Left Termination proof of /tmp/tmpdnu2yw/pl1.2

Left Termination of the query pattern perm_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

append2(parts([], Y), is(sum(Y))).
append2(parts(.(H, X), Y), is(sum(.(H, Z)))) :- append2(parts(X, Y), is(sum(Z))).
append1(parts([], Y), is(sum(Y))).
append1(parts(.(H, X), Y), is(sum(.(H, Z)))) :- append1(parts(X, Y), is(sum(Z))).
perm([], []).
perm(L, .(H, T)) :- ','(append2(parts(V, .(H, U)), is(sum(L))), ','(append1(parts(V, U), is(sum(W))), perm(W, T))).

Queries:

perm(g,a).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
perm_in: (b,f)
append2_in: (f,b)
append1_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
append2_in_ag(parts([], Y), is(sum(Y))) → append2_out_ag(parts([], Y), is(sum(Y)))
append2_in_ag(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_ag(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
U1_ag(H, X, Y, Z, append2_out_ag(parts(X, Y), is(sum(Z)))) → append2_out_ag(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
append1_in_ga(parts([], Y), is(sum(Y))) → append1_out_ga(parts([], Y), is(sum(Y)))
append1_in_ga(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_ga(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
U2_ga(H, X, Y, Z, append1_out_ga(parts(X, Y), is(sum(Z)))) → append1_out_ga(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_ga(W, T))
U5_ga(L, H, T, perm_out_ga(W, T)) → perm_out_ga(L, .(H, T))

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:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
append2_in_ag(parts([], Y), is(sum(Y))) → append2_out_ag(parts([], Y), is(sum(Y)))
append2_in_ag(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_ag(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
U1_ag(H, X, Y, Z, append2_out_ag(parts(X, Y), is(sum(Z)))) → append2_out_ag(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
append1_in_ga(parts([], Y), is(sum(Y))) → append1_out_ga(parts([], Y), is(sum(Y)))
append1_in_ga(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_ga(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
U2_ga(H, X, Y, Z, append1_out_ga(parts(X, Y), is(sum(Z)))) → append1_out_ga(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_ga(W, T))
U5_ga(L, H, T, perm_out_ga(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2) = perm_in_ga(x1)
[] = []
perm_out_ga(x1, x2) = perm_out_ga(x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
append2_in_ag(x1, x2) = append2_in_ag(x2)
.(x1, x2) = .(x2)
is(x1) = is(x1)
sum(x1) = sum(x1)
append2_out_ag(x1, x2) = append2_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x5)
parts(x1, x2) = parts(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6)
append1_in_ga(x1, x2) = append1_in_ga(x1)
append1_out_ga(x1, x2) = append1_out_ga(x2)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
U5_ga(x1, x2, x3, x4) = U5_ga(x4)

Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

PERM_IN_GA(L, .(H, T)) → U3_GA(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
PERM_IN_GA(L, .(H, T)) → APPEND2_IN_AG(parts(V, .(H, U)), is(sum(L)))
APPEND2_IN_AG(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_AG(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
APPEND2_IN_AG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND2_IN_AG(parts(X, Y), is(sum(Z)))
U3_GA(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) → U4_GA(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
U3_GA(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) → APPEND1_IN_GA(parts(V, U), is(sum(W)))
APPEND1_IN_GA(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_GA(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
APPEND1_IN_GA(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND1_IN_GA(parts(X, Y), is(sum(Z)))
U4_GA(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) → U5_GA(L, H, T, perm_in_ga(W, T))
U4_GA(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) → PERM_IN_GA(W, T)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
append2_in_ag(parts([], Y), is(sum(Y))) → append2_out_ag(parts([], Y), is(sum(Y)))
append2_in_ag(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_ag(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
U1_ag(H, X, Y, Z, append2_out_ag(parts(X, Y), is(sum(Z)))) → append2_out_ag(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
append1_in_ga(parts([], Y), is(sum(Y))) → append1_out_ga(parts([], Y), is(sum(Y)))
append1_in_ga(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_ga(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
U2_ga(H, X, Y, Z, append1_out_ga(parts(X, Y), is(sum(Z)))) → append1_out_ga(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_ga(W, T))
U5_ga(L, H, T, perm_out_ga(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2) = perm_in_ga(x1)
[] = []
perm_out_ga(x1, x2) = perm_out_ga(x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
append2_in_ag(x1, x2) = append2_in_ag(x2)
.(x1, x2) = .(x2)
is(x1) = is(x1)
sum(x1) = sum(x1)
append2_out_ag(x1, x2) = append2_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x5)
parts(x1, x2) = parts(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6)
append1_in_ga(x1, x2) = append1_in_ga(x1)
append1_out_ga(x1, x2) = append1_out_ga(x2)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
U5_ga(x1, x2, x3, x4) = U5_ga(x4)
U5_GA(x1, x2, x3, x4) = U5_GA(x4)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x5)
U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x6)
U3_GA(x1, x2, x3, x4) = U3_GA(x4)
APPEND1_IN_GA(x1, x2) = APPEND1_IN_GA(x1)
U1_AG(x1, x2, x3, x4, x5) = U1_AG(x5)
PERM_IN_GA(x1, x2) = PERM_IN_GA(x1)
APPEND2_IN_AG(x1, x2) = APPEND2_IN_AG(x2)

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:

PERM_IN_GA(L, .(H, T)) → U3_GA(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
PERM_IN_GA(L, .(H, T)) → APPEND2_IN_AG(parts(V, .(H, U)), is(sum(L)))
APPEND2_IN_AG(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_AG(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
APPEND2_IN_AG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND2_IN_AG(parts(X, Y), is(sum(Z)))
U3_GA(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) → U4_GA(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
U3_GA(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) → APPEND1_IN_GA(parts(V, U), is(sum(W)))
APPEND1_IN_GA(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_GA(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
APPEND1_IN_GA(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND1_IN_GA(parts(X, Y), is(sum(Z)))
U4_GA(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) → U5_GA(L, H, T, perm_in_ga(W, T))
U4_GA(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) → PERM_IN_GA(W, T)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
append2_in_ag(parts([], Y), is(sum(Y))) → append2_out_ag(parts([], Y), is(sum(Y)))
append2_in_ag(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_ag(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
U1_ag(H, X, Y, Z, append2_out_ag(parts(X, Y), is(sum(Z)))) → append2_out_ag(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
append1_in_ga(parts([], Y), is(sum(Y))) → append1_out_ga(parts([], Y), is(sum(Y)))
append1_in_ga(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_ga(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
U2_ga(H, X, Y, Z, append1_out_ga(parts(X, Y), is(sum(Z)))) → append1_out_ga(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_ga(W, T))
U5_ga(L, H, T, perm_out_ga(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2) = perm_in_ga(x1)
[] = []
perm_out_ga(x1, x2) = perm_out_ga(x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
append2_in_ag(x1, x2) = append2_in_ag(x2)
.(x1, x2) = .(x2)
is(x1) = is(x1)
sum(x1) = sum(x1)
append2_out_ag(x1, x2) = append2_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x5)
parts(x1, x2) = parts(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6)
append1_in_ga(x1, x2) = append1_in_ga(x1)
append1_out_ga(x1, x2) = append1_out_ga(x2)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
U5_ga(x1, x2, x3, x4) = U5_ga(x4)
U5_GA(x1, x2, x3, x4) = U5_GA(x4)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x5)
U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x6)
U3_GA(x1, x2, x3, x4) = U3_GA(x4)
APPEND1_IN_GA(x1, x2) = APPEND1_IN_GA(x1)
U1_AG(x1, x2, x3, x4, x5) = U1_AG(x5)
PERM_IN_GA(x1, x2) = PERM_IN_GA(x1)
APPEND2_IN_AG(x1, x2) = APPEND2_IN_AG(x2)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 5 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

APPEND1_IN_GA(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND1_IN_GA(parts(X, Y), is(sum(Z)))

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
append2_in_ag(parts([], Y), is(sum(Y))) → append2_out_ag(parts([], Y), is(sum(Y)))
append2_in_ag(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_ag(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
U1_ag(H, X, Y, Z, append2_out_ag(parts(X, Y), is(sum(Z)))) → append2_out_ag(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
append1_in_ga(parts([], Y), is(sum(Y))) → append1_out_ga(parts([], Y), is(sum(Y)))
append1_in_ga(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_ga(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
U2_ga(H, X, Y, Z, append1_out_ga(parts(X, Y), is(sum(Z)))) → append1_out_ga(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_ga(W, T))
U5_ga(L, H, T, perm_out_ga(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2) = perm_in_ga(x1)
[] = []
perm_out_ga(x1, x2) = perm_out_ga(x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
append2_in_ag(x1, x2) = append2_in_ag(x2)
.(x1, x2) = .(x2)
is(x1) = is(x1)
sum(x1) = sum(x1)
append2_out_ag(x1, x2) = append2_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x5)
parts(x1, x2) = parts(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6)
append1_in_ga(x1, x2) = append1_in_ga(x1)
append1_out_ga(x1, x2) = append1_out_ga(x2)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
U5_ga(x1, x2, x3, x4) = U5_ga(x4)
APPEND1_IN_GA(x1, x2) = APPEND1_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

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

APPEND1_IN_GA(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND1_IN_GA(parts(X, Y), is(sum(Z)))

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
is(x1) = is(x1)
sum(x1) = sum(x1)
parts(x1, x2) = parts(x1, x2)
APPEND1_IN_GA(x1, x2) = APPEND1_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ PiDP

              ↳ PiDP

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

APPEND1_IN_GA(parts(.(X), Y)) → APPEND1_IN_GA(parts(X, Y))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

APPEND1_IN_GA(parts(.(X), Y)) → APPEND1_IN_GA(parts(X, Y))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(.(x₁)) = 2·x₁
POL(APPEND1_IN_GA(x₁)) = 2·x₁
POL(parts(x₁, x₂)) = 2·x₁ + x₂

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ PiDP

              ↳ PiDP

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

APPEND2_IN_AG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND2_IN_AG(parts(X, Y), is(sum(Z)))

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
append2_in_ag(parts([], Y), is(sum(Y))) → append2_out_ag(parts([], Y), is(sum(Y)))
append2_in_ag(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_ag(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
U1_ag(H, X, Y, Z, append2_out_ag(parts(X, Y), is(sum(Z)))) → append2_out_ag(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
append1_in_ga(parts([], Y), is(sum(Y))) → append1_out_ga(parts([], Y), is(sum(Y)))
append1_in_ga(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_ga(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
U2_ga(H, X, Y, Z, append1_out_ga(parts(X, Y), is(sum(Z)))) → append1_out_ga(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_ga(W, T))
U5_ga(L, H, T, perm_out_ga(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2) = perm_in_ga(x1)
[] = []
perm_out_ga(x1, x2) = perm_out_ga(x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
append2_in_ag(x1, x2) = append2_in_ag(x2)
.(x1, x2) = .(x2)
is(x1) = is(x1)
sum(x1) = sum(x1)
append2_out_ag(x1, x2) = append2_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x5)
parts(x1, x2) = parts(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6)
append1_in_ga(x1, x2) = append1_in_ga(x1)
append1_out_ga(x1, x2) = append1_out_ga(x2)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
U5_ga(x1, x2, x3, x4) = U5_ga(x4)
APPEND2_IN_AG(x1, x2) = APPEND2_IN_AG(x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

APPEND2_IN_AG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND2_IN_AG(parts(X, Y), is(sum(Z)))

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
is(x1) = is(x1)
sum(x1) = sum(x1)
parts(x1, x2) = parts(x1, x2)
APPEND2_IN_AG(x1, x2) = APPEND2_IN_AG(x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ PiDP

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

APPEND2_IN_AG(is(sum(.(Z)))) → APPEND2_IN_AG(is(sum(Z)))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(.(x₁)) = 2·x₁
POL(APPEND2_IN_AG(x₁)) = 2·x₁
POL(is(x₁)) = 2·x₁
POL(sum(x₁)) = 2·x₁

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ PiDP

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

U3_GA(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) → U4_GA(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
PERM_IN_GA(L, .(H, T)) → U3_GA(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
U4_GA(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) → PERM_IN_GA(W, T)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
append2_in_ag(parts([], Y), is(sum(Y))) → append2_out_ag(parts([], Y), is(sum(Y)))
append2_in_ag(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_ag(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
U1_ag(H, X, Y, Z, append2_out_ag(parts(X, Y), is(sum(Z)))) → append2_out_ag(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
append1_in_ga(parts([], Y), is(sum(Y))) → append1_out_ga(parts([], Y), is(sum(Y)))
append1_in_ga(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_ga(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
U2_ga(H, X, Y, Z, append1_out_ga(parts(X, Y), is(sum(Z)))) → append1_out_ga(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_ga(W, T))
U5_ga(L, H, T, perm_out_ga(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2) = perm_in_ga(x1)
[] = []
perm_out_ga(x1, x2) = perm_out_ga(x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
append2_in_ag(x1, x2) = append2_in_ag(x2)
.(x1, x2) = .(x2)
is(x1) = is(x1)
sum(x1) = sum(x1)
append2_out_ag(x1, x2) = append2_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x5)
parts(x1, x2) = parts(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6) = U4_ga(x6)
append1_in_ga(x1, x2) = append1_in_ga(x1)
append1_out_ga(x1, x2) = append1_out_ga(x2)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
U5_ga(x1, x2, x3, x4) = U5_ga(x4)
U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x6)
U3_GA(x1, x2, x3, x4) = U3_GA(x4)
PERM_IN_GA(x1, x2) = PERM_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

U3_GA(L, H, T, append2_out_ag(parts(V, .(H, U)), is(sum(L)))) → U4_GA(L, H, T, V, U, append1_in_ga(parts(V, U), is(sum(W))))
PERM_IN_GA(L, .(H, T)) → U3_GA(L, H, T, append2_in_ag(parts(V, .(H, U)), is(sum(L))))
U4_GA(L, H, T, V, U, append1_out_ga(parts(V, U), is(sum(W)))) → PERM_IN_GA(W, T)

