f(0) -> 1

f(s(

g(0,

g(s(

g(s(

+(

+(

R

↳Dependency Pair Analysis

F(s(x)) -> G(x, s(x))

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

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

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

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

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Remaining Obligation(s)

→DP Problem 2

↳Remaining Obligation(s)

The following remains to be proven:

**Dependency Pair:****+'(***x*, s(*y*)) -> +'(*x*,*y*)**Rules:**

f(0) -> 1

f(s(*x*)) -> g(*x*, s(*x*))

g(0,*y*) ->*y*

g(s(*x*),*y*) -> g(*x*, +(*y*, s(*x*)))

g(s(*x*),*y*) -> g(*x*, s(+(*y*,*x*)))

+(*x*, 0) ->*x*

+(*x*, s(*y*)) -> s(+(*x*,*y*))**Dependency Pairs:****G(s(***x*),*y*) -> G(*x*, s(+(*y*,*x*)))**G(s(***x*),*y*) -> G(*x*, +(*y*, s(*x*)))**Rules:**

f(0) -> 1

f(s(*x*)) -> g(*x*, s(*x*))

g(0,*y*) ->*y*

g(s(*x*),*y*) -> g(*x*, +(*y*, s(*x*)))

g(s(*x*),*y*) -> g(*x*, s(+(*y*,*x*)))

+(*x*, 0) ->*x*

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

R

↳DPs

→DP Problem 1

↳Remaining Obligation(s)

→DP Problem 2

↳Remaining Obligation(s)

The following remains to be proven:

**Dependency Pair:****+'(***x*, s(*y*)) -> +'(*x*,*y*)**Rules:**

f(0) -> 1

f(s(*x*)) -> g(*x*, s(*x*))

g(0,*y*) ->*y*

g(s(*x*),*y*) -> g(*x*, +(*y*, s(*x*)))

g(s(*x*),*y*) -> g(*x*, s(+(*y*,*x*)))

+(*x*, 0) ->*x*

+(*x*, s(*y*)) -> s(+(*x*,*y*))**Dependency Pairs:****G(s(***x*),*y*) -> G(*x*, s(+(*y*,*x*)))**G(s(***x*),*y*) -> G(*x*, +(*y*, s(*x*)))**Rules:**

f(0) -> 1

f(s(*x*)) -> g(*x*, s(*x*))

g(0,*y*) ->*y*

g(s(*x*),*y*) -> g(*x*, +(*y*, s(*x*)))

g(s(*x*),*y*) -> g(*x*, s(+(*y*,*x*)))

+(*x*, 0) ->*x*

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

Duration:

0:00 minutes