Term Rewriting System R:
[X, Y, DX, DY, DDX]
din(der(plus(X, Y))) -> u21(din(der(X)), X, Y)
din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
din(der(der(X))) -> u41(din(der(X)), X)
u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX)
u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY))
u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
u41(dout(DX), X) -> u42(din(der(DX)), X, DX)
u42(dout(DDX), X, DX) -> dout(DDX)

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

DIN(der(plus(X, Y))) -> U21(din(der(X)), X, Y)
DIN(der(plus(X, Y))) -> DIN(der(X))
DIN(der(times(X, Y))) -> U31(din(der(X)), X, Y)
DIN(der(times(X, Y))) -> DIN(der(X))
DIN(der(der(X))) -> U41(din(der(X)), X)
DIN(der(der(X))) -> DIN(der(X))
U21(dout(DX), X, Y) -> U22(din(der(Y)), X, Y, DX)
U21(dout(DX), X, Y) -> DIN(der(Y))
U31(dout(DX), X, Y) -> U32(din(der(Y)), X, Y, DX)
U31(dout(DX), X, Y) -> DIN(der(Y))
U41(dout(DX), X) -> U42(din(der(DX)), X, DX)
U41(dout(DX), X) -> DIN(der(DX))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

DIN(der(der(X))) -> DIN(der(X))
U41(dout(DX), X) -> DIN(der(DX))
DIN(der(der(X))) -> U41(din(der(X)), X)
DIN(der(times(X, Y))) -> DIN(der(X))
U31(dout(DX), X, Y) -> DIN(der(Y))
DIN(der(times(X, Y))) -> U31(din(der(X)), X, Y)
DIN(der(plus(X, Y))) -> DIN(der(X))
U21(dout(DX), X, Y) -> DIN(der(Y))
DIN(der(plus(X, Y))) -> U21(din(der(X)), X, Y)


Rules:


din(der(plus(X, Y))) -> u21(din(der(X)), X, Y)
din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
din(der(der(X))) -> u41(din(der(X)), X)
u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX)
u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY))
u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
u41(dout(DX), X) -> u42(din(der(DX)), X, DX)
u42(dout(DDX), X, DX) -> dout(DDX)




Termination of R could not be shown.
Duration:
0:18 minutes