/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) Overlay + Local Confluence [EQUIVALENT, 0 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) QDP
(5) TransformationProof [EQUIVALENT, 0 ms]
(6) QDP
(7) TransformationProof [EQUIVALENT, 0 ms]
(8) QDP
(9) QDPOrderProof [EQUIVALENT, 354 ms]
(10) QDP
(11) PisEmptyProof [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   h(f(x, y)) -> f(y, f(h(h(x)), a))

Q is empty.

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

(1) Overlay + Local Confluence (EQUIVALENT)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   h(f(x, y)) -> f(y, f(h(h(x)), a))

The set Q consists of the following terms:

   h(f(x0, x1))


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

(3) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   H(f(x, y)) -> H(h(x))
   H(f(x, y)) -> H(x)

The TRS R consists of the following rules:

   h(f(x, y)) -> f(y, f(h(h(x)), a))

The set Q consists of the following terms:

   h(f(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(5) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule H(f(x, y)) -> H(h(x)) at position [0] we obtained the following new rules [LPAR04]:

   (H(f(f(x0, x1), y1)) -> H(f(x1, f(h(h(x0)), a))),H(f(f(x0, x1), y1)) -> H(f(x1, f(h(h(x0)), a))))


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

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   H(f(x, y)) -> H(x)
   H(f(f(x0, x1), y1)) -> H(f(x1, f(h(h(x0)), a)))

The TRS R consists of the following rules:

   h(f(x, y)) -> f(y, f(h(h(x)), a))

The set Q consists of the following terms:

   h(f(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(7) TransformationProof (EQUIVALENT)
By forward instantiating [JAR06] the rule H(f(x, y)) -> H(x) we obtained the following new rules [LPAR04]:

   (H(f(f(y_0, y_1), x1)) -> H(f(y_0, y_1)),H(f(f(y_0, y_1), x1)) -> H(f(y_0, y_1)))
   (H(f(f(f(y_0, y_1), y_2), x1)) -> H(f(f(y_0, y_1), y_2)),H(f(f(f(y_0, y_1), y_2), x1)) -> H(f(f(y_0, y_1), y_2)))


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

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   H(f(f(x0, x1), y1)) -> H(f(x1, f(h(h(x0)), a)))
   H(f(f(y_0, y_1), x1)) -> H(f(y_0, y_1))
   H(f(f(f(y_0, y_1), y_2), x1)) -> H(f(f(y_0, y_1), y_2))

The TRS R consists of the following rules:

   h(f(x, y)) -> f(y, f(h(h(x)), a))

The set Q consists of the following terms:

   h(f(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(9) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   H(f(f(x0, x1), y1)) -> H(f(x1, f(h(h(x0)), a)))
   H(f(f(y_0, y_1), x1)) -> H(f(y_0, y_1))
   H(f(f(f(y_0, y_1), y_2), x1)) -> H(f(f(y_0, y_1), y_2))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

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

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

   <<<
 POL(h(x_1)) =  	[[0], [0]] 	 +  	[[0, 1], [3, 0]] 	* 	x_1
>>>

   <<<
 POL(a) =  	[[0], [0]]
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   h(f(x, y)) -> f(y, f(h(h(x)), a))


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(10)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   h(f(x, y)) -> f(y, f(h(h(x)), a))

The set Q consists of the following terms:

   h(f(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(11) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
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(12)
YES