Left Termination proof of /tmp/tmpSZMNEf/evenodd.pl

Left Termination of the query pattern even(b) w.r.t. the given Prolog program could successfully be proven:

↳ PROLOG

  ↳ PrologToPiTRSProof

even₁(0₀).
even₁(s₁²(0₀)).
even₁(s₁³(X)) :- odd₁(X).
odd₁(s₁(0₀)).
odd₁(s₁(X)) :- even₁(s₁²(X)).

With regard to the inferred argument filtering the predicates were used in the following modes:
even₁: (b)
odd₁: (b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

even_1_in_g₁(0_0) -> even_1_out_g₁(0_0)
even_1_in_g₁(s_1₁(s_1₁(0_0))) -> even_1_out_g₁(s_1₁(s_1₁(0_0)))
even_1_in_g₁(s_1₁(s_1₁(s_1₁(X)))) -> if_even_1_in_1_g₂(X, odd_1_in_g₁(X))
odd_1_in_g₁(s_1₁(0_0)) -> odd_1_out_g₁(s_1₁(0_0))
odd_1_in_g₁(s_1₁(X)) -> if_odd_1_in_1_g₂(X, even_1_in_g₁(s_1₁(s_1₁(X))))
if_odd_1_in_1_g₂(X, even_1_out_g₁(s_1₁(s_1₁(X)))) -> odd_1_out_g₁(s_1₁(X))
if_even_1_in_1_g₂(X, odd_1_out_g₁(X)) -> even_1_out_g₁(s_1₁(s_1₁(s_1₁(X))))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

even_1_in_g₁(0_0) -> even_1_out_g₁(0_0)
even_1_in_g₁(s_1₁(s_1₁(0_0))) -> even_1_out_g₁(s_1₁(s_1₁(0_0)))
even_1_in_g₁(s_1₁(s_1₁(s_1₁(X)))) -> if_even_1_in_1_g₂(X, odd_1_in_g₁(X))
odd_1_in_g₁(s_1₁(0_0)) -> odd_1_out_g₁(s_1₁(0_0))
odd_1_in_g₁(s_1₁(X)) -> if_odd_1_in_1_g₂(X, even_1_in_g₁(s_1₁(s_1₁(X))))
if_odd_1_in_1_g₂(X, even_1_out_g₁(s_1₁(s_1₁(X)))) -> odd_1_out_g₁(s_1₁(X))
if_even_1_in_1_g₂(X, odd_1_out_g₁(X)) -> even_1_out_g₁(s_1₁(s_1₁(s_1₁(X))))

EVEN_1_IN_G₁(s_1₁(s_1₁(s_1₁(X)))) -> IF_EVEN_1_IN_1_G₂(X, odd_1_in_g₁(X))
EVEN_1_IN_G₁(s_1₁(s_1₁(s_1₁(X)))) -> ODD_1_IN_G₁(X)
ODD_1_IN_G₁(s_1₁(X)) -> IF_ODD_1_IN_1_G₂(X, even_1_in_g₁(s_1₁(s_1₁(X))))
ODD_1_IN_G₁(s_1₁(X)) -> EVEN_1_IN_G₁(s_1₁(s_1₁(X)))

The TRS R consists of the following rules:

even_1_in_g₁(0_0) -> even_1_out_g₁(0_0)
even_1_in_g₁(s_1₁(s_1₁(0_0))) -> even_1_out_g₁(s_1₁(s_1₁(0_0)))
even_1_in_g₁(s_1₁(s_1₁(s_1₁(X)))) -> if_even_1_in_1_g₂(X, odd_1_in_g₁(X))
odd_1_in_g₁(s_1₁(0_0)) -> odd_1_out_g₁(s_1₁(0_0))
odd_1_in_g₁(s_1₁(X)) -> if_odd_1_in_1_g₂(X, even_1_in_g₁(s_1₁(s_1₁(X))))
if_odd_1_in_1_g₂(X, even_1_out_g₁(s_1₁(s_1₁(X)))) -> odd_1_out_g₁(s_1₁(X))
if_even_1_in_1_g₂(X, odd_1_out_g₁(X)) -> even_1_out_g₁(s_1₁(s_1₁(s_1₁(X))))

The argument filtering Pi contains the following mapping:
even_1_in_g₁(x1) = even_1_in_g₁(x1)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
even_1_out_g₁(x1) = even_1_out_g
if_even_1_in_1_g₂(x1, x2) = if_even_1_in_1_g₁(x2)
odd_1_in_g₁(x1) = odd_1_in_g₁(x1)
odd_1_out_g₁(x1) = odd_1_out_g
if_odd_1_in_1_g₂(x1, x2) = if_odd_1_in_1_g₁(x2)
IF_ODD_1_IN_1_G₂(x1, x2) = IF_ODD_1_IN_1_G₁(x2)
ODD_1_IN_G₁(x1) = ODD_1_IN_G₁(x1)
IF_EVEN_1_IN_1_G₂(x1, x2) = IF_EVEN_1_IN_1_G₁(x2)
EVEN_1_IN_G₁(x1) = EVEN_1_IN_G₁(x1)

We have to consider all (P,R,Pi)-chains

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

EVEN_1_IN_G₁(s_1₁(s_1₁(s_1₁(X)))) -> IF_EVEN_1_IN_1_G₂(X, odd_1_in_g₁(X))
EVEN_1_IN_G₁(s_1₁(s_1₁(s_1₁(X)))) -> ODD_1_IN_G₁(X)
ODD_1_IN_G₁(s_1₁(X)) -> IF_ODD_1_IN_1_G₂(X, even_1_in_g₁(s_1₁(s_1₁(X))))
ODD_1_IN_G₁(s_1₁(X)) -> EVEN_1_IN_G₁(s_1₁(s_1₁(X)))

The TRS R consists of the following rules:

even_1_in_g₁(0_0) -> even_1_out_g₁(0_0)
even_1_in_g₁(s_1₁(s_1₁(0_0))) -> even_1_out_g₁(s_1₁(s_1₁(0_0)))
even_1_in_g₁(s_1₁(s_1₁(s_1₁(X)))) -> if_even_1_in_1_g₂(X, odd_1_in_g₁(X))
odd_1_in_g₁(s_1₁(0_0)) -> odd_1_out_g₁(s_1₁(0_0))
odd_1_in_g₁(s_1₁(X)) -> if_odd_1_in_1_g₂(X, even_1_in_g₁(s_1₁(s_1₁(X))))
if_odd_1_in_1_g₂(X, even_1_out_g₁(s_1₁(s_1₁(X)))) -> odd_1_out_g₁(s_1₁(X))
if_even_1_in_1_g₂(X, odd_1_out_g₁(X)) -> even_1_out_g₁(s_1₁(s_1₁(s_1₁(X))))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

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

EVEN_1_IN_G₁(s_1₁(s_1₁(s_1₁(X)))) -> ODD_1_IN_G₁(X)
ODD_1_IN_G₁(s_1₁(X)) -> EVEN_1_IN_G₁(s_1₁(s_1₁(X)))

The TRS R consists of the following rules:

even_1_in_g₁(0_0) -> even_1_out_g₁(0_0)
even_1_in_g₁(s_1₁(s_1₁(0_0))) -> even_1_out_g₁(s_1₁(s_1₁(0_0)))
even_1_in_g₁(s_1₁(s_1₁(s_1₁(X)))) -> if_even_1_in_1_g₂(X, odd_1_in_g₁(X))
odd_1_in_g₁(s_1₁(0_0)) -> odd_1_out_g₁(s_1₁(0_0))
odd_1_in_g₁(s_1₁(X)) -> if_odd_1_in_1_g₂(X, even_1_in_g₁(s_1₁(s_1₁(X))))
if_odd_1_in_1_g₂(X, even_1_out_g₁(s_1₁(s_1₁(X)))) -> odd_1_out_g₁(s_1₁(X))
if_even_1_in_1_g₂(X, odd_1_out_g₁(X)) -> even_1_out_g₁(s_1₁(s_1₁(s_1₁(X))))

The argument filtering Pi contains the following mapping:
even_1_in_g₁(x1) = even_1_in_g₁(x1)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
even_1_out_g₁(x1) = even_1_out_g
if_even_1_in_1_g₂(x1, x2) = if_even_1_in_1_g₁(x2)
odd_1_in_g₁(x1) = odd_1_in_g₁(x1)
odd_1_out_g₁(x1) = odd_1_out_g
if_odd_1_in_1_g₂(x1, x2) = if_odd_1_in_1_g₁(x2)
ODD_1_IN_G₁(x1) = ODD_1_IN_G₁(x1)
EVEN_1_IN_G₁(x1) = EVEN_1_IN_G₁(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting we can delete all non-usable rules from R.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

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

EVEN_1_IN_G₁(s_1₁(s_1₁(s_1₁(X)))) -> ODD_1_IN_G₁(X)
ODD_1_IN_G₁(s_1₁(X)) -> EVEN_1_IN_G₁(s_1₁(s_1₁(X)))

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

                      ↳ RuleRemovalProof

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

EVEN_1_IN_G₁(s_1₁(s_1₁(s_1₁(X)))) -> ODD_1_IN_G₁(X)
ODD_1_IN_G₁(s_1₁(X)) -> EVEN_1_IN_G₁(s_1₁(s_1₁(X)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {ODD_1_IN_G₁, EVEN_1_IN_G₁}.
By using a polynomial ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

EVEN_1_IN_G₁(s_1₁(s_1₁(s_1₁(X)))) -> ODD_1_IN_G₁(X)

Used ordering: POLO with Polynomial interpretation:

POL(ODD_1_IN_G₁(x₁)) = 1 + x₁
POL(EVEN_1_IN_G₁(x₁)) = x₁
POL(s_1₁(x₁)) = 1 + x₁

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ DependencyGraphProof

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

ODD_1_IN_G₁(s_1₁(X)) -> EVEN_1_IN_G₁(s_1₁(s_1₁(X)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {EVEN_1_IN_G₁, ODD_1_IN_G₁}.
The approximation of the Dependency Graph contains 0 SCCs with 1 less node.