/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRS Reverse [EQUIVALENT, 0 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 52 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 2869 ms]
(6) RelTRS
(7) RelTRSRRRProof [EQUIVALENT, 0 ms]
(8) RelTRS
(9) RelTRSRRRProof [EQUIVALENT, 370 ms]
(10) RelTRS
(11) RelTRSRRRProof [EQUIVALENT, 269 ms]
(12) RelTRS
(13) RelTRSRRRProof [EQUIVALENT, 870 ms]
(14) RelTRS
(15) RIsEmptyProof [EQUIVALENT, 0 ms]
(16) YES


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

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

   d(d(n(n(x1)))) -> d(d(x1))
   d(d(o(o(x1)))) -> d(d(x1))
   o(o(u(u(x1)))) -> u(u(x1))

The relative TRS consists of the following S rules:

   t(t(u(u(x1)))) -> t(t(c(c(d(d(x1))))))
   d(d(f(f(x1)))) -> f(f(d(d(x1))))
   d(d(g(g(x1)))) -> u(u(g(g(x1))))
   f(f(u(u(x1)))) -> u(u(f(f(x1))))
   n(n(u(u(x1)))) -> u(u(x1))
   f(f(x1)) -> f(f(n(n(x1))))
   t(t(x1)) -> t(t(c(c(n(n(x1))))))
   c(c(n(n(x1)))) -> n(n(c(c(x1))))
   c(c(o(o(x1)))) -> o(o(c(c(x1))))
   c(c(o(o(x1)))) -> o(o(x1))
   c(c(f(f(x1)))) -> f(f(c(c(x1))))
   c(c(u(u(x1)))) -> u(u(c(c(x1))))
   c(c(d(d(x1)))) -> d(d(c(c(x1))))


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

(1) RelTRS Reverse (EQUIVALENT)
We have reversed the following relative TRS [REVERSE]:
The set of rules R is 
   d(d(n(n(x1)))) -> d(d(x1))
   d(d(o(o(x1)))) -> d(d(x1))
   o(o(u(u(x1)))) -> u(u(x1))

The set of rules S is 
   t(t(u(u(x1)))) -> t(t(c(c(d(d(x1))))))
   d(d(f(f(x1)))) -> f(f(d(d(x1))))
   d(d(g(g(x1)))) -> u(u(g(g(x1))))
   f(f(u(u(x1)))) -> u(u(f(f(x1))))
   n(n(u(u(x1)))) -> u(u(x1))
   f(f(x1)) -> f(f(n(n(x1))))
   t(t(x1)) -> t(t(c(c(n(n(x1))))))
   c(c(n(n(x1)))) -> n(n(c(c(x1))))
   c(c(o(o(x1)))) -> o(o(c(c(x1))))
   c(c(o(o(x1)))) -> o(o(x1))
   c(c(f(f(x1)))) -> f(f(c(c(x1))))
   c(c(u(u(x1)))) -> u(u(c(c(x1))))
   c(c(d(d(x1)))) -> d(d(c(c(x1))))

We have obtained the following relative TRS:
The set of rules R is 
   n(n(d(d(x1)))) -> d(d(x1))
   o(o(d(d(x1)))) -> d(d(x1))
   u(u(o(o(x1)))) -> u(u(x1))

The set of rules S is 
   u(u(t(t(x1)))) -> d(d(c(c(t(t(x1))))))
   f(f(d(d(x1)))) -> d(d(f(f(x1))))
   g(g(d(d(x1)))) -> g(g(u(u(x1))))
   u(u(f(f(x1)))) -> f(f(u(u(x1))))
   u(u(n(n(x1)))) -> u(u(x1))
   f(f(x1)) -> n(n(f(f(x1))))
   t(t(x1)) -> n(n(c(c(t(t(x1))))))
   n(n(c(c(x1)))) -> c(c(n(n(x1))))
   o(o(c(c(x1)))) -> c(c(o(o(x1))))
   o(o(c(c(x1)))) -> o(o(x1))
   f(f(c(c(x1)))) -> c(c(f(f(x1))))
   u(u(c(c(x1)))) -> c(c(u(u(x1))))
   d(d(c(c(x1)))) -> c(c(d(d(x1))))


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

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

   n(n(d(d(x1)))) -> d(d(x1))
   o(o(d(d(x1)))) -> d(d(x1))
   u(u(o(o(x1)))) -> u(u(x1))

The relative TRS consists of the following S rules:

   u(u(t(t(x1)))) -> d(d(c(c(t(t(x1))))))
   f(f(d(d(x1)))) -> d(d(f(f(x1))))
   g(g(d(d(x1)))) -> g(g(u(u(x1))))
   u(u(f(f(x1)))) -> f(f(u(u(x1))))
   u(u(n(n(x1)))) -> u(u(x1))
   f(f(x1)) -> n(n(f(f(x1))))
   t(t(x1)) -> n(n(c(c(t(t(x1))))))
   n(n(c(c(x1)))) -> c(c(n(n(x1))))
   o(o(c(c(x1)))) -> c(c(o(o(x1))))
   o(o(c(c(x1)))) -> o(o(x1))
   f(f(c(c(x1)))) -> c(c(f(f(x1))))
   u(u(c(c(x1)))) -> c(c(u(u(x1))))
   d(d(c(c(x1)))) -> c(c(d(d(x1))))


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

(3) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(c(x_1)) = x_1
   POL(d(x_1)) = 1 + x_1
   POL(f(x_1)) = x_1
   POL(g(x_1)) = x_1
   POL(n(x_1)) = x_1
   POL(o(x_1)) = 1 + x_1
   POL(t(x_1)) = x_1
   POL(u(x_1)) = 1 + x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   o(o(d(d(x1)))) -> d(d(x1))
   u(u(o(o(x1)))) -> u(u(x1))
Rules from S:
none




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

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

   n(n(d(d(x1)))) -> d(d(x1))

The relative TRS consists of the following S rules:

   u(u(t(t(x1)))) -> d(d(c(c(t(t(x1))))))
   f(f(d(d(x1)))) -> d(d(f(f(x1))))
   g(g(d(d(x1)))) -> g(g(u(u(x1))))
   u(u(f(f(x1)))) -> f(f(u(u(x1))))
   u(u(n(n(x1)))) -> u(u(x1))
   f(f(x1)) -> n(n(f(f(x1))))
   t(t(x1)) -> n(n(c(c(t(t(x1))))))
   n(n(c(c(x1)))) -> c(c(n(n(x1))))
   o(o(c(c(x1)))) -> c(c(o(o(x1))))
   o(o(c(c(x1)))) -> o(o(x1))
   f(f(c(c(x1)))) -> c(c(f(f(x1))))
   u(u(c(c(x1)))) -> c(c(u(u(x1))))
   d(d(c(c(x1)))) -> c(c(d(d(x1))))


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

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

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

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

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

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

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

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

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

   <<<
 POL(o(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 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:

   u(u(t(t(x1)))) -> d(d(c(c(t(t(x1))))))




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

(6)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   n(n(d(d(x1)))) -> d(d(x1))

The relative TRS consists of the following S rules:

   f(f(d(d(x1)))) -> d(d(f(f(x1))))
   g(g(d(d(x1)))) -> g(g(u(u(x1))))
   u(u(f(f(x1)))) -> f(f(u(u(x1))))
   u(u(n(n(x1)))) -> u(u(x1))
   f(f(x1)) -> n(n(f(f(x1))))
   t(t(x1)) -> n(n(c(c(t(t(x1))))))
   n(n(c(c(x1)))) -> c(c(n(n(x1))))
   o(o(c(c(x1)))) -> c(c(o(o(x1))))
   o(o(c(c(x1)))) -> o(o(x1))
   f(f(c(c(x1)))) -> c(c(f(f(x1))))
   u(u(c(c(x1)))) -> c(c(u(u(x1))))
   d(d(c(c(x1)))) -> c(c(d(d(x1))))


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

(7) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(c(x_1)) = x_1
   POL(d(x_1)) = 1 + x_1
   POL(f(x_1)) = x_1
   POL(g(x_1)) = x_1
   POL(n(x_1)) = x_1
   POL(o(x_1)) = x_1
   POL(t(x_1)) = x_1
   POL(u(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
none
Rules from S:

   g(g(d(d(x1)))) -> g(g(u(u(x1))))




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

(8)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   n(n(d(d(x1)))) -> d(d(x1))

The relative TRS consists of the following S rules:

   f(f(d(d(x1)))) -> d(d(f(f(x1))))
   u(u(f(f(x1)))) -> f(f(u(u(x1))))
   u(u(n(n(x1)))) -> u(u(x1))
   f(f(x1)) -> n(n(f(f(x1))))
   t(t(x1)) -> n(n(c(c(t(t(x1))))))
   n(n(c(c(x1)))) -> c(c(n(n(x1))))
   o(o(c(c(x1)))) -> c(c(o(o(x1))))
   o(o(c(c(x1)))) -> o(o(x1))
   f(f(c(c(x1)))) -> c(c(f(f(x1))))
   u(u(c(c(x1)))) -> c(c(u(u(x1))))
   d(d(c(c(x1)))) -> c(c(d(d(x1))))


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

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

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

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

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

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

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

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

   <<<
 POL(o(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 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:

   f(f(d(d(x1)))) -> d(d(f(f(x1))))




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

(10)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   n(n(d(d(x1)))) -> d(d(x1))

The relative TRS consists of the following S rules:

   u(u(f(f(x1)))) -> f(f(u(u(x1))))
   u(u(n(n(x1)))) -> u(u(x1))
   f(f(x1)) -> n(n(f(f(x1))))
   t(t(x1)) -> n(n(c(c(t(t(x1))))))
   n(n(c(c(x1)))) -> c(c(n(n(x1))))
   o(o(c(c(x1)))) -> c(c(o(o(x1))))
   o(o(c(c(x1)))) -> o(o(x1))
   f(f(c(c(x1)))) -> c(c(f(f(x1))))
   u(u(c(c(x1)))) -> c(c(u(u(x1))))
   d(d(c(c(x1)))) -> c(c(d(d(x1))))


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

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

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

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

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

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

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

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

   <<<
 POL(o(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:
none
Rules from S:

   u(u(f(f(x1)))) -> f(f(u(u(x1))))




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

(12)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   n(n(d(d(x1)))) -> d(d(x1))

The relative TRS consists of the following S rules:

   u(u(n(n(x1)))) -> u(u(x1))
   f(f(x1)) -> n(n(f(f(x1))))
   t(t(x1)) -> n(n(c(c(t(t(x1))))))
   n(n(c(c(x1)))) -> c(c(n(n(x1))))
   o(o(c(c(x1)))) -> c(c(o(o(x1))))
   o(o(c(c(x1)))) -> o(o(x1))
   f(f(c(c(x1)))) -> c(c(f(f(x1))))
   u(u(c(c(x1)))) -> c(c(u(u(x1))))
   d(d(c(c(x1)))) -> c(c(d(d(x1))))


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

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

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

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

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

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

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

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

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

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

   n(n(d(d(x1)))) -> d(d(x1))
Rules from S:
none




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

(14)
Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:

   u(u(n(n(x1)))) -> u(u(x1))
   f(f(x1)) -> n(n(f(f(x1))))
   t(t(x1)) -> n(n(c(c(t(t(x1))))))
   n(n(c(c(x1)))) -> c(c(n(n(x1))))
   o(o(c(c(x1)))) -> c(c(o(o(x1))))
   o(o(c(c(x1)))) -> o(o(x1))
   f(f(c(c(x1)))) -> c(c(f(f(x1))))
   u(u(c(c(x1)))) -> c(c(u(u(x1))))
   d(d(c(c(x1)))) -> c(c(d(d(x1))))


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

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

(16)
YES