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


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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRS Reverse [SOUND, 0 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 29 ms]
(4) RelTRS
(5) RIsEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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

   f(g(x)) -> x

The relative TRS consists of the following S rules:

   a -> h(g(f(a)))


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(1) RelTRS Reverse (SOUND)
We have reversed the following relative TRS [REVERSE]:
The set of rules R is 
   f(g(x)) -> x

The set of rules S is 
   a -> h(g(f(a)))

We have obtained the following relative TRS:
The set of rules R is 
   g(f(x)) -> x

The set of rules S is 
   a'(x) -> a'(f(g(h(x))))


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

   g(f(x)) -> x

The relative TRS consists of the following S rules:

   a'(x) -> a'(f(g(h(x))))


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

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

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

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

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

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

   g(f(x)) -> x
Rules from S:
none




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

   a'(x) -> a'(f(g(h(x))))


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(5) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(6)
YES