[isabelle] Proving something is an instance of a locale



Dear Isabelle Users,

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 â  ('a set à 'a) set Ring" where
   "stalk_ring x =
   âcarrier = stalk x,
     pop = stalk_pop x,
     mop = stalk_mop x,
     zero = stalk_zero x,
     tp = stalk_tp x,
     un = stalk_un xâ"
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 writing

lemma stalk_set_is_ring:
assumes P: "x ââT"
shows "Ring (stalk_ring x)"
proof unfold_locales

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) â T"
shows "Ring (objectsmap U)"
proof unfold_locales

and failed. This should show for each U that objectsmap U is a Ring, and here objectsmap :: "'a set â ('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 such.

I'm a mathematician recently introduced to Isabelle, so I'd appreciate any orientation on the matter.

Sincerely,

Josà Siqueira




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