How to get the boolean value of an inequality involving an InterpolatingFunction?

Question

Here's the code:

yan = FunctionInterpolation[x^2, {x, -1, 1}];
FullSimplify[yan[x] > -1, -1 < x < 1]

Needless to say, what I expect to see in the output is "True", but FullSimplify doesn't seem to work. What function should I turn to?

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@J.M. @belisarius @acl

囧…A very simple solution suddenly struck me, it is:

yan = FunctionInterpolation[x^2, {x, -1, 1}];
MinValue[{yan[x], -1 < x < 1},x]>-1

Answer 1

The following is basically the same @acl did, but using the package InterpolatingFunctionAnatomy which (in principle) will behave better than peeking at the internal structures when Mma version changes.

Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"];
yan = FunctionInterpolation[x^2, {x, -1, 1}];

yin = InterpolatingPolynomial[Transpose[Flatten /@
                                {InterpolatingFunctionCoordinates@yan, 
                                 InterpolatingFunctionValuesOnGrid@yan}], x];
FullSimplify[yin > -1, -1 < x < 1]
(*
  True
*)

Answer 2

You can do this by explicitly constructing the InterpolatingPolynomial corresponding to yan, and then using FullSimplify:

yin = InterpolatingPolynomial[Transpose[Flatten /@ {yan[[3]],yan[[4]]}],x];
FullSimplify[yin > -1, -1 < x < 1]
(*True*)

Why does this work? Because yan actually has a list of points:

FullForm[yan]

so I can extract them with Transpose[Flatten /@ {yan[[3]],yan[[4]]}] and use them to construct a polynomial, which does the same thing as the interpolation function but which FullSimplify can now handle.

Maybe there's a better way to construct the InterpolatingPolynomial but this works.