/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 11 ms]
(4) QTRS
(5) DependencyPairsProof [EQUIVALENT, 27 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) AND
    (9) QDP
        (10) UsableRulesProof [EQUIVALENT, 0 ms]
        (11) QDP
        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) QDP
        (15) UsableRulesProof [EQUIVALENT, 0 ms]
        (16) QDP
        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (18) YES
    (19) QDP
        (20) UsableRulesProof [EQUIVALENT, 0 ms]
        (21) QDP
        (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (23) YES
    (24) QDP
        (25) UsableRulesProof [EQUIVALENT, 0 ms]
        (26) QDP
        (27) QDPOrderProof [EQUIVALENT, 23 ms]
        (28) QDP
        (29) PisEmptyProof [EQUIVALENT, 0 ms]
        (30) YES


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

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

   top(free(x)) -> top(check(new(x)))
   new(free(x)) -> free(new(x))
   old(free(x)) -> free(old(x))
   new(serve) -> free(serve)
   old(serve) -> free(serve)
   check(free(x)) -> free(check(x))
   check(new(x)) -> new(check(x))
   check(old(x)) -> old(check(x))
   check(old(x)) -> old(x)

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:

   free(top(x)) -> new(check(top(x)))
   free(new(x)) -> new(free(x))
   free(old(x)) -> old(free(x))
   serve'(new(x)) -> serve'(free(x))
   serve'(old(x)) -> serve'(free(x))
   free(check(x)) -> check(free(x))
   new(check(x)) -> check(new(x))
   old(check(x)) -> check(old(x))
   old(check(x)) -> old(x)

Q is empty.

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

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

   POL(check(x_1)) = x_1
   POL(free(x_1)) = x_1
   POL(new(x_1)) = x_1
   POL(old(x_1)) = 1 + x_1
   POL(serve'(x_1)) = x_1
   POL(top(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   serve'(old(x)) -> serve'(free(x))




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

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

   free(top(x)) -> new(check(top(x)))
   free(new(x)) -> new(free(x))
   free(old(x)) -> old(free(x))
   serve'(new(x)) -> serve'(free(x))
   free(check(x)) -> check(free(x))
   new(check(x)) -> check(new(x))
   old(check(x)) -> check(old(x))
   old(check(x)) -> old(x)

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:

   FREE(top(x)) -> NEW(check(top(x)))
   FREE(new(x)) -> NEW(free(x))
   FREE(new(x)) -> FREE(x)
   FREE(old(x)) -> OLD(free(x))
   FREE(old(x)) -> FREE(x)
   SERVE'(new(x)) -> SERVE'(free(x))
   SERVE'(new(x)) -> FREE(x)
   FREE(check(x)) -> FREE(x)
   NEW(check(x)) -> NEW(x)
   OLD(check(x)) -> OLD(x)

The TRS R consists of the following rules:

   free(top(x)) -> new(check(top(x)))
   free(new(x)) -> new(free(x))
   free(old(x)) -> old(free(x))
   serve'(new(x)) -> serve'(free(x))
   free(check(x)) -> check(free(x))
   new(check(x)) -> check(new(x))
   old(check(x)) -> check(old(x))
   old(check(x)) -> old(x)

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 4 SCCs with 4 less nodes.
----------------------------------------

(8)
Complex Obligation (AND)

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

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

   OLD(check(x)) -> OLD(x)

The TRS R consists of the following rules:

   free(top(x)) -> new(check(top(x)))
   free(new(x)) -> new(free(x))
   free(old(x)) -> old(free(x))
   serve'(new(x)) -> serve'(free(x))
   free(check(x)) -> check(free(x))
   new(check(x)) -> check(new(x))
   old(check(x)) -> check(old(x))
   old(check(x)) -> old(x)

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:

   OLD(check(x)) -> OLD(x)

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

(12) 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:
*OLD(check(x)) -> OLD(x)
The graph contains the following edges 1 > 1


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

(13)
YES

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

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   NEW(check(x)) -> NEW(x)

The TRS R consists of the following rules:

   free(top(x)) -> new(check(top(x)))
   free(new(x)) -> new(free(x))
   free(old(x)) -> old(free(x))
   serve'(new(x)) -> serve'(free(x))
   free(check(x)) -> check(free(x))
   new(check(x)) -> check(new(x))
   old(check(x)) -> check(old(x))
   old(check(x)) -> old(x)

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

(15) 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.
----------------------------------------

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   NEW(check(x)) -> NEW(x)

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

(17) 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:
*NEW(check(x)) -> NEW(x)
The graph contains the following edges 1 > 1


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

(18)
YES

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

(19)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   FREE(old(x)) -> FREE(x)
   FREE(new(x)) -> FREE(x)
   FREE(check(x)) -> FREE(x)

The TRS R consists of the following rules:

   free(top(x)) -> new(check(top(x)))
   free(new(x)) -> new(free(x))
   free(old(x)) -> old(free(x))
   serve'(new(x)) -> serve'(free(x))
   free(check(x)) -> check(free(x))
   new(check(x)) -> check(new(x))
   old(check(x)) -> check(old(x))
   old(check(x)) -> old(x)

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

(20) 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.
----------------------------------------

(21)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   FREE(old(x)) -> FREE(x)
   FREE(new(x)) -> FREE(x)
   FREE(check(x)) -> FREE(x)

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

(22) 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:
*FREE(old(x)) -> FREE(x)
The graph contains the following edges 1 > 1


*FREE(new(x)) -> FREE(x)
The graph contains the following edges 1 > 1


*FREE(check(x)) -> FREE(x)
The graph contains the following edges 1 > 1


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

(23)
YES

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

(24)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   SERVE'(new(x)) -> SERVE'(free(x))

The TRS R consists of the following rules:

   free(top(x)) -> new(check(top(x)))
   free(new(x)) -> new(free(x))
   free(old(x)) -> old(free(x))
   serve'(new(x)) -> serve'(free(x))
   free(check(x)) -> check(free(x))
   new(check(x)) -> check(new(x))
   old(check(x)) -> check(old(x))
   old(check(x)) -> old(x)

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

(25) 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.
----------------------------------------

(26)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   SERVE'(new(x)) -> SERVE'(free(x))

The TRS R consists of the following rules:

   free(top(x)) -> new(check(top(x)))
   free(new(x)) -> new(free(x))
   free(old(x)) -> old(free(x))
   free(check(x)) -> check(free(x))
   old(check(x)) -> check(old(x))
   old(check(x)) -> old(x)
   new(check(x)) -> check(new(x))

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

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


The following pairs can be oriented strictly and are deleted.

   SERVE'(new(x)) -> SERVE'(free(x))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(SERVE'(x_1)) = 2*x_1
   POL(check(x_1)) = 1
   POL(free(x_1)) = 2*x_1
   POL(new(x_1)) = 4 + 5*x_1
   POL(old(x_1)) = 2
   POL(top(x_1)) = 5

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

   free(top(x)) -> new(check(top(x)))
   free(new(x)) -> new(free(x))
   free(old(x)) -> old(free(x))
   free(check(x)) -> check(free(x))
   old(check(x)) -> check(old(x))
   old(check(x)) -> old(x)
   new(check(x)) -> check(new(x))


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

(28)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   free(top(x)) -> new(check(top(x)))
   free(new(x)) -> new(free(x))
   free(old(x)) -> old(free(x))
   free(check(x)) -> check(free(x))
   old(check(x)) -> check(old(x))
   old(check(x)) -> old(x)
   new(check(x)) -> check(new(x))

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

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

(30)
YES