f(

f(

R

↳Dependency Pair Analysis

F(y, f(x, f(a,x))) -> F(f(f(a,x), f(x, a)), f(a,y))

F(y, f(x, f(a,x))) -> F(f(a,x), f(x, a))

F(y, f(x, f(a,x))) -> F(x, a)

F(y, f(x, f(a,x))) -> F(a,y)

F(x, f(x,y)) -> F(f(f(x, a), a), a)

F(x, f(x,y)) -> F(f(x, a), a)

F(x, f(x,y)) -> F(x, a)

Furthermore,

R

↳DPs

→DP Problem 1

↳Narrowing Transformation

**F( y, f(x, f(a, x))) -> F(a, y)**

f(y, f(x, f(a,x))) -> f(f(f(a,x), f(x, a)), f(a,y))

f(x, f(x,y)) -> f(f(f(x, a), a), a)

innermost

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

As a result of transforming the rule

no new Dependency Pairs are created.

F(y, f(x, f(a,x))) -> F(f(a,x), f(x, a))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Narrowing Transformation

**F( y, f(x, f(a, x))) -> F(f(f(a, x), f(x, a)), f(a, y))**

f(y, f(x, f(a,x))) -> f(f(f(a,x), f(x, a)), f(a,y))

f(x, f(x,y)) -> f(f(f(x, a), a), a)

innermost

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

As a result of transforming the rule

two new Dependency Pairs are created:

F(y, f(x, f(a,x))) -> F(f(f(a,x), f(x, a)), f(a,y))

F(f(x'', f(a,x'')), f(x, f(a,x))) -> F(f(f(a,x), f(x, a)), f(f(f(a,x''), f(x'', a)), f(a, a)))

F(f(a,y''), f(x, f(a,x))) -> F(f(f(a,x), f(x, a)), f(f(f(a, a), a), a))

The transformation is resulting in two new DP problems:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 3

↳Instantiation Transformation

**F( y, f(x, f(a, x))) -> F(a, y)**

f(y, f(x, f(a,x))) -> f(f(f(a,x), f(x, a)), f(a,y))

f(x, f(x,y)) -> f(f(f(x, a), a), a)

innermost

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(y, f(x, f(a,x))) -> F(a,y)

F(a, f(x', f(a,x'))) -> F(a, a)

The transformation is resulting in no new DP problems.

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 4

↳Narrowing Transformation

**F(f(a, y''), f(x, f(a, x))) -> F(f(f(a, x), f(x, a)), f(f(f(a, a), a), a))**

f(y, f(x, f(a,x))) -> f(f(f(a,x), f(x, a)), f(a,y))

f(x, f(x,y)) -> f(f(f(x, a), a), a)

innermost

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

As a result of transforming the rule

no new Dependency Pairs are created.

F(f(x'', f(a,x'')), f(x, f(a,x))) -> F(f(f(a,x), f(x, a)), f(f(f(a,x''), f(x'', a)), f(a, a)))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 5

↳Narrowing Transformation

**F(f(a, y''), f(x, f(a, x))) -> F(f(f(a, x), f(x, a)), f(f(f(a, a), a), a))**

f(y, f(x, f(a,x))) -> f(f(f(a,x), f(x, a)), f(a,y))

f(x, f(x,y)) -> f(f(f(x, a), a), a)

innermost

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

As a result of transforming the rule

no new Dependency Pairs are created.

F(f(a,y''), f(x, f(a,x))) -> F(f(f(a,x), f(x, a)), f(f(f(a, a), a), a))

The transformation is resulting in no new DP problems.

Duration:

0:00 minutes