/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 73 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) AAECC Innermost [EQUIVALENT, 0 ms]
(8) QTRS
(9) DependencyPairsProof [EQUIVALENT, 0 ms]
(10) QDP
(11) DependencyGraphProof [EQUIVALENT, 0 ms]
(12) 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)))

Q is empty.

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

(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)))

Q is empty.

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

(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
f/1)YES(
s/1)YES(
g/1)YES(
cons/2(YES,YES)
h/1(YES)

Quasi precedence:
[cons_2, h_1]


Status:
cons_2: multiset status
h_1: multiset status

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)




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

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

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

Q is empty.

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

(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
f/1)YES(
s/1(YES)
h/1(YES)
cons/2(YES,YES)
g/1)YES(

Quasi precedence:
s_1 > cons_2
h_1 > cons_2


Status:
s_1: multiset status
h_1: multiset status
cons_2: multiset status

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)




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

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

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

Q is empty.

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

(7) AAECC Innermost (EQUIVALENT)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none

The TRS R 2 is 
   h(cons(X, Y)) -> h(g(cons(X, Y)))

The signature Sigma is {h_1}
----------------------------------------

(8)
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:

   h(cons(x0, x1))


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

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

(10)
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:

   h(cons(x0, x1))

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

(11) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
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(12)
TRUE