Mma10 mystery: Failure of pure function substitution with Interpretation?

Question

Bug introduced in 10.0.0 and fixed in 10.0.1

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I had code to display with subscripts from Displaying index as subscript on output: e.g. C[i] -> C_i with Notation[...] or Interpretation[..]? which is now breaking an innocuous pure function substitution as in Looking for an elegant way to solve a (system of) ODEs/functional equations with undetermined coefficients The substitution itself has nothing to do with any actual display of subscripts.

I am having a little trouble creating a minimal example, but here goes. First, in Mma10 try the pre-side effect code:

ODE = F'[z] == g F[z]
Fguess = {F-> (Exp[a #]&)}
ODE /. Fguess //FullSimplify

=> e^(a z)(a - g)== 0, the proper output

Now, add in the following code, stripped down from that previous example. This isn't intended to actually have anything to do with the substitution, but is causing side effects.

NotScriptedVarQ[z_] := ! MemberQ[$scriptedfunctionsvars, z, Infinity];
makeDef[pat_, body_] := (MakeBoxes[a : pat, fmt_] := 
ToBoxes[Interpretation[body, a], fmt] /; 
 MemberQ[Union[$scriptedconstants, $scriptedfunctions], 
  Unevaluated@h];
  )
makeDef[h_Symbol[argssub__?NotScriptedVarQ], h[argssub]];
$scriptedconstants = {};
$scriptedfunctions = {};
$scriptedfunctionsvars = {};

Now, try the substitution and a few permutations:

(*The ODE and guess are*)
ODE = F'[z] == g F[z]
Fguess = {F -> (Exp[a #] &)}
(*Substituting NOW DOESN'T work*)
ODE /. Fguess // FullSimplify
(*But if the guess is NOT displayed before the substitution(i.e., no ;) WORKS!*)
Fguess = {F -> (Exp[a #] &)};
ODE /. Fguess // FullSimplify
(*And if the guess is substituted directly, IT WORKS!*)
F'[z] == g F[z] /. {F -> (Exp[a #] &)} // FullSimplify

As this used to work fine in Mma9, I imagine this is a bug. If so, any idea of a workaround?

One last thing I learned: If you change the definition so that NotScriptedVarQ[z_] := True, or change it to always be False, the bug is no longer there!

Answer 1

I can't yet explain the behavior but I can greatly reduce your minimum example:

MakeBoxes[a_ /; FreeQ["foo", a], fmt_] := "bar" /; False

ODE = F'[z] == g F[z];
Fguess = {F -> (Exp[a #] &)}

ODE /. {F -> (Exp[a #] &)}
ODE /. Fguess

{F -> (Exp[a #1] &)}

a E^(a z) == E^(a z) g

Derivative[1][F][z] == g F[z]

Critically:

• If I suppress the output of Fguess = {F -> (Exp[a #] &)} by appending a ; the problem exhibited in the final line of code goes away.

• If I replace FreeQ["foo", a] with FreeQ["foo", HoldPattern @ a] the problem goes away.
  (Kernel restart required.)

Apparently the evaluation of a within FreeQ (or MemberQ) when MakeBoxes is performed on the output of the line Fguess = {F -> (Exp[a #] &)} breaks something but at this point I have no idea what.