/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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 of the given CSR could be proven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 76 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 0 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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(0)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   eq(0, 0) -> true
   eq(s(X), s(Y)) -> eq(X, Y)
   eq(X, Y) -> false
   inf(X) -> cons(X, inf(s(X)))
   take(0, X) -> nil
   take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   length(nil) -> 0
   length(cons(X, L)) -> s(length(L))

The replacement map contains the following entries:

eq: empty set
0: empty set
true: empty set
s: empty set
false: empty set
inf: {1}
cons: empty set
take: {1, 2}
nil: empty set
length: {1}

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(1) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   eq(0, 0) -> true
   eq(s(X), s(Y)) -> eq(X, Y)
   eq(X, Y) -> false
   inf(X) -> cons(X, inf(s(X)))
   take(0, X) -> nil
   take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   length(nil) -> 0
   length(cons(X, L)) -> s(length(L))

The replacement map contains the following entries:

eq: empty set
0: empty set
true: empty set
s: empty set
false: empty set
inf: {1}
cons: empty set
take: {1, 2}
nil: empty set
length: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 1
   POL(cons(x_1, x_2)) = 1 + x_1
   POL(eq(x_1, x_2)) = 0
   POL(false) = 0
   POL(inf(x_1)) = 2 + x_1
   POL(length(x_1)) = 2 + x_1
   POL(nil) = 1
   POL(s(x_1)) = 0
   POL(take(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(true) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   inf(X) -> cons(X, inf(s(X)))
   take(0, X) -> nil
   take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   length(nil) -> 0
   length(cons(X, L)) -> s(length(L))




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(2)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   eq(0, 0) -> true
   eq(s(X), s(Y)) -> eq(X, Y)
   eq(X, Y) -> false

The replacement map contains the following entries:

eq: empty set
0: empty set
true: empty set
s: empty set
false: empty set

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

(3) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   eq(0, 0) -> true
   eq(s(X), s(Y)) -> eq(X, Y)
   eq(X, Y) -> false

The replacement map contains the following entries:

eq: empty set
0: empty set
true: empty set
s: empty set
false: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 2
   POL(eq(x_1, x_2)) = 2 + 2*x_1 + x_2
   POL(false) = 1
   POL(s(x_1)) = 2*x_1
   POL(true) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   eq(0, 0) -> true
   eq(X, Y) -> false




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(4)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

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

The replacement map contains the following entries:

eq: empty set
s: empty set

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(5) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

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

The replacement map contains the following entries:

eq: empty set
s: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(eq(x_1, x_2)) = x_1 + 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:

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




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(6)
Obligation:
Context-sensitive rewrite system:
R is empty.

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