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


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Termination of the given CSR could be proven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 116 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 10 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 6 ms]
(6) CSR
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

   U11(tt, M, N) -> U12(tt, M, N)
   U12(tt, M, N) -> s(plus(N, M))
   plus(N, 0) -> N
   plus(N, s(M)) -> U11(tt, M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
s: {1}
plus: {1, 2}
0: empty set

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

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

   U11(tt, M, N) -> U12(tt, M, N)
   U12(tt, M, N) -> s(plus(N, M))
   plus(N, 0) -> N
   plus(N, s(M)) -> U11(tt, M, N)

The replacement map contains the following entries:

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

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[1], [0]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2 	 +  	[[1, 1], [1, 1]] 	* 	x_3
>>>

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

   <<<
 POL(U12(x_1, x_2, x_3)) =  	[[1], [0]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2 	 +  	[[1, 1], [1, 1]] 	* 	x_3
>>>

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

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

   <<<
 POL(0) =  	[[0], [1]]
>>>

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

   U12(tt, M, N) -> s(plus(N, M))
   plus(N, 0) -> N




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

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

   U11(tt, M, N) -> U12(tt, M, N)
   plus(N, s(M)) -> U11(tt, M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
s: {1}
plus: {1, 2}

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

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

   U11(tt, M, N) -> U12(tt, M, N)
   plus(N, s(M)) -> U11(tt, M, N)

The replacement map contains the following entries:

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

   POL(U11(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3
   POL(U12(x_1, x_2, x_3)) = x_1 + 2*x_3
   POL(plus(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(s(x_1)) = 2 + 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:

   plus(N, s(M)) -> U11(tt, M, N)




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

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

   U11(tt, M, N) -> U12(tt, M, N)

The replacement map contains the following entries:

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:

   U11(tt, M, N) -> U12(tt, M, N)

The replacement map contains the following entries:

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

   <<<
 POL(U11(x_1, x_2, x_3)) =  	[[1]] 	 +  	[[1, 1]] 	* 	x_1 	 +  	[[1, 1]] 	* 	x_2 	 +  	[[1, 1]] 	* 	x_3
>>>

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

   <<<
 POL(U12(x_1, x_2, x_3)) =  	[[1]] 	 +  	[[1, 0]] 	* 	x_1 	 +  	[[1, 1]] 	* 	x_2 	 +  	[[1, 1]] 	* 	x_3
>>>

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

   U11(tt, M, N) -> U12(tt, M, N)




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

(6)
Obligation:
Context-sensitive rewrite system:
R is empty.

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

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

(8)
YES