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:

f1(0) -> true
f1(1) -> false
f1(s1(x)) -> f1(x)
if3(true, s1(x), s1(y)) -> s1(x)
if3(false, s1(x), s1(y)) -> s1(y)
g2(x, c1(y)) -> c1(g2(x, y))
g2(x, c1(y)) -> g2(x, if3(f1(x), c1(g2(s1(x), y)), c1(y)))

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

f1(0) -> true
f1(1) -> false
f1(s1(x)) -> f1(x)
if3(true, s1(x), s1(y)) -> s1(x)
if3(false, s1(x), s1(y)) -> s1(y)
g2(x, c1(y)) -> c1(g2(x, y))
g2(x, c1(y)) -> g2(x, if3(f1(x), c1(g2(s1(x), y)), c1(y)))

Q is empty.

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

G2(x, c1(y)) -> G2(x, if3(f1(x), c1(g2(s1(x), y)), c1(y)))
G2(x, c1(y)) -> G2(x, y)
G2(x, c1(y)) -> IF3(f1(x), c1(g2(s1(x), y)), c1(y))
G2(x, c1(y)) -> G2(s1(x), y)
G2(x, c1(y)) -> F1(x)
F1(s1(x)) -> F1(x)

The TRS R consists of the following rules:

f1(0) -> true
f1(1) -> false
f1(s1(x)) -> f1(x)
if3(true, s1(x), s1(y)) -> s1(x)
if3(false, s1(x), s1(y)) -> s1(y)
g2(x, c1(y)) -> c1(g2(x, y))
g2(x, c1(y)) -> g2(x, if3(f1(x), c1(g2(s1(x), y)), c1(y)))

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

↳ QTRS
  ↳ DependencyPairsProof
QDP
      ↳ DependencyGraphProof

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

G2(x, c1(y)) -> G2(x, if3(f1(x), c1(g2(s1(x), y)), c1(y)))
G2(x, c1(y)) -> G2(x, y)
G2(x, c1(y)) -> IF3(f1(x), c1(g2(s1(x), y)), c1(y))
G2(x, c1(y)) -> G2(s1(x), y)
G2(x, c1(y)) -> F1(x)
F1(s1(x)) -> F1(x)

The TRS R consists of the following rules:

f1(0) -> true
f1(1) -> false
f1(s1(x)) -> f1(x)
if3(true, s1(x), s1(y)) -> s1(x)
if3(false, s1(x), s1(y)) -> s1(y)
g2(x, c1(y)) -> c1(g2(x, y))
g2(x, c1(y)) -> g2(x, if3(f1(x), c1(g2(s1(x), y)), c1(y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 2 SCCs with 3 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
QDP
            ↳ QDPAfsSolverProof
          ↳ QDP

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

F1(s1(x)) -> F1(x)

The TRS R consists of the following rules:

f1(0) -> true
f1(1) -> false
f1(s1(x)) -> f1(x)
if3(true, s1(x), s1(y)) -> s1(x)
if3(false, s1(x), s1(y)) -> s1(y)
g2(x, c1(y)) -> c1(g2(x, y))
g2(x, c1(y)) -> g2(x, if3(f1(x), c1(g2(s1(x), y)), c1(y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

F1(s1(x)) -> F1(x)
Used argument filtering: F1(x1)  =  x1
s1(x1)  =  s1(x1)
Used ordering: Quasi Precedence: trivial


↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPAfsSolverProof
QDP
                ↳ PisEmptyProof
          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

f1(0) -> true
f1(1) -> false
f1(s1(x)) -> f1(x)
if3(true, s1(x), s1(y)) -> s1(x)
if3(false, s1(x), s1(y)) -> s1(y)
g2(x, c1(y)) -> c1(g2(x, y))
g2(x, c1(y)) -> g2(x, if3(f1(x), c1(g2(s1(x), y)), c1(y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
QDP
            ↳ QDPAfsSolverProof

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

G2(x, c1(y)) -> G2(x, y)
G2(x, c1(y)) -> G2(s1(x), y)

The TRS R consists of the following rules:

f1(0) -> true
f1(1) -> false
f1(s1(x)) -> f1(x)
if3(true, s1(x), s1(y)) -> s1(x)
if3(false, s1(x), s1(y)) -> s1(y)
g2(x, c1(y)) -> c1(g2(x, y))
g2(x, c1(y)) -> g2(x, if3(f1(x), c1(g2(s1(x), y)), c1(y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

G2(x, c1(y)) -> G2(x, y)
G2(x, c1(y)) -> G2(s1(x), y)
Used argument filtering: G2(x1, x2)  =  x2
c1(x1)  =  c1(x1)
Used ordering: Quasi Precedence: trivial


↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPAfsSolverProof
QDP
                ↳ PisEmptyProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

f1(0) -> true
f1(1) -> false
f1(s1(x)) -> f1(x)
if3(true, s1(x), s1(y)) -> s1(x)
if3(false, s1(x), s1(y)) -> s1(y)
g2(x, c1(y)) -> c1(g2(x, y))
g2(x, c1(y)) -> g2(x, if3(f1(x), c1(g2(s1(x), y)), c1(y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.