/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 [SOUND, 0 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 7 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 52 ms]
(6) RelTRS
(7) RelTRSRRRProof [EQUIVALENT, 18 ms]
(8) RelTRS
(9) RelTRSRRRProof [EQUIVALENT, 4 ms]
(10) RelTRS
(11) RIsEmptyProof [EQUIVALENT, 0 ms]
(12) YES


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

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

   f(0) -> s(0)
   f(s(0)) -> s(0)
   f(s(s(x))) -> f(f(s(x)))

The relative TRS consists of the following S rules:

   rand(x) -> x
   rand(x) -> rand(s(x))


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

(1) RelTRS Reverse (SOUND)
We have reversed the following relative TRS [REVERSE]:
The set of rules R is 
   f(0) -> s(0)
   f(s(0)) -> s(0)
   f(s(s(x))) -> f(f(s(x)))

The set of rules S is 
   rand(x) -> x
   rand(x) -> rand(s(x))

We have obtained the following relative TRS:
The set of rules R is 
   0'(f(x)) -> 0'(s(x))
   0'(s(f(x))) -> 0'(s(x))
   s(s(f(x))) -> s(f(f(x)))

The set of rules S is 
   rand(x) -> x
   rand(x) -> s(rand(x))


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

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

   0'(f(x)) -> 0'(s(x))
   0'(s(f(x))) -> 0'(s(x))
   s(s(f(x))) -> s(f(f(x)))

The relative TRS consists of the following S rules:

   rand(x) -> x
   rand(x) -> s(rand(x))


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

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

   POL(0'(x_1)) = x_1
   POL(f(x_1)) = x_1
   POL(rand(x_1)) = 1 + x_1
   POL(s(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:

   rand(x) -> x




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

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

   0'(f(x)) -> 0'(s(x))
   0'(s(f(x))) -> 0'(s(x))
   s(s(f(x))) -> s(f(f(x)))

The relative TRS consists of the following S rules:

   rand(x) -> s(rand(x))


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

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

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

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

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

   <<<
 POL(rand(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:

   0'(s(f(x))) -> 0'(s(x))
Rules from S:
none




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

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

   0'(f(x)) -> 0'(s(x))
   s(s(f(x))) -> s(f(f(x)))

The relative TRS consists of the following S rules:

   rand(x) -> s(rand(x))


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

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

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

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

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

   <<<
 POL(rand(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:

   0'(f(x)) -> 0'(s(x))
Rules from S:
none




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

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

   s(s(f(x))) -> s(f(f(x)))

The relative TRS consists of the following S rules:

   rand(x) -> s(rand(x))


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

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

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

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

   <<<
 POL(rand(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:

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




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

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

   rand(x) -> s(rand(x))


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

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

(12)
YES