Left Termination proof of /tmp/tmpjL9No9/delete-fbf.pl

Left Termination of the query pattern delete_in_3(a, g, a) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

Clauses:

delete(X, tree(X, void, Right), Right).
delete(X, tree(X, Left, void), Left).
delete(X, tree(X, Left, Right), tree(Y, Left, Right1)) :- delmin(Right, Y, Right1).
delete(X, tree(Y, Left, Right), tree(Y, Left1, Right)) :- ','(less(X, Y), delete(X, Left, Left1)).
delete(X, tree(Y, Left, Right), tree(Y, Left, Right1)) :- ','(less(Y, X), delete(X, Right, Right1)).
delmin(tree(Y, void, Right), Y, Right).
delmin(tree(X, Left, X1), Y, tree(X, Left1, X2)) :- delmin(Left, Y, Left1).
less(0, s(X)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

delete(a,g,a).

We use the technique of [30].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

delete_in(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4(X, Y, Left, Right, Right1, less_in(Y, X))
less_in(s(X), s(Y)) → U7(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U7(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U4(X, Y, Left, Right, Right1, less_out(Y, X)) → U5(X, Y, Left, Right, Right1, delete_in(X, Right, Right1))
delete_in(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2(X, Y, Left, Right, Left1, less_in(X, Y))
U2(X, Y, Left, Right, Left1, less_out(X, Y)) → U3(X, Y, Left, Right, Left1, delete_in(X, Left, Left1))
delete_in(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1(X, Left, Right, Y, Right1, delmin_in(Right, Y, Right1))
delmin_in(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6(X, Left, X1, Y, Left1, X2, delmin_in(Left, Y, Left1))
delmin_in(tree(Y, void, Right), Y, Right) → delmin_out(tree(Y, void, Right), Y, Right)
U6(X, Left, X1, Y, Left1, X2, delmin_out(Left, Y, Left1)) → delmin_out(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1(X, Left, Right, Y, Right1, delmin_out(Right, Y, Right1)) → delete_out(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in(X, tree(X, Left, void), Left) → delete_out(X, tree(X, Left, void), Left)
delete_in(X, tree(X, void, Right), Right) → delete_out(X, tree(X, void, Right), Right)
U3(X, Y, Left, Right, Left1, delete_out(X, Left, Left1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U5(X, Y, Left, Right, Right1, delete_out(X, Right, Right1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left, Right1))

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:

delete_in(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4(X, Y, Left, Right, Right1, less_in(Y, X))
less_in(s(X), s(Y)) → U7(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U7(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U4(X, Y, Left, Right, Right1, less_out(Y, X)) → U5(X, Y, Left, Right, Right1, delete_in(X, Right, Right1))
delete_in(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2(X, Y, Left, Right, Left1, less_in(X, Y))
U2(X, Y, Left, Right, Left1, less_out(X, Y)) → U3(X, Y, Left, Right, Left1, delete_in(X, Left, Left1))
delete_in(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1(X, Left, Right, Y, Right1, delmin_in(Right, Y, Right1))
delmin_in(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6(X, Left, X1, Y, Left1, X2, delmin_in(Left, Y, Left1))
delmin_in(tree(Y, void, Right), Y, Right) → delmin_out(tree(Y, void, Right), Y, Right)
U6(X, Left, X1, Y, Left1, X2, delmin_out(Left, Y, Left1)) → delmin_out(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1(X, Left, Right, Y, Right1, delmin_out(Right, Y, Right1)) → delete_out(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in(X, tree(X, Left, void), Left) → delete_out(X, tree(X, Left, void), Left)
delete_in(X, tree(X, void, Right), Right) → delete_out(X, tree(X, void, Right), Right)
U3(X, Y, Left, Right, Left1, delete_out(X, Left, Left1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U5(X, Y, Left, Right, Right1, delete_out(X, Right, Right1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left, Right1))

DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4¹(X, Y, Left, Right, Right1, less_in(Y, X))
DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN(Y, X)
LESS_IN(s(X), s(Y)) → U7¹(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U4¹(X, Y, Left, Right, Right1, less_out(Y, X)) → U5¹(X, Y, Left, Right, Right1, delete_in(X, Right, Right1))
U4¹(X, Y, Left, Right, Right1, less_out(Y, X)) → DELETE_IN(X, Right, Right1)
DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2¹(X, Y, Left, Right, Left1, less_in(X, Y))
DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN(X, Y)
U2¹(X, Y, Left, Right, Left1, less_out(X, Y)) → U3¹(X, Y, Left, Right, Left1, delete_in(X, Left, Left1))
U2¹(X, Y, Left, Right, Left1, less_out(X, Y)) → DELETE_IN(X, Left, Left1)
DELETE_IN(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1¹(X, Left, Right, Y, Right1, delmin_in(Right, Y, Right1))
DELETE_IN(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN(Right, Y, Right1)
DELMIN_IN(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6¹(X, Left, X1, Y, Left1, X2, delmin_in(Left, Y, Left1))
DELMIN_IN(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN(Left, Y, Left1)

The TRS R consists of the following rules:

delete_in(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4(X, Y, Left, Right, Right1, less_in(Y, X))
less_in(s(X), s(Y)) → U7(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U7(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U4(X, Y, Left, Right, Right1, less_out(Y, X)) → U5(X, Y, Left, Right, Right1, delete_in(X, Right, Right1))
delete_in(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2(X, Y, Left, Right, Left1, less_in(X, Y))
U2(X, Y, Left, Right, Left1, less_out(X, Y)) → U3(X, Y, Left, Right, Left1, delete_in(X, Left, Left1))
delete_in(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1(X, Left, Right, Y, Right1, delmin_in(Right, Y, Right1))
delmin_in(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6(X, Left, X1, Y, Left1, X2, delmin_in(Left, Y, Left1))
delmin_in(tree(Y, void, Right), Y, Right) → delmin_out(tree(Y, void, Right), Y, Right)
U6(X, Left, X1, Y, Left1, X2, delmin_out(Left, Y, Left1)) → delmin_out(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1(X, Left, Right, Y, Right1, delmin_out(Right, Y, Right1)) → delete_out(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in(X, tree(X, Left, void), Left) → delete_out(X, tree(X, Left, void), Left)
delete_in(X, tree(X, void, Right), Right) → delete_out(X, tree(X, void, Right), Right)
U3(X, Y, Left, Right, Left1, delete_out(X, Left, Left1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U5(X, Y, Left, Right, Right1, delete_out(X, Right, Right1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in(x1, x2, x3) = delete_in(x2)
tree(x1, x2, x3) = tree(x1, x2, x3)
U4(x1, x2, x3, x4, x5, x6) = U4(x4, x6)
less_in(x1, x2) = less_in
s(x1) = s(x1)
U7(x1, x2, x3) = U7(x3)
0 = 0
less_out(x1, x2) = less_out(x1)
U5(x1, x2, x3, x4, x5, x6) = U5(x6)
U2(x1, x2, x3, x4, x5, x6) = U2(x3, x6)
U3(x1, x2, x3, x4, x5, x6) = U3(x1, x6)
U1(x1, x2, x3, x4, x5, x6) = U1(x1, x6)
delmin_in(x1, x2, x3) = delmin_in(x1)
U6(x1, x2, x3, x4, x5, x6, x7) = U6(x7)
void = void
delmin_out(x1, x2, x3) = delmin_out(x2)
delete_out(x1, x2, x3) = delete_out(x1)
DELETE_IN(x1, x2, x3) = DELETE_IN(x2)
U7¹(x1, x2, x3) = U7¹(x3)
LESS_IN(x1, x2) = LESS_IN
U5¹(x1, x2, x3, x4, x5, x6) = U5¹(x6)
DELMIN_IN(x1, x2, x3) = DELMIN_IN(x1)
U2¹(x1, x2, x3, x4, x5, x6) = U2¹(x3, x6)
U4¹(x1, x2, x3, x4, x5, x6) = U4¹(x4, x6)
U1¹(x1, x2, x3, x4, x5, x6) = U1¹(x1, x6)
U6¹(x1, x2, x3, x4, x5, x6, x7) = U6¹(x7)
U3¹(x1, x2, x3, x4, x5, x6) = U3¹(x1, x6)

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:

DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4¹(X, Y, Left, Right, Right1, less_in(Y, X))
DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN(Y, X)
LESS_IN(s(X), s(Y)) → U7¹(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U4¹(X, Y, Left, Right, Right1, less_out(Y, X)) → U5¹(X, Y, Left, Right, Right1, delete_in(X, Right, Right1))
U4¹(X, Y, Left, Right, Right1, less_out(Y, X)) → DELETE_IN(X, Right, Right1)
DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2¹(X, Y, Left, Right, Left1, less_in(X, Y))
DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN(X, Y)
U2¹(X, Y, Left, Right, Left1, less_out(X, Y)) → U3¹(X, Y, Left, Right, Left1, delete_in(X, Left, Left1))
U2¹(X, Y, Left, Right, Left1, less_out(X, Y)) → DELETE_IN(X, Left, Left1)
DELETE_IN(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1¹(X, Left, Right, Y, Right1, delmin_in(Right, Y, Right1))
DELETE_IN(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN(Right, Y, Right1)
DELMIN_IN(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6¹(X, Left, X1, Y, Left1, X2, delmin_in(Left, Y, Left1))
DELMIN_IN(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN(Left, Y, Left1)

The TRS R consists of the following rules:

delete_in(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4(X, Y, Left, Right, Right1, less_in(Y, X))
less_in(s(X), s(Y)) → U7(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U7(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U4(X, Y, Left, Right, Right1, less_out(Y, X)) → U5(X, Y, Left, Right, Right1, delete_in(X, Right, Right1))
delete_in(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2(X, Y, Left, Right, Left1, less_in(X, Y))
U2(X, Y, Left, Right, Left1, less_out(X, Y)) → U3(X, Y, Left, Right, Left1, delete_in(X, Left, Left1))
delete_in(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1(X, Left, Right, Y, Right1, delmin_in(Right, Y, Right1))
delmin_in(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6(X, Left, X1, Y, Left1, X2, delmin_in(Left, Y, Left1))
delmin_in(tree(Y, void, Right), Y, Right) → delmin_out(tree(Y, void, Right), Y, Right)
U6(X, Left, X1, Y, Left1, X2, delmin_out(Left, Y, Left1)) → delmin_out(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1(X, Left, Right, Y, Right1, delmin_out(Right, Y, Right1)) → delete_out(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in(X, tree(X, Left, void), Left) → delete_out(X, tree(X, Left, void), Left)
delete_in(X, tree(X, void, Right), Right) → delete_out(X, tree(X, void, Right), Right)
U3(X, Y, Left, Right, Left1, delete_out(X, Left, Left1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U5(X, Y, Left, Right, Right1, delete_out(X, Right, Right1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left, Right1))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

DELMIN_IN(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN(Left, Y, Left1)

The TRS R consists of the following rules:

delete_in(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4(X, Y, Left, Right, Right1, less_in(Y, X))
less_in(s(X), s(Y)) → U7(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U7(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U4(X, Y, Left, Right, Right1, less_out(Y, X)) → U5(X, Y, Left, Right, Right1, delete_in(X, Right, Right1))
delete_in(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2(X, Y, Left, Right, Left1, less_in(X, Y))
U2(X, Y, Left, Right, Left1, less_out(X, Y)) → U3(X, Y, Left, Right, Left1, delete_in(X, Left, Left1))
delete_in(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1(X, Left, Right, Y, Right1, delmin_in(Right, Y, Right1))
delmin_in(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6(X, Left, X1, Y, Left1, X2, delmin_in(Left, Y, Left1))
delmin_in(tree(Y, void, Right), Y, Right) → delmin_out(tree(Y, void, Right), Y, Right)
U6(X, Left, X1, Y, Left1, X2, delmin_out(Left, Y, Left1)) → delmin_out(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1(X, Left, Right, Y, Right1, delmin_out(Right, Y, Right1)) → delete_out(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in(X, tree(X, Left, void), Left) → delete_out(X, tree(X, Left, void), Left)
delete_in(X, tree(X, void, Right), Right) → delete_out(X, tree(X, void, Right), Right)
U3(X, Y, Left, Right, Left1, delete_out(X, Left, Left1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U5(X, Y, Left, Right, Right1, delete_out(X, Right, Right1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left, Right1))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

DELMIN_IN(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN(Left, Y, Left1)

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

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

DELMIN_IN(tree(X, Left, X1)) → DELMIN_IN(Left)

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:

DELMIN_IN(tree(X, Left, X1)) → DELMIN_IN(Left)
The graph contains the following edges 1 > 1

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)

The TRS R consists of the following rules:

delete_in(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4(X, Y, Left, Right, Right1, less_in(Y, X))
less_in(s(X), s(Y)) → U7(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U7(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U4(X, Y, Left, Right, Right1, less_out(Y, X)) → U5(X, Y, Left, Right, Right1, delete_in(X, Right, Right1))
delete_in(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2(X, Y, Left, Right, Left1, less_in(X, Y))
U2(X, Y, Left, Right, Left1, less_out(X, Y)) → U3(X, Y, Left, Right, Left1, delete_in(X, Left, Left1))
delete_in(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1(X, Left, Right, Y, Right1, delmin_in(Right, Y, Right1))
delmin_in(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6(X, Left, X1, Y, Left1, X2, delmin_in(Left, Y, Left1))
delmin_in(tree(Y, void, Right), Y, Right) → delmin_out(tree(Y, void, Right), Y, Right)
U6(X, Left, X1, Y, Left1, X2, delmin_out(Left, Y, Left1)) → delmin_out(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1(X, Left, Right, Y, Right1, delmin_out(Right, Y, Right1)) → delete_out(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in(X, tree(X, Left, void), Left) → delete_out(X, tree(X, Left, void), Left)
delete_in(X, tree(X, void, Right), Right) → delete_out(X, tree(X, void, Right), Right)
U3(X, Y, Left, Right, Left1, delete_out(X, Left, Left1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U5(X, Y, Left, Right, Right1, delete_out(X, Right, Right1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left, Right1))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)

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

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

                        ↳ NonTerminationProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LESS_IN → LESS_IN

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

LESS_IN → LESS_IN

The TRS R consists of the following rules:none

s = LESS_IN evaluates to t =LESS_IN

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LESS_IN to LESS_IN.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2¹(X, Y, Left, Right, Left1, less_in(X, Y))
U2¹(X, Y, Left, Right, Left1, less_out(X, Y)) → DELETE_IN(X, Left, Left1)
U4¹(X, Y, Left, Right, Right1, less_out(Y, X)) → DELETE_IN(X, Right, Right1)
DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4¹(X, Y, Left, Right, Right1, less_in(Y, X))

The TRS R consists of the following rules:

delete_in(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4(X, Y, Left, Right, Right1, less_in(Y, X))
less_in(s(X), s(Y)) → U7(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U7(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U4(X, Y, Left, Right, Right1, less_out(Y, X)) → U5(X, Y, Left, Right, Right1, delete_in(X, Right, Right1))
delete_in(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2(X, Y, Left, Right, Left1, less_in(X, Y))
U2(X, Y, Left, Right, Left1, less_out(X, Y)) → U3(X, Y, Left, Right, Left1, delete_in(X, Left, Left1))
delete_in(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1(X, Left, Right, Y, Right1, delmin_in(Right, Y, Right1))
delmin_in(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6(X, Left, X1, Y, Left1, X2, delmin_in(Left, Y, Left1))
delmin_in(tree(Y, void, Right), Y, Right) → delmin_out(tree(Y, void, Right), Y, Right)
U6(X, Left, X1, Y, Left1, X2, delmin_out(Left, Y, Left1)) → delmin_out(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1(X, Left, Right, Y, Right1, delmin_out(Right, Y, Right1)) → delete_out(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in(X, tree(X, Left, void), Left) → delete_out(X, tree(X, Left, void), Left)
delete_in(X, tree(X, void, Right), Right) → delete_out(X, tree(X, void, Right), Right)
U3(X, Y, Left, Right, Left1, delete_out(X, Left, Left1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U5(X, Y, Left, Right, Right1, delete_out(X, Right, Right1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in(x1, x2, x3) = delete_in(x2)
tree(x1, x2, x3) = tree(x1, x2, x3)
U4(x1, x2, x3, x4, x5, x6) = U4(x4, x6)
less_in(x1, x2) = less_in
s(x1) = s(x1)
U7(x1, x2, x3) = U7(x3)
0 = 0
less_out(x1, x2) = less_out(x1)
U5(x1, x2, x3, x4, x5, x6) = U5(x6)
U2(x1, x2, x3, x4, x5, x6) = U2(x3, x6)
U3(x1, x2, x3, x4, x5, x6) = U3(x1, x6)
U1(x1, x2, x3, x4, x5, x6) = U1(x1, x6)
delmin_in(x1, x2, x3) = delmin_in(x1)
U6(x1, x2, x3, x4, x5, x6, x7) = U6(x7)
void = void
delmin_out(x1, x2, x3) = delmin_out(x2)
delete_out(x1, x2, x3) = delete_out(x1)
DELETE_IN(x1, x2, x3) = DELETE_IN(x2)
U2¹(x1, x2, x3, x4, x5, x6) = U2¹(x3, x6)
U4¹(x1, x2, x3, x4, x5, x6) = U4¹(x4, x6)

