Closed LeventErkok closed 4 years ago
Let me first rewrite the original formula:
sin²x + cos²x ≠ 1
= ¬(sin²x + cos²x = 1)
= ¬[(sin²x + cos²x ≥ 1) ∧ (sin²x + cos²x ≤ 1)]
= (sin²x + cos²x < 1) ∨ (sin²x + cos²x > 1)
The meaning of delta-sat with a model is that any point in the model satisfies the delta-perturbation of the original formula:
(sin²x + cos²x ≤ 1 + δ) ∨ (sin²x + cos²x ≥ 1 - δ)
Since this is a trigonometric identity, LHS is always 1. So the formula reduces to:
(1 ≤ 1 + δ) ∨ (1 ≥ 1 - δ)]
= (0 ≤ δ) ∨ (0 ≥ -δ)
= (0 ≤ δ)
Since δ is a positive rational, this formula always holds regardless of the choice of δ.
Reference:
Thanks for the explanation. I need to read up on delta-perturbation more. Much appreciated.
This is a question on how to interpret this output. For this benchmark:
dReal produces:
The benchmark is asking dreal to find a value between
0
and1
such thatsin^2 x + cos^2 x
is not equal to1
. Since this is a trigonometric identity, I'd have expected dReal to sayunsat
, but perhaps I'm misinterpreting the output here.If I do a simple calculation on the lower-bound with an IEEE-754 double, I get:
that is, I do not get a numeric error. My understanding is that the proof is over arbitrary reals backed by IEEE-doubles in models.
So, how should I interpret this output?