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


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


YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given CSR could be proven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 46 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 0 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 5 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 0 ms]
(8) CSR
(9) RisEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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

(0)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   __(X, nil) -> X
   __(nil, X) -> X
   U11(tt) -> U12(tt)
   U12(tt) -> tt
   isNePal(__(I, __(P, I))) -> U11(tt)

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U12: {1}
isNePal: {1}

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

(1) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   __(X, nil) -> X
   __(nil, X) -> X
   U11(tt) -> U12(tt)
   U12(tt) -> tt
   isNePal(__(I, __(P, I))) -> U11(tt)

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
U11: {1}
tt: empty set
U12: {1}
isNePal: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(U11(x_1)) = x_1
   POL(U12(x_1)) = x_1
   POL(__(x_1, x_2)) = x_1 + x_2
   POL(isNePal(x_1)) = 1 + x_1
   POL(nil) = 1
   POL(tt) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   __(X, nil) -> X
   __(nil, X) -> X




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

(2)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   U11(tt) -> U12(tt)
   U12(tt) -> tt
   isNePal(__(I, __(P, I))) -> U11(tt)

The replacement map contains the following entries:

__: {1, 2}
U11: {1}
tt: empty set
U12: {1}
isNePal: {1}

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

(3) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   U11(tt) -> U12(tt)
   U12(tt) -> tt
   isNePal(__(I, __(P, I))) -> U11(tt)

The replacement map contains the following entries:

__: {1, 2}
U11: {1}
tt: empty set
U12: {1}
isNePal: {1}
Used ordering:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

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

   <<<
 POL(U11(x_1)) =  	[[1], [0]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1
>>>

   <<<
 POL(tt) =  	[[1], [1]]
>>>

   <<<
 POL(U12(x_1)) =  	[[1], [0]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(isNePal(x_1)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
>>>

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U12(tt) -> tt
   isNePal(__(I, __(P, I))) -> U11(tt)




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

(4)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   U11(tt) -> U12(tt)

The replacement map contains the following entries:

__: {1, 2}
U11: {1}
tt: empty set
U12: {1}

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

(5) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   U11(tt) -> U12(tt)

The replacement map contains the following entries:

__: {1, 2}
U11: {1}
tt: empty set
U12: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(U11(x_1)) = 2*x_1
   POL(U12(x_1)) = x_1
   POL(__(x_1, x_2)) = 2 + 2*x_1 + x_2
   POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   __(__(X, Y), Z) -> __(X, __(Y, Z))




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

(6)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U11(tt) -> U12(tt)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}

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

(7) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U11(tt) -> U12(tt)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(U11(x_1)) = 1 + x_1
   POL(U12(x_1)) = x_1
   POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U11(tt) -> U12(tt)




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

(8)
Obligation:
Context-sensitive rewrite system:
R is empty.

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

(9) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
----------------------------------------

(10)
YES