NO
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 66 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) AND
    (7) QDP
        (8) UsableRulesProof [EQUIVALENT, 0 ms]
        (9) QDP
        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QDP
        (13) NonLoopProof [COMPLETE, 0 ms]
        (14) NO


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

   h(X, Z) -> f(X, s(X), Z)
   f(X, Y, g(X, Y)) -> h(0, g(X, Y))
   g(0, Y) -> 0
   g(X, s(Y)) -> g(X, Y)

Q is empty.

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

   POL(0) = 0
   POL(f(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(g(x_1, x_2)) = 2 + 2*x_1 + x_2
   POL(h(x_1, x_2)) = 2*x_1 + x_2
   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:

   g(0, Y) -> 0




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

   h(X, Z) -> f(X, s(X), Z)
   f(X, Y, g(X, Y)) -> h(0, g(X, Y))
   g(X, s(Y)) -> g(X, Y)

Q is empty.

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(3) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
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(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   H(X, Z) -> F(X, s(X), Z)
   F(X, Y, g(X, Y)) -> H(0, g(X, Y))
   G(X, s(Y)) -> G(X, Y)

The TRS R consists of the following rules:

   h(X, Z) -> f(X, s(X), Z)
   f(X, Y, g(X, Y)) -> h(0, g(X, Y))
   g(X, s(Y)) -> g(X, Y)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
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(6)
Complex Obligation (AND)

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

   G(X, s(Y)) -> G(X, Y)

The TRS R consists of the following rules:

   h(X, Z) -> f(X, s(X), Z)
   f(X, Y, g(X, Y)) -> h(0, g(X, Y))
   g(X, s(Y)) -> g(X, Y)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(8) 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.
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(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   G(X, s(Y)) -> G(X, Y)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(10) 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:
*G(X, s(Y)) -> G(X, Y)
The graph contains the following edges 1 >= 1, 2 > 2


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(11)
YES

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(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(X, Y, g(X, Y)) -> H(0, g(X, Y))
   H(X, Z) -> F(X, s(X), Z)

The TRS R consists of the following rules:

   h(X, Z) -> f(X, s(X), Z)
   f(X, Y, g(X, Y)) -> h(0, g(X, Y))
   g(X, s(Y)) -> g(X, Y)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(13) NonLoopProof (COMPLETE)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 0, 
σ' = [ ], and μ' = [ ] on the rule
H(0, g(0, s(0)))[ ]^n[x0 / 0] -> H(0, g(0, s(0)))[ ]^n[x0 / 0]
This rule is correct for the QDP as the following derivation shows:

H(0, g(0, s(0)))[ ]^n[x0 / 0] -> H(0, g(0, s(0)))[ ]^n[x0 / 0]
    by Equivalency by Simplifying Mu with mu1: [x0 / 0] mu2: [x0 / 0]
    intermediate steps: Equiv Smu (lhs) - Instantiate mu - Instantiation
    H(x1, g(x1, s(x1)))[ ]^n[ ] -> H(0, g(x1, s(x1)))[ ]^n[ ]
        by Narrowing at position: []
        intermediate steps: Instantiation - Instantiation
        H(X, Z)[ ]^n[ ] -> F(X, s(X), Z)[ ]^n[ ]
            by Rule from TRS P

        intermediate steps: Instantiation - Instantiation - Instantiation - Instantiation
        F(X, Y, g(X, Y))[ ]^n[ ] -> H(0, g(X, Y))[ ]^n[ ]
            by Rule from TRS P
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(14)
NO