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))

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
         ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳AFS


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))





The following dependency pair can be strictly oriented:

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


There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
+'(x1, x2) -> +'(x1, x2)
s(x1) -> s(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 3
             ↳Dependency Graph
       →DP Problem 2
         ↳AFS


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))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Argument Filtering and Ordering


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))





The following dependency pair can be strictly oriented:

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


There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
G(x1) -> G(x1)
s(x1) -> s(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
           →DP Problem 4
             ↳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))





Using the Dependency Graph resulted in no new DP problems.

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