YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRSRRRProof [EQUIVALENT, 16 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 14 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 18 ms]
(6) RelTRS
(7) RelTRSRRRProof [EQUIVALENT, 12 ms]
(8) RelTRS
(9) RIsEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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

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

   f(g(x)) -> g(f(f(x)))
   f(h(x)) -> h(g(x))
   f'(s(x), y, y) -> f'(y, x, s(x))

The relative TRS consists of the following S rules:

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


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

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

   POL(f(x_1)) = x_1
   POL(f'(x_1, x_2, x_3)) = 3*x_1 + x_2 + 2*x_3
   POL(g(x_1)) = x_1
   POL(h(x_1)) = 2*x_1
   POL(rand(x_1)) = 2 + 7*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




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

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

   f(g(x)) -> g(f(f(x)))
   f(h(x)) -> h(g(x))
   f'(s(x), y, y) -> f'(y, x, s(x))

The relative TRS consists of the following S rules:

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


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

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

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

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

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

   <<<
 POL(f'(x_1, x_2, x_3)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [1, 0]] 	* 	x_3
>>>

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

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

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

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




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

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

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

The relative TRS consists of the following S rules:

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


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

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

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

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

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

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

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

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

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




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

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

   f(g(x)) -> g(f(f(x)))

The relative TRS consists of the following S rules:

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


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

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

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

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

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

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

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

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




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

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

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


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

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

(10)
YES