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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given CSR could be proven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 50 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 10 ms]
(4) CSR
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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

   fst(0, Z) -> nil
   fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
   from(X) -> cons(X, from(s(X)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   len(nil) -> 0
   len(cons(X, Z)) -> s(len(Z))

The replacement map contains the following entries:

fst: {1, 2}
0: empty set
nil: empty set
s: empty set
cons: {1}
from: {1}
add: {1, 2}
len: {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:

   fst(0, Z) -> nil
   fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
   from(X) -> cons(X, from(s(X)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   len(nil) -> 0
   len(cons(X, Z)) -> s(len(Z))

The replacement map contains the following entries:

fst: {1, 2}
0: empty set
nil: empty set
s: empty set
cons: {1}
from: {1}
add: {1, 2}
len: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 1
   POL(add(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = 1 + x_1
   POL(from(x_1)) = 1 + x_1
   POL(fst(x_1, x_2)) = x_1 + x_2
   POL(len(x_1)) = x_1
   POL(nil) = 1
   POL(s(x_1)) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
   add(0, X) -> X




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

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

   fst(0, Z) -> nil
   from(X) -> cons(X, from(s(X)))
   add(s(X), Y) -> s(add(X, Y))
   len(nil) -> 0
   len(cons(X, Z)) -> s(len(Z))

The replacement map contains the following entries:

fst: {1, 2}
0: empty set
nil: empty set
s: empty set
cons: {1}
from: {1}
add: {1, 2}
len: {1}

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

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

   fst(0, Z) -> nil
   from(X) -> cons(X, from(s(X)))
   add(s(X), Y) -> s(add(X, Y))
   len(nil) -> 0
   len(cons(X, Z)) -> s(len(Z))

The replacement map contains the following entries:

fst: {1, 2}
0: empty set
nil: empty set
s: empty set
cons: {1}
from: {1}
add: {1, 2}
len: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 1
   POL(add(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(cons(x_1, x_2)) = 1 + x_1
   POL(from(x_1)) = 2 + 2*x_1
   POL(fst(x_1, x_2)) = 1 + 2*x_1 + x_2
   POL(len(x_1)) = 2 + x_1
   POL(nil) = 0
   POL(s(x_1)) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   fst(0, Z) -> nil
   from(X) -> cons(X, from(s(X)))
   add(s(X), Y) -> s(add(X, Y))
   len(nil) -> 0
   len(cons(X, Z)) -> s(len(Z))




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(4)
Obligation:
Context-sensitive rewrite system:
R is empty.

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(5) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
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(6)
YES