/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 25 ms]
(4) QTRS
(5) DependencyPairsProof [EQUIVALENT, 26 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) AND
    (9) QDP
        (10) UsableRulesProof [EQUIVALENT, 0 ms]
        (11) QDP
        (12) QDPOrderProof [EQUIVALENT, 22 ms]
        (13) QDP
        (14) QDPOrderProof [EQUIVALENT, 0 ms]
        (15) QDP
        (16) QDPOrderProof [EQUIVALENT, 504 ms]
        (17) QDP
        (18) PisEmptyProof [EQUIVALENT, 0 ms]
        (19) YES
    (20) QDP
        (21) UsableRulesProof [EQUIVALENT, 0 ms]
        (22) QDP
        (23) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (24) YES


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

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

   log(s(x1)) -> s(log(half(s(x1))))
   half(0(x1)) -> 0(s(s(half(x1))))
   half(s(0(x1))) -> 0(x1)
   half(s(s(x1))) -> s(half(p(s(s(x1)))))
   half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1))))
   p(s(s(s(x1)))) -> s(p(s(s(x1))))
   s(s(p(s(x1)))) -> s(s(x1))
   0(x1) -> x1

Q is empty.

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

(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
----------------------------------------

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

   s(log(x1)) -> s(half(log(s(x1))))
   0(half(x1)) -> half(s(s(0(x1))))
   0(s(half(x1))) -> 0(x1)
   s(s(half(x1))) -> s(s(p(half(s(x1)))))
   s(s(s(s(half(half(x1)))))) -> half(half(s(s(x1))))
   s(s(s(p(x1)))) -> s(s(p(s(x1))))
   s(p(s(s(x1)))) -> s(s(x1))
   0(x1) -> x1

Q is empty.

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

(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(0(x_1)) = 1 + x_1
   POL(half(x_1)) = x_1
   POL(log(x_1)) = x_1
   POL(p(x_1)) = x_1
   POL(s(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   0(x1) -> x1




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

(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   s(log(x1)) -> s(half(log(s(x1))))
   0(half(x1)) -> half(s(s(0(x1))))
   0(s(half(x1))) -> 0(x1)
   s(s(half(x1))) -> s(s(p(half(s(x1)))))
   s(s(s(s(half(half(x1)))))) -> half(half(s(s(x1))))
   s(s(s(p(x1)))) -> s(s(p(s(x1))))
   s(p(s(s(x1)))) -> s(s(x1))

Q is empty.

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

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

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

   S(log(x1)) -> S(half(log(s(x1))))
   S(log(x1)) -> S(x1)
   0^1(half(x1)) -> S(s(0(x1)))
   0^1(half(x1)) -> S(0(x1))
   0^1(half(x1)) -> 0^1(x1)
   0^1(s(half(x1))) -> 0^1(x1)
   S(s(half(x1))) -> S(s(p(half(s(x1)))))
   S(s(half(x1))) -> S(p(half(s(x1))))
   S(s(half(x1))) -> S(x1)
   S(s(s(s(half(half(x1)))))) -> S(s(x1))
   S(s(s(s(half(half(x1)))))) -> S(x1)
   S(s(s(p(x1)))) -> S(s(p(s(x1))))
   S(s(s(p(x1)))) -> S(p(s(x1)))
   S(s(s(p(x1)))) -> S(x1)

The TRS R consists of the following rules:

   s(log(x1)) -> s(half(log(s(x1))))
   0(half(x1)) -> half(s(s(0(x1))))
   0(s(half(x1))) -> 0(x1)
   s(s(half(x1))) -> s(s(p(half(s(x1)))))
   s(s(s(s(half(half(x1)))))) -> half(half(s(s(x1))))
   s(s(s(p(x1)))) -> s(s(p(s(x1))))
   s(p(s(s(x1)))) -> s(s(x1))

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

(7) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 6 less nodes.
----------------------------------------

(8)
Complex Obligation (AND)

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

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(half(x1))) -> S(x1)
   S(log(x1)) -> S(x1)
   S(s(s(s(half(half(x1)))))) -> S(s(x1))
   S(s(s(s(half(half(x1)))))) -> S(x1)
   S(s(s(p(x1)))) -> S(s(p(s(x1))))
   S(s(s(p(x1)))) -> S(x1)

The TRS R consists of the following rules:

   s(log(x1)) -> s(half(log(s(x1))))
   0(half(x1)) -> half(s(s(0(x1))))
   0(s(half(x1))) -> 0(x1)
   s(s(half(x1))) -> s(s(p(half(s(x1)))))
   s(s(s(s(half(half(x1)))))) -> half(half(s(s(x1))))
   s(s(s(p(x1)))) -> s(s(p(s(x1))))
   s(p(s(s(x1)))) -> s(s(x1))

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

(10) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(half(x1))) -> S(x1)
   S(log(x1)) -> S(x1)
   S(s(s(s(half(half(x1)))))) -> S(s(x1))
   S(s(s(s(half(half(x1)))))) -> S(x1)
   S(s(s(p(x1)))) -> S(s(p(s(x1))))
   S(s(s(p(x1)))) -> S(x1)

The TRS R consists of the following rules:

   s(log(x1)) -> s(half(log(s(x1))))
   s(s(half(x1))) -> s(s(p(half(s(x1)))))
   s(s(s(s(half(half(x1)))))) -> half(half(s(s(x1))))
   s(p(s(s(x1)))) -> s(s(x1))
   s(s(s(p(x1)))) -> s(s(p(s(x1))))

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

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


The following pairs can be oriented strictly and are deleted.

   S(log(x1)) -> S(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(S(x_1)) = x_1
   POL(half(x_1)) = x_1
   POL(log(x_1)) = 1 + x_1
   POL(p(x_1)) = x_1
   POL(s(x_1)) = x_1

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

   s(log(x1)) -> s(half(log(s(x1))))
   s(s(half(x1))) -> s(s(p(half(s(x1)))))
   s(s(s(s(half(half(x1)))))) -> half(half(s(s(x1))))
   s(s(s(p(x1)))) -> s(s(p(s(x1))))
   s(p(s(s(x1)))) -> s(s(x1))


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

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(half(x1))) -> S(x1)
   S(s(s(s(half(half(x1)))))) -> S(s(x1))
   S(s(s(s(half(half(x1)))))) -> S(x1)
   S(s(s(p(x1)))) -> S(s(p(s(x1))))
   S(s(s(p(x1)))) -> S(x1)

The TRS R consists of the following rules:

   s(log(x1)) -> s(half(log(s(x1))))
   s(s(half(x1))) -> s(s(p(half(s(x1)))))
   s(s(s(s(half(half(x1)))))) -> half(half(s(s(x1))))
   s(p(s(s(x1)))) -> s(s(x1))
   s(s(s(p(x1)))) -> s(s(p(s(x1))))

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

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


The following pairs can be oriented strictly and are deleted.

   S(s(half(x1))) -> S(x1)
   S(s(s(s(half(half(x1)))))) -> S(s(x1))
   S(s(s(s(half(half(x1)))))) -> S(x1)
   S(s(s(p(x1)))) -> S(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( S_1(x_1) ) = 2x_1 + 2
POL( s_1(x_1) ) = x_1 + 2
POL( log_1(x_1) ) = max{0, -2}
POL( half_1(x_1) ) = x_1
POL( p_1(x_1) ) = max{0, x_1 - 2}

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

   s(log(x1)) -> s(half(log(s(x1))))
   s(s(half(x1))) -> s(s(p(half(s(x1)))))
   s(s(s(s(half(half(x1)))))) -> half(half(s(s(x1))))
   s(s(s(p(x1)))) -> s(s(p(s(x1))))
   s(p(s(s(x1)))) -> s(s(x1))


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

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(s(p(x1)))) -> S(s(p(s(x1))))

The TRS R consists of the following rules:

   s(log(x1)) -> s(half(log(s(x1))))
   s(s(half(x1))) -> s(s(p(half(s(x1)))))
   s(s(s(s(half(half(x1)))))) -> half(half(s(s(x1))))
   s(p(s(s(x1)))) -> s(s(x1))
   s(s(s(p(x1)))) -> s(s(p(s(x1))))

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

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


The following pairs can be oriented strictly and are deleted.

   S(s(s(p(x1)))) -> S(s(p(s(x1))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(S(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
>>>

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

   <<<
 POL(p(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [-I, -I, 0A], [1A, 1A, 0A]] 	* 	x_1
>>>

   <<<
 POL(log(x_1)) =  	[[-I], [-I], [1A]] 	 +  	[[0A, -I, 0A], [1A, -I, -I], [-I, 0A, -I]] 	* 	x_1
>>>

   <<<
 POL(half(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
>>>


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

   s(log(x1)) -> s(half(log(s(x1))))
   s(s(half(x1))) -> s(s(p(half(s(x1)))))
   s(s(s(s(half(half(x1)))))) -> half(half(s(s(x1))))
   s(s(s(p(x1)))) -> s(s(p(s(x1))))
   s(p(s(s(x1)))) -> s(s(x1))


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

(17)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   s(log(x1)) -> s(half(log(s(x1))))
   s(s(half(x1))) -> s(s(p(half(s(x1)))))
   s(s(s(s(half(half(x1)))))) -> half(half(s(s(x1))))
   s(p(s(s(x1)))) -> s(s(x1))
   s(s(s(p(x1)))) -> s(s(p(s(x1))))

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

(18) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(19)
YES

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

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(s(half(x1))) -> 0^1(x1)
   0^1(half(x1)) -> 0^1(x1)

The TRS R consists of the following rules:

   s(log(x1)) -> s(half(log(s(x1))))
   0(half(x1)) -> half(s(s(0(x1))))
   0(s(half(x1))) -> 0(x1)
   s(s(half(x1))) -> s(s(p(half(s(x1)))))
   s(s(s(s(half(half(x1)))))) -> half(half(s(s(x1))))
   s(s(s(p(x1)))) -> s(s(p(s(x1))))
   s(p(s(s(x1)))) -> s(s(x1))

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

(21) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(s(half(x1))) -> 0^1(x1)
   0^1(half(x1)) -> 0^1(x1)

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

(23) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*0^1(s(half(x1))) -> 0^1(x1)
The graph contains the following edges 1 > 1


*0^1(half(x1)) -> 0^1(x1)
The graph contains the following edges 1 > 1


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

(24)
YES