Termination w.r.t. Q of the following Term Rewriting System could be proven:

Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.


QTRS
  ↳ Overlay + Local Confluence

Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

The TRS is overlay and locally confluent. By [15] we can switch to innermost.

↳ QTRS
  ↳ Overlay + Local Confluence
QTRS
      ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

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

The set Q consists of the following terms:

din(der(plus(x0, x1)))
u21(dout(x0), x1, x2)
u22(dout(x0), x1, x2, x3)
din(der(times(x0, x1)))
u31(dout(x0), x1, x2)
u32(dout(x0), x1, x2, x3)
din(der(der(x0)))
u41(dout(x0), x1)
u42(dout(x0), x1, x2)


Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

din(der(plus(x0, x1)))
u21(dout(x0), x1, x2)
u22(dout(x0), x1, x2, x3)
din(der(times(x0, x1)))
u31(dout(x0), x1, x2)
u32(dout(x0), x1, x2, x3)
din(der(der(x0)))
u41(dout(x0), x1)
u42(dout(x0), x1, x2)

We have to consider all minimal (P,Q,R)-chains.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
QDP
          ↳ EdgeDeletionProof

Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

din(der(plus(x0, x1)))
u21(dout(x0), x1, x2)
u22(dout(x0), x1, x2, x3)
din(der(times(x0, x1)))
u31(dout(x0), x1, x2)
u32(dout(x0), x1, x2, x3)
din(der(der(x0)))
u41(dout(x0), x1)
u42(dout(x0), x1, x2)

We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
QDP
              ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

din(der(plus(x0, x1)))
u21(dout(x0), x1, x2)
u22(dout(x0), x1, x2, x3)
din(der(times(x0, x1)))
u31(dout(x0), x1, x2)
u32(dout(x0), x1, x2, x3)
din(der(der(x0)))
u41(dout(x0), x1)
u42(dout(x0), x1, x2)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 3 less nodes.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
QDP
                  ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

din(der(plus(x0, x1)))
u21(dout(x0), x1, x2)
u22(dout(x0), x1, x2, x3)
din(der(times(x0, x1)))
u31(dout(x0), x1, x2)
u32(dout(x0), x1, x2, x3)
din(der(der(x0)))
u41(dout(x0), x1)
u42(dout(x0), x1, x2)

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


DIN(der(plus(X, Y))) → U21(din(der(X)), X, Y)
U21(dout(DX), X, Y) → DIN(der(Y))
U41(dout(DX), X) → DIN(der(DX))
DIN(der(der(X))) → U41(din(der(X)), X)
The remaining pairs can at least be oriented weakly.

DIN(der(der(X))) → DIN(der(X))
DIN(der(times(X, Y))) → DIN(der(X))
DIN(der(plus(X, Y))) → DIN(der(X))
DIN(der(times(X, Y))) → U31(din(der(X)), X, Y)
U31(dout(DX), X, Y) → DIN(der(Y))
Used ordering: Combined order from the following AFS and order.
DIN(x1)  =  x1
der(x1)  =  der
plus(x1, x2)  =  plus
U21(x1, x2, x3)  =  U21(x1)
din(x1)  =  din
dout(x1)  =  dout(x1)
times(x1, x2)  =  times
U31(x1, x2, x3)  =  U31
U41(x1, x2)  =  x1
u21(x1, x2, x3)  =  x1
u22(x1, x2, x3, x4)  =  x1
u32(x1, x2, x3, x4)  =  x1
u41(x1, x2)  =  x1
u42(x1, x2, x3)  =  x1
u31(x1, x2, x3)  =  x1

Recursive path order with status [2].
Quasi-Precedence:
[der, dout1, U31] > U211 > [plus, times]
[der, dout1, U31] > din > [plus, times]

Status:
U31: multiset
U211: multiset
der: multiset
din: []
plus: []
times: multiset
dout1: multiset


The following usable rules [14] were oriented:

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



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof
QDP
                      ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

DIN(der(der(X))) → DIN(der(X))
DIN(der(times(X, Y))) → DIN(der(X))
DIN(der(plus(X, Y))) → DIN(der(X))
DIN(der(times(X, Y))) → U31(din(der(X)), X, Y)
U31(dout(DX), X, Y) → DIN(der(Y))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

din(der(plus(x0, x1)))
u21(dout(x0), x1, x2)
u22(dout(x0), x1, x2, x3)
din(der(times(x0, x1)))
u31(dout(x0), x1, x2)
u32(dout(x0), x1, x2, x3)
din(der(der(x0)))
u41(dout(x0), x1)
u42(dout(x0), x1, x2)

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


DIN(der(times(X, Y))) → DIN(der(X))
DIN(der(plus(X, Y))) → DIN(der(X))
DIN(der(times(X, Y))) → U31(din(der(X)), X, Y)
U31(dout(DX), X, Y) → DIN(der(Y))
The remaining pairs can at least be oriented weakly.

DIN(der(der(X))) → DIN(der(X))
Used ordering: Combined order from the following AFS and order.
DIN(x1)  =  x1
der(x1)  =  x1
times(x1, x2)  =  times(x1, x2)
plus(x1, x2)  =  plus(x1)
U31(x1, x2, x3)  =  U31(x1, x2, x3)
din(x1)  =  x1
dout(x1)  =  dout
u21(x1, x2, x3)  =  u21(x1)
u22(x1, x2, x3, x4)  =  u22
u32(x1, x2, x3, x4)  =  u32(x1, x3)
u41(x1, x2)  =  x1
u42(x1, x2, x3)  =  u42
u31(x1, x2, x3)  =  u31(x1, x3)

Recursive path order with status [2].
Quasi-Precedence:
times2 > U313
times2 > u312 > u322
plus1 > u211 > u22 > [dout, u42] > u322

Status:
U313: multiset
u322: multiset
plus1: multiset
u22: multiset
u42: []
dout: []
times2: [1,2]
u211: [1]
u312: multiset


The following usable rules [14] were oriented:

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



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof
                    ↳ QDP
                      ↳ QDPOrderProof
QDP
                          ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

DIN(der(der(X))) → DIN(der(X))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

din(der(plus(x0, x1)))
u21(dout(x0), x1, x2)
u22(dout(x0), x1, x2, x3)
din(der(times(x0, x1)))
u31(dout(x0), x1, x2)
u32(dout(x0), x1, x2, x3)
din(der(der(x0)))
u41(dout(x0), x1)
u42(dout(x0), x1, x2)

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


DIN(der(der(X))) → DIN(der(X))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
DIN(x1)  =  DIN(x1)
der(x1)  =  der(x1)

Recursive path order with status [2].
Quasi-Precedence:
[DIN1, der1]

Status:
DIN1: multiset
der1: multiset


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof
                    ↳ QDP
                      ↳ QDPOrderProof
                        ↳ QDP
                          ↳ QDPOrderProof
QDP
                              ↳ PisEmptyProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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

The set Q consists of the following terms:

din(der(plus(x0, x1)))
u21(dout(x0), x1, x2)
u22(dout(x0), x1, x2, x3)
din(der(times(x0, x1)))
u31(dout(x0), x1, x2)
u32(dout(x0), x1, x2, x3)
din(der(der(x0)))
u41(dout(x0), x1)
u42(dout(x0), x1, x2)

We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.