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

  ↳ PrologToPiTRSProof

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

DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2¹(X, Y, Left, Right, Left1, less_in(X, Y))
U2¹(X, Y, Left, Right, Left1, less_out(X, Y)) → DELETE_IN(X, Left, Left1)
U4¹(X, Y, Left, Right, Right1, less_out(Y, X)) → DELETE_IN(X, Right, Right1)
DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4¹(X, Y, Left, Right, Right1, less_in(Y, X))

The TRS R consists of the following rules:

less_in(s(X), s(Y)) → U7(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U7(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3) = tree(x1, x2, x3)
less_in(x1, x2) = less_in
s(x1) = s(x1)
U7(x1, x2, x3) = U7(x3)
0 = 0
less_out(x1, x2) = less_out(x1)
DELETE_IN(x1, x2, x3) = DELETE_IN(x2)
U2¹(x1, x2, x3, x4, x5, x6) = U2¹(x3, x6)
U4¹(x1, x2, x3, x4, x5, x6) = U4¹(x4, x6)

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

                        ↳ QDPSizeChangeProof

  ↳ PrologToPiTRSProof

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

U2¹(Left, less_out(X)) → DELETE_IN(Left)
DELETE_IN(tree(Y, Left, Right)) → U4¹(Right, less_in)
DELETE_IN(tree(Y, Left, Right)) → U2¹(Left, less_in)
U4¹(Right, less_out(Y)) → DELETE_IN(Right)

The TRS R consists of the following rules:

less_in → U7(less_in)
less_in → less_out(0)
U7(less_out(X)) → less_out(s(X))

The set Q consists of the following terms:

less_in
U7(x0)

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:

U2¹(Left, less_out(X)) → DELETE_IN(Left)
The graph contains the following edges 1 >= 1
U4¹(Right, less_out(Y)) → DELETE_IN(Right)
The graph contains the following edges 1 >= 1
DELETE_IN(tree(Y, Left, Right)) → U2¹(Left, less_in)
The graph contains the following edges 1 > 1
DELETE_IN(tree(Y, Left, Right)) → U4¹(Right, less_in)
The graph contains the following edges 1 > 1

We use the technique of [30].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

delete_in(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4(X, Y, Left, Right, Right1, less_in(Y, X))
less_in(s(X), s(Y)) → U7(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U7(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U4(X, Y, Left, Right, Right1, less_out(Y, X)) → U5(X, Y, Left, Right, Right1, delete_in(X, Right, Right1))
delete_in(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2(X, Y, Left, Right, Left1, less_in(X, Y))
U2(X, Y, Left, Right, Left1, less_out(X, Y)) → U3(X, Y, Left, Right, Left1, delete_in(X, Left, Left1))
delete_in(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1(X, Left, Right, Y, Right1, delmin_in(Right, Y, Right1))
delmin_in(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6(X, Left, X1, Y, Left1, X2, delmin_in(Left, Y, Left1))
delmin_in(tree(Y, void, Right), Y, Right) → delmin_out(tree(Y, void, Right), Y, Right)
U6(X, Left, X1, Y, Left1, X2, delmin_out(Left, Y, Left1)) → delmin_out(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1(X, Left, Right, Y, Right1, delmin_out(Right, Y, Right1)) → delete_out(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in(X, tree(X, Left, void), Left) → delete_out(X, tree(X, Left, void), Left)
delete_in(X, tree(X, void, Right), Right) → delete_out(X, tree(X, void, Right), Right)
U3(X, Y, Left, Right, Left1, delete_out(X, Left, Left1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U5(X, Y, Left, Right, Right1, delete_out(X, Right, Right1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left, Right1))

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:

delete_in(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4(X, Y, Left, Right, Right1, less_in(Y, X))
less_in(s(X), s(Y)) → U7(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U7(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U4(X, Y, Left, Right, Right1, less_out(Y, X)) → U5(X, Y, Left, Right, Right1, delete_in(X, Right, Right1))
delete_in(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2(X, Y, Left, Right, Left1, less_in(X, Y))
U2(X, Y, Left, Right, Left1, less_out(X, Y)) → U3(X, Y, Left, Right, Left1, delete_in(X, Left, Left1))
delete_in(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1(X, Left, Right, Y, Right1, delmin_in(Right, Y, Right1))
delmin_in(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6(X, Left, X1, Y, Left1, X2, delmin_in(Left, Y, Left1))
delmin_in(tree(Y, void, Right), Y, Right) → delmin_out(tree(Y, void, Right), Y, Right)
U6(X, Left, X1, Y, Left1, X2, delmin_out(Left, Y, Left1)) → delmin_out(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1(X, Left, Right, Y, Right1, delmin_out(Right, Y, Right1)) → delete_out(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in(X, tree(X, Left, void), Left) → delete_out(X, tree(X, Left, void), Left)
delete_in(X, tree(X, void, Right), Right) → delete_out(X, tree(X, void, Right), Right)
U3(X, Y, Left, Right, Left1, delete_out(X, Left, Left1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U5(X, Y, Left, Right, Right1, delete_out(X, Right, Right1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left, Right1))

DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4¹(X, Y, Left, Right, Right1, less_in(Y, X))
DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN(Y, X)
LESS_IN(s(X), s(Y)) → U7¹(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U4¹(X, Y, Left, Right, Right1, less_out(Y, X)) → U5¹(X, Y, Left, Right, Right1, delete_in(X, Right, Right1))
U4¹(X, Y, Left, Right, Right1, less_out(Y, X)) → DELETE_IN(X, Right, Right1)
DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2¹(X, Y, Left, Right, Left1, less_in(X, Y))
DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN(X, Y)
U2¹(X, Y, Left, Right, Left1, less_out(X, Y)) → U3¹(X, Y, Left, Right, Left1, delete_in(X, Left, Left1))
U2¹(X, Y, Left, Right, Left1, less_out(X, Y)) → DELETE_IN(X, Left, Left1)
DELETE_IN(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1¹(X, Left, Right, Y, Right1, delmin_in(Right, Y, Right1))
DELETE_IN(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN(Right, Y, Right1)
DELMIN_IN(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6¹(X, Left, X1, Y, Left1, X2, delmin_in(Left, Y, Left1))
DELMIN_IN(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN(Left, Y, Left1)

The TRS R consists of the following rules:

delete_in(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4(X, Y, Left, Right, Right1, less_in(Y, X))
less_in(s(X), s(Y)) → U7(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U7(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U4(X, Y, Left, Right, Right1, less_out(Y, X)) → U5(X, Y, Left, Right, Right1, delete_in(X, Right, Right1))
delete_in(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2(X, Y, Left, Right, Left1, less_in(X, Y))
U2(X, Y, Left, Right, Left1, less_out(X, Y)) → U3(X, Y, Left, Right, Left1, delete_in(X, Left, Left1))
delete_in(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1(X, Left, Right, Y, Right1, delmin_in(Right, Y, Right1))
delmin_in(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6(X, Left, X1, Y, Left1, X2, delmin_in(Left, Y, Left1))
delmin_in(tree(Y, void, Right), Y, Right) → delmin_out(tree(Y, void, Right), Y, Right)
U6(X, Left, X1, Y, Left1, X2, delmin_out(Left, Y, Left1)) → delmin_out(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1(X, Left, Right, Y, Right1, delmin_out(Right, Y, Right1)) → delete_out(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in(X, tree(X, Left, void), Left) → delete_out(X, tree(X, Left, void), Left)
delete_in(X, tree(X, void, Right), Right) → delete_out(X, tree(X, void, Right), Right)
U3(X, Y, Left, Right, Left1, delete_out(X, Left, Left1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U5(X, Y, Left, Right, Right1, delete_out(X, Right, Right1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in(x1, x2, x3) = delete_in(x2)
tree(x1, x2, x3) = tree(x1, x2, x3)
U4(x1, x2, x3, x4, x5, x6) = U4(x2, x3, x4, x6)
less_in(x1, x2) = less_in
s(x1) = s(x1)
U7(x1, x2, x3) = U7(x3)
0 = 0
less_out(x1, x2) = less_out(x1)
U5(x1, x2, x3, x4, x5, x6) = U5(x2, x3, x4, x6)
U2(x1, x2, x3, x4, x5, x6) = U2(x2, x3, x4, x6)
U3(x1, x2, x3, x4, x5, x6) = U3(x1, x2, x3, x4, x6)
U1(x1, x2, x3, x4, x5, x6) = U1(x1, x2, x3, x6)
delmin_in(x1, x2, x3) = delmin_in(x1)
U6(x1, x2, x3, x4, x5, x6, x7) = U6(x1, x2, x3, x7)
void = void
delmin_out(x1, x2, x3) = delmin_out(x1, x2)
delete_out(x1, x2, x3) = delete_out(x1, x2)
DELETE_IN(x1, x2, x3) = DELETE_IN(x2)
U7¹(x1, x2, x3) = U7¹(x3)
LESS_IN(x1, x2) = LESS_IN
U5¹(x1, x2, x3, x4, x5, x6) = U5¹(x2, x3, x4, x6)
DELMIN_IN(x1, x2, x3) = DELMIN_IN(x1)
U2¹(x1, x2, x3, x4, x5, x6) = U2¹(x2, x3, x4, x6)
U4¹(x1, x2, x3, x4, x5, x6) = U4¹(x2, x3, x4, x6)
U1¹(x1, x2, x3, x4, x5, x6) = U1¹(x1, x2, x3, x6)
U6¹(x1, x2, x3, x4, x5, x6, x7) = U6¹(x1, x2, x3, x7)
U3¹(x1, x2, x3, x4, x5, x6) = U3¹(x1, x2, x3, x4, x6)

