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)abbrev category DIFEXT DifferentialExtension
++ Description:
++ Differential extensions of a ring R.
++ Given a differentiation on R, extend it to a differentiation on %.

DifferentialExtension(R) : Category == SIG where
  R : Ring

  SIG ==> Ring with

    differentiate : (%, R -> R) -> %
      ++ differentiate(x, deriv) differentiates x extending
      ++ the derivation deriv on R.

    differentiate : (%, R -> R, NonNegativeInteger) -> %
      ++ differentiate(x, deriv, n) differentiate x n times
      ++ using a derivation which extends deriv on R.

    D : (%, R -> R) -> %
      ++ D(x, deriv) differentiates x extending
      ++ the derivation deriv on R.

    D : (%, R -> R, NonNegativeInteger) -> %
      ++ D(x, deriv, n) differentiate x n times
      ++ using a derivation which extends deriv on R.

    if R has DifferentialRing then DifferentialRing

    if R has PartialDifferentialRing(Symbol) then
             PartialDifferentialRing(Symbol)

   add

     differentiate(x:%, derivation: R -> R, n:NonNegativeInteger):% ==
       for i in 1..n repeat x := differentiate(x, derivation)
       x
 
     D(x:%, derivation: R -> R) == differentiate(x, derivation)
 
     D(x:%, derivation: R -> R, n:NonNegativeInteger) ==
             differentiate(x, derivation, n)
 
     if R has DifferentialRing then
 
       differentiate x == differentiate(x, differentiate$R)
 
     if R has PartialDifferentialRing Symbol then
 
       differentiate(x:%, v:Symbol):% ==
         differentiate(x, s +-> differentiate(s, v)$R)