The TRS R consists of the following rules:

append1_in_ga(parts([], Y), is(sum(Y))) → append1_out_ga(parts([], Y), is(sum(Y)))
append1_in_ga(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_ga(H, X, Y, Z, append1_in_ga(parts(X, Y), is(sum(Z))))
append2_in_ag(parts([], Y), is(sum(Y))) → append2_out_ag(parts([], Y), is(sum(Y)))
append2_in_ag(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_ag(H, X, Y, Z, append2_in_ag(parts(X, Y), is(sum(Z))))
U2_ga(H, X, Y, Z, append1_out_ga(parts(X, Y), is(sum(Z)))) → append1_out_ga(parts(.(H, X), Y), is(sum(.(H, Z))))
U1_ag(H, X, Y, Z, append2_out_ag(parts(X, Y), is(sum(Z)))) → append2_out_ag(parts(.(H, X), Y), is(sum(.(H, Z))))

The argument filtering Pi contains the following mapping:
[] = []
append2_in_ag(x1, x2) = append2_in_ag(x2)
.(x1, x2) = .(x2)
is(x1) = is(x1)
sum(x1) = sum(x1)
append2_out_ag(x1, x2) = append2_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x5)
parts(x1, x2) = parts(x1, x2)
append1_in_ga(x1, x2) = append1_in_ga(x1)
append1_out_ga(x1, x2) = append1_out_ga(x2)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x6)
U3_GA(x1, x2, x3, x4) = U3_GA(x4)
PERM_IN_GA(x1, x2) = PERM_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

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

PERM_IN_GA(L) → U3_GA(append2_in_ag(is(sum(L))))
U3_GA(append2_out_ag(parts(V, .(U)))) → U4_GA(append1_in_ga(parts(V, U)))
U4_GA(append1_out_ga(is(sum(W)))) → PERM_IN_GA(W)

The TRS R consists of the following rules:

append1_in_ga(parts([], Y)) → append1_out_ga(is(sum(Y)))
append1_in_ga(parts(.(X), Y)) → U2_ga(append1_in_ga(parts(X, Y)))
append2_in_ag(is(sum(Y))) → append2_out_ag(parts([], Y))
append2_in_ag(is(sum(.(Z)))) → U1_ag(append2_in_ag(is(sum(Z))))
U2_ga(append1_out_ga(is(sum(Z)))) → append1_out_ga(is(sum(.(Z))))
U1_ag(append2_out_ag(parts(X, Y))) → append2_out_ag(parts(.(X), Y))

The set Q consists of the following terms:

append1_in_ga(x0)
append2_in_ag(x0)
U2_ga(x0)
U1_ag(x0)

We have to consider all (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

U3_GA(append2_out_ag(parts(V, .(U)))) → U4_GA(append1_in_ga(parts(V, U)))

Used ordering: POLO with Polynomial interpretation [25]:

POL(.(x₁)) = 1 + x₁
POL(PERM_IN_GA(x₁)) = 2·x₁
POL(U1_ag(x₁)) = 1 + x₁
POL(U2_ga(x₁)) = 1 + x₁
POL(U3_GA(x₁)) = 2·x₁
POL(U4_GA(x₁)) = 2·x₁
POL([]) = 0
POL(append1_in_ga(x₁)) = x₁
POL(append1_out_ga(x₁)) = x₁
POL(append2_in_ag(x₁)) = x₁
POL(append2_out_ag(x₁)) = x₁
POL(is(x₁)) = x₁
POL(parts(x₁, x₂)) = x₁ + x₂
POL(sum(x₁)) = x₁

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ DependencyGraphProof

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

PERM_IN_GA(L) → U3_GA(append2_in_ag(is(sum(L))))
U4_GA(append1_out_ga(is(sum(W)))) → PERM_IN_GA(W)

The TRS R consists of the following rules:

append1_in_ga(parts([], Y)) → append1_out_ga(is(sum(Y)))
append1_in_ga(parts(.(X), Y)) → U2_ga(append1_in_ga(parts(X, Y)))
append2_in_ag(is(sum(Y))) → append2_out_ag(parts([], Y))
append2_in_ag(is(sum(.(Z)))) → U1_ag(append2_in_ag(is(sum(Z))))
U2_ga(append1_out_ga(is(sum(Z)))) → append1_out_ga(is(sum(.(Z))))
U1_ag(append2_out_ag(parts(X, Y))) → append2_out_ag(parts(.(X), Y))

The set Q consists of the following terms:

append1_in_ga(x0)
append2_in_ag(x0)
U2_ga(x0)
U1_ag(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.