/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 18 ms]
(2) QDP
(3) QDPOrderProof [EQUIVALENT, 103 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 32 ms]
(6) QDP
(7) QDPOrderProof [EQUIVALENT, 43 ms]
(8) QDP
(9) QDPOrderProof [EQUIVALENT, 49 ms]
(10) QDP
(11) QDPOrderProof [EQUIVALENT, 66 ms]
(12) QDP
(13) DependencyGraphProof [EQUIVALENT, 0 ms]
(14) AND
    (15) QDP
        (16) QDPOrderProof [EQUIVALENT, 12 ms]
        (17) QDP
        (18) PisEmptyProof [EQUIVALENT, 0 ms]
        (19) YES
    (20) QDP
        (21) QDPOrderProof [EQUIVALENT, 7 ms]
        (22) QDP
        (23) QDPOrderProof [EQUIVALENT, 23 ms]
        (24) QDP
        (25) DependencyGraphProof [EQUIVALENT, 0 ms]
        (26) QDP
        (27) UsableRulesProof [EQUIVALENT, 0 ms]
        (28) QDP
        (29) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (30) YES


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

Q is empty.

----------------------------------------

(1) DependencyPairsProof (EQUIVALENT)
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:

   A__FROM(X) -> MARK(X)
   A__AFTER(0, XS) -> MARK(XS)
   A__AFTER(s(N), cons(X, XS)) -> A__AFTER(mark(N), mark(XS))
   A__AFTER(s(N), cons(X, XS)) -> MARK(N)
   A__AFTER(s(N), cons(X, XS)) -> MARK(XS)
   MARK(from(X)) -> A__FROM(mark(X))
   MARK(from(X)) -> MARK(X)
   MARK(after(X1, X2)) -> A__AFTER(mark(X1), mark(X2))
   MARK(after(X1, X2)) -> MARK(X1)
   MARK(after(X1, X2)) -> MARK(X2)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)

The TRS R consists of the following rules:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

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

(3) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   A__AFTER(s(N), cons(X, XS)) -> MARK(N)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(A__FROM(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(MARK(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(A__AFTER(x_1, x_2)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(0) =  	[[0A]]
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(from(x_1)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(after(x_1, x_2)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(a__from(x_1)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(a__after(x_1, x_2)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)
   a__from(X) -> cons(mark(X), from(s(X)))


----------------------------------------

(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A__FROM(X) -> MARK(X)
   A__AFTER(0, XS) -> MARK(XS)
   A__AFTER(s(N), cons(X, XS)) -> A__AFTER(mark(N), mark(XS))
   A__AFTER(s(N), cons(X, XS)) -> MARK(XS)
   MARK(from(X)) -> A__FROM(mark(X))
   MARK(from(X)) -> MARK(X)
   MARK(after(X1, X2)) -> A__AFTER(mark(X1), mark(X2))
   MARK(after(X1, X2)) -> MARK(X1)
   MARK(after(X1, X2)) -> MARK(X2)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)

The TRS R consists of the following rules:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

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

(5) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(after(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(A__FROM(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(MARK(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(A__AFTER(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(0) =  	[[0A]]
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(from(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(after(x_1, x_2)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(a__from(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(a__after(x_1, x_2)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)
   a__from(X) -> cons(mark(X), from(s(X)))


----------------------------------------

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A__FROM(X) -> MARK(X)
   A__AFTER(0, XS) -> MARK(XS)
   A__AFTER(s(N), cons(X, XS)) -> A__AFTER(mark(N), mark(XS))
   A__AFTER(s(N), cons(X, XS)) -> MARK(XS)
   MARK(from(X)) -> A__FROM(mark(X))
   MARK(from(X)) -> MARK(X)
   MARK(after(X1, X2)) -> A__AFTER(mark(X1), mark(X2))
   MARK(after(X1, X2)) -> MARK(X2)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)

The TRS R consists of the following rules:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

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

(7) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(after(X1, X2)) -> MARK(X2)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(A__FROM(x_1)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(MARK(x_1)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(A__AFTER(x_1, x_2)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(0) =  	[[3A]]
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(from(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(after(x_1, x_2)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(a__from(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(a__after(x_1, x_2)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)
   a__from(X) -> cons(mark(X), from(s(X)))


----------------------------------------

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A__FROM(X) -> MARK(X)
   A__AFTER(0, XS) -> MARK(XS)
   A__AFTER(s(N), cons(X, XS)) -> A__AFTER(mark(N), mark(XS))
   A__AFTER(s(N), cons(X, XS)) -> MARK(XS)
   MARK(from(X)) -> A__FROM(mark(X))
   MARK(from(X)) -> MARK(X)
   MARK(after(X1, X2)) -> A__AFTER(mark(X1), mark(X2))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)

