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

↳Argument Filtering and 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))))

The following usable rules for innermost w.r.t. to the 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: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

a > c

a > b

resulting in one new DP problem.

Used Argument Filtering System:

F(x,_{1}x) ->_{2}x_{1}

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Remaining Obligation(s)

The following remains to be proven:

**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

Duration:

0:00 minutes