/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 [EQUIVALENT, 0 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 32 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 316 ms]
(6) RelTRS
(7) RelTRSRRRProof [EQUIVALENT, 0 ms]
(8) RelTRS
(9) RelTRSRRRProof [EQUIVALENT, 34 ms]
(10) RelTRS
(11) RelTRSRRRProof [EQUIVALENT, 30 ms]
(12) RelTRS
(13) RelTRSRRRProof [EQUIVALENT, 53 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:

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

The relative TRS consists of the following S rules:

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


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

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

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

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

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


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

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

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

The relative TRS consists of the following S rules:

   u(n(x1)) -> u(x1)
   f(x1) -> n(f(x1))
   t(x1) -> n(c(t(x1)))
   n(c(x1)) -> c(n(x1))
   o(c(x1)) -> c(o(x1))
   o(c(x1)) -> o(x1)
   f(c(x1)) -> c(f(x1))
   u(c(x1)) -> c(u(x1))
   d(c(x1)) -> c(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)) = 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)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

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




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

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

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

The relative TRS consists of the following S rules:

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


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

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

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

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

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

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

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

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

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




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

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

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

The relative TRS consists of the following S rules:

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

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




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

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

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

The relative TRS consists of the following S rules:

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


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

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

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

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

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

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

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

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




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

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

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

The relative TRS consists of the following S rules:

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


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

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

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

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

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

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

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

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

   u(f(x1)) -> f(u(x1))
Rules from S:
none




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

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

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

The relative TRS consists of the following S rules:

   u(n(x1)) -> u(x1)
   f(x1) -> n(f(x1))
   t(x1) -> n(c(t(x1)))
   n(c(x1)) -> c(n(x1))
   o(c(x1)) -> c(o(x1))
   o(c(x1)) -> o(x1)
   f(c(x1)) -> c(f(x1))
   u(c(x1)) -> c(u(x1))
   d(c(x1)) -> c(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, 1], [0, 1]] 	* 	x_1
>>>

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

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

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

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


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

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

(16)
YES