I usually think of also have
as working like this:
have "P r Q1" by simp
also have "... r Q2" by simp
also have "... r Q3" by simp
...
also have "... r Qn" by simp
finally have "P r Qn+1" by simp
where "... r Qm"
means "Qm-1 r Qm"
and r
is some transitive relation.
With r
meaning =
this seems to really be the case, however, when using ≥
I have found a counterexample to this description:
...
have "1- 1/(2^(n+1))≥1/(2::real)" by simp
also have "... ≥ 0" (* here when I check the 'output' it seems to be
considering "0 ≤ 1 - 1 / 2 ^ (n + 1)" which in the previous notation
would be Qn r Qn+2 !*)
My question is, how does also have
work, in particular, how do I predict what ...
is going to refer to?
AFAIR,
a ≥ b
is an abbreviation forb ≤ a
. With this in mind, you see that this does not fit the pattern expected byalso
.I suggest that you state your chain of inequations the other way, from lowest to highest. You can still state the final result using
≥
if you want – after all, it is just an abbreviation.