/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 161 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

   U11(tt, N) -> activate(N)
   U21(tt, M, N) -> s(plus(activate(N), activate(M)))
   and(tt, X) -> activate(X)
   isNat(n__0) -> tt
   isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2)))
   isNat(n__s(V1)) -> isNat(activate(V1))
   plus(N, 0) -> U11(isNat(N), N)
   plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N)
   0 -> n__0
   plus(X1, X2) -> n__plus(X1, X2)
   isNat(X) -> n__isNat(X)
   s(X) -> n__s(X)
   activate(n__0) -> 0
   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
   activate(n__isNat(X)) -> isNat(X)
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
U11/2(YES,YES)
tt/0)
activate/1)YES(
U21/3(YES,YES,YES)
s/1(YES)
plus/2(YES,YES)
and/2(YES,YES)
isNat/1)YES(
n__0/0)
n__plus/2(YES,YES)
n__isNat/1)YES(
n__s/1(YES)
0/0)

Quasi precedence:
[U21_3, plus_2, n__plus_2] > U11_2
[U21_3, plus_2, n__plus_2] > [s_1, n__s_1] > and_2
[n__0, 0] > tt


Status:
U11_2: multiset status
tt: multiset status
U21_3: [3,2,1]
s_1: multiset status
plus_2: [1,2]
and_2: multiset status
n__0: multiset status
n__plus_2: [1,2]
n__s_1: multiset status
0: multiset status

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U11(tt, N) -> activate(N)
   U21(tt, M, N) -> s(plus(activate(N), activate(M)))
   and(tt, X) -> activate(X)
   isNat(n__0) -> tt
   isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2)))
   isNat(n__s(V1)) -> isNat(activate(V1))
   plus(N, 0) -> U11(isNat(N), N)
   plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N)




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

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

   0 -> n__0
   plus(X1, X2) -> n__plus(X1, X2)
   isNat(X) -> n__isNat(X)
   s(X) -> n__s(X)
   activate(n__0) -> 0
   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
   activate(n__isNat(X)) -> isNat(X)
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

Q is empty.

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

   POL(0) = 1
   POL(activate(x_1)) = 2*x_1
   POL(isNat(x_1)) = 2*x_1
   POL(n__0) = 1
   POL(n__isNat(x_1)) = 2*x_1
   POL(n__plus(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(n__s(x_1)) = 1 + x_1
   POL(plus(x_1, x_2)) = 2*x_1 + 2*x_2
   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:

   activate(n__0) -> 0
   activate(n__s(X)) -> s(activate(X))




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

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

   0 -> n__0
   plus(X1, X2) -> n__plus(X1, X2)
   isNat(X) -> n__isNat(X)
   s(X) -> n__s(X)
   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
   activate(n__isNat(X)) -> isNat(X)
   activate(X) -> X

Q is empty.

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(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:activate_1 > n__s_1 > plus_2 > n__0 > n__plus_2 > s_1 > isNat_1 > n__isNat_1 > 0

and weight map:

   0=2
   n__0=1
   isNat_1=1
   n__isNat_1=1
   s_1=2
   n__s_1=1
   activate_1=0
   plus_2=0
   n__plus_2=0

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   0 -> n__0
   plus(X1, X2) -> n__plus(X1, X2)
   isNat(X) -> n__isNat(X)
   s(X) -> n__s(X)
   activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
   activate(n__isNat(X)) -> isNat(X)
   activate(X) -> X




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(6)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(7) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(8)
YES