/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) QTRSRRRProof [EQUIVALENT, 74 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) AND
    (7) QDP
        (8) MNOCProof [EQUIVALENT, 0 ms]
        (9) QDP
        (10) UsableRulesProof [EQUIVALENT, 0 ms]
        (11) QDP
        (12) QReductionProof [EQUIVALENT, 0 ms]
        (13) QDP
        (14) MRRProof [EQUIVALENT, 0 ms]
        (15) QDP
        (16) PisEmptyProof [EQUIVALENT, 0 ms]
        (17) YES
    (18) QDP
        (19) MNOCProof [EQUIVALENT, 0 ms]
        (20) QDP
        (21) UsableRulesProof [EQUIVALENT, 0 ms]
        (22) QDP
        (23) QReductionProof [EQUIVALENT, 0 ms]
        (24) QDP
        (25) MRRProof [EQUIVALENT, 0 ms]
        (26) QDP
        (27) PisEmptyProof [EQUIVALENT, 0 ms]
        (28) YES


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

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

   g(c(x, s(y))) -> g(c(s(x), y))
   f(c(s(x), y)) -> f(c(x, s(y)))
   f(f(x)) -> f(d(f(x)))
   f(x) -> x

Q is empty.

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

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

   POL(c(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(d(x_1)) = x_1
   POL(f(x_1)) = 2 + 2*x_1
   POL(g(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:

   f(x) -> x




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

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

   g(c(x, s(y))) -> g(c(s(x), y))
   f(c(s(x), y)) -> f(c(x, s(y)))
   f(f(x)) -> f(d(f(x)))

Q is empty.

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

(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:

   G(c(x, s(y))) -> G(c(s(x), y))
   F(c(s(x), y)) -> F(c(x, s(y)))
   F(f(x)) -> F(d(f(x)))

The TRS R consists of the following rules:

   g(c(x, s(y))) -> g(c(s(x), y))
   f(c(s(x), y)) -> f(c(x, s(y)))
   f(f(x)) -> f(d(f(x)))

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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
----------------------------------------

(6)
Complex Obligation (AND)

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

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(c(s(x), y)) -> F(c(x, s(y)))

The TRS R consists of the following rules:

   g(c(x, s(y))) -> g(c(s(x), y))
   f(c(s(x), y)) -> f(c(x, s(y)))
   f(f(x)) -> f(d(f(x)))

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

(8) MNOCProof (EQUIVALENT)
We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
----------------------------------------

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

   F(c(s(x), y)) -> F(c(x, s(y)))

The TRS R consists of the following rules:

   g(c(x, s(y))) -> g(c(s(x), y))
   f(c(s(x), y)) -> f(c(x, s(y)))
   f(f(x)) -> f(d(f(x)))

The set Q consists of the following terms:

   g(c(x0, s(x1)))
   f(c(s(x0), x1))
   f(f(x0))

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

(10) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

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

   F(c(s(x), y)) -> F(c(x, s(y)))

R is empty.
The set Q consists of the following terms:

   g(c(x0, s(x1)))
   f(c(s(x0), x1))
   f(f(x0))

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

(12) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   g(c(x0, s(x1)))
   f(c(s(x0), x1))
   f(f(x0))


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

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

   F(c(s(x), y)) -> F(c(x, s(y)))

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

(14) MRRProof (EQUIVALENT)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented dependency pairs:

   F(c(s(x), y)) -> F(c(x, s(y)))


Used ordering: Knuth-Bendix order [KBO] with precedence:s_1 > c_2 > F_1

and weight map:

   F_1=1
   s_1=1
   c_2=0

The variable weight is 1

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

(15)
Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

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

(17)
YES

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

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   G(c(x, s(y))) -> G(c(s(x), y))

The TRS R consists of the following rules:

   g(c(x, s(y))) -> g(c(s(x), y))
   f(c(s(x), y)) -> f(c(x, s(y)))
   f(f(x)) -> f(d(f(x)))

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

(19) MNOCProof (EQUIVALENT)
We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
----------------------------------------

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

   G(c(x, s(y))) -> G(c(s(x), y))

The TRS R consists of the following rules:

   g(c(x, s(y))) -> g(c(s(x), y))
   f(c(s(x), y)) -> f(c(x, s(y)))
   f(f(x)) -> f(d(f(x)))

The set Q consists of the following terms:

   g(c(x0, s(x1)))
   f(c(s(x0), x1))
   f(f(x0))

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

(21) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

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

   G(c(x, s(y))) -> G(c(s(x), y))

R is empty.
The set Q consists of the following terms:

   g(c(x0, s(x1)))
   f(c(s(x0), x1))
   f(f(x0))

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

(23) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   g(c(x0, s(x1)))
   f(c(s(x0), x1))
   f(f(x0))


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

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

   G(c(x, s(y))) -> G(c(s(x), y))

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

(25) MRRProof (EQUIVALENT)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented dependency pairs:

   G(c(x, s(y))) -> G(c(s(x), y))


Used ordering: Polynomial interpretation [POLO]:

   POL(G(x_1)) = 2*x_1
   POL(c(x_1, x_2)) = x_1 + 2*x_2
   POL(s(x_1)) = 2 + x_1


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

(26)
Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

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

(28)
YES