f(f(f(a, b), c),

f(

R

↳Dependency Pair Analysis

F(f(f(a, b), c),x) -> F(b, f(a, f(c, f(b,x))))

F(f(f(a, b), c),x) -> F(a, f(c, f(b,x)))

F(f(f(a, b), c),x) -> F(c, f(b,x))

F(f(f(a, b), c),x) -> F(b,x)

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

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Polynomial Ordering

**F(f(f(a, b), c), x) -> F(b, x)**

f(f(f(a, b), c),x) -> f(b, f(a, f(c, f(b,x))))

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

innermost

The following dependency pairs can be strictly oriented:

F(f(f(a, b), c),x) -> F(b,x)

F(f(f(a, b), c),x) -> F(c, f(b,x))

F(f(f(a, b), c),x) -> F(b, f(a, f(c, f(b,x))))

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

f(f(f(a, b), c),x) -> f(b, f(a, f(c, f(b,x))))

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

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(c)= 0 _{ }^{ }_{ }^{ }POL(b)= 0 _{ }^{ }_{ }^{ }POL(a)= 1 _{ }^{ }_{ }^{ }POL(f(x)_{1}, x_{2})= x _{1}_{ }^{ }_{ }^{ }POL(F(x)_{1}, x_{2})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Narrowing Transformation

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

f(f(f(a, b), c),x) -> f(b, f(a, f(c, f(b,x))))

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

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(f(f(a, b), c),x) -> F(a, f(c, f(b,x)))

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

F(f(f(a, b), c), f(y',z')) -> F(a, f(c, f(f(b,y'),z')))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Nar

...

→DP Problem 3

↳Forward Instantiation Transformation

**F(f(f(a, b), c), f( y', z')) -> F(a, f(c, f(f(b, y'), z')))**

f(f(f(a, b), c),x) -> f(b, f(a, f(c, f(b,x))))

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

innermost

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

As a result of transforming the rule

three new Dependency Pairs are created:

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

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

F(f(f(a, b), c), f(y',z)) -> F(f(f(a, b), c),y')

F(f(f(a, b), c), f(f(y''',z'''),z)) -> F(f(f(a, b), c), f(y''',z'''))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Nar

...

→DP Problem 4

↳Remaining Obligation(s)

The following remains to be proven:

**F(f(f(a, b), c), f(f( y''', z'''), z)) -> F(f(f(a, b), c), f(y''', z'''))**

f(f(f(a, b), c),x) -> f(b, f(a, f(c, f(b,x))))

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

innermost

Duration:

0:00 minutes