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

↳Forward Instantiation Transformation

**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)

innermost

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

As a result of transforming the rule

one new Dependency Pair is created:

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

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

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Forward Instantiation Transformation

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

innermost

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

As a result of transforming the rule

one new Dependency Pair is created:

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

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

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳FwdInst

...

→DP Problem 3

↳Argument Filtering and Ordering

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

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Homeomorphic Embedding Order with EMB

resulting in one new DP problem.

Used Argument Filtering System:

G(x,_{1}x) -> G(_{2}x,_{1}x)_{2}

s(x) -> s(_{1}x)_{1}

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳FwdInst

...

→DP Problem 4

↳Dependency Graph

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)

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes