Left Termination proof of /tmp/tmpzT4rFu/subset-bf.pl

Left Termination of the query pattern subset(b,f) w.r.t. the given Prolog program could not be shown:

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

subset₂({}₀, underscore).
subset₂(.₂(X, Xs), Ys) :- member₂(X, Ys), subset₂(Xs, Ys).
member₂(X, .₂(X, underscore1)).
member₂(X, .₂(underscore2, Xs)) :- member₂(X, Xs).

With regard to the inferred argument filtering the predicates were used in the following modes:
subset₂: (b,f) (f,b)
member₂: (f,f) (f,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

subset_2_in_ga₂([]_0, underscore) -> subset_2_out_ga₂([]_0, underscore)
subset_2_in_ga₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_in_aa₂(X, Ys))
member_2_in_aa₂(X, ._2₂(X, underscore1)) -> member_2_out_aa₂(X, ._2₂(X, underscore1))
member_2_in_aa₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_aa₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
subset_2_in_ag₂([]_0, underscore) -> subset_2_out_ag₂([]_0, underscore)
subset_2_in_ag₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))
member_2_in_ag₂(X, ._2₂(X, underscore1)) -> member_2_out_ag₂(X, ._2₂(X, underscore1))
member_2_in_ag₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_ag₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ag₂(._2₂(X, Xs), Ys)
if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ga₂(._2₂(X, Xs), Ys)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

  ↳ PrologToPiTRSProof

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

subset_2_in_ga₂([]_0, underscore) -> subset_2_out_ga₂([]_0, underscore)
subset_2_in_ga₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_in_aa₂(X, Ys))
member_2_in_aa₂(X, ._2₂(X, underscore1)) -> member_2_out_aa₂(X, ._2₂(X, underscore1))
member_2_in_aa₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_aa₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
subset_2_in_ag₂([]_0, underscore) -> subset_2_out_ag₂([]_0, underscore)
subset_2_in_ag₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))
member_2_in_ag₂(X, ._2₂(X, underscore1)) -> member_2_out_ag₂(X, ._2₂(X, underscore1))
member_2_in_ag₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_ag₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ag₂(._2₂(X, Xs), Ys)
if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ga₂(._2₂(X, Xs), Ys)

SUBSET_2_IN_GA₂(._2₂(X, Xs), Ys) -> IF_SUBSET_2_IN_1_GA₄(X, Xs, Ys, member_2_in_aa₂(X, Ys))
SUBSET_2_IN_GA₂(._2₂(X, Xs), Ys) -> MEMBER_2_IN_AA₂(X, Ys)
MEMBER_2_IN_AA₂(X, ._2₂(underscore2, Xs)) -> IF_MEMBER_2_IN_1_AA₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
MEMBER_2_IN_AA₂(X, ._2₂(underscore2, Xs)) -> MEMBER_2_IN_AA₂(X, Xs)
IF_SUBSET_2_IN_1_GA₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> IF_SUBSET_2_IN_2_GA₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
IF_SUBSET_2_IN_1_GA₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> SUBSET_2_IN_AG₂(Xs, Ys)
SUBSET_2_IN_AG₂(._2₂(X, Xs), Ys) -> IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))
SUBSET_2_IN_AG₂(._2₂(X, Xs), Ys) -> MEMBER_2_IN_AG₂(X, Ys)
MEMBER_2_IN_AG₂(X, ._2₂(underscore2, Xs)) -> IF_MEMBER_2_IN_1_AG₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
MEMBER_2_IN_AG₂(X, ._2₂(underscore2, Xs)) -> MEMBER_2_IN_AA₂(X, Xs)
IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> IF_SUBSET_2_IN_2_AG₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> SUBSET_2_IN_AG₂(Xs, Ys)

The TRS R consists of the following rules:

subset_2_in_ga₂([]_0, underscore) -> subset_2_out_ga₂([]_0, underscore)
subset_2_in_ga₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_in_aa₂(X, Ys))
member_2_in_aa₂(X, ._2₂(X, underscore1)) -> member_2_out_aa₂(X, ._2₂(X, underscore1))
member_2_in_aa₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_aa₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
subset_2_in_ag₂([]_0, underscore) -> subset_2_out_ag₂([]_0, underscore)
subset_2_in_ag₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))
member_2_in_ag₂(X, ._2₂(X, underscore1)) -> member_2_out_ag₂(X, ._2₂(X, underscore1))
member_2_in_ag₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_ag₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ag₂(._2₂(X, Xs), Ys)
if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ga₂(._2₂(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_2_in_ga₂(x1, x2) = subset_2_in_ga₁(x1)
[]_0 = []_0
._2₂(x1, x2) = ._2
subset_2_out_ga₂(x1, x2) = subset_2_out_ga
if_subset_2_in_1_ga₄(x1, x2, x3, x4) = if_subset_2_in_1_ga₁(x4)
member_2_in_aa₂(x1, x2) = member_2_in_aa
member_2_out_aa₂(x1, x2) = member_2_out_aa₁(x2)
if_member_2_in_1_aa₄(x1, x2, x3, x4) = if_member_2_in_1_aa₁(x4)
if_subset_2_in_2_ga₄(x1, x2, x3, x4) = if_subset_2_in_2_ga₁(x4)
subset_2_in_ag₂(x1, x2) = subset_2_in_ag₁(x2)
subset_2_out_ag₂(x1, x2) = subset_2_out_ag₁(x1)
if_subset_2_in_1_ag₄(x1, x2, x3, x4) = if_subset_2_in_1_ag₂(x3, x4)
member_2_in_ag₂(x1, x2) = member_2_in_ag₁(x2)
member_2_out_ag₂(x1, x2) = member_2_out_ag
if_member_2_in_1_ag₄(x1, x2, x3, x4) = if_member_2_in_1_ag₁(x4)
if_subset_2_in_2_ag₄(x1, x2, x3, x4) = if_subset_2_in_2_ag₁(x4)
IF_SUBSET_2_IN_2_AG₄(x1, x2, x3, x4) = IF_SUBSET_2_IN_2_AG₁(x4)
IF_MEMBER_2_IN_1_AG₄(x1, x2, x3, x4) = IF_MEMBER_2_IN_1_AG₁(x4)
MEMBER_2_IN_AA₂(x1, x2) = MEMBER_2_IN_AA
IF_SUBSET_2_IN_1_GA₄(x1, x2, x3, x4) = IF_SUBSET_2_IN_1_GA₁(x4)
SUBSET_2_IN_GA₂(x1, x2) = SUBSET_2_IN_GA₁(x1)
MEMBER_2_IN_AG₂(x1, x2) = MEMBER_2_IN_AG₁(x2)
IF_SUBSET_2_IN_1_AG₄(x1, x2, x3, x4) = IF_SUBSET_2_IN_1_AG₂(x3, x4)
IF_SUBSET_2_IN_2_GA₄(x1, x2, x3, x4) = IF_SUBSET_2_IN_2_GA₁(x4)
IF_MEMBER_2_IN_1_AA₄(x1, x2, x3, x4) = IF_MEMBER_2_IN_1_AA₁(x4)
SUBSET_2_IN_AG₂(x1, x2) = SUBSET_2_IN_AG₁(x2)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

SUBSET_2_IN_GA₂(._2₂(X, Xs), Ys) -> IF_SUBSET_2_IN_1_GA₄(X, Xs, Ys, member_2_in_aa₂(X, Ys))
SUBSET_2_IN_GA₂(._2₂(X, Xs), Ys) -> MEMBER_2_IN_AA₂(X, Ys)
MEMBER_2_IN_AA₂(X, ._2₂(underscore2, Xs)) -> IF_MEMBER_2_IN_1_AA₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
MEMBER_2_IN_AA₂(X, ._2₂(underscore2, Xs)) -> MEMBER_2_IN_AA₂(X, Xs)
IF_SUBSET_2_IN_1_GA₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> IF_SUBSET_2_IN_2_GA₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
IF_SUBSET_2_IN_1_GA₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> SUBSET_2_IN_AG₂(Xs, Ys)
SUBSET_2_IN_AG₂(._2₂(X, Xs), Ys) -> IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))
SUBSET_2_IN_AG₂(._2₂(X, Xs), Ys) -> MEMBER_2_IN_AG₂(X, Ys)
MEMBER_2_IN_AG₂(X, ._2₂(underscore2, Xs)) -> IF_MEMBER_2_IN_1_AG₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
MEMBER_2_IN_AG₂(X, ._2₂(underscore2, Xs)) -> MEMBER_2_IN_AA₂(X, Xs)
IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> IF_SUBSET_2_IN_2_AG₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> SUBSET_2_IN_AG₂(Xs, Ys)

The TRS R consists of the following rules:

subset_2_in_ga₂([]_0, underscore) -> subset_2_out_ga₂([]_0, underscore)
subset_2_in_ga₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_in_aa₂(X, Ys))
member_2_in_aa₂(X, ._2₂(X, underscore1)) -> member_2_out_aa₂(X, ._2₂(X, underscore1))
member_2_in_aa₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_aa₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
subset_2_in_ag₂([]_0, underscore) -> subset_2_out_ag₂([]_0, underscore)
subset_2_in_ag₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))
member_2_in_ag₂(X, ._2₂(X, underscore1)) -> member_2_out_ag₂(X, ._2₂(X, underscore1))
member_2_in_ag₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_ag₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ag₂(._2₂(X, Xs), Ys)
if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ga₂(._2₂(X, Xs), Ys)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

MEMBER_2_IN_AA₂(X, ._2₂(underscore2, Xs)) -> MEMBER_2_IN_AA₂(X, Xs)

The TRS R consists of the following rules:

subset_2_in_ga₂([]_0, underscore) -> subset_2_out_ga₂([]_0, underscore)
subset_2_in_ga₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_in_aa₂(X, Ys))
member_2_in_aa₂(X, ._2₂(X, underscore1)) -> member_2_out_aa₂(X, ._2₂(X, underscore1))
member_2_in_aa₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_aa₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
subset_2_in_ag₂([]_0, underscore) -> subset_2_out_ag₂([]_0, underscore)
subset_2_in_ag₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))
member_2_in_ag₂(X, ._2₂(X, underscore1)) -> member_2_out_ag₂(X, ._2₂(X, underscore1))
member_2_in_ag₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_ag₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ag₂(._2₂(X, Xs), Ys)
if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ga₂(._2₂(X, Xs), Ys)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

MEMBER_2_IN_AA₂(X, ._2₂(underscore2, Xs)) -> MEMBER_2_IN_AA₂(X, Xs)

R is empty.
The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2
MEMBER_2_IN_AA₂(x1, x2) = MEMBER_2_IN_AA

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

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

MEMBER_2_IN_AA -> MEMBER_2_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {MEMBER_2_IN_AA}.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> SUBSET_2_IN_AG₂(Xs, Ys)
SUBSET_2_IN_AG₂(._2₂(X, Xs), Ys) -> IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))

The TRS R consists of the following rules:

subset_2_in_ga₂([]_0, underscore) -> subset_2_out_ga₂([]_0, underscore)
subset_2_in_ga₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_in_aa₂(X, Ys))
member_2_in_aa₂(X, ._2₂(X, underscore1)) -> member_2_out_aa₂(X, ._2₂(X, underscore1))
member_2_in_aa₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_aa₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
subset_2_in_ag₂([]_0, underscore) -> subset_2_out_ag₂([]_0, underscore)
subset_2_in_ag₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))
member_2_in_ag₂(X, ._2₂(X, underscore1)) -> member_2_out_ag₂(X, ._2₂(X, underscore1))
member_2_in_ag₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_ag₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ag₂(._2₂(X, Xs), Ys)
if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ga₂(._2₂(X, Xs), Ys)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

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

IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> SUBSET_2_IN_AG₂(Xs, Ys)
SUBSET_2_IN_AG₂(._2₂(X, Xs), Ys) -> IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))

The TRS R consists of the following rules:

member_2_in_ag₂(X, ._2₂(X, underscore1)) -> member_2_out_ag₂(X, ._2₂(X, underscore1))
member_2_in_ag₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_ag₂(X, ._2₂(underscore2, Xs))
member_2_in_aa₂(X, ._2₂(X, underscore1)) -> member_2_out_aa₂(X, ._2₂(X, underscore1))
member_2_in_aa₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_aa₂(X, ._2₂(underscore2, Xs))

The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2
member_2_in_aa₂(x1, x2) = member_2_in_aa
member_2_out_aa₂(x1, x2) = member_2_out_aa₁(x2)
if_member_2_in_1_aa₄(x1, x2, x3, x4) = if_member_2_in_1_aa₁(x4)
member_2_in_ag₂(x1, x2) = member_2_in_ag₁(x2)
member_2_out_ag₂(x1, x2) = member_2_out_ag
if_member_2_in_1_ag₄(x1, x2, x3, x4) = if_member_2_in_1_ag₁(x4)
IF_SUBSET_2_IN_1_AG₄(x1, x2, x3, x4) = IF_SUBSET_2_IN_1_AG₂(x3, x4)
SUBSET_2_IN_AG₂(x1, x2) = SUBSET_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

  ↳ PrologToPiTRSProof

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

IF_SUBSET_2_IN_1_AG₂(Ys, member_2_out_ag) -> SUBSET_2_IN_AG₁(Ys)
SUBSET_2_IN_AG₁(Ys) -> IF_SUBSET_2_IN_1_AG₂(Ys, member_2_in_ag₁(Ys))

The TRS R consists of the following rules:

member_2_in_ag₁(._2) -> member_2_out_ag
member_2_in_ag₁(._2) -> if_member_2_in_1_ag₁(member_2_in_aa)
if_member_2_in_1_ag₁(member_2_out_aa₁(Xs)) -> member_2_out_ag
member_2_in_aa -> member_2_out_aa₁(._2)
member_2_in_aa -> if_member_2_in_1_aa₁(member_2_in_aa)
if_member_2_in_1_aa₁(member_2_out_aa₁(Xs)) -> member_2_out_aa₁(._2)

The set Q consists of the following terms:

member_2_in_ag₁(x0)
if_member_2_in_1_ag₁(x0)
member_2_in_aa
if_member_2_in_1_aa₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {SUBSET_2_IN_AG₁, IF_SUBSET_2_IN_1_AG₂}.
With regard to the inferred argument filtering the predicates were used in the following modes:
subset₂: (b,f) (f,b)
member₂: (f,f) (f,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

subset_2_in_ga₂([]_0, underscore) -> subset_2_out_ga₂([]_0, underscore)
subset_2_in_ga₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_in_aa₂(X, Ys))
member_2_in_aa₂(X, ._2₂(X, underscore1)) -> member_2_out_aa₂(X, ._2₂(X, underscore1))
member_2_in_aa₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_aa₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
subset_2_in_ag₂([]_0, underscore) -> subset_2_out_ag₂([]_0, underscore)
subset_2_in_ag₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))
member_2_in_ag₂(X, ._2₂(X, underscore1)) -> member_2_out_ag₂(X, ._2₂(X, underscore1))
member_2_in_ag₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_ag₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ag₂(._2₂(X, Xs), Ys)
if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ga₂(._2₂(X, Xs), Ys)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

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

subset_2_in_ga₂([]_0, underscore) -> subset_2_out_ga₂([]_0, underscore)
subset_2_in_ga₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_in_aa₂(X, Ys))
member_2_in_aa₂(X, ._2₂(X, underscore1)) -> member_2_out_aa₂(X, ._2₂(X, underscore1))
member_2_in_aa₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_aa₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
subset_2_in_ag₂([]_0, underscore) -> subset_2_out_ag₂([]_0, underscore)
subset_2_in_ag₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))
member_2_in_ag₂(X, ._2₂(X, underscore1)) -> member_2_out_ag₂(X, ._2₂(X, underscore1))
member_2_in_ag₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_ag₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ag₂(._2₂(X, Xs), Ys)
if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ga₂(._2₂(X, Xs), Ys)

SUBSET_2_IN_GA₂(._2₂(X, Xs), Ys) -> IF_SUBSET_2_IN_1_GA₄(X, Xs, Ys, member_2_in_aa₂(X, Ys))
SUBSET_2_IN_GA₂(._2₂(X, Xs), Ys) -> MEMBER_2_IN_AA₂(X, Ys)
MEMBER_2_IN_AA₂(X, ._2₂(underscore2, Xs)) -> IF_MEMBER_2_IN_1_AA₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
MEMBER_2_IN_AA₂(X, ._2₂(underscore2, Xs)) -> MEMBER_2_IN_AA₂(X, Xs)
IF_SUBSET_2_IN_1_GA₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> IF_SUBSET_2_IN_2_GA₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
IF_SUBSET_2_IN_1_GA₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> SUBSET_2_IN_AG₂(Xs, Ys)
SUBSET_2_IN_AG₂(._2₂(X, Xs), Ys) -> IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))
SUBSET_2_IN_AG₂(._2₂(X, Xs), Ys) -> MEMBER_2_IN_AG₂(X, Ys)
MEMBER_2_IN_AG₂(X, ._2₂(underscore2, Xs)) -> IF_MEMBER_2_IN_1_AG₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
MEMBER_2_IN_AG₂(X, ._2₂(underscore2, Xs)) -> MEMBER_2_IN_AA₂(X, Xs)
IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> IF_SUBSET_2_IN_2_AG₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> SUBSET_2_IN_AG₂(Xs, Ys)

The TRS R consists of the following rules:

subset_2_in_ga₂([]_0, underscore) -> subset_2_out_ga₂([]_0, underscore)
subset_2_in_ga₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_in_aa₂(X, Ys))
member_2_in_aa₂(X, ._2₂(X, underscore1)) -> member_2_out_aa₂(X, ._2₂(X, underscore1))
member_2_in_aa₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_aa₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
subset_2_in_ag₂([]_0, underscore) -> subset_2_out_ag₂([]_0, underscore)
subset_2_in_ag₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))
member_2_in_ag₂(X, ._2₂(X, underscore1)) -> member_2_out_ag₂(X, ._2₂(X, underscore1))
member_2_in_ag₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_ag₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ag₂(._2₂(X, Xs), Ys)
if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ga₂(._2₂(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_2_in_ga₂(x1, x2) = subset_2_in_ga₁(x1)
[]_0 = []_0
._2₂(x1, x2) = ._2
subset_2_out_ga₂(x1, x2) = subset_2_out_ga₁(x1)
if_subset_2_in_1_ga₄(x1, x2, x3, x4) = if_subset_2_in_1_ga₁(x4)
member_2_in_aa₂(x1, x2) = member_2_in_aa
member_2_out_aa₂(x1, x2) = member_2_out_aa₁(x2)
if_member_2_in_1_aa₄(x1, x2, x3, x4) = if_member_2_in_1_aa₁(x4)
if_subset_2_in_2_ga₄(x1, x2, x3, x4) = if_subset_2_in_2_ga₁(x4)
subset_2_in_ag₂(x1, x2) = subset_2_in_ag₁(x2)
subset_2_out_ag₂(x1, x2) = subset_2_out_ag₂(x1, x2)
if_subset_2_in_1_ag₄(x1, x2, x3, x4) = if_subset_2_in_1_ag₂(x3, x4)
member_2_in_ag₂(x1, x2) = member_2_in_ag₁(x2)
member_2_out_ag₂(x1, x2) = member_2_out_ag₁(x2)
if_member_2_in_1_ag₄(x1, x2, x3, x4) = if_member_2_in_1_ag₁(x4)
if_subset_2_in_2_ag₄(x1, x2, x3, x4) = if_subset_2_in_2_ag₂(x3, x4)
IF_SUBSET_2_IN_2_AG₄(x1, x2, x3, x4) = IF_SUBSET_2_IN_2_AG₂(x3, x4)
IF_MEMBER_2_IN_1_AG₄(x1, x2, x3, x4) = IF_MEMBER_2_IN_1_AG₁(x4)
MEMBER_2_IN_AA₂(x1, x2) = MEMBER_2_IN_AA
IF_SUBSET_2_IN_1_GA₄(x1, x2, x3, x4) = IF_SUBSET_2_IN_1_GA₁(x4)
SUBSET_2_IN_GA₂(x1, x2) = SUBSET_2_IN_GA₁(x1)
MEMBER_2_IN_AG₂(x1, x2) = MEMBER_2_IN_AG₁(x2)
IF_SUBSET_2_IN_1_AG₄(x1, x2, x3, x4) = IF_SUBSET_2_IN_1_AG₂(x3, x4)
IF_SUBSET_2_IN_2_GA₄(x1, x2, x3, x4) = IF_SUBSET_2_IN_2_GA₁(x4)
IF_MEMBER_2_IN_1_AA₄(x1, x2, x3, x4) = IF_MEMBER_2_IN_1_AA₁(x4)
SUBSET_2_IN_AG₂(x1, x2) = SUBSET_2_IN_AG₁(x2)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

SUBSET_2_IN_GA₂(._2₂(X, Xs), Ys) -> IF_SUBSET_2_IN_1_GA₄(X, Xs, Ys, member_2_in_aa₂(X, Ys))
SUBSET_2_IN_GA₂(._2₂(X, Xs), Ys) -> MEMBER_2_IN_AA₂(X, Ys)
MEMBER_2_IN_AA₂(X, ._2₂(underscore2, Xs)) -> IF_MEMBER_2_IN_1_AA₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
MEMBER_2_IN_AA₂(X, ._2₂(underscore2, Xs)) -> MEMBER_2_IN_AA₂(X, Xs)
IF_SUBSET_2_IN_1_GA₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> IF_SUBSET_2_IN_2_GA₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
IF_SUBSET_2_IN_1_GA₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> SUBSET_2_IN_AG₂(Xs, Ys)
SUBSET_2_IN_AG₂(._2₂(X, Xs), Ys) -> IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))
SUBSET_2_IN_AG₂(._2₂(X, Xs), Ys) -> MEMBER_2_IN_AG₂(X, Ys)
MEMBER_2_IN_AG₂(X, ._2₂(underscore2, Xs)) -> IF_MEMBER_2_IN_1_AG₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
MEMBER_2_IN_AG₂(X, ._2₂(underscore2, Xs)) -> MEMBER_2_IN_AA₂(X, Xs)
IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> IF_SUBSET_2_IN_2_AG₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> SUBSET_2_IN_AG₂(Xs, Ys)

The TRS R consists of the following rules:

subset_2_in_ga₂([]_0, underscore) -> subset_2_out_ga₂([]_0, underscore)
subset_2_in_ga₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_in_aa₂(X, Ys))
member_2_in_aa₂(X, ._2₂(X, underscore1)) -> member_2_out_aa₂(X, ._2₂(X, underscore1))
member_2_in_aa₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_aa₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
subset_2_in_ag₂([]_0, underscore) -> subset_2_out_ag₂([]_0, underscore)
subset_2_in_ag₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))
member_2_in_ag₂(X, ._2₂(X, underscore1)) -> member_2_out_ag₂(X, ._2₂(X, underscore1))
member_2_in_ag₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_ag₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ag₂(._2₂(X, Xs), Ys)
if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ga₂(._2₂(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_2_in_ga₂(x1, x2) = subset_2_in_ga₁(x1)
[]_0 = []_0
._2₂(x1, x2) = ._2
subset_2_out_ga₂(x1, x2) = subset_2_out_ga₁(x1)
if_subset_2_in_1_ga₄(x1, x2, x3, x4) = if_subset_2_in_1_ga₁(x4)
member_2_in_aa₂(x1, x2) = member_2_in_aa
member_2_out_aa₂(x1, x2) = member_2_out_aa₁(x2)
if_member_2_in_1_aa₄(x1, x2, x3, x4) = if_member_2_in_1_aa₁(x4)
if_subset_2_in_2_ga₄(x1, x2, x3, x4) = if_subset_2_in_2_ga₁(x4)
subset_2_in_ag₂(x1, x2) = subset_2_in_ag₁(x2)
subset_2_out_ag₂(x1, x2) = subset_2_out_ag₂(x1, x2)
if_subset_2_in_1_ag₄(x1, x2, x3, x4) = if_subset_2_in_1_ag₂(x3, x4)
member_2_in_ag₂(x1, x2) = member_2_in_ag₁(x2)
member_2_out_ag₂(x1, x2) = member_2_out_ag₁(x2)
if_member_2_in_1_ag₄(x1, x2, x3, x4) = if_member_2_in_1_ag₁(x4)
if_subset_2_in_2_ag₄(x1, x2, x3, x4) = if_subset_2_in_2_ag₂(x3, x4)
IF_SUBSET_2_IN_2_AG₄(x1, x2, x3, x4) = IF_SUBSET_2_IN_2_AG₂(x3, x4)
IF_MEMBER_2_IN_1_AG₄(x1, x2, x3, x4) = IF_MEMBER_2_IN_1_AG₁(x4)
MEMBER_2_IN_AA₂(x1, x2) = MEMBER_2_IN_AA
IF_SUBSET_2_IN_1_GA₄(x1, x2, x3, x4) = IF_SUBSET_2_IN_1_GA₁(x4)
SUBSET_2_IN_GA₂(x1, x2) = SUBSET_2_IN_GA₁(x1)
MEMBER_2_IN_AG₂(x1, x2) = MEMBER_2_IN_AG₁(x2)
IF_SUBSET_2_IN_1_AG₄(x1, x2, x3, x4) = IF_SUBSET_2_IN_1_AG₂(x3, x4)
IF_SUBSET_2_IN_2_GA₄(x1, x2, x3, x4) = IF_SUBSET_2_IN_2_GA₁(x4)
IF_MEMBER_2_IN_1_AA₄(x1, x2, x3, x4) = IF_MEMBER_2_IN_1_AA₁(x4)
SUBSET_2_IN_AG₂(x1, x2) = SUBSET_2_IN_AG₁(x2)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 2 SCCs with 9 less nodes.

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

MEMBER_2_IN_AA₂(X, ._2₂(underscore2, Xs)) -> MEMBER_2_IN_AA₂(X, Xs)

The TRS R consists of the following rules:

subset_2_in_ga₂([]_0, underscore) -> subset_2_out_ga₂([]_0, underscore)
subset_2_in_ga₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_in_aa₂(X, Ys))
member_2_in_aa₂(X, ._2₂(X, underscore1)) -> member_2_out_aa₂(X, ._2₂(X, underscore1))
member_2_in_aa₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_aa₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
subset_2_in_ag₂([]_0, underscore) -> subset_2_out_ag₂([]_0, underscore)
subset_2_in_ag₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))
member_2_in_ag₂(X, ._2₂(X, underscore1)) -> member_2_out_ag₂(X, ._2₂(X, underscore1))
member_2_in_ag₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_ag₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ag₂(._2₂(X, Xs), Ys)
if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ga₂(._2₂(X, Xs), Ys)

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

              ↳ PiDP

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

MEMBER_2_IN_AA₂(X, ._2₂(underscore2, Xs)) -> MEMBER_2_IN_AA₂(X, Xs)

R is empty.
The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2
MEMBER_2_IN_AA₂(x1, x2) = MEMBER_2_IN_AA

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

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> SUBSET_2_IN_AG₂(Xs, Ys)
SUBSET_2_IN_AG₂(._2₂(X, Xs), Ys) -> IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))

The TRS R consists of the following rules:

subset_2_in_ga₂([]_0, underscore) -> subset_2_out_ga₂([]_0, underscore)
subset_2_in_ga₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_in_aa₂(X, Ys))
member_2_in_aa₂(X, ._2₂(X, underscore1)) -> member_2_out_aa₂(X, ._2₂(X, underscore1))
member_2_in_aa₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_aa₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ga₄(X, Xs, Ys, member_2_out_aa₂(X, Ys)) -> if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
subset_2_in_ag₂([]_0, underscore) -> subset_2_out_ag₂([]_0, underscore)
subset_2_in_ag₂(._2₂(X, Xs), Ys) -> if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))
member_2_in_ag₂(X, ._2₂(X, underscore1)) -> member_2_out_ag₂(X, ._2₂(X, underscore1))
member_2_in_ag₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_ag₂(X, ._2₂(underscore2, Xs))
if_subset_2_in_1_ag₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_in_ag₂(Xs, Ys))
if_subset_2_in_2_ag₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ag₂(._2₂(X, Xs), Ys)
if_subset_2_in_2_ga₄(X, Xs, Ys, subset_2_out_ag₂(Xs, Ys)) -> subset_2_out_ga₂(._2₂(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
subset_2_in_ga₂(x1, x2) = subset_2_in_ga₁(x1)
[]_0 = []_0
._2₂(x1, x2) = ._2
subset_2_out_ga₂(x1, x2) = subset_2_out_ga₁(x1)
if_subset_2_in_1_ga₄(x1, x2, x3, x4) = if_subset_2_in_1_ga₁(x4)
member_2_in_aa₂(x1, x2) = member_2_in_aa
member_2_out_aa₂(x1, x2) = member_2_out_aa₁(x2)
if_member_2_in_1_aa₄(x1, x2, x3, x4) = if_member_2_in_1_aa₁(x4)
if_subset_2_in_2_ga₄(x1, x2, x3, x4) = if_subset_2_in_2_ga₁(x4)
subset_2_in_ag₂(x1, x2) = subset_2_in_ag₁(x2)
subset_2_out_ag₂(x1, x2) = subset_2_out_ag₂(x1, x2)
if_subset_2_in_1_ag₄(x1, x2, x3, x4) = if_subset_2_in_1_ag₂(x3, x4)
member_2_in_ag₂(x1, x2) = member_2_in_ag₁(x2)
member_2_out_ag₂(x1, x2) = member_2_out_ag₁(x2)
if_member_2_in_1_ag₄(x1, x2, x3, x4) = if_member_2_in_1_ag₁(x4)
if_subset_2_in_2_ag₄(x1, x2, x3, x4) = if_subset_2_in_2_ag₂(x3, x4)
IF_SUBSET_2_IN_1_AG₄(x1, x2, x3, x4) = IF_SUBSET_2_IN_1_AG₂(x3, x4)
SUBSET_2_IN_AG₂(x1, x2) = SUBSET_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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

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

IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_out_ag₂(X, Ys)) -> SUBSET_2_IN_AG₂(Xs, Ys)
SUBSET_2_IN_AG₂(._2₂(X, Xs), Ys) -> IF_SUBSET_2_IN_1_AG₄(X, Xs, Ys, member_2_in_ag₂(X, Ys))

The TRS R consists of the following rules:

member_2_in_ag₂(X, ._2₂(X, underscore1)) -> member_2_out_ag₂(X, ._2₂(X, underscore1))
member_2_in_ag₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_ag₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_ag₂(X, ._2₂(underscore2, Xs))
member_2_in_aa₂(X, ._2₂(X, underscore1)) -> member_2_out_aa₂(X, ._2₂(X, underscore1))
member_2_in_aa₂(X, ._2₂(underscore2, Xs)) -> if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_in_aa₂(X, Xs))
if_member_2_in_1_aa₄(X, underscore2, Xs, member_2_out_aa₂(X, Xs)) -> member_2_out_aa₂(X, ._2₂(underscore2, Xs))

The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2
member_2_in_aa₂(x1, x2) = member_2_in_aa
member_2_out_aa₂(x1, x2) = member_2_out_aa₁(x2)
if_member_2_in_1_aa₄(x1, x2, x3, x4) = if_member_2_in_1_aa₁(x4)
member_2_in_ag₂(x1, x2) = member_2_in_ag₁(x2)
member_2_out_ag₂(x1, x2) = member_2_out_ag₁(x2)
if_member_2_in_1_ag₄(x1, x2, x3, x4) = if_member_2_in_1_ag₁(x4)
IF_SUBSET_2_IN_1_AG₄(x1, x2, x3, x4) = IF_SUBSET_2_IN_1_AG₂(x3, x4)
SUBSET_2_IN_AG₂(x1, x2) = SUBSET_2_IN_AG₁(x2)

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