≤ 2 * pi"
shows "sin ((x - y) / 2) = - sin ((y - x) / 2)"
apply (smt minus_divide_left sin_minus)
oops

text ‹
In fact, the assumptions are not required for the proof
and the issue can be reproduced in a slightly smaller
lemma without the use of assumptions.
›

lemma sin_diff_minus:
fixes x y :: real
shows "sin ((x - y) / 2) = - sin ((y - x) / 2)"
apply (smt minus_divide_left sin_minus)
oops

text ‹
Fortunately, an alternative proof is quickly found.
The lemma is proved with a few simplification steps.
›

lemma sin_diff_minus:
fixes x y :: real
shows "sin ((x - y) / 2) = - sin ((y - x) / 2)"
by (simp only: sin_minus[symmetric] minus_divide_left minus_diff_eq)

end


--
Best regards,

Lukas

[Lawrence Paulson]

This precise sort of thing has always annoyed me. I was under the impression that we could perform such simplifications (when there is a gcd; simply distributing the /2 is trivial), but upon looking, it seems that no such simprocs ever existed.

Larry

In your particular case, the obvious simplification for „(2x +2y) / 2“ is not performed, because some rule or simproc is missing in linarith’s simpset. Extending the simpset appropriately would be a solution. I’ll see what I can do, such that your issue is resolved in the next release of Isabelle.