+(a, b) -> +(b, a)

+(a, +(b,

+(+(

f(a,

f(b,

f(+(

R

↳Dependency Pair Analysis

+'(a, b) -> +'(b, a)

+'(a, +(b,z)) -> +'(b, +(a,z))

+'(a, +(b,z)) -> +'(a,z)

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

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

F(+(x,y),z) -> +'(f(x,z), f(y,z))

F(+(x,y),z) -> F(x,z)

F(+(x,y),z) -> F(y,z)

Furthermore,

R

↳DPs

→DP Problem 1

↳Remaining Obligation(s)

→DP Problem 2

↳Remaining Obligation(s)

→DP Problem 3

↳Remaining Obligation(s)

The following remains to be proven:

**Dependency Pair:****+'(a, +(b,***z*)) -> +'(a,*z*)**Rules:**

+(a, b) -> +(b, a)

+(a, +(b,*z*)) -> +(b, +(a,*z*))

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

f(a,*y*) -> a

f(b,*y*) -> b

f(+(*x*,*y*),*z*) -> +(f(*x*,*z*), f(*y*,*z*))**Dependency Pairs:****+'(+(***x*,*y*),*z*) -> +'(*y*,*z*)**+'(+(***x*,*y*),*z*) -> +'(*x*, +(*y*,*z*))**Rules:**

+(a, b) -> +(b, a)

+(a, +(b,*z*)) -> +(b, +(a,*z*))

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

f(a,*y*) -> a

f(b,*y*) -> b

f(+(*x*,*y*),*z*) -> +(f(*x*,*z*), f(*y*,*z*))**Dependency Pairs:****F(+(***x*,*y*),*z*) -> F(*y*,*z*)**F(+(***x*,*y*),*z*) -> F(*x*,*z*)**Rules:**

+(a, b) -> +(b, a)

+(a, +(b,*z*)) -> +(b, +(a,*z*))

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

f(a,*y*) -> a

f(b,*y*) -> b

f(+(*x*,*y*),*z*) -> +(f(*x*,*z*), f(*y*,*z*))

R

↳DPs

→DP Problem 1

↳Remaining Obligation(s)

→DP Problem 2

↳Remaining Obligation(s)

→DP Problem 3

↳Remaining Obligation(s)

The following remains to be proven:

**Dependency Pair:****+'(a, +(b,***z*)) -> +'(a,*z*)**Rules:**

+(a, b) -> +(b, a)

+(a, +(b,*z*)) -> +(b, +(a,*z*))

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

f(a,*y*) -> a

f(b,*y*) -> b

f(+(*x*,*y*),*z*) -> +(f(*x*,*z*), f(*y*,*z*))**Dependency Pairs:****+'(+(***x*,*y*),*z*) -> +'(*y*,*z*)**+'(+(***x*,*y*),*z*) -> +'(*x*, +(*y*,*z*))**Rules:**

+(a, b) -> +(b, a)

+(a, +(b,*z*)) -> +(b, +(a,*z*))

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

f(a,*y*) -> a

f(b,*y*) -> b

f(+(*x*,*y*),*z*) -> +(f(*x*,*z*), f(*y*,*z*))**Dependency Pairs:****F(+(***x*,*y*),*z*) -> F(*y*,*z*)**F(+(***x*,*y*),*z*) -> F(*x*,*z*)**Rules:**

+(a, b) -> +(b, a)

+(a, +(b,*z*)) -> +(b, +(a,*z*))

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

f(a,*y*) -> a

f(b,*y*) -> b

f(+(*x*,*y*),*z*) -> +(f(*x*,*z*), f(*y*,*z*))

R

↳DPs

→DP Problem 1

↳Remaining Obligation(s)

→DP Problem 2

↳Remaining Obligation(s)

→DP Problem 3

↳Remaining Obligation(s)

The following remains to be proven:

**Dependency Pair:****+'(a, +(b,***z*)) -> +'(a,*z*)**Rules:**

+(a, b) -> +(b, a)

+(a, +(b,*z*)) -> +(b, +(a,*z*))

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

f(a,*y*) -> a

f(b,*y*) -> b

f(+(*x*,*y*),*z*) -> +(f(*x*,*z*), f(*y*,*z*))**Dependency Pairs:****+'(+(***x*,*y*),*z*) -> +'(*y*,*z*)**+'(+(***x*,*y*),*z*) -> +'(*x*, +(*y*,*z*))**Rules:**

+(a, b) -> +(b, a)

+(a, +(b,*z*)) -> +(b, +(a,*z*))

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

f(a,*y*) -> a

f(b,*y*) -> b

f(+(*x*,*y*),*z*) -> +(f(*x*,*z*), f(*y*,*z*))**Dependency Pairs:****F(+(***x*,*y*),*z*) -> F(*y*,*z*)**F(+(***x*,*y*),*z*) -> F(*x*,*z*)**Rules:**

+(a, b) -> +(b, a)

+(a, +(b,*z*)) -> +(b, +(a,*z*))

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

f(a,*y*) -> a

f(b,*y*) -> b

f(+(*x*,*y*),*z*) -> +(f(*x*,*z*), f(*y*,*z*))

Duration:

0:00 minutes