Left Termination proof of /tmp/tmpCtrkxY/bappend.pl

Left Termination of the query pattern goal_in_1(g) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

append([], XS, XS).
append(.(X, XS), YS, .(X, ZS)) :- append(XS, YS, ZS).
s2l(s(X), .(Y, Xs)) :- s2l(X, Xs).
s2l(0, []).
goal(X) :- ','(s2l(X, XS), append(XS, YS, ZS)).

Queries:

goal(g).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal_in: (b)
s2l_in: (b,f)
append_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_in_g(X) → U3_g(X, s2l_in_ga(X, XS))
s2l_in_ga(s(X), .(Y, Xs)) → U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
s2l_in_ga(0, []) → s2l_out_ga(0, [])
U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) → s2l_out_ga(s(X), .(Y, Xs))
U3_g(X, s2l_out_ga(X, XS)) → U4_g(X, append_in_gaa(XS, YS, ZS))
append_in_gaa([], XS, XS) → append_out_gaa([], XS, XS)
append_in_gaa(.(X, XS), YS, .(X, ZS)) → U1_gaa(X, XS, YS, ZS, append_in_gaa(XS, YS, ZS))
U1_gaa(X, XS, YS, ZS, append_out_gaa(XS, YS, ZS)) → append_out_gaa(.(X, XS), YS, .(X, ZS))
U4_g(X, append_out_gaa(XS, YS, ZS)) → goal_out_g(X)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

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

goal_in_g(X) → U3_g(X, s2l_in_ga(X, XS))
s2l_in_ga(s(X), .(Y, Xs)) → U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
s2l_in_ga(0, []) → s2l_out_ga(0, [])
U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) → s2l_out_ga(s(X), .(Y, Xs))
U3_g(X, s2l_out_ga(X, XS)) → U4_g(X, append_in_gaa(XS, YS, ZS))
append_in_gaa([], XS, XS) → append_out_gaa([], XS, XS)
append_in_gaa(.(X, XS), YS, .(X, ZS)) → U1_gaa(X, XS, YS, ZS, append_in_gaa(XS, YS, ZS))
U1_gaa(X, XS, YS, ZS, append_out_gaa(XS, YS, ZS)) → append_out_gaa(.(X, XS), YS, .(X, ZS))
U4_g(X, append_out_gaa(XS, YS, ZS)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1) = goal_in_g(x1)
U3_g(x1, x2) = U3_g(x2)
s2l_in_ga(x1, x2) = s2l_in_ga(x1)
s(x1) = s(x1)
U2_ga(x1, x2, x3, x4) = U2_ga(x4)
0 = 0
s2l_out_ga(x1, x2) = s2l_out_ga(x2)
.(x1, x2) = .(x2)
U4_g(x1, x2) = U4_g(x2)
append_in_gaa(x1, x2, x3) = append_in_gaa(x1)
[] = []
append_out_gaa(x1, x2, x3) = append_out_gaa
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5)
goal_out_g(x1) = goal_out_g

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

GOAL_IN_G(X) → U3_G(X, s2l_in_ga(X, XS))
GOAL_IN_G(X) → S2L_IN_GA(X, XS)
S2L_IN_GA(s(X), .(Y, Xs)) → U2_GA(X, Y, Xs, s2l_in_ga(X, Xs))
S2L_IN_GA(s(X), .(Y, Xs)) → S2L_IN_GA(X, Xs)
U3_G(X, s2l_out_ga(X, XS)) → U4_G(X, append_in_gaa(XS, YS, ZS))
U3_G(X, s2l_out_ga(X, XS)) → APPEND_IN_GAA(XS, YS, ZS)
APPEND_IN_GAA(.(X, XS), YS, .(X, ZS)) → U1_GAA(X, XS, YS, ZS, append_in_gaa(XS, YS, ZS))
APPEND_IN_GAA(.(X, XS), YS, .(X, ZS)) → APPEND_IN_GAA(XS, YS, ZS)

