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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 35 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) RisEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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

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

   and(true, X) -> X
   and(false, Y) -> false
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

and: {1}
true: empty set
false: empty set
if: {1}
add: {1}
0: empty set
s: empty set
first: {1, 2}
nil: empty set
cons: empty set
from: empty set

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

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

   and(true, X) -> X
   and(false, Y) -> false
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

and: {1}
true: empty set
false: empty set
if: {1}
add: {1}
0: empty set
s: empty set
first: {1, 2}
nil: empty set
cons: empty set
from: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 1
   POL(add(x_1, x_2)) = 1 + x_1 + x_2
   POL(and(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = x_2
   POL(false) = 0
   POL(first(x_1, x_2)) = 1 + x_1 + x_2
   POL(from(x_1)) = 1 + x_1
   POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(nil) = 0
   POL(s(x_1)) = 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:

   add(0, X) -> X
   first(0, X) -> nil




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

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

   and(true, X) -> X
   and(false, Y) -> false
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   add(s(X), Y) -> s(add(X, Y))
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

and: {1}
true: empty set
false: empty set
if: {1}
add: {1}
s: empty set
first: {1, 2}
cons: empty set
from: empty set

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

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

   and(true, X) -> X
   and(false, Y) -> false
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   add(s(X), Y) -> s(add(X, Y))
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

and: {1}
true: empty set
false: empty set
if: {1}
add: {1}
s: empty set
first: {1, 2}
cons: empty set
from: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(add(x_1, x_2)) = 1 + x_1 + x_2
   POL(and(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = x_2
   POL(false) = 1
   POL(first(x_1, x_2)) = 1 + x_1 + x_2
   POL(from(x_1)) = 1 + x_1
   POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(s(x_1)) = 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:

   if(false, X, Y) -> Y




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

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

   and(true, X) -> X
   and(false, Y) -> false
   if(true, X, Y) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

and: {1}
true: empty set
false: empty set
if: {1}
add: {1}
s: empty set
first: {1, 2}
cons: empty set
from: empty set

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

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

   and(true, X) -> X
   and(false, Y) -> false
   if(true, X, Y) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

and: {1}
true: empty set
false: empty set
if: {1}
add: {1}
s: empty set
first: {1, 2}
cons: empty set
from: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(add(x_1, x_2)) = 1 + x_1 + x_2
   POL(and(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = x_2
   POL(false) = 0
   POL(first(x_1, x_2)) = 1 + x_1 + x_2
   POL(from(x_1)) = 1 + x_1
   POL(if(x_1, x_2, x_3)) = x_1 + x_2
   POL(s(x_1)) = x_1
   POL(true) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   and(true, X) -> X
   if(true, X, Y) -> X




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

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

   and(false, Y) -> false
   add(s(X), Y) -> s(add(X, Y))
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

and: {1}
false: empty set
add: {1}
s: empty set
first: {1, 2}
cons: empty set
from: empty set

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

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

   and(false, Y) -> false
   add(s(X), Y) -> s(add(X, Y))
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

and: {1}
false: empty set
add: {1}
s: empty set
first: {1, 2}
cons: empty set
from: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(add(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(and(x_1, x_2)) = 2*x_1
   POL(cons(x_1, x_2)) = 2*x_1
   POL(false) = 1
   POL(first(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(from(x_1)) = 1 + 2*x_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:

   and(false, Y) -> false
   add(s(X), Y) -> s(add(X, Y))
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
   from(X) -> cons(X, from(s(X)))




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

(8)
Obligation:
Context-sensitive rewrite system:
R is empty.

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

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

(10)
YES