Termination w.r.t. Q of the following Term Rewriting System could be proven:

Q restricted rewrite system:
The TRS R consists of the following rules:

f2(g2(i3(a, b, b'), c), d) -> if3(e, f2(.2(b, c), d'), f2(.2(b', c), d'))
f2(g2(h2(a, b), c), d) -> if3(e, f2(.2(b, g2(h2(a, b), c)), d), f2(c, d'))

Q is empty.


↳ QTRS
  ↳ Non-Overlap Check

Q restricted rewrite system:
The TRS R consists of the following rules:

f2(g2(i3(a, b, b'), c), d) -> if3(e, f2(.2(b, c), d'), f2(.2(b', c), d'))
f2(g2(h2(a, b), c), d) -> if3(e, f2(.2(b, g2(h2(a, b), c)), d), f2(c, d'))

Q is empty.

The TRS is non-overlapping. Hence, we can switch to innermost.

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

f2(g2(i3(a, b, b'), c), d) -> if3(e, f2(.2(b, c), d'), f2(.2(b', c), d'))
f2(g2(h2(a, b), c), d) -> if3(e, f2(.2(b, g2(h2(a, b), c)), d), f2(c, d'))

The set Q consists of the following terms:

f2(g2(i3(a, b, b'), c), d)
f2(g2(h2(a, b), c), d)


Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

F2(g2(h2(a, b), c), d) -> F2(c, d')
F2(g2(h2(a, b), c), d) -> F2(.2(b, g2(h2(a, b), c)), d)
F2(g2(i3(a, b, b'), c), d) -> F2(.2(b', c), d')
F2(g2(i3(a, b, b'), c), d) -> F2(.2(b, c), d')

The TRS R consists of the following rules:

f2(g2(i3(a, b, b'), c), d) -> if3(e, f2(.2(b, c), d'), f2(.2(b', c), d'))
f2(g2(h2(a, b), c), d) -> if3(e, f2(.2(b, g2(h2(a, b), c)), d), f2(c, d'))

The set Q consists of the following terms:

f2(g2(i3(a, b, b'), c), d)
f2(g2(h2(a, b), c), d)

We have to consider all minimal (P,Q,R)-chains.

↳ QTRS
  ↳ Non-Overlap Check
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

F2(g2(h2(a, b), c), d) -> F2(c, d')
F2(g2(h2(a, b), c), d) -> F2(.2(b, g2(h2(a, b), c)), d)
F2(g2(i3(a, b, b'), c), d) -> F2(.2(b', c), d')
F2(g2(i3(a, b, b'), c), d) -> F2(.2(b, c), d')

The TRS R consists of the following rules:

f2(g2(i3(a, b, b'), c), d) -> if3(e, f2(.2(b, c), d'), f2(.2(b', c), d'))
f2(g2(h2(a, b), c), d) -> if3(e, f2(.2(b, g2(h2(a, b), c)), d), f2(c, d'))

The set Q consists of the following terms:

f2(g2(i3(a, b, b'), c), d)
f2(g2(h2(a, b), c), d)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 4 less nodes.