Left Termination proof of /tmp/tmpovfIFN/intlist.pl

Left Termination of the query pattern int_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

intlist([], []).
intlist(.(X, XS), .(s(X), YS)) :- intlist(XS, YS).
int(0, 0, .(0, [])).
int(0, s(Y), .(0, XS)) :- int(s(0), s(Y), XS).
int(s(X), 0, []).
int(s(X), s(Y), XS) :- ','(int(X, Y, ZS), intlist(ZS, XS)).

Queries:

int(g,g,a).

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

int_in_gga(0, 0, .(0, [])) → int_out_gga(0, 0, .(0, []))
int_in_gga(0, s(Y), .(0, XS)) → U2_gga(Y, XS, int_in_gga(s(0), s(Y), XS))
int_in_gga(s(X), 0, []) → int_out_gga(s(X), 0, [])
int_in_gga(s(X), s(Y), XS) → U3_gga(X, Y, XS, int_in_gga(X, Y, ZS))
U3_gga(X, Y, XS, int_out_gga(X, Y, ZS)) → U4_gga(X, Y, XS, intlist_in_ga(ZS, XS))
intlist_in_ga([], []) → intlist_out_ga([], [])
intlist_in_ga(.(X, XS), .(s(X), YS)) → U1_ga(X, XS, YS, intlist_in_ga(XS, YS))
U1_ga(X, XS, YS, intlist_out_ga(XS, YS)) → intlist_out_ga(.(X, XS), .(s(X), YS))
U4_gga(X, Y, XS, intlist_out_ga(ZS, XS)) → int_out_gga(s(X), s(Y), XS)
U2_gga(Y, XS, int_out_gga(s(0), s(Y), XS)) → int_out_gga(0, s(Y), .(0, XS))

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:

int_in_gga(0, 0, .(0, [])) → int_out_gga(0, 0, .(0, []))
int_in_gga(0, s(Y), .(0, XS)) → U2_gga(Y, XS, int_in_gga(s(0), s(Y), XS))
int_in_gga(s(X), 0, []) → int_out_gga(s(X), 0, [])
int_in_gga(s(X), s(Y), XS) → U3_gga(X, Y, XS, int_in_gga(X, Y, ZS))
U3_gga(X, Y, XS, int_out_gga(X, Y, ZS)) → U4_gga(X, Y, XS, intlist_in_ga(ZS, XS))
intlist_in_ga([], []) → intlist_out_ga([], [])
intlist_in_ga(.(X, XS), .(s(X), YS)) → U1_ga(X, XS, YS, intlist_in_ga(XS, YS))
U1_ga(X, XS, YS, intlist_out_ga(XS, YS)) → intlist_out_ga(.(X, XS), .(s(X), YS))
U4_gga(X, Y, XS, intlist_out_ga(ZS, XS)) → int_out_gga(s(X), s(Y), XS)
U2_gga(Y, XS, int_out_gga(s(0), s(Y), XS)) → int_out_gga(0, s(Y), .(0, XS))

The argument filtering Pi contains the following mapping:
int_in_gga(x1, x2, x3) = int_in_gga(x1, x2)
0 = 0
int_out_gga(x1, x2, x3) = int_out_gga(x3)
s(x1) = s(x1)
U2_gga(x1, x2, x3) = U2_gga(x3)
U3_gga(x1, x2, x3, x4) = U3_gga(x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x4)
intlist_in_ga(x1, x2) = intlist_in_ga(x1)
[] = []
intlist_out_ga(x1, x2) = intlist_out_ga(x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, 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:

INT_IN_GGA(0, s(Y), .(0, XS)) → U2_GGA(Y, XS, int_in_gga(s(0), s(Y), XS))
INT_IN_GGA(0, s(Y), .(0, XS)) → INT_IN_GGA(s(0), s(Y), XS)
INT_IN_GGA(s(X), s(Y), XS) → U3_GGA(X, Y, XS, int_in_gga(X, Y, ZS))
INT_IN_GGA(s(X), s(Y), XS) → INT_IN_GGA(X, Y, ZS)
U3_GGA(X, Y, XS, int_out_gga(X, Y, ZS)) → U4_GGA(X, Y, XS, intlist_in_ga(ZS, XS))
U3_GGA(X, Y, XS, int_out_gga(X, Y, ZS)) → INTLIST_IN_GA(ZS, XS)
INTLIST_IN_GA(.(X, XS), .(s(X), YS)) → U1_GA(X, XS, YS, intlist_in_ga(XS, YS))
INTLIST_IN_GA(.(X, XS), .(s(X), YS)) → INTLIST_IN_GA(XS, YS)

The TRS R consists of the following rules:

int_in_gga(0, 0, .(0, [])) → int_out_gga(0, 0, .(0, []))
int_in_gga(0, s(Y), .(0, XS)) → U2_gga(Y, XS, int_in_gga(s(0), s(Y), XS))
int_in_gga(s(X), 0, []) → int_out_gga(s(X), 0, [])
int_in_gga(s(X), s(Y), XS) → U3_gga(X, Y, XS, int_in_gga(X, Y, ZS))
U3_gga(X, Y, XS, int_out_gga(X, Y, ZS)) → U4_gga(X, Y, XS, intlist_in_ga(ZS, XS))
intlist_in_ga([], []) → intlist_out_ga([], [])
intlist_in_ga(.(X, XS), .(s(X), YS)) → U1_ga(X, XS, YS, intlist_in_ga(XS, YS))
U1_ga(X, XS, YS, intlist_out_ga(XS, YS)) → intlist_out_ga(.(X, XS), .(s(X), YS))
U4_gga(X, Y, XS, intlist_out_ga(ZS, XS)) → int_out_gga(s(X), s(Y), XS)
U2_gga(Y, XS, int_out_gga(s(0), s(Y), XS)) → int_out_gga(0, s(Y), .(0, XS))

The argument filtering Pi contains the following mapping:
int_in_gga(x1, x2, x3) = int_in_gga(x1, x2)
0 = 0
int_out_gga(x1, x2, x3) = int_out_gga(x3)
s(x1) = s(x1)
U2_gga(x1, x2, x3) = U2_gga(x3)
U3_gga(x1, x2, x3, x4) = U3_gga(x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x4)
intlist_in_ga(x1, x2) = intlist_in_ga(x1)
[] = []
intlist_out_ga(x1, x2) = intlist_out_ga(x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
INT_IN_GGA(x1, x2, x3) = INT_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x4)
U2_GGA(x1, x2, x3) = U2_GGA(x3)
INTLIST_IN_GA(x1, x2) = INTLIST_IN_GA(x1)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x4)

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:

INT_IN_GGA(0, s(Y), .(0, XS)) → U2_GGA(Y, XS, int_in_gga(s(0), s(Y), XS))
INT_IN_GGA(0, s(Y), .(0, XS)) → INT_IN_GGA(s(0), s(Y), XS)
INT_IN_GGA(s(X), s(Y), XS) → U3_GGA(X, Y, XS, int_in_gga(X, Y, ZS))
INT_IN_GGA(s(X), s(Y), XS) → INT_IN_GGA(X, Y, ZS)
U3_GGA(X, Y, XS, int_out_gga(X, Y, ZS)) → U4_GGA(X, Y, XS, intlist_in_ga(ZS, XS))
U3_GGA(X, Y, XS, int_out_gga(X, Y, ZS)) → INTLIST_IN_GA(ZS, XS)
INTLIST_IN_GA(.(X, XS), .(s(X), YS)) → U1_GA(X, XS, YS, intlist_in_ga(XS, YS))
INTLIST_IN_GA(.(X, XS), .(s(X), YS)) → INTLIST_IN_GA(XS, YS)

The TRS R consists of the following rules:

int_in_gga(0, 0, .(0, [])) → int_out_gga(0, 0, .(0, []))
int_in_gga(0, s(Y), .(0, XS)) → U2_gga(Y, XS, int_in_gga(s(0), s(Y), XS))
int_in_gga(s(X), 0, []) → int_out_gga(s(X), 0, [])
int_in_gga(s(X), s(Y), XS) → U3_gga(X, Y, XS, int_in_gga(X, Y, ZS))
U3_gga(X, Y, XS, int_out_gga(X, Y, ZS)) → U4_gga(X, Y, XS, intlist_in_ga(ZS, XS))
intlist_in_ga([], []) → intlist_out_ga([], [])
intlist_in_ga(.(X, XS), .(s(X), YS)) → U1_ga(X, XS, YS, intlist_in_ga(XS, YS))
U1_ga(X, XS, YS, intlist_out_ga(XS, YS)) → intlist_out_ga(.(X, XS), .(s(X), YS))
U4_gga(X, Y, XS, intlist_out_ga(ZS, XS)) → int_out_gga(s(X), s(Y), XS)
U2_gga(Y, XS, int_out_gga(s(0), s(Y), XS)) → int_out_gga(0, s(Y), .(0, XS))

The argument filtering Pi contains the following mapping:
int_in_gga(x1, x2, x3) = int_in_gga(x1, x2)
0 = 0
int_out_gga(x1, x2, x3) = int_out_gga(x3)
s(x1) = s(x1)
U2_gga(x1, x2, x3) = U2_gga(x3)
U3_gga(x1, x2, x3, x4) = U3_gga(x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x4)
intlist_in_ga(x1, x2) = intlist_in_ga(x1)
[] = []
intlist_out_ga(x1, x2) = intlist_out_ga(x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
INT_IN_GGA(x1, x2, x3) = INT_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x4)
U2_GGA(x1, x2, x3) = U2_GGA(x3)
INTLIST_IN_GA(x1, x2) = INTLIST_IN_GA(x1)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 5 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

INTLIST_IN_GA(.(X, XS), .(s(X), YS)) → INTLIST_IN_GA(XS, YS)

The TRS R consists of the following rules:

int_in_gga(0, 0, .(0, [])) → int_out_gga(0, 0, .(0, []))
int_in_gga(0, s(Y), .(0, XS)) → U2_gga(Y, XS, int_in_gga(s(0), s(Y), XS))
int_in_gga(s(X), 0, []) → int_out_gga(s(X), 0, [])
int_in_gga(s(X), s(Y), XS) → U3_gga(X, Y, XS, int_in_gga(X, Y, ZS))
U3_gga(X, Y, XS, int_out_gga(X, Y, ZS)) → U4_gga(X, Y, XS, intlist_in_ga(ZS, XS))
intlist_in_ga([], []) → intlist_out_ga([], [])
intlist_in_ga(.(X, XS), .(s(X), YS)) → U1_ga(X, XS, YS, intlist_in_ga(XS, YS))
U1_ga(X, XS, YS, intlist_out_ga(XS, YS)) → intlist_out_ga(.(X, XS), .(s(X), YS))
U4_gga(X, Y, XS, intlist_out_ga(ZS, XS)) → int_out_gga(s(X), s(Y), XS)
U2_gga(Y, XS, int_out_gga(s(0), s(Y), XS)) → int_out_gga(0, s(Y), .(0, XS))

The argument filtering Pi contains the following mapping:
int_in_gga(x1, x2, x3) = int_in_gga(x1, x2)
0 = 0
int_out_gga(x1, x2, x3) = int_out_gga(x3)
s(x1) = s(x1)
U2_gga(x1, x2, x3) = U2_gga(x3)
U3_gga(x1, x2, x3, x4) = U3_gga(x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x4)
intlist_in_ga(x1, x2) = intlist_in_ga(x1)
[] = []
intlist_out_ga(x1, x2) = intlist_out_ga(x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
INTLIST_IN_GA(x1, x2) = INTLIST_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

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

INTLIST_IN_GA(.(X, XS), .(s(X), YS)) → INTLIST_IN_GA(XS, YS)

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

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

INTLIST_IN_GA(.(X, XS)) → INTLIST_IN_GA(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:

INTLIST_IN_GA(.(X, XS)) → INTLIST_IN_GA(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:

INT_IN_GGA(0, s(Y), .(0, XS)) → INT_IN_GGA(s(0), s(Y), XS)
INT_IN_GGA(s(X), s(Y), XS) → INT_IN_GGA(X, Y, ZS)

The TRS R consists of the following rules:

int_in_gga(0, 0, .(0, [])) → int_out_gga(0, 0, .(0, []))
int_in_gga(0, s(Y), .(0, XS)) → U2_gga(Y, XS, int_in_gga(s(0), s(Y), XS))
int_in_gga(s(X), 0, []) → int_out_gga(s(X), 0, [])
int_in_gga(s(X), s(Y), XS) → U3_gga(X, Y, XS, int_in_gga(X, Y, ZS))
U3_gga(X, Y, XS, int_out_gga(X, Y, ZS)) → U4_gga(X, Y, XS, intlist_in_ga(ZS, XS))
intlist_in_ga([], []) → intlist_out_ga([], [])
intlist_in_ga(.(X, XS), .(s(X), YS)) → U1_ga(X, XS, YS, intlist_in_ga(XS, YS))
U1_ga(X, XS, YS, intlist_out_ga(XS, YS)) → intlist_out_ga(.(X, XS), .(s(X), YS))
U4_gga(X, Y, XS, intlist_out_ga(ZS, XS)) → int_out_gga(s(X), s(Y), XS)
U2_gga(Y, XS, int_out_gga(s(0), s(Y), XS)) → int_out_gga(0, s(Y), .(0, XS))

The argument filtering Pi contains the following mapping:
int_in_gga(x1, x2, x3) = int_in_gga(x1, x2)
0 = 0
int_out_gga(x1, x2, x3) = int_out_gga(x3)
s(x1) = s(x1)
U2_gga(x1, x2, x3) = U2_gga(x3)
U3_gga(x1, x2, x3, x4) = U3_gga(x4)
U4_gga(x1, x2, x3, x4) = U4_gga(x4)
intlist_in_ga(x1, x2) = intlist_in_ga(x1)
[] = []
intlist_out_ga(x1, x2) = intlist_out_ga(x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
INT_IN_GGA(x1, x2, x3) = INT_IN_GGA(x1, 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

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

INT_IN_GGA(0, s(Y), .(0, XS)) → INT_IN_GGA(s(0), s(Y), XS)
INT_IN_GGA(s(X), s(Y), XS) → INT_IN_GGA(X, Y, ZS)

R is empty.
The argument filtering Pi contains the following mapping:
0 = 0
s(x1) = s(x1)
.(x1, x2) = .(x1, x2)
INT_IN_GGA(x1, x2, x3) = INT_IN_GGA(x1, 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

                        ↳ QDPSizeChangeProof

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

INT_IN_GGA(0, s(Y)) → INT_IN_GGA(s(0), s(Y))
INT_IN_GGA(s(X), s(Y)) → INT_IN_GGA(X, Y)

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

INT_IN_GGA(s(X), s(Y)) → INT_IN_GGA(X, Y)
The graph contains the following edges 1 > 1, 2 > 2
INT_IN_GGA(0, s(Y)) → INT_IN_GGA(s(0), s(Y))
The graph contains the following edges 2 >= 2