Termination w.r.t. Q of the given QTRS could be proven:

(0) Obligation:

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.

(1) Overlay + Local Confluence (EQUIVALENT transformation)

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

(2) Obligation:

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)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(4) Obligation:

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

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(6) Obligation:

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

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

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21(x1, x2, x3)  =  U21(x1)
dout(x1)  =  dout(x1)
DIN(x1)  =  DIN(x1)
der(x1)  =  der
plus(x1, x2)  =  plus
din(x1)  =  din(x1)
times(x1, x2)  =  x2
U31(x1, x2, x3)  =  U31(x1)
U41(x1, x2)  =  x1
u21(x1, x2, x3)  =  x1
u22(x1, x2, x3, x4)  =  u22(x1, x4)
u31(x1, x2, x3)  =  x1
u32(x1, x2, x3, x4)  =  x1
u41(x1, x2)  =  x1
u42(x1, x2, x3)  =  u42(x1)

Recursive path order with status [RPO].
Precedence:

dout₁ > DIN₁ > U21₁ > der > plus
dout₁ > DIN₁ > din₁ > der > plus
dout₁ > DIN₁ > U31₁ > der > plus
dout₁ > u22₂ > plus
dout₁ > u42₁ > plus

Status:

U21₁: multiset
dout₁: multiset
DIN₁: multiset
der: multiset
plus: multiset
din₁: multiset
U31₁: multiset
u22₂: multiset
u42₁: multiset

The following usable rules [FROCOS05] were oriented:

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)

(8) Obligation:

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

DIN(der(plus(X, Y))) → DIN(der(X))
DIN(der(times(X, Y))) → DIN(der(X))
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.

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

DIN(der(plus(X, Y))) → DIN(der(X))
DIN(der(times(X, Y))) → DIN(der(X))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
DIN(x1)  =  DIN(x1)
der(x1)  =  x1
plus(x1, x2)  =  plus(x1)
times(x1, x2)  =  times(x1, x2)
din(x1)  =  din(x1)
u21(x1, x2, x3)  =  x2
dout(x1)  =  dout
u22(x1, x2, x3, x4)  =  x2
u31(x1, x2, x3)  =  u31(x2)
u32(x1, x2, x3, x4)  =  x2
u41(x1, x2)  =  u41(x2)
u42(x1, x2, x3)  =  u42

Recursive path order with status [RPO].
Precedence:

plus₁ > DIN₁ > dout
plus₁ > din₁ > u41₁ > u42 > dout
times₂ > DIN₁ > dout
times₂ > u31₁ > din₁ > u41₁ > u42 > dout

Status:

DIN₁: multiset
plus₁: multiset
times₂: multiset
din₁: [1]
dout: []
u31₁: multiset
u41₁: multiset
u42: multiset

The following usable rules [FROCOS05] were oriented:

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)

(10) Obligation:

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.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

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

Recursive path order with status [RPO].
Precedence:

DIN₁ > times
dout₁ > der₁ > din > times
dout₁ > u32₁ > times

Status:

DIN₁: [1]
der₁: [1]
din: []
dout₁: [1]
times: multiset
u32₁: multiset

The following usable rules [FROCOS05] were oriented:

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)

(12) Obligation:

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.

(13) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(14) TRUE