The TRS R consists of the following rules:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

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

(9) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   A__AFTER(s(N), cons(X, XS)) -> MARK(XS)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(A__FROM(x_1)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(MARK(x_1)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(A__AFTER(x_1, x_2)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(0) =  	[[0A]]
>>>

   <<<
 POL(s(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(mark(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(from(x_1)) =  	[[1A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(after(x_1, x_2)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(a__from(x_1)) =  	[[1A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(a__after(x_1, x_2)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)
   a__from(X) -> cons(mark(X), from(s(X)))


----------------------------------------

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A__FROM(X) -> MARK(X)
   A__AFTER(0, XS) -> MARK(XS)
   A__AFTER(s(N), cons(X, XS)) -> A__AFTER(mark(N), mark(XS))
   MARK(from(X)) -> A__FROM(mark(X))
   MARK(from(X)) -> MARK(X)
   MARK(after(X1, X2)) -> A__AFTER(mark(X1), mark(X2))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)

The TRS R consists of the following rules:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

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

(11) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   A__AFTER(0, XS) -> MARK(XS)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(A__FROM(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(MARK(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(A__AFTER(x_1, x_2)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(0) =  	[[0A]]
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(mark(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(from(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(after(x_1, x_2)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(a__from(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(a__after(x_1, x_2)) =  	[[1A]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)
   a__from(X) -> cons(mark(X), from(s(X)))


----------------------------------------

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A__FROM(X) -> MARK(X)
   A__AFTER(s(N), cons(X, XS)) -> A__AFTER(mark(N), mark(XS))
   MARK(from(X)) -> A__FROM(mark(X))
   MARK(from(X)) -> MARK(X)
   MARK(after(X1, X2)) -> A__AFTER(mark(X1), mark(X2))
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)

The TRS R consists of the following rules:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

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

(13) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
----------------------------------------

(14)
Complex Obligation (AND)

----------------------------------------

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A__AFTER(s(N), cons(X, XS)) -> A__AFTER(mark(N), mark(XS))

The TRS R consists of the following rules:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

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

(16) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   A__AFTER(s(N), cons(X, XS)) -> A__AFTER(mark(N), mark(XS))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( A__AFTER_2(x_1, x_2) ) = 2x_1 + x_2 + 2
POL( mark_1(x_1) ) = x_1
POL( from_1(x_1) ) = max{0, -2}
POL( a__from_1(x_1) ) = max{0, -2}
POL( after_2(x_1, x_2) ) = 2x_2
POL( a__after_2(x_1, x_2) ) = 2x_2
POL( 0 ) = 0
POL( s_1(x_1) ) = 2x_1 + 2
POL( cons_2(x_1, x_2) ) = 2x_2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)
   a__from(X) -> cons(mark(X), from(s(X)))


----------------------------------------

(17)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

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

(18) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(19)
YES

----------------------------------------

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(from(X)) -> A__FROM(mark(X))
   A__FROM(X) -> MARK(X)
   MARK(from(X)) -> MARK(X)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)

The TRS R consists of the following rules:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

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

(21) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(from(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(MARK(x_1)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(from(x_1)) =  	[[3A]] 	 +  	[[3A]] 	* 	x_1
>>>

   <<<
 POL(A__FROM(x_1)) =  	[[2A]] 	 +  	[[3A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(a__from(x_1)) =  	[[3A]] 	 +  	[[3A]] 	* 	x_1
>>>

   <<<
 POL(after(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(a__after(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(0) =  	[[0A]]
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)
   a__from(X) -> cons(mark(X), from(s(X)))


----------------------------------------

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(from(X)) -> A__FROM(mark(X))
   A__FROM(X) -> MARK(X)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)

The TRS R consists of the following rules:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

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

(23) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(from(X)) -> A__FROM(mark(X))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(MARK(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(from(x_1)) =  	[[0A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(A__FROM(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(a__from(x_1)) =  	[[0A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(after(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(a__after(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(0) =  	[[2A]]
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)
   a__from(X) -> cons(mark(X), from(s(X)))


----------------------------------------

(24)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A__FROM(X) -> MARK(X)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)

The TRS R consists of the following rules:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

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

(25) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(26)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(s(X)) -> MARK(X)
   MARK(cons(X1, X2)) -> MARK(X1)

The TRS R consists of the following rules:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

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

(27) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(28)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(s(X)) -> MARK(X)
   MARK(cons(X1, X2)) -> MARK(X1)

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

(29) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*MARK(s(X)) -> MARK(X)
The graph contains the following edges 1 > 1


*MARK(cons(X1, X2)) -> MARK(X1)
The graph contains the following edges 1 > 1


----------------------------------------

(30)
YES