The TRS R consists of the following rules:

goal_in_g(X) → U3_g(X, s2l_in_ga(X, XS))
s2l_in_ga(s(X), .(Y, Xs)) → U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
s2l_in_ga(0, []) → s2l_out_ga(0, [])
U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) → s2l_out_ga(s(X), .(Y, Xs))
U3_g(X, s2l_out_ga(X, XS)) → U4_g(X, append_in_gaa(XS, YS, ZS))
append_in_gaa([], XS, XS) → append_out_gaa([], XS, XS)
append_in_gaa(.(X, XS), YS, .(X, ZS)) → U1_gaa(X, XS, YS, ZS, append_in_gaa(XS, YS, ZS))
U1_gaa(X, XS, YS, ZS, append_out_gaa(XS, YS, ZS)) → append_out_gaa(.(X, XS), YS, .(X, ZS))
U4_g(X, append_out_gaa(XS, YS, ZS)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1) = goal_in_g(x1)
U3_g(x1, x2) = U3_g(x2)
s2l_in_ga(x1, x2) = s2l_in_ga(x1)
s(x1) = s(x1)
U2_ga(x1, x2, x3, x4) = U2_ga(x4)
0 = 0
s2l_out_ga(x1, x2) = s2l_out_ga(x2)
.(x1, x2) = .(x2)
U4_g(x1, x2) = U4_g(x2)
append_in_gaa(x1, x2, x3) = append_in_gaa(x1)
[] = []
append_out_gaa(x1, x2, x3) = append_out_gaa
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5)
goal_out_g(x1) = goal_out_g
APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1)
U4_G(x1, x2) = U4_G(x2)
GOAL_IN_G(x1) = GOAL_IN_G(x1)
S2L_IN_GA(x1, x2) = S2L_IN_GA(x1)
U2_GA(x1, x2, x3, x4) = U2_GA(x4)
U3_G(x1, x2) = U3_G(x2)
U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x5)

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_IN_G(X) → U3_G(X, s2l_in_ga(X, XS))
GOAL_IN_G(X) → S2L_IN_GA(X, XS)
S2L_IN_GA(s(X), .(Y, Xs)) → U2_GA(X, Y, Xs, s2l_in_ga(X, Xs))
S2L_IN_GA(s(X), .(Y, Xs)) → S2L_IN_GA(X, Xs)
U3_G(X, s2l_out_ga(X, XS)) → U4_G(X, append_in_gaa(XS, YS, ZS))
U3_G(X, s2l_out_ga(X, XS)) → APPEND_IN_GAA(XS, YS, ZS)
APPEND_IN_GAA(.(X, XS), YS, .(X, ZS)) → U1_GAA(X, XS, YS, ZS, append_in_gaa(XS, YS, ZS))
APPEND_IN_GAA(.(X, XS), YS, .(X, ZS)) → APPEND_IN_GAA(XS, YS, ZS)

The TRS R consists of the following rules:

goal_in_g(X) → U3_g(X, s2l_in_ga(X, XS))
s2l_in_ga(s(X), .(Y, Xs)) → U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
s2l_in_ga(0, []) → s2l_out_ga(0, [])
U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) → s2l_out_ga(s(X), .(Y, Xs))
U3_g(X, s2l_out_ga(X, XS)) → U4_g(X, append_in_gaa(XS, YS, ZS))
append_in_gaa([], XS, XS) → append_out_gaa([], XS, XS)
append_in_gaa(.(X, XS), YS, .(X, ZS)) → U1_gaa(X, XS, YS, ZS, append_in_gaa(XS, YS, ZS))
U1_gaa(X, XS, YS, ZS, append_out_gaa(XS, YS, ZS)) → append_out_gaa(.(X, XS), YS, .(X, ZS))
U4_g(X, append_out_gaa(XS, YS, ZS)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1) = goal_in_g(x1)
U3_g(x1, x2) = U3_g(x2)
s2l_in_ga(x1, x2) = s2l_in_ga(x1)
s(x1) = s(x1)
U2_ga(x1, x2, x3, x4) = U2_ga(x4)
0 = 0
s2l_out_ga(x1, x2) = s2l_out_ga(x2)
.(x1, x2) = .(x2)
U4_g(x1, x2) = U4_g(x2)
append_in_gaa(x1, x2, x3) = append_in_gaa(x1)
[] = []
append_out_gaa(x1, x2, x3) = append_out_gaa
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5)
goal_out_g(x1) = goal_out_g
APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1)
U4_G(x1, x2) = U4_G(x2)
GOAL_IN_G(x1) = GOAL_IN_G(x1)
S2L_IN_GA(x1, x2) = S2L_IN_GA(x1)
U2_GA(x1, x2, x3, x4) = U2_GA(x4)
U3_G(x1, x2) = U3_G(x2)
U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 6 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

