Term Rewriting System R:
[x, y]
fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(0))) -> s(0)
fib(s(s(x))) -> sp(g(x))
g(0) -> pair(s(0), 0)
g(s(0)) -> pair(s(0), s(0))
g(s(x)) -> np(g(x))
sp(pair(x, y)) -> +(x, y)
np(pair(x, y)) -> pair(+(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

FIB(s(s(x))) -> SP(g(x))
FIB(s(s(x))) -> G(x)
G(s(x)) -> NP(g(x))
G(s(x)) -> G(x)
SP(pair(x, y)) -> +'(x, y)
NP(pair(x, y)) -> +'(x, y)
+'(x, s(y)) -> +'(x, y)

Furthermore, R contains two SCCs.


   R
DPs
       →DP Problem 1
Forward Instantiation Transformation
       →DP Problem 2
FwdInst


Dependency Pair:

+'(x, s(y)) -> +'(x, y)


Rules:


fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(0))) -> s(0)
fib(s(s(x))) -> sp(g(x))
g(0) -> pair(s(0), 0)
g(s(0)) -> pair(s(0), s(0))
g(s(x)) -> np(g(x))
sp(pair(x, y)) -> +(x, y)
np(pair(x, y)) -> pair(+(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

+'(x, s(y)) -> +'(x, y)
one new Dependency Pair is created:

+'(x'', s(s(y''))) -> +'(x'', s(y''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 3
Forward Instantiation Transformation
       →DP Problem 2
FwdInst


Dependency Pair:

+'(x'', s(s(y''))) -> +'(x'', s(y''))


Rules:


fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(0))) -> s(0)
fib(s(s(x))) -> sp(g(x))
g(0) -> pair(s(0), 0)
g(s(0)) -> pair(s(0), s(0))
g(s(x)) -> np(g(x))
sp(pair(x, y)) -> +(x, y)
np(pair(x, y)) -> pair(+(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

+'(x'', s(s(y''))) -> +'(x'', s(y''))
one new Dependency Pair is created:

+'(x'''', s(s(s(y'''')))) -> +'(x'''', s(s(y'''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 3
FwdInst
             ...
               →DP Problem 4
Polynomial Ordering
       →DP Problem 2
FwdInst


Dependency Pair:

+'(x'''', s(s(s(y'''')))) -> +'(x'''', s(s(y'''')))


Rules:


fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(0))) -> s(0)
fib(s(s(x))) -> sp(g(x))
g(0) -> pair(s(0), 0)
g(s(0)) -> pair(s(0), s(0))
g(s(x)) -> np(g(x))
sp(pair(x, y)) -> +(x, y)
np(pair(x, y)) -> pair(+(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




The following dependency pair can be strictly oriented:

+'(x'''', s(s(s(y'''')))) -> +'(x'''', s(s(y'''')))


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(s(x1))=  1 + x1  
  POL(+'(x1, x2))=  1 + x2  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 3
FwdInst
             ...
               →DP Problem 5
Dependency Graph
       →DP Problem 2
FwdInst


Dependency Pair:


Rules:


fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(0))) -> s(0)
fib(s(s(x))) -> sp(g(x))
g(0) -> pair(s(0), 0)
g(s(0)) -> pair(s(0), s(0))
g(s(x)) -> np(g(x))
sp(pair(x, y)) -> +(x, y)
np(pair(x, y)) -> pair(+(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Forward Instantiation Transformation


Dependency Pair:

G(s(x)) -> G(x)


Rules:


fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(0))) -> s(0)
fib(s(s(x))) -> sp(g(x))
g(0) -> pair(s(0), 0)
g(s(0)) -> pair(s(0), s(0))
g(s(x)) -> np(g(x))
sp(pair(x, y)) -> +(x, y)
np(pair(x, y)) -> pair(+(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

G(s(x)) -> G(x)
one new Dependency Pair is created:

G(s(s(x''))) -> G(s(x''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
           →DP Problem 6
Forward Instantiation Transformation


Dependency Pair:

G(s(s(x''))) -> G(s(x''))


Rules:


fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(0))) -> s(0)
fib(s(s(x))) -> sp(g(x))
g(0) -> pair(s(0), 0)
g(s(0)) -> pair(s(0), s(0))
g(s(x)) -> np(g(x))
sp(pair(x, y)) -> +(x, y)
np(pair(x, y)) -> pair(+(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

G(s(s(x''))) -> G(s(x''))
one new Dependency Pair is created:

G(s(s(s(x'''')))) -> G(s(s(x'''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
           →DP Problem 6
FwdInst
             ...
               →DP Problem 7
Polynomial Ordering


Dependency Pair:

G(s(s(s(x'''')))) -> G(s(s(x'''')))


Rules:


fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(0))) -> s(0)
fib(s(s(x))) -> sp(g(x))
g(0) -> pair(s(0), 0)
g(s(0)) -> pair(s(0), s(0))
g(s(x)) -> np(g(x))
sp(pair(x, y)) -> +(x, y)
np(pair(x, y)) -> pair(+(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




The following dependency pair can be strictly oriented:

G(s(s(s(x'''')))) -> G(s(s(x'''')))


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(G(x1))=  1 + x1  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
           →DP Problem 6
FwdInst
             ...
               →DP Problem 8
Dependency Graph


Dependency Pair:


Rules:


fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(0))) -> s(0)
fib(s(s(x))) -> sp(g(x))
g(0) -> pair(s(0), 0)
g(s(0)) -> pair(s(0), s(0))
g(s(x)) -> np(g(x))
sp(pair(x, y)) -> +(x, y)
np(pair(x, y)) -> pair(+(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes