Left Termination proof of /tmp/tmpoW_

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

↳ PROLOG

  ↳ PrologToPiTRSProof

normal₂(F, N) :- rewrite₂(F, F1), normal₂(F1, N).
normal₂(F, F).
rewrite₂(op₂(op₂(A, B), C), op₂(A, op₂(B, C))).
rewrite₂(op₂(A, op₂(B, C)), op₂(A, L)) :- rewrite₂(op₂(B, C), L).

With regard to the inferred argument filtering the predicates were used in the following modes:
normal₂: (b,f)
rewrite₂: (b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

normal_2_in_ga₂(F, N) -> if_normal_2_in_1_ga₃(F, N, rewrite_2_in_ga₂(F, F1))
rewrite_2_in_ga₂(op_2₂(op_2₂(A, B), C), op_2₂(A, op_2₂(B, C))) -> rewrite_2_out_ga₂(op_2₂(op_2₂(A, B), C), op_2₂(A, op_2₂(B, C)))
rewrite_2_in_ga₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L)) -> if_rewrite_2_in_1_ga₅(A, B, C, L, rewrite_2_in_ga₂(op_2₂(B, C), L))
if_rewrite_2_in_1_ga₅(A, B, C, L, rewrite_2_out_ga₂(op_2₂(B, C), L)) -> rewrite_2_out_ga₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L))
if_normal_2_in_1_ga₃(F, N, rewrite_2_out_ga₂(F, F1)) -> if_normal_2_in_2_ga₄(F, N, F1, normal_2_in_ga₂(F1, N))
normal_2_in_ga₂(F, F) -> normal_2_out_ga₂(F, F)
if_normal_2_in_2_ga₄(F, N, F1, normal_2_out_ga₂(F1, N)) -> normal_2_out_ga₂(F, N)

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:

normal_2_in_ga₂(F, N) -> if_normal_2_in_1_ga₃(F, N, rewrite_2_in_ga₂(F, F1))
rewrite_2_in_ga₂(op_2₂(op_2₂(A, B), C), op_2₂(A, op_2₂(B, C))) -> rewrite_2_out_ga₂(op_2₂(op_2₂(A, B), C), op_2₂(A, op_2₂(B, C)))
rewrite_2_in_ga₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L)) -> if_rewrite_2_in_1_ga₅(A, B, C, L, rewrite_2_in_ga₂(op_2₂(B, C), L))
if_rewrite_2_in_1_ga₅(A, B, C, L, rewrite_2_out_ga₂(op_2₂(B, C), L)) -> rewrite_2_out_ga₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L))
if_normal_2_in_1_ga₃(F, N, rewrite_2_out_ga₂(F, F1)) -> if_normal_2_in_2_ga₄(F, N, F1, normal_2_in_ga₂(F1, N))
normal_2_in_ga₂(F, F) -> normal_2_out_ga₂(F, F)
if_normal_2_in_2_ga₄(F, N, F1, normal_2_out_ga₂(F1, N)) -> normal_2_out_ga₂(F, N)

NORMAL_2_IN_GA₂(F, N) -> IF_NORMAL_2_IN_1_GA₃(F, N, rewrite_2_in_ga₂(F, F1))
NORMAL_2_IN_GA₂(F, N) -> REWRITE_2_IN_GA₂(F, F1)
REWRITE_2_IN_GA₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L)) -> IF_REWRITE_2_IN_1_GA₅(A, B, C, L, rewrite_2_in_ga₂(op_2₂(B, C), L))
REWRITE_2_IN_GA₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L)) -> REWRITE_2_IN_GA₂(op_2₂(B, C), L)
IF_NORMAL_2_IN_1_GA₃(F, N, rewrite_2_out_ga₂(F, F1)) -> IF_NORMAL_2_IN_2_GA₄(F, N, F1, normal_2_in_ga₂(F1, N))
IF_NORMAL_2_IN_1_GA₃(F, N, rewrite_2_out_ga₂(F, F1)) -> NORMAL_2_IN_GA₂(F1, N)

The TRS R consists of the following rules:

normal_2_in_ga₂(F, N) -> if_normal_2_in_1_ga₃(F, N, rewrite_2_in_ga₂(F, F1))
rewrite_2_in_ga₂(op_2₂(op_2₂(A, B), C), op_2₂(A, op_2₂(B, C))) -> rewrite_2_out_ga₂(op_2₂(op_2₂(A, B), C), op_2₂(A, op_2₂(B, C)))
rewrite_2_in_ga₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L)) -> if_rewrite_2_in_1_ga₅(A, B, C, L, rewrite_2_in_ga₂(op_2₂(B, C), L))
if_rewrite_2_in_1_ga₅(A, B, C, L, rewrite_2_out_ga₂(op_2₂(B, C), L)) -> rewrite_2_out_ga₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L))
if_normal_2_in_1_ga₃(F, N, rewrite_2_out_ga₂(F, F1)) -> if_normal_2_in_2_ga₄(F, N, F1, normal_2_in_ga₂(F1, N))
normal_2_in_ga₂(F, F) -> normal_2_out_ga₂(F, F)
if_normal_2_in_2_ga₄(F, N, F1, normal_2_out_ga₂(F1, N)) -> normal_2_out_ga₂(F, N)

The argument filtering Pi contains the following mapping:
normal_2_in_ga₂(x1, x2) = normal_2_in_ga₁(x1)
op_2₂(x1, x2) = op_2₂(x1, x2)
if_normal_2_in_1_ga₃(x1, x2, x3) = if_normal_2_in_1_ga₁(x3)
rewrite_2_in_ga₂(x1, x2) = rewrite_2_in_ga₁(x1)
rewrite_2_out_ga₂(x1, x2) = rewrite_2_out_ga₁(x2)
if_rewrite_2_in_1_ga₅(x1, x2, x3, x4, x5) = if_rewrite_2_in_1_ga₂(x1, x5)
if_normal_2_in_2_ga₄(x1, x2, x3, x4) = if_normal_2_in_2_ga₁(x4)
normal_2_out_ga₂(x1, x2) = normal_2_out_ga₁(x2)
IF_REWRITE_2_IN_1_GA₅(x1, x2, x3, x4, x5) = IF_REWRITE_2_IN_1_GA₂(x1, x5)
REWRITE_2_IN_GA₂(x1, x2) = REWRITE_2_IN_GA₁(x1)
IF_NORMAL_2_IN_1_GA₃(x1, x2, x3) = IF_NORMAL_2_IN_1_GA₁(x3)
IF_NORMAL_2_IN_2_GA₄(x1, x2, x3, x4) = IF_NORMAL_2_IN_2_GA₁(x4)
NORMAL_2_IN_GA₂(x1, x2) = NORMAL_2_IN_GA₁(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:

NORMAL_2_IN_GA₂(F, N) -> IF_NORMAL_2_IN_1_GA₃(F, N, rewrite_2_in_ga₂(F, F1))
NORMAL_2_IN_GA₂(F, N) -> REWRITE_2_IN_GA₂(F, F1)
REWRITE_2_IN_GA₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L)) -> IF_REWRITE_2_IN_1_GA₅(A, B, C, L, rewrite_2_in_ga₂(op_2₂(B, C), L))
REWRITE_2_IN_GA₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L)) -> REWRITE_2_IN_GA₂(op_2₂(B, C), L)
IF_NORMAL_2_IN_1_GA₃(F, N, rewrite_2_out_ga₂(F, F1)) -> IF_NORMAL_2_IN_2_GA₄(F, N, F1, normal_2_in_ga₂(F1, N))
IF_NORMAL_2_IN_1_GA₃(F, N, rewrite_2_out_ga₂(F, F1)) -> NORMAL_2_IN_GA₂(F1, N)

The TRS R consists of the following rules:

normal_2_in_ga₂(F, N) -> if_normal_2_in_1_ga₃(F, N, rewrite_2_in_ga₂(F, F1))
rewrite_2_in_ga₂(op_2₂(op_2₂(A, B), C), op_2₂(A, op_2₂(B, C))) -> rewrite_2_out_ga₂(op_2₂(op_2₂(A, B), C), op_2₂(A, op_2₂(B, C)))
rewrite_2_in_ga₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L)) -> if_rewrite_2_in_1_ga₅(A, B, C, L, rewrite_2_in_ga₂(op_2₂(B, C), L))
if_rewrite_2_in_1_ga₅(A, B, C, L, rewrite_2_out_ga₂(op_2₂(B, C), L)) -> rewrite_2_out_ga₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L))
if_normal_2_in_1_ga₃(F, N, rewrite_2_out_ga₂(F, F1)) -> if_normal_2_in_2_ga₄(F, N, F1, normal_2_in_ga₂(F1, N))
normal_2_in_ga₂(F, F) -> normal_2_out_ga₂(F, F)
if_normal_2_in_2_ga₄(F, N, F1, normal_2_out_ga₂(F1, N)) -> normal_2_out_ga₂(F, N)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

REWRITE_2_IN_GA₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L)) -> REWRITE_2_IN_GA₂(op_2₂(B, C), L)

The TRS R consists of the following rules:

normal_2_in_ga₂(F, N) -> if_normal_2_in_1_ga₃(F, N, rewrite_2_in_ga₂(F, F1))
rewrite_2_in_ga₂(op_2₂(op_2₂(A, B), C), op_2₂(A, op_2₂(B, C))) -> rewrite_2_out_ga₂(op_2₂(op_2₂(A, B), C), op_2₂(A, op_2₂(B, C)))
rewrite_2_in_ga₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L)) -> if_rewrite_2_in_1_ga₅(A, B, C, L, rewrite_2_in_ga₂(op_2₂(B, C), L))
if_rewrite_2_in_1_ga₅(A, B, C, L, rewrite_2_out_ga₂(op_2₂(B, C), L)) -> rewrite_2_out_ga₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L))
if_normal_2_in_1_ga₃(F, N, rewrite_2_out_ga₂(F, F1)) -> if_normal_2_in_2_ga₄(F, N, F1, normal_2_in_ga₂(F1, N))
normal_2_in_ga₂(F, F) -> normal_2_out_ga₂(F, F)
if_normal_2_in_2_ga₄(F, N, F1, normal_2_out_ga₂(F1, N)) -> normal_2_out_ga₂(F, N)

The argument filtering Pi contains the following mapping:
normal_2_in_ga₂(x1, x2) = normal_2_in_ga₁(x1)
op_2₂(x1, x2) = op_2₂(x1, x2)
if_normal_2_in_1_ga₃(x1, x2, x3) = if_normal_2_in_1_ga₁(x3)
rewrite_2_in_ga₂(x1, x2) = rewrite_2_in_ga₁(x1)
rewrite_2_out_ga₂(x1, x2) = rewrite_2_out_ga₁(x2)
if_rewrite_2_in_1_ga₅(x1, x2, x3, x4, x5) = if_rewrite_2_in_1_ga₂(x1, x5)
if_normal_2_in_2_ga₄(x1, x2, x3, x4) = if_normal_2_in_2_ga₁(x4)
normal_2_out_ga₂(x1, x2) = normal_2_out_ga₁(x2)
REWRITE_2_IN_GA₂(x1, x2) = REWRITE_2_IN_GA₁(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:

REWRITE_2_IN_GA₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L)) -> REWRITE_2_IN_GA₂(op_2₂(B, C), L)

R is empty.
The argument filtering Pi contains the following mapping:
op_2₂(x1, x2) = op_2₂(x1, x2)
REWRITE_2_IN_GA₂(x1, x2) = REWRITE_2_IN_GA₁(x1)

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:

REWRITE_2_IN_GA₁(op_2₂(A, op_2₂(B, C))) -> REWRITE_2_IN_GA₁(op_2₂(B, C))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {REWRITE_2_IN_GA₁}.
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:

REWRITE_2_IN_GA₁(op_2₂(A, op_2₂(B, C))) -> REWRITE_2_IN_GA₁(op_2₂(B, C))
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:

NORMAL_2_IN_GA₂(F, N) -> IF_NORMAL_2_IN_1_GA₃(F, N, rewrite_2_in_ga₂(F, F1))
IF_NORMAL_2_IN_1_GA₃(F, N, rewrite_2_out_ga₂(F, F1)) -> NORMAL_2_IN_GA₂(F1, N)

The TRS R consists of the following rules:

normal_2_in_ga₂(F, N) -> if_normal_2_in_1_ga₃(F, N, rewrite_2_in_ga₂(F, F1))
rewrite_2_in_ga₂(op_2₂(op_2₂(A, B), C), op_2₂(A, op_2₂(B, C))) -> rewrite_2_out_ga₂(op_2₂(op_2₂(A, B), C), op_2₂(A, op_2₂(B, C)))
rewrite_2_in_ga₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L)) -> if_rewrite_2_in_1_ga₅(A, B, C, L, rewrite_2_in_ga₂(op_2₂(B, C), L))
if_rewrite_2_in_1_ga₅(A, B, C, L, rewrite_2_out_ga₂(op_2₂(B, C), L)) -> rewrite_2_out_ga₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L))
if_normal_2_in_1_ga₃(F, N, rewrite_2_out_ga₂(F, F1)) -> if_normal_2_in_2_ga₄(F, N, F1, normal_2_in_ga₂(F1, N))
normal_2_in_ga₂(F, F) -> normal_2_out_ga₂(F, F)
if_normal_2_in_2_ga₄(F, N, F1, normal_2_out_ga₂(F1, N)) -> normal_2_out_ga₂(F, N)

The argument filtering Pi contains the following mapping:
normal_2_in_ga₂(x1, x2) = normal_2_in_ga₁(x1)
op_2₂(x1, x2) = op_2₂(x1, x2)
if_normal_2_in_1_ga₃(x1, x2, x3) = if_normal_2_in_1_ga₁(x3)
rewrite_2_in_ga₂(x1, x2) = rewrite_2_in_ga₁(x1)
rewrite_2_out_ga₂(x1, x2) = rewrite_2_out_ga₁(x2)
if_rewrite_2_in_1_ga₅(x1, x2, x3, x4, x5) = if_rewrite_2_in_1_ga₂(x1, x5)
if_normal_2_in_2_ga₄(x1, x2, x3, x4) = if_normal_2_in_2_ga₁(x4)
normal_2_out_ga₂(x1, x2) = normal_2_out_ga₁(x2)
IF_NORMAL_2_IN_1_GA₃(x1, x2, x3) = IF_NORMAL_2_IN_1_GA₁(x3)
NORMAL_2_IN_GA₂(x1, x2) = NORMAL_2_IN_GA₁(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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

NORMAL_2_IN_GA₂(F, N) -> IF_NORMAL_2_IN_1_GA₃(F, N, rewrite_2_in_ga₂(F, F1))
IF_NORMAL_2_IN_1_GA₃(F, N, rewrite_2_out_ga₂(F, F1)) -> NORMAL_2_IN_GA₂(F1, N)

The TRS R consists of the following rules:

rewrite_2_in_ga₂(op_2₂(op_2₂(A, B), C), op_2₂(A, op_2₂(B, C))) -> rewrite_2_out_ga₂(op_2₂(op_2₂(A, B), C), op_2₂(A, op_2₂(B, C)))
rewrite_2_in_ga₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L)) -> if_rewrite_2_in_1_ga₅(A, B, C, L, rewrite_2_in_ga₂(op_2₂(B, C), L))
if_rewrite_2_in_1_ga₅(A, B, C, L, rewrite_2_out_ga₂(op_2₂(B, C), L)) -> rewrite_2_out_ga₂(op_2₂(A, op_2₂(B, C)), op_2₂(A, L))

The argument filtering Pi contains the following mapping:
op_2₂(x1, x2) = op_2₂(x1, x2)
rewrite_2_in_ga₂(x1, x2) = rewrite_2_in_ga₁(x1)
rewrite_2_out_ga₂(x1, x2) = rewrite_2_out_ga₁(x2)
if_rewrite_2_in_1_ga₅(x1, x2, x3, x4, x5) = if_rewrite_2_in_1_ga₂(x1, x5)
IF_NORMAL_2_IN_1_GA₃(x1, x2, x3) = IF_NORMAL_2_IN_1_GA₁(x3)
NORMAL_2_IN_GA₂(x1, x2) = NORMAL_2_IN_GA₁(x1)

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

                        ↳ RuleRemovalProof

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

NORMAL_2_IN_GA₁(F) -> IF_NORMAL_2_IN_1_GA₁(rewrite_2_in_ga₁(F))
IF_NORMAL_2_IN_1_GA₁(rewrite_2_out_ga₁(F1)) -> NORMAL_2_IN_GA₁(F1)

The TRS R consists of the following rules:

rewrite_2_in_ga₁(op_2₂(op_2₂(A, B), C)) -> rewrite_2_out_ga₁(op_2₂(A, op_2₂(B, C)))
rewrite_2_in_ga₁(op_2₂(A, op_2₂(B, C))) -> if_rewrite_2_in_1_ga₂(A, rewrite_2_in_ga₁(op_2₂(B, C)))
if_rewrite_2_in_1_ga₂(A, rewrite_2_out_ga₁(L)) -> rewrite_2_out_ga₁(op_2₂(A, L))

The set Q consists of the following terms:

rewrite_2_in_ga₁(x0)
if_rewrite_2_in_1_ga₂(x0, x1)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_NORMAL_2_IN_1_GA₁, NORMAL_2_IN_GA₁}.
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:

IF_NORMAL_2_IN_1_GA₁(rewrite_2_out_ga₁(F1)) -> NORMAL_2_IN_GA₁(F1)

Strictly oriented rules of the TRS R:

rewrite_2_in_ga₁(op_2₂(op_2₂(A, B), C)) -> rewrite_2_out_ga₁(op_2₂(A, op_2₂(B, C)))

Used ordering: POLO with Polynomial interpretation:

POL(IF_NORMAL_2_IN_1_GA₁(x₁)) = x₁
POL(if_rewrite_2_in_1_ga₂(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(op_2₂(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(rewrite_2_out_ga₁(x₁)) = 1 + x₁
POL(rewrite_2_in_ga₁(x₁)) = x₁
POL(NORMAL_2_IN_GA₁(x₁)) = x₁

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ RuleRemovalProof

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

NORMAL_2_IN_GA₁(F) -> IF_NORMAL_2_IN_1_GA₁(rewrite_2_in_ga₁(F))

The TRS R consists of the following rules:

rewrite_2_in_ga₁(op_2₂(A, op_2₂(B, C))) -> if_rewrite_2_in_1_ga₂(A, rewrite_2_in_ga₁(op_2₂(B, C)))
if_rewrite_2_in_1_ga₂(A, rewrite_2_out_ga₁(L)) -> rewrite_2_out_ga₁(op_2₂(A, L))

The set Q consists of the following terms:

rewrite_2_in_ga₁(x0)
if_rewrite_2_in_1_ga₂(x0, x1)

rewrite_2_in_ga₁(op_2₂(A, op_2₂(B, C))) -> if_rewrite_2_in_1_ga₂(A, rewrite_2_in_ga₁(op_2₂(B, C)))

Used ordering: POLO with Polynomial interpretation:

POL(IF_NORMAL_2_IN_1_GA₁(x₁)) = x₁
POL(if_rewrite_2_in_1_ga₂(x₁, x₂)) = 1 + x₁ + x₂
POL(op_2₂(x₁, x₂)) = 1 + x₁ + x₂
POL(rewrite_2_out_ga₁(x₁)) = x₁
POL(rewrite_2_in_ga₁(x₁)) = 1 + 2·x₁
POL(NORMAL_2_IN_GA₁(x₁)) = 1 + 2·x₁

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ RuleRemovalProof

                              ↳ QDP

                                ↳ DependencyGraphProof

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

NORMAL_2_IN_GA₁(F) -> IF_NORMAL_2_IN_1_GA₁(rewrite_2_in_ga₁(F))

The TRS R consists of the following rules:

if_rewrite_2_in_1_ga₂(A, rewrite_2_out_ga₁(L)) -> rewrite_2_out_ga₁(op_2₂(A, L))

The set Q consists of the following terms:

rewrite_2_in_ga₁(x0)
if_rewrite_2_in_1_ga₂(x0, x1)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_NORMAL_2_IN_1_GA₁, NORMAL_2_IN_GA₁}.
The approximation of the Dependency Graph contains 0 SCCs with 1 less node.