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:

DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4¹(X, Y, Left, Right, Right1, less_in(Y, X))
DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN(Y, X)
LESS_IN(s(X), s(Y)) → U7¹(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U4¹(X, Y, Left, Right, Right1, less_out(Y, X)) → U5¹(X, Y, Left, Right, Right1, delete_in(X, Right, Right1))
U4¹(X, Y, Left, Right, Right1, less_out(Y, X)) → DELETE_IN(X, Right, Right1)
DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2¹(X, Y, Left, Right, Left1, less_in(X, Y))
DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN(X, Y)
U2¹(X, Y, Left, Right, Left1, less_out(X, Y)) → U3¹(X, Y, Left, Right, Left1, delete_in(X, Left, Left1))
U2¹(X, Y, Left, Right, Left1, less_out(X, Y)) → DELETE_IN(X, Left, Left1)
DELETE_IN(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1¹(X, Left, Right, Y, Right1, delmin_in(Right, Y, Right1))
DELETE_IN(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN(Right, Y, Right1)
DELMIN_IN(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6¹(X, Left, X1, Y, Left1, X2, delmin_in(Left, Y, Left1))
DELMIN_IN(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN(Left, Y, Left1)

The TRS R consists of the following rules:

delete_in(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4(X, Y, Left, Right, Right1, less_in(Y, X))
less_in(s(X), s(Y)) → U7(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U7(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U4(X, Y, Left, Right, Right1, less_out(Y, X)) → U5(X, Y, Left, Right, Right1, delete_in(X, Right, Right1))
delete_in(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2(X, Y, Left, Right, Left1, less_in(X, Y))
U2(X, Y, Left, Right, Left1, less_out(X, Y)) → U3(X, Y, Left, Right, Left1, delete_in(X, Left, Left1))
delete_in(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1(X, Left, Right, Y, Right1, delmin_in(Right, Y, Right1))
delmin_in(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6(X, Left, X1, Y, Left1, X2, delmin_in(Left, Y, Left1))
delmin_in(tree(Y, void, Right), Y, Right) → delmin_out(tree(Y, void, Right), Y, Right)
U6(X, Left, X1, Y, Left1, X2, delmin_out(Left, Y, Left1)) → delmin_out(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1(X, Left, Right, Y, Right1, delmin_out(Right, Y, Right1)) → delete_out(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in(X, tree(X, Left, void), Left) → delete_out(X, tree(X, Left, void), Left)
delete_in(X, tree(X, void, Right), Right) → delete_out(X, tree(X, void, Right), Right)
U3(X, Y, Left, Right, Left1, delete_out(X, Left, Left1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U5(X, Y, Left, Right, Right1, delete_out(X, Right, Right1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left, Right1))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

DELMIN_IN(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN(Left, Y, Left1)

The TRS R consists of the following rules:

delete_in(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4(X, Y, Left, Right, Right1, less_in(Y, X))
less_in(s(X), s(Y)) → U7(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U7(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U4(X, Y, Left, Right, Right1, less_out(Y, X)) → U5(X, Y, Left, Right, Right1, delete_in(X, Right, Right1))
delete_in(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2(X, Y, Left, Right, Left1, less_in(X, Y))
U2(X, Y, Left, Right, Left1, less_out(X, Y)) → U3(X, Y, Left, Right, Left1, delete_in(X, Left, Left1))
delete_in(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1(X, Left, Right, Y, Right1, delmin_in(Right, Y, Right1))
delmin_in(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6(X, Left, X1, Y, Left1, X2, delmin_in(Left, Y, Left1))
delmin_in(tree(Y, void, Right), Y, Right) → delmin_out(tree(Y, void, Right), Y, Right)
U6(X, Left, X1, Y, Left1, X2, delmin_out(Left, Y, Left1)) → delmin_out(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1(X, Left, Right, Y, Right1, delmin_out(Right, Y, Right1)) → delete_out(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in(X, tree(X, Left, void), Left) → delete_out(X, tree(X, Left, void), Left)
delete_in(X, tree(X, void, Right), Right) → delete_out(X, tree(X, void, Right), Right)
U3(X, Y, Left, Right, Left1, delete_out(X, Left, Left1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U5(X, Y, Left, Right, Right1, delete_out(X, Right, Right1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left, Right1))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

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

DELMIN_IN(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN(Left, Y, Left1)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

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

DELMIN_IN(tree(X, Left, X1)) → DELMIN_IN(Left)

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

DELMIN_IN(tree(X, Left, X1)) → DELMIN_IN(Left)
The graph contains the following edges 1 > 1

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)

The TRS R consists of the following rules:

delete_in(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4(X, Y, Left, Right, Right1, less_in(Y, X))
less_in(s(X), s(Y)) → U7(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U7(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U4(X, Y, Left, Right, Right1, less_out(Y, X)) → U5(X, Y, Left, Right, Right1, delete_in(X, Right, Right1))
delete_in(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2(X, Y, Left, Right, Left1, less_in(X, Y))
U2(X, Y, Left, Right, Left1, less_out(X, Y)) → U3(X, Y, Left, Right, Left1, delete_in(X, Left, Left1))
delete_in(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1(X, Left, Right, Y, Right1, delmin_in(Right, Y, Right1))
delmin_in(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6(X, Left, X1, Y, Left1, X2, delmin_in(Left, Y, Left1))
delmin_in(tree(Y, void, Right), Y, Right) → delmin_out(tree(Y, void, Right), Y, Right)
U6(X, Left, X1, Y, Left1, X2, delmin_out(Left, Y, Left1)) → delmin_out(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1(X, Left, Right, Y, Right1, delmin_out(Right, Y, Right1)) → delete_out(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in(X, tree(X, Left, void), Left) → delete_out(X, tree(X, Left, void), Left)
delete_in(X, tree(X, void, Right), Right) → delete_out(X, tree(X, void, Right), Right)
U3(X, Y, Left, Right, Left1, delete_out(X, Left, Left1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U5(X, Y, Left, Right, Right1, delete_out(X, Right, Right1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left, Right1))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

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

LESS_IN → LESS_IN

The TRS R consists of the following rules:none

s = LESS_IN evaluates to t =LESS_IN

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LESS_IN to LESS_IN.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2¹(X, Y, Left, Right, Left1, less_in(X, Y))
U2¹(X, Y, Left, Right, Left1, less_out(X, Y)) → DELETE_IN(X, Left, Left1)
U4¹(X, Y, Left, Right, Right1, less_out(Y, X)) → DELETE_IN(X, Right, Right1)
DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4¹(X, Y, Left, Right, Right1, less_in(Y, X))

The TRS R consists of the following rules:

delete_in(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4(X, Y, Left, Right, Right1, less_in(Y, X))
less_in(s(X), s(Y)) → U7(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U7(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U4(X, Y, Left, Right, Right1, less_out(Y, X)) → U5(X, Y, Left, Right, Right1, delete_in(X, Right, Right1))
delete_in(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2(X, Y, Left, Right, Left1, less_in(X, Y))
U2(X, Y, Left, Right, Left1, less_out(X, Y)) → U3(X, Y, Left, Right, Left1, delete_in(X, Left, Left1))
delete_in(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1(X, Left, Right, Y, Right1, delmin_in(Right, Y, Right1))
delmin_in(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6(X, Left, X1, Y, Left1, X2, delmin_in(Left, Y, Left1))
delmin_in(tree(Y, void, Right), Y, Right) → delmin_out(tree(Y, void, Right), Y, Right)
U6(X, Left, X1, Y, Left1, X2, delmin_out(Left, Y, Left1)) → delmin_out(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1(X, Left, Right, Y, Right