APPEND_IN_GAA(.(X, XS), YS, .(X, ZS)) → APPEND_IN_GAA(XS, YS, ZS)

The TRS R consists of the following rules:

goal_in_g(X) → U3_g(X, s2l_in_ga(X, XS))
s2l_in_ga(s(X), .(Y, Xs)) → U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
s2l_in_ga(0, []) → s2l_out_ga(0, [])
U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) → s2l_out_ga(s(X), .(Y, Xs))
U3_g(X, s2l_out_ga(X, XS)) → U4_g(X, append_in_gaa(XS, YS, ZS))
append_in_gaa([], XS, XS) → append_out_gaa([], XS, XS)
append_in_gaa(.(X, XS), YS, .(X, ZS)) → U1_gaa(X, XS, YS, ZS, append_in_gaa(XS, YS, ZS))
U1_gaa(X, XS, YS, ZS, append_out_gaa(XS, YS, ZS)) → append_out_gaa(.(X, XS), YS, .(X, ZS))
U4_g(X, append_out_gaa(XS, YS, ZS)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1) = goal_in_g(x1)
U3_g(x1, x2) = U3_g(x2)
s2l_in_ga(x1, x2) = s2l_in_ga(x1)
s(x1) = s(x1)
U2_ga(x1, x2, x3, x4) = U2_ga(x4)
0 = 0
s2l_out_ga(x1, x2) = s2l_out_ga(x2)
.(x1, x2) = .(x2)
U4_g(x1, x2) = U4_g(x2)
append_in_gaa(x1, x2, x3) = append_in_gaa(x1)
[] = []
append_out_gaa(x1, x2, x3) = append_out_gaa
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5)
goal_out_g(x1) = goal_out_g
APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(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

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

APPEND_IN_GAA(.(X, XS), YS, .(X, ZS)) → APPEND_IN_GAA(XS, YS, ZS)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(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

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

APPEND_IN_GAA(.(XS)) → APPEND_IN_GAA(XS)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:

APPEND_IN_GAA(.(XS)) → APPEND_IN_GAA(XS)
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:

S2L_IN_GA(s(X), .(Y, Xs)) → S2L_IN_GA(X, Xs)

The TRS R consists of the following rules:

goal_in_g(X) → U3_g(X, s2l_in_ga(X, XS))
s2l_in_ga(s(X), .(Y, Xs)) → U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
s2l_in_ga(0, []) → s2l_out_ga(0, [])
U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) → s2l_out_ga(s(X), .(Y, Xs))
U3_g(X, s2l_out_ga(X, XS)) → U4_g(X, append_in_gaa(XS, YS, ZS))
append_in_gaa([], XS, XS) → append_out_gaa([], XS, XS)
append_in_gaa(.(X, XS), YS, .(X, ZS)) → U1_gaa(X, XS, YS, ZS, append_in_gaa(XS, YS, ZS))
U1_gaa(X, XS, YS, ZS, append_out_gaa(XS, YS, ZS)) → append_out_gaa(.(X, XS), YS, .(X, ZS))
U4_g(X, append_out_gaa(XS, YS, ZS)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1) = goal_in_g(x1)
U3_g(x1, x2) = U3_g(x2)
s2l_in_ga(x1, x2) = s2l_in_ga(x1)
s(x1) = s(x1)
U2_ga(x1, x2, x3, x4) = U2_ga(x4)
0 = 0
s2l_out_ga(x1, x2) = s2l_out_ga(x2)
.(x1, x2) = .(x2)
U4_g(x1, x2) = U4_g(x2)
append_in_gaa(x1, x2, x3) = append_in_gaa(x1)
[] = []
append_out_gaa(x1, x2, x3) = append_out_gaa
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5)
goal_out_g(x1) = goal_out_g
S2L_IN_GA(x1, x2) = S2L_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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

S2L_IN_GA(s(X), .(Y, Xs)) → S2L_IN_GA(X, Xs)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
.(x1, x2) = .(x2)
S2L_IN_GA(x1, x2) = S2L_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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

S2L_IN_GA(s(X)) → S2L_IN_GA(X)

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

S2L_IN_GA(s(X)) → S2L_IN_GA(X)
The graph contains the following edges 1 > 1