YES
proof of Transformed_CSR_04_Ex6_9_Luc02c_GM.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished


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

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 0 ms]
(2) QDP
(3) QDPOrderProof [EQUIVALENT, 49 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) QDP
(7) QDPOrderProof [EQUIVALENT, 20 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) QDP
(11) QDPOrderProof [EQUIVALENT, 30 ms]
(12) QDP
(13) QDPOrderProof [EQUIVALENT, 0 ms]
(14) QDP
(15) QDPOrderProof [EQUIVALENT, 17 ms]
(16) QDP
(17) DependencyGraphProof [EQUIVALENT, 0 ms]
(18) QDP
(19) UsableRulesProof [EQUIVALENT, 0 ms]
(20) QDP
(21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(22) YES


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

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

   a__2nd(cons1(X, cons(Y, Z))) -> mark(Y)
   a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1)))
   a__from(X) -> cons(mark(X), from(s(X)))
   mark(2nd(X)) -> a__2nd(mark(X))
   mark(from(X)) -> a__from(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2))
   a__2nd(X) -> 2nd(X)
   a__from(X) -> from(X)

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__2ND(cons1(X, cons(Y, Z))) -> MARK(Y)
   A__2ND(cons(X, X1)) -> A__2ND(cons1(mark(X), mark(X1)))
   A__2ND(cons(X, X1)) -> MARK(X)
   A__2ND(cons(X, X1)) -> MARK(X1)
   A__FROM(X) -> MARK(X)
   MARK(2nd(X)) -> A__2ND(mark(X))
   MARK(2nd(X)) -> MARK(X)
   MARK(from(X)) -> A__FROM(mark(X))
   MARK(from(X)) -> MARK(X)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(cons1(X1, X2)) -> MARK(X1)
   MARK(cons1(X1, X2)) -> MARK(X2)

The TRS R consists of the following rules:

   a__2nd(cons1(X, cons(Y, Z))) -> mark(Y)
   a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1)))
   a__from(X) -> cons(mark(X), from(s(X)))
   mark(2nd(X)) -> a__2nd(mark(X))
   mark(from(X)) -> a__from(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2))
   a__2nd(X) -> 2nd(X)
   a__from(X) -> from(X)

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__2ND(cons1(X, cons(Y, Z))) -> MARK(Y)
   A__2ND(cons(X, X1)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(A__2ND(x_1)) =  	[[3A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons1(x_1, x_2)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[-I]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

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

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

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

   <<<
 POL(2nd(x_1)) =  	[[3A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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

   <<<
 POL(a__2nd(x_1)) =  	[[3A]] 	 +  	[[0A]] 	* 	x_1
>>>

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


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

   mark(2nd(X)) -> a__2nd(mark(X))
   a__2nd(cons1(X, cons(Y, Z))) -> mark(Y)
   a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1)))
   mark(from(X)) -> a__from(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2))
   a__2nd(X) -> 2nd(X)
   a__from(X) -> from(X)
   a__from(X) -> cons(mark(X), from(s(X)))


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

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

   A__2ND(cons(X, X1)) -> A__2ND(cons1(mark(X), mark(X1)))
   A__2ND(cons(X, X1)) -> MARK(X1)
   A__FROM(X) -> MARK(X)
   MARK(2nd(X)) -> A__2ND(mark(X))
   MARK(2nd(X)) -> MARK(X)
   MARK(from(X)) -> A__FROM(mark(X))
   MARK(from(X)) -> MARK(X)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(cons1(X1, X2)) -> MARK(X1)
   MARK(cons1(X1, X2)) -> MARK(X2)

The TRS R consists of the following rules:

   a__2nd(cons1(X, cons(Y, Z))) -> mark(Y)
   a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1)))
   a__from(X) -> cons(mark(X), from(s(X)))
   mark(2nd(X)) -> a__2nd(mark(X))
   mark(from(X)) -> a__from(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2))
   a__2nd(X) -> 2nd(X)
   a__from(X) -> from(X)

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

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

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

   A__2ND(cons(X, X1)) -> MARK(X1)
   MARK(2nd(X)) -> A__2ND(mark(X))
   MARK(2nd(X)) -> MARK(X)
   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)
   MARK(cons1(X1, X2)) -> MARK(X1)
   MARK(cons1(X1, X2)) -> MARK(X2)

The TRS R consists of the following rules:

   a__2nd(cons1(X, cons(Y, Z))) -> mark(Y)
   a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1)))
   a__from(X) -> cons(mark(X), from(s(X)))
   mark(2nd(X)) -> a__2nd(mark(X))
   mark(from(X)) -> a__from(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2))
   a__2nd(X) -> 2nd(X)
   a__from(X) -> from(X)

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(from(X)) -> A__FROM(mark(X))
   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(A__2ND(x_1)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1
>>>

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

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

   <<<
 POL(2nd(x_1)) =  	[[1A]] 	 +  	[[3A]] 	* 	x_1
>>>

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

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

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

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

   <<<
 POL(cons1(x_1, x_2)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(a__2nd(x_1)) =  	[[1A]] 	 +  	[[3A]] 	* 	x_1
>>>

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


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

   mark(2nd(X)) -> a__2nd(mark(X))
   a__2nd(cons1(X, cons(Y, Z))) -> mark(Y)
   a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1)))
   mark(from(X)) -> a__from(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2))
   a__2nd(X) -> 2nd(X)
   a__from(X) -> from(X)
   a__from(X) -> cons(mark(X), from(s(X)))


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

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

   A__2ND(cons(X, X1)) -> MARK(X1)
   MARK(2nd(X)) -> A__2ND(mark(X))
   MARK(2nd(X)) -> MARK(X)
   A__FROM(X) -> MARK(X)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(cons1(X1, X2)) -> MARK(X1)
   MARK(cons1(X1, X2)) -> MARK(X2)

The TRS R consists of the following rules:

   a__2nd(cons1(X, cons(Y, Z))) -> mark(Y)
   a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1)))
   a__from(X) -> cons(mark(X), from(s(X)))
   mark(2nd(X)) -> a__2nd(mark(X))
   mark(from(X)) -> a__from(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2))
   a__2nd(X) -> 2nd(X)
   a__from(X) -> from(X)

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

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

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

   MARK(2nd(X)) -> A__2ND(mark(X))
   A__2ND(cons(X, X1)) -> MARK(X1)
   MARK(2nd(X)) -> MARK(X)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(cons1(X1, X2)) -> MARK(X1)
   MARK(cons1(X1, X2)) -> MARK(X2)

The TRS R consists of the following rules:

   a__2nd(cons1(X, cons(Y, Z))) -> mark(Y)
   a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1)))
   a__from(X) -> cons(mark(X), from(s(X)))
   mark(2nd(X)) -> a__2nd(mark(X))
   mark(from(X)) -> a__from(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2))
   a__2nd(X) -> 2nd(X)
   a__from(X) -> from(X)

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.

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

   <<<
 POL(2nd(x_1)) =  	[[1A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(A__2ND(x_1)) =  	[[1A]] 	 +  	[[0A]] 	* 	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(cons1(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(a__2nd(x_1)) =  	[[1A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(from(x_1)) =  	[[4A]] 	 +  	[[3A]] 	* 	x_1
>>>

   <<<
 POL(a__from(x_1)) =  	[[4A]] 	 +  	[[3A]] 	* 	x_1
>>>


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

   mark(2nd(X)) -> a__2nd(mark(X))
   a__2nd(cons1(X, cons(Y, Z))) -> mark(Y)
   a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1)))
   mark(from(X)) -> a__from(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2))
   a__2nd(X) -> 2nd(X)
   a__from(X) -> from(X)
   a__from(X) -> cons(mark(X), from(s(X)))


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

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

   MARK(2nd(X)) -> A__2ND(mark(X))
   A__2ND(cons(X, X1)) -> MARK(X1)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(cons1(X1, X2)) -> MARK(X1)
   MARK(cons1(X1, X2)) -> MARK(X2)

The TRS R consists of the following rules:

   a__2nd(cons1(X, cons(Y, Z))) -> mark(Y)
   a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1)))
   a__from(X) -> cons(mark(X), from(s(X)))
   mark(2nd(X)) -> a__2nd(mark(X))
   mark(from(X)) -> a__from(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2))
   a__2nd(X) -> 2nd(X)
   a__from(X) -> from(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(cons1(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(MARK(x_1)) =  	[[4A]] 	 +  	[[4A]] 	* 	x_1
>>>

   <<<
 POL(2nd(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(A__2ND(x_1)) =  	[[4A]] 	 +  	[[4A]] 	* 	x_1
>>>

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

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

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

   <<<
 POL(cons1(x_1, x_2)) =  	[[1A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(a__2nd(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(from(x_1)) =  	[[5A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(a__from(x_1)) =  	[[5A]] 	 +  	[[1A]] 	* 	x_1
>>>


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

   mark(2nd(X)) -> a__2nd(mark(X))
   a__2nd(cons1(X, cons(Y, Z))) -> mark(Y)
   a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1)))
   mark(from(X)) -> a__from(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2))
   a__2nd(X) -> 2nd(X)
   a__from(X) -> from(X)
   a__from(X) -> cons(mark(X), from(s(X)))


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

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(2nd(X)) -> A__2ND(mark(X))
   A__2ND(cons(X, X1)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(cons1(X1, X2)) -> MARK(X2)

The TRS R consists of the following rules:

   a__2nd(cons1(X, cons(Y, Z))) -> mark(Y)
   a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1)))
   a__from(X) -> cons(mark(X), from(s(X)))
   mark(2nd(X)) -> a__2nd(mark(X))
   mark(from(X)) -> a__from(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2))
   a__2nd(X) -> 2nd(X)
   a__from(X) -> from(X)

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

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


The following pairs can be oriented strictly and are deleted.

   MARK(2nd(X)) -> A__2ND(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)) =  	[[5A]] 	 +  	[[5A]] 	* 	x_1
>>>

   <<<
 POL(2nd(x_1)) =  	[[4A]] 	 +  	[[2A]] 	* 	x_1
>>>

   <<<
 POL(A__2ND(x_1)) =  	[[5A]] 	 +  	[[5A]] 	* 	x_1
>>>

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

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

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

   <<<
 POL(cons1(x_1, x_2)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(a__2nd(x_1)) =  	[[4A]] 	 +  	[[2A]] 	* 	x_1
>>>

   <<<
 POL(from(x_1)) =  	[[5A]] 	 +  	[[3A]] 	* 	x_1
>>>

   <<<
 POL(a__from(x_1)) =  	[[5A]] 	 +  	[[3A]] 	* 	x_1
>>>


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

   mark(2nd(X)) -> a__2nd(mark(X))
   a__2nd(cons1(X, cons(Y, Z))) -> mark(Y)
   a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1)))
   mark(from(X)) -> a__from(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2))
   a__2nd(X) -> 2nd(X)
   a__from(X) -> from(X)
   a__from(X) -> cons(mark(X), from(s(X)))


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

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A__2ND(cons(X, X1)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)
   MARK(cons1(X1, X2)) -> MARK(X2)

The TRS R consists of the following rules:

   a__2nd(cons1(X, cons(Y, Z))) -> mark(Y)
   a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1)))
   a__from(X) -> cons(mark(X), from(s(X)))
   mark(2nd(X)) -> a__2nd(mark(X))
   mark(from(X)) -> a__from(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2))
   a__2nd(X) -> 2nd(X)
   a__from(X) -> from(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(17) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
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(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(cons1(X1, X2)) -> MARK(X2)
   MARK(s(X)) -> MARK(X)

The TRS R consists of the following rules:

   a__2nd(cons1(X, cons(Y, Z))) -> mark(Y)
   a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1)))
   a__from(X) -> cons(mark(X), from(s(X)))
   mark(2nd(X)) -> a__2nd(mark(X))
   mark(from(X)) -> a__from(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2))
   a__2nd(X) -> 2nd(X)
   a__from(X) -> from(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(19) 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.
----------------------------------------

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

   MARK(cons1(X1, X2)) -> MARK(X2)
   MARK(s(X)) -> MARK(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(21) 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(cons1(X1, X2)) -> MARK(X2)
The graph contains the following edges 1 > 1


*MARK(s(X)) -> MARK(X)
The graph contains the following edges 1 > 1


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(22)
YES