YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 63 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 8 ms]
(6) QTRS
(7) DependencyPairsProof [EQUIVALENT, 0 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) TRUE


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

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

The set Q consists of the following terms:

   f(s(x0))
   g(cons(0, x0))
   g(cons(s(x0), x1))
   h(cons(x0, x1))


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

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

   POL(0) = 2
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(f(x_1)) = 2*x_1
   POL(g(x_1)) = x_1
   POL(h(x_1)) = 2*x_1
   POL(s(x_1)) = 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   g(cons(0, Y)) -> g(Y)




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

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

   f(s(X)) -> f(X)
   g(cons(s(X), Y)) -> s(X)
   h(cons(X, Y)) -> h(g(cons(X, Y)))

The set Q consists of the following terms:

   f(s(x0))
   g(cons(0, x0))
   g(cons(s(x0), x1))
   h(cons(x0, x1))


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

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

   POL(cons(x_1, x_2)) = x_1 + 2*x_2
   POL(f(x_1)) = 2*x_1
   POL(g(x_1)) = x_1
   POL(h(x_1)) = x_1
   POL(s(x_1)) = 1 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f(s(X)) -> f(X)




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

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

   g(cons(s(X), Y)) -> s(X)
   h(cons(X, Y)) -> h(g(cons(X, Y)))

The set Q consists of the following terms:

   f(s(x0))
   g(cons(0, x0))
   g(cons(s(x0), x1))
   h(cons(x0, x1))


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

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

   POL(cons(x_1, x_2)) = 1 + x_1 + x_2
   POL(g(x_1)) = x_1
   POL(h(x_1)) = x_1
   POL(s(x_1)) = 1 + 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   g(cons(s(X), Y)) -> s(X)




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

(6)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   h(cons(X, Y)) -> h(g(cons(X, Y)))

The set Q consists of the following terms:

   f(s(x0))
   g(cons(0, x0))
   g(cons(s(x0), x1))
   h(cons(x0, x1))


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

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

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

   H(cons(X, Y)) -> H(g(cons(X, Y)))

The TRS R consists of the following rules:

   h(cons(X, Y)) -> h(g(cons(X, Y)))

The set Q consists of the following terms:

   f(s(x0))
   g(cons(0, x0))
   g(cons(s(x0), x1))
   h(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
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(9) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
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(10)
TRUE