[isabelle] Infinity - infinity = infinity

Dear all,

As noted before on this mailing list, automation for "enat" ("Library/Extended_Nat.thy") is quite poor. Often, the only way to proceed is to perform case distinctions on all "enat" and use auto on the emerging subgoals.

My impression is that many type classes are not available because of the definition of subtraction. Because "â - â = â" (where "â" is the infinity symbol), we lack one of the two properties required by "cancel_comm_monoid_add":

 1. âa b. a + b - a = b
 2. âa b c. a - b - c = a - (b + c)

and we lack the third property required by "comm_semiring_1_cancel":

 3. âa b c. a * (b - c) = a * b - a * c

Counterexample for 1: a = â, b = 0.
Counterexample for 3: a = â, b = c = 1.

These omissions affect further layers in the type class hierarchy -- e.g. we cannot use "ordered_cancel_comm_monoid_diff", even though some of its theorems (e.g. "add_diff_assoc2") turn out to hold.

My proposal is to change the definition of subtraction so that "â - â = 0" and to instantiate the missing type classes. I believe this would make "enat" much less painful to use, and mathematically I'm not so convinced that "â - â = â" is such a great idea anyway. Indeed, I have recently implemented ordinals below Î_0 in Isabelle and was able to have much better automation than with "enat", and there we have Ï - Ï = 0.

"enat" occurs in about 70 ".thy" files in Isabelle and the AFP, so this change (including the type class instantiations) seems quite manageable. We (= Mathias and I) would wait until after the 2016-1 release to avoid any interference.

Any opinions for or against?


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