/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Termination of the given CSR could be proven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 48 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 0 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

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

   from(X) -> cons(X, from(s(X)))
   length(nil) -> 0
   length(cons(X, Y)) -> s(length1(Y))
   length1(X) -> length(X)

The replacement map contains the following entries:

from: {1}
cons: {1}
s: {1}
length: empty set
nil: empty set
0: empty set
length1: empty set

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

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

   from(X) -> cons(X, from(s(X)))
   length(nil) -> 0
   length(cons(X, Y)) -> s(length1(Y))
   length1(X) -> length(X)

The replacement map contains the following entries:

from: {1}
cons: {1}
s: {1}
length: empty set
nil: empty set
0: empty set
length1: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(cons(x_1, x_2)) = x_1
   POL(from(x_1)) = 1 + 2*x_1
   POL(length(x_1)) = 0
   POL(length1(x_1)) = 0
   POL(nil) = 2
   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:

   from(X) -> cons(X, from(s(X)))




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

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

   length(nil) -> 0
   length(cons(X, Y)) -> s(length1(Y))
   length1(X) -> length(X)

The replacement map contains the following entries:

cons: {1}
s: {1}
length: empty set
nil: empty set
0: empty set
length1: empty set

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

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

   length(nil) -> 0
   length(cons(X, Y)) -> s(length1(Y))
   length1(X) -> length(X)

The replacement map contains the following entries:

cons: {1}
s: {1}
length: empty set
nil: empty set
0: empty set
length1: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 1
   POL(cons(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(length(x_1)) = 1 + 2*x_1
   POL(length1(x_1)) = 2 + 2*x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   length(cons(X, Y)) -> s(length1(Y))
   length1(X) -> length(X)




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

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

   length(nil) -> 0

The replacement map contains the following entries:

length: empty set
nil: empty set
0: empty set

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

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

   length(nil) -> 0

The replacement map contains the following entries:

length: empty set
nil: empty set
0: empty set
Used ordering:
Polynomial interpretation [POLO]:

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

   length(nil) -> 0




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

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

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

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

(8)
YES