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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given CSR could be proven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 42 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 0 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 0 ms]
(8) CSR
(9) CSRRRRProof [EQUIVALENT, 0 ms]
(10) CSR
(11) CSRRRRProof [EQUIVALENT, 0 ms]
(12) CSR
(13) RisEmptyProof [EQUIVALENT, 3 ms]
(14) YES


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

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

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), incr(L))
   adx(nil) -> nil
   adx(cons(X, L)) -> incr(cons(X, adx(L)))
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   head(cons(X, L)) -> X
   tail(cons(X, L)) -> L

The replacement map contains the following entries:

incr: {1}
nil: empty set
cons: {1}
s: {1}
adx: {1}
nats: empty set
zeros: empty set
0: empty set
head: {1}
tail: {1}

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

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

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), incr(L))
   adx(nil) -> nil
   adx(cons(X, L)) -> incr(cons(X, adx(L)))
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   head(cons(X, L)) -> X
   tail(cons(X, L)) -> L

The replacement map contains the following entries:

incr: {1}
nil: empty set
cons: {1}
s: {1}
adx: {1}
nats: empty set
zeros: empty set
0: empty set
head: {1}
tail: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(adx(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(head(x_1)) = 1 + x_1
   POL(incr(x_1)) = x_1
   POL(nats) = 1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(tail(x_1)) = x_1
   POL(zeros) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   head(cons(X, L)) -> X




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

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

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), incr(L))
   adx(nil) -> nil
   adx(cons(X, L)) -> incr(cons(X, adx(L)))
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   tail(cons(X, L)) -> L

The replacement map contains the following entries:

incr: {1}
nil: empty set
cons: {1}
s: {1}
adx: {1}
nats: empty set
zeros: empty set
0: empty set
tail: {1}

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

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

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), incr(L))
   adx(nil) -> nil
   adx(cons(X, L)) -> incr(cons(X, adx(L)))
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   tail(cons(X, L)) -> L

The replacement map contains the following entries:

incr: {1}
nil: empty set
cons: {1}
s: {1}
adx: {1}
nats: empty set
zeros: empty set
0: empty set
tail: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(adx(x_1)) = 1 + x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(incr(x_1)) = x_1
   POL(nats) = 1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(tail(x_1)) = x_1
   POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   adx(nil) -> nil




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

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

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), incr(L))
   adx(cons(X, L)) -> incr(cons(X, adx(L)))
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   tail(cons(X, L)) -> L

The replacement map contains the following entries:

incr: {1}
nil: empty set
cons: {1}
s: {1}
adx: {1}
nats: empty set
zeros: empty set
0: empty set
tail: {1}

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

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

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), incr(L))
   adx(cons(X, L)) -> incr(cons(X, adx(L)))
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   tail(cons(X, L)) -> L

The replacement map contains the following entries:

incr: {1}
nil: empty set
cons: {1}
s: {1}
adx: {1}
nats: empty set
zeros: empty set
0: empty set
tail: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(adx(x_1)) = 1 + x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(incr(x_1)) = x_1
   POL(nats) = 1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(tail(x_1)) = 1 + x_1
   POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   tail(cons(X, L)) -> L




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

(6)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), incr(L))
   adx(cons(X, L)) -> incr(cons(X, adx(L)))
   nats -> adx(zeros)
   zeros -> cons(0, zeros)

The replacement map contains the following entries:

incr: {1}
nil: empty set
cons: {1}
s: {1}
adx: {1}
nats: empty set
zeros: empty set
0: empty set

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

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

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), incr(L))
   adx(cons(X, L)) -> incr(cons(X, adx(L)))
   nats -> adx(zeros)
   zeros -> cons(0, zeros)

The replacement map contains the following entries:

incr: {1}
nil: empty set
cons: {1}
s: {1}
adx: {1}
nats: empty set
zeros: empty set
0: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(adx(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1
   POL(incr(x_1)) = x_1
   POL(nats) = 1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(zeros) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   zeros -> cons(0, zeros)




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

(8)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), incr(L))
   adx(cons(X, L)) -> incr(cons(X, adx(L)))
   nats -> adx(zeros)

The replacement map contains the following entries:

incr: {1}
nil: empty set
cons: {1}
s: {1}
adx: {1}
nats: empty set
zeros: empty set

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

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

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), incr(L))
   adx(cons(X, L)) -> incr(cons(X, adx(L)))
   nats -> adx(zeros)

The replacement map contains the following entries:

incr: {1}
nil: empty set
cons: {1}
s: {1}
adx: {1}
nats: empty set
zeros: empty set
Used ordering:
Polynomial interpretation [POLO]:

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

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), incr(L))
   adx(cons(X, L)) -> incr(cons(X, adx(L)))




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

(10)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   nats -> adx(zeros)

The replacement map contains the following entries:

adx: {1}
nats: empty set
zeros: empty set

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

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

   nats -> adx(zeros)

The replacement map contains the following entries:

adx: {1}
nats: empty set
zeros: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(adx(x_1)) = x_1
   POL(nats) = 1
   POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   nats -> adx(zeros)




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

(12)
Obligation:
Context-sensitive rewrite system:
R is empty.

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

(13) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
----------------------------------------

(14)
YES