/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRSRRRProof [EQUIVALENT, 943 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 14 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 1 ms]
(6) RelTRS
(7) RIsEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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(0)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   b(a(a(x1))) -> c(a(c(x1)))
   a(c(a(x1))) -> b(c(a(x1)))

The relative TRS consists of the following S rules:

   b(b(c(x1))) -> c(c(a(x1)))
   a(b(b(x1))) -> b(c(b(x1)))
   c(c(b(x1))) -> b(a(b(x1)))
   b(a(a(x1))) -> a(c(a(x1)))


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

(1) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :

   <<<
 POL(b(x_1)) =  	[[0], [0], [0]] 	 +  	[[2, 0, 1], [2, 0, 1], [2, 0, 1]] 	* 	x_1
>>>

   <<<
 POL(a(x_1)) =  	[[0], [2], [0]] 	 +  	[[3, 0, 0], [3, 0, 0], [2, 1, 0]] 	* 	x_1
>>>

   <<<
 POL(c(x_1)) =  	[[0], [0], [0]] 	 +  	[[3, 0, 0], [3, 0, 0], [3, 0, 0]] 	* 	x_1
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   b(a(a(x1))) -> c(a(c(x1)))
Rules from S:

   b(a(a(x1))) -> a(c(a(x1)))




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

(2)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a(c(a(x1))) -> b(c(a(x1)))

The relative TRS consists of the following S rules:

   b(b(c(x1))) -> c(c(a(x1)))
   a(b(b(x1))) -> b(c(b(x1)))
   c(c(b(x1))) -> b(a(b(x1)))


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

(3) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

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

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

   <<<
 POL(b(x_1)) =  	[[1], [0]] 	 +  	[[2, 0], [2, 0]] 	* 	x_1
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
none
Rules from S:

   b(b(c(x1))) -> c(c(a(x1)))
   a(b(b(x1))) -> b(c(b(x1)))




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

(4)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a(c(a(x1))) -> b(c(a(x1)))

The relative TRS consists of the following S rules:

   c(c(b(x1))) -> b(a(b(x1)))


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

(5) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Knuth-Bendix order [KBO] with precedence:c_1 > a_1 > b_1

and weight map:

   a_1=3
   c_1=2
   b_1=1

The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   a(c(a(x1))) -> b(c(a(x1)))
Rules from S:

   c(c(b(x1))) -> b(a(b(x1)))




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

(6)
Obligation:
Relative term rewrite system:
R is empty.
S is empty.

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(7) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------

(8)
YES