Left Termination proof of /tmp/tmpy1Upw5/lte.pl

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

↳ PROLOG

  ↳ PrologToPiTRSProof

even₁(s₁²(X)) :- even₁(X).
even₁(0₀).
lte₂(s₁(X), s₁(Y)) :- lte₂(X, Y).
lte₂(0₀, Y).
goal₀ :- lte₂(X, s₁⁴(0₀)), even₁(X).

With regard to the inferred argument filtering the predicates were used in the following modes:
lte₂: (f,b)
even₁: (b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_0_in_ -> if_goal_0_in_1_₁(lte_2_in_ag₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))
lte_2_in_ag₂(s_1₁(X), s_1₁(Y)) -> if_lte_2_in_1_ag₃(X, Y, lte_2_in_ag₂(X, Y))
lte_2_in_ag₂(0_0, Y) -> lte_2_out_ag₂(0_0, Y)
if_lte_2_in_1_ag₃(X, Y, lte_2_out_ag₂(X, Y)) -> lte_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_goal_0_in_1_₁(lte_2_out_ag₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))) -> if_goal_0_in_2_₂(X, even_1_in_g₁(X))
even_1_in_g₁(s_1₁(s_1₁(X))) -> if_even_1_in_1_g₂(X, even_1_in_g₁(X))
even_1_in_g₁(0_0) -> even_1_out_g₁(0_0)
if_even_1_in_1_g₂(X, even_1_out_g₁(X)) -> even_1_out_g₁(s_1₁(s_1₁(X)))
if_goal_0_in_2_₂(X, even_1_out_g₁(X)) -> goal_0_out_

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:

goal_0_in_ -> if_goal_0_in_1_₁(lte_2_in_ag₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))
lte_2_in_ag₂(s_1₁(X), s_1₁(Y)) -> if_lte_2_in_1_ag₃(X, Y, lte_2_in_ag₂(X, Y))
lte_2_in_ag₂(0_0, Y) -> lte_2_out_ag₂(0_0, Y)
if_lte_2_in_1_ag₃(X, Y, lte_2_out_ag₂(X, Y)) -> lte_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_goal_0_in_1_₁(lte_2_out_ag₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))) -> if_goal_0_in_2_₂(X, even_1_in_g₁(X))
even_1_in_g₁(s_1₁(s_1₁(X))) -> if_even_1_in_1_g₂(X, even_1_in_g₁(X))
even_1_in_g₁(0_0) -> even_1_out_g₁(0_0)
if_even_1_in_1_g₂(X, even_1_out_g₁(X)) -> even_1_out_g₁(s_1₁(s_1₁(X)))
if_goal_0_in_2_₂(X, even_1_out_g₁(X)) -> goal_0_out_

GOAL_0_IN_ -> IF_GOAL_0_IN_1_₁(lte_2_in_ag₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))
GOAL_0_IN_ -> LTE_2_IN_AG₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))
LTE_2_IN_AG₂(s_1₁(X), s_1₁(Y)) -> IF_LTE_2_IN_1_AG₃(X, Y, lte_2_in_ag₂(X, Y))
LTE_2_IN_AG₂(s_1₁(X), s_1₁(Y)) -> LTE_2_IN_AG₂(X, Y)
IF_GOAL_0_IN_1_₁(lte_2_out_ag₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))) -> IF_GOAL_0_IN_2_₂(X, even_1_in_g₁(X))
IF_GOAL_0_IN_1_₁(lte_2_out_ag₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))) -> EVEN_1_IN_G₁(X)
EVEN_1_IN_G₁(s_1₁(s_1₁(X))) -> IF_EVEN_1_IN_1_G₂(X, even_1_in_g₁(X))
EVEN_1_IN_G₁(s_1₁(s_1₁(X))) -> EVEN_1_IN_G₁(X)

The TRS R consists of the following rules:

goal_0_in_ -> if_goal_0_in_1_₁(lte_2_in_ag₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))
lte_2_in_ag₂(s_1₁(X), s_1₁(Y)) -> if_lte_2_in_1_ag₃(X, Y, lte_2_in_ag₂(X, Y))
lte_2_in_ag₂(0_0, Y) -> lte_2_out_ag₂(0_0, Y)
if_lte_2_in_1_ag₃(X, Y, lte_2_out_ag₂(X, Y)) -> lte_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_goal_0_in_1_₁(lte_2_out_ag₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))) -> if_goal_0_in_2_₂(X, even_1_in_g₁(X))
even_1_in_g₁(s_1₁(s_1₁(X))) -> if_even_1_in_1_g₂(X, even_1_in_g₁(X))
even_1_in_g₁(0_0) -> even_1_out_g₁(0_0)
if_even_1_in_1_g₂(X, even_1_out_g₁(X)) -> even_1_out_g₁(s_1₁(s_1₁(X)))
if_goal_0_in_2_₂(X, even_1_out_g₁(X)) -> goal_0_out_

The argument filtering Pi contains the following mapping:
goal_0_in_ = goal_0_in_
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
if_goal_0_in_1_₁(x1) = if_goal_0_in_1_₁(x1)
lte_2_in_ag₂(x1, x2) = lte_2_in_ag₁(x2)
if_lte_2_in_1_ag₃(x1, x2, x3) = if_lte_2_in_1_ag₁(x3)
lte_2_out_ag₂(x1, x2) = lte_2_out_ag₁(x1)
if_goal_0_in_2_₂(x1, x2) = if_goal_0_in_2_₁(x2)
even_1_in_g₁(x1) = even_1_in_g₁(x1)
if_even_1_in_1_g₂(x1, x2) = if_even_1_in_1_g₁(x2)
even_1_out_g₁(x1) = even_1_out_g
goal_0_out_ = goal_0_out_
LTE_2_IN_AG₂(x1, x2) = LTE_2_IN_AG₁(x2)
IF_LTE_2_IN_1_AG₃(x1, x2, x3) = IF_LTE_2_IN_1_AG₁(x3)
IF_GOAL_0_IN_2_₂(x1, x2) = IF_GOAL_0_IN_2_₁(x2)
IF_GOAL_0_IN_1_₁(x1) = IF_GOAL_0_IN_1_₁(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)
GOAL_0_IN_ = GOAL_0_IN_

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:

GOAL_0_IN_ -> IF_GOAL_0_IN_1_₁(lte_2_in_ag₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))
GOAL_0_IN_ -> LTE_2_IN_AG₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))
LTE_2_IN_AG₂(s_1₁(X), s_1₁(Y)) -> IF_LTE_2_IN_1_AG₃(X, Y, lte_2_in_ag₂(X, Y))
LTE_2_IN_AG₂(s_1₁(X), s_1₁(Y)) -> LTE_2_IN_AG₂(X, Y)
IF_GOAL_0_IN_1_₁(lte_2_out_ag₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))) -> IF_GOAL_0_IN_2_₂(X, even_1_in_g₁(X))
IF_GOAL_0_IN_1_₁(lte_2_out_ag₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))) -> EVEN_1_IN_G₁(X)
EVEN_1_IN_G₁(s_1₁(s_1₁(X))) -> IF_EVEN_1_IN_1_G₂(X, even_1_in_g₁(X))
EVEN_1_IN_G₁(s_1₁(s_1₁(X))) -> EVEN_1_IN_G₁(X)

The TRS R consists of the following rules:

goal_0_in_ -> if_goal_0_in_1_₁(lte_2_in_ag₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))
lte_2_in_ag₂(s_1₁(X), s_1₁(Y)) -> if_lte_2_in_1_ag₃(X, Y, lte_2_in_ag₂(X, Y))
lte_2_in_ag₂(0_0, Y) -> lte_2_out_ag₂(0_0, Y)
if_lte_2_in_1_ag₃(X, Y, lte_2_out_ag₂(X, Y)) -> lte_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_goal_0_in_1_₁(lte_2_out_ag₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))) -> if_goal_0_in_2_₂(X, even_1_in_g₁(X))
even_1_in_g₁(s_1₁(s_1₁(X))) -> if_even_1_in_1_g₂(X, even_1_in_g₁(X))
even_1_in_g₁(0_0) -> even_1_out_g₁(0_0)
if_even_1_in_1_g₂(X, even_1_out_g₁(X)) -> even_1_out_g₁(s_1₁(s_1₁(X)))
if_goal_0_in_2_₂(X, even_1_out_g₁(X)) -> goal_0_out_

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

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

The TRS R consists of the following rules:

goal_0_in_ -> if_goal_0_in_1_₁(lte_2_in_ag₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))
lte_2_in_ag₂(s_1₁(X), s_1₁(Y)) -> if_lte_2_in_1_ag₃(X, Y, lte_2_in_ag₂(X, Y))
lte_2_in_ag₂(0_0, Y) -> lte_2_out_ag₂(0_0, Y)
if_lte_2_in_1_ag₃(X, Y, lte_2_out_ag₂(X, Y)) -> lte_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_goal_0_in_1_₁(lte_2_out_ag₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))) -> if_goal_0_in_2_₂(X, even_1_in_g₁(X))
even_1_in_g₁(s_1₁(s_1₁(X))) -> if_even_1_in_1_g₂(X, even_1_in_g₁(X))
even_1_in_g₁(0_0) -> even_1_out_g₁(0_0)
if_even_1_in_1_g₂(X, even_1_out_g₁(X)) -> even_1_out_g₁(s_1₁(s_1₁(X)))
if_goal_0_in_2_₂(X, even_1_out_g₁(X)) -> goal_0_out_

The argument filtering Pi contains the following mapping:
goal_0_in_ = goal_0_in_
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
if_goal_0_in_1_₁(x1) = if_goal_0_in_1_₁(x1)
lte_2_in_ag₂(x1, x2) = lte_2_in_ag₁(x2)
if_lte_2_in_1_ag₃(x1, x2, x3) = if_lte_2_in_1_ag₁(x3)
lte_2_out_ag₂(x1, x2) = lte_2_out_ag₁(x1)
if_goal_0_in_2_₂(x1, x2) = if_goal_0_in_2_₁(x2)
even_1_in_g₁(x1) = even_1_in_g₁(x1)
if_even_1_in_1_g₂(x1, x2) = if_even_1_in_1_g₁(x2)
even_1_out_g₁(x1) = even_1_out_g
goal_0_out_ = goal_0_out_
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

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

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

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

EVEN_1_IN_G₁(s_1₁(s_1₁(X))) -> EVEN_1_IN_G₁(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}.
By using the subterm criterion together with the size-change analysis we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

EVEN_1_IN_G₁(s_1₁(s_1₁(X))) -> EVEN_1_IN_G₁(X)
The graph contains the following edges 1 > 1

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

LTE_2_IN_AG₂(s_1₁(X), s_1₁(Y)) -> LTE_2_IN_AG₂(X, Y)

The TRS R consists of the following rules:

goal_0_in_ -> if_goal_0_in_1_₁(lte_2_in_ag₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0))))))
lte_2_in_ag₂(s_1₁(X), s_1₁(Y)) -> if_lte_2_in_1_ag₃(X, Y, lte_2_in_ag₂(X, Y))
lte_2_in_ag₂(0_0, Y) -> lte_2_out_ag₂(0_0, Y)
if_lte_2_in_1_ag₃(X, Y, lte_2_out_ag₂(X, Y)) -> lte_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_goal_0_in_1_₁(lte_2_out_ag₂(X, s_1₁(s_1₁(s_1₁(s_1₁(0_0)))))) -> if_goal_0_in_2_₂(X, even_1_in_g₁(X))
even_1_in_g₁(s_1₁(s_1₁(X))) -> if_even_1_in_1_g₂(X, even_1_in_g₁(X))
even_1_in_g₁(0_0) -> even_1_out_g₁(0_0)
if_even_1_in_1_g₂(X, even_1_out_g₁(X)) -> even_1_out_g₁(s_1₁(s_1₁(X)))
if_goal_0_in_2_₂(X, even_1_out_g₁(X)) -> goal_0_out_

The argument filtering Pi contains the following mapping:
goal_0_in_ = goal_0_in_
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
if_goal_0_in_1_₁(x1) = if_goal_0_in_1_₁(x1)
lte_2_in_ag₂(x1, x2) = lte_2_in_ag₁(x2)
if_lte_2_in_1_ag₃(x1, x2, x3) = if_lte_2_in_1_ag₁(x3)
lte_2_out_ag₂(x1, x2) = lte_2_out_ag₁(x1)
if_goal_0_in_2_₂(x1, x2) = if_goal_0_in_2_₁(x2)
even_1_in_g₁(x1) = even_1_in_g₁(x1)
if_even_1_in_1_g₂(x1, x2) = if_even_1_in_1_g₁(x2)
even_1_out_g₁(x1) = even_1_out_g
goal_0_out_ = goal_0_out_
LTE_2_IN_AG₂(x1, x2) = LTE_2_IN_AG₁(x2)

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

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

LTE_2_IN_AG₂(s_1₁(X), s_1₁(Y)) -> LTE_2_IN_AG₂(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s_1₁(x1) = s_1₁(x1)
LTE_2_IN_AG₂(x1, x2) = LTE_2_IN_AG₁(x2)

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

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

LTE_2_IN_AG₁(s_1₁(Y)) -> LTE_2_IN_AG₁(Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {LTE_2_IN_AG₁}.
By using the subterm criterion together with the size-change analysis we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

LTE_2_IN_AG₁(s_1₁(Y)) -> LTE_2_IN_AG₁(Y)
The graph contains the following edges 1 > 1