(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) DependencyPairsProof (EQUIVALENT transformation)

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

(2) 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)

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

(3) DependencyGraphProof (EQUIVALENT transformation)

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

(4) 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)

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

(5) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
U21(x0, x1, x2, x3)  =  U21(x1)
DIN(x0, x1)  =  DIN(x0, x1)
U31(x0, x1, x2, x3)  =  U31(x0)
U41(x0, x1, x2)  =  U41(x0)

Tags:
U21 has argument tags [9,1,15,14] and root tag 2
DIN has argument tags [2,8] and root tag 0
U31 has argument tags [8,7,12,12] and root tag 0
U41 has argument tags [8,0,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(DIN(x1)) = 0   
POL(U21(x1, x2, x3)) = 1 + x2 + x3   
POL(U31(x1, x2, x3)) = 0   
POL(U41(x1, x2)) = 0   
POL(der(x1)) = 0   
POL(din(x1)) = 0   
POL(dout(x1)) = 1   
POL(plus(x1, x2)) = 0   
POL(times(x1, x2)) = 0   
POL(u21(x1, x2, x3)) = 0   
POL(u22(x1, x2, x3, x4)) = x1   
POL(u31(x1, x2, x3)) = 0   
POL(u32(x1, x2, x3, x4)) = x1   
POL(u41(x1, x2)) = x1   
POL(u42(x1, x2, x3)) = 1 + x1   

The following usable rules [FROCOS05] were oriented: none

(6) 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))) → 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)

Q is empty.
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.


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: SCNP Order with the following components:
Level mapping:
Top level AFS:
DIN(x0, x1)  =  DIN(x1)
U31(x0, x1, x2, x3)  =  U31(x0, x1)
U41(x0, x1, x2)  =  U41(x0, x1)

Tags:
DIN has argument tags [0,0] and root tag 0
U31 has argument tags [0,0,8,0] and root tag 0
U41 has argument tags [0,0,12] and root tag 1

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(DIN(x1)) = 1   
POL(U31(x1, x2, x3)) = 1   
POL(U41(x1, x2)) = 0   
POL(der(x1)) = 1   
POL(din(x1)) = 0   
POL(dout(x1)) = 1   
POL(plus(x1, x2)) = 0   
POL(times(x1, x2)) = 0   
POL(u21(x1, x2, x3)) = x1   
POL(u22(x1, x2, x3, x4)) = 1   
POL(u31(x1, x2, x3)) = x1   
POL(u32(x1, x2, x3, x4)) = 1   
POL(u41(x1, x2)) = x1   
POL(u42(x1, x2, x3)) = 1   

The following usable rules [FROCOS05] were oriented: none

(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))) → 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))) → 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)

Q is empty.
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))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
DIN(x0, x1)  =  DIN(x1)
U31(x0, x1, x2, x3)  =  U31(x3)

Tags:
DIN has argument tags [0,1] and root tag 0
U31 has argument tags [7,6,2,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(DIN(x1)) = 0   
POL(U31(x1, x2, x3)) = 1 + x2 + x3   
POL(der(x1)) = x1   
POL(din(x1)) = 0   
POL(dout(x1)) = 1   
POL(plus(x1, x2)) = 1 + x1   
POL(times(x1, x2)) = x1 + x2   
POL(u21(x1, x2, x3)) = 0   
POL(u22(x1, x2, x3, x4)) = 1 + x2 + x3 + x4   
POL(u31(x1, x2, x3)) = 0   
POL(u32(x1, x2, x3, x4)) = 0   
POL(u41(x1, x2)) = x2   
POL(u42(x1, x2, x3)) = x3   

The following usable rules [FROCOS05] were oriented: none

(10) Obligation:

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

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))) → 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)

Q is empty.
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(times(X, Y))) → U31(din(der(X)), X, Y)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
DIN(x0, x1)  =  DIN(x1)
U31(x0, x1, x2, x3)  =  U31(x0, x1)

Tags:
DIN has argument tags [0,1] and root tag 0
U31 has argument tags [4,1,6,6] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(DIN(x1)) = 0   
POL(U31(x1, x2, x3)) = 0   
POL(der(x1)) = 1   
POL(din(x1)) = 0   
POL(dout(x1)) = 1   
POL(plus(x1, x2)) = 0   
POL(times(x1, x2)) = 0   
POL(u21(x1, x2, x3)) = 0   
POL(u22(x1, x2, x3, x4)) = x1   
POL(u31(x1, x2, x3)) = x1   
POL(u32(x1, x2, x3, x4)) = x1   
POL(u41(x1, x2)) = x1   
POL(u42(x1, x2, x3)) = 1 + x1   

The following usable rules [FROCOS05] were oriented: none

(12) Obligation:

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

U31(dout(DX), X, Y) → DIN(der(Y))
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)

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

(13) DependencyGraphProof (EQUIVALENT transformation)

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

(14) Obligation:

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))

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


DIN(der(times(X, Y))) → DIN(der(X))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
DIN(x0, x1)  =  DIN(x0, x1)

Tags:
DIN has argument tags [0,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(DIN(x1)) = 0   
POL(der(x1)) = x1   
POL(times(x1, x2)) = 1 + x1   

The following usable rules [FROCOS05] were oriented: none

(16) 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)

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

(17) 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: SCNP Order with the following components:
Level mapping:
Top level AFS:
DIN(x0, x1)  =  DIN(x1)

Tags:
DIN has argument tags [1,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(DIN(x1)) = 0   
POL(der(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented: none

(18) 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)

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

(19) PisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE