g(0, f(

g(

g(s(

g(f(

R

↳Dependency Pair Analysis

G(x, s(y)) -> G(f(x,y), 0)

G(s(x),y) -> G(f(x,y), 0)

G(f(x,y), 0) -> G(x, 0)

G(f(x,y), 0) -> G(y, 0)

Furthermore,

R

↳DPs

→DP Problem 1

↳Instantiation Transformation

**G(f( x, y), 0) -> G(y, 0)**

g(0, f(x,x)) ->x

g(x, s(y)) -> g(f(x,y), 0)

g(s(x),y) -> g(f(x,y), 0)

g(f(x,y), 0) -> f(g(x, 0), g(y, 0))

innermost

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

As a result of transforming the rule

two new Dependency Pairs are created:

G(s(x),y) -> G(f(x,y), 0)

G(s(x''), 0) -> G(f(x'', 0), 0)

G(s(x'), 0) -> G(f(x', 0), 0)

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳Argument Filtering and Ordering

**G(s( x'), 0) -> G(f(x', 0), 0)**

g(0, f(x,x)) ->x

g(x, s(y)) -> g(f(x,y), 0)

g(s(x),y) -> g(f(x,y), 0)

g(f(x,y), 0) -> f(g(x, 0), g(y, 0))

innermost

The following dependency pairs can be strictly oriented:

G(s(x'), 0) -> G(f(x', 0), 0)

G(s(x''), 0) -> G(f(x'', 0), 0)

G(f(x,y), 0) -> G(x, 0)

G(f(x,y), 0) -> G(y, 0)

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{0, s} > f

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}

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

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳AFS

...

→DP Problem 3

↳Dependency Graph

g(0, f(x,x)) ->x

g(x, s(y)) -> g(f(x,y), 0)

g(s(x),y) -> g(f(x,y), 0)

g(f(x,y), 0) -> f(g(x, 0), g(y, 0))

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes