2016-08-01 09:47:17 UTC
I seem to be missing something while trying to show a certain
construction is a Ring (a locale in an imported theory).
I defined all the necessary operations and this
definition stalk_ring :: "'a ⇒ ('a set × 'a) set Ring" where
"stalk_ring x =
⦇carrier = stalk x,
pop = stalk_pop x,
mop = stalk_mop x,
zero = stalk_zero x,
tp = stalk_tp x,
un = stalk_un x⦈"
which should give me a Ring stalk_ring x for each x. I then tried to
show it's a ring, but have been unable to show any of the subgoals after
assumes P: "x ∈⋃T"
shows "Ring (stalk_ring x)"
although at least most of them must be trivial (I built this ring out of
an existing one). There must be a gap in my understanding of this
process, since I tried to prove this apparently trivial
lemma (in presheaf) objecstmapringvalued:
assumes L: "(U:: 'a set) ∈ T"
shows "Ring (objectsmap U)"
and failed. This should show for each U that objectsmap U is a Ring, and
here objectsmap :: "'a set ⇒ ('a, 'm) Ring_scheme" is one of the
parameters of the locale presheaf, so wouldn't that be immediate since
the last type is ('a,'m) Ring_schemes? I also noticed that although I
can invoke presheaf_axioms and presheaf_def, I can't directly use
something like the definition of objectsmap itself in a proof. I'm still
not sure about how to tell Isabelle to use an instance of the ring
axioms or theorems for a particular ring during a proof -- in my
situation, for example, all the new ring operations are built out of
those of rings objectsmap U, so it would be great to invoke facts for
I'm a mathematician recently introduced to Isabelle, so I'd appreciate
any orientation on the matter.