h(

f(

g(0,

g(

R

↳Dependency Pair Analysis

H(X,Z) -> F(X, s(X),Z)

F(X,Y, g(X,Y)) -> H(0, g(X,Y))

G(X, s(Y)) -> G(X,Y)

Furthermore,

R

↳DPs

→DP Problem 1

↳Polynomial Ordering

→DP Problem 2

↳Inst

**G( X, s(Y)) -> G(X, Y)**

h(X,Z) -> f(X, s(X),Z)

f(X,Y, g(X,Y)) -> h(0, g(X,Y))

g(0,Y) -> 0

g(X, s(Y)) -> g(X,Y)

The following dependency pair can be strictly oriented:

G(X, s(Y)) -> G(X,Y)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(G(x)_{1}, x_{2})= x _{2}_{ }^{ }_{ }^{ }POL(s(x)_{1})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 3

↳Dependency Graph

→DP Problem 2

↳Inst

h(X,Z) -> f(X, s(X),Z)

f(X,Y, g(X,Y)) -> h(0, g(X,Y))

g(0,Y) -> 0

g(X, s(Y)) -> g(X,Y)

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Instantiation Transformation

**F( X, Y, g(X, Y)) -> H(0, g(X, Y))**

h(X,Z) -> f(X, s(X),Z)

f(X,Y, g(X,Y)) -> h(0, g(X,Y))

g(0,Y) -> 0

g(X, s(Y)) -> g(X,Y)

On this DP problem, an Instantiation SCC transformation can be performed.

As a result of transforming the rule

one new Dependency Pair is created:

H(X,Z) -> F(X, s(X),Z)

H(0,Z') -> F(0, s(0),Z')

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Inst

→DP Problem 4

↳Instantiation Transformation

**H(0, Z') -> F(0, s(0), Z')**

h(X,Z) -> f(X, s(X),Z)

f(X,Y, g(X,Y)) -> h(0, g(X,Y))

g(0,Y) -> 0

g(X, s(Y)) -> g(X,Y)

On this DP problem, an Instantiation SCC transformation can be performed.

As a result of transforming the rule

one new Dependency Pair is created:

F(X,Y, g(X,Y)) -> H(0, g(X,Y))

F(0, s(0), g(0, s(0))) -> H(0, g(0, s(0)))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Inst

→DP Problem 4

↳Inst

...

→DP Problem 5

↳Remaining Obligation(s)

The following remains to be proven:

**F(0, s(0), g(0, s(0))) -> H(0, g(0, s(0)))****H(0, Z') -> F(0, s(0), Z')**

h(X,Z) -> f(X, s(X),Z)

f(X,Y, g(X,Y)) -> h(0, g(X,Y))

g(0,Y) -> 0

g(X, s(Y)) -> g(X,Y)

Duration:

0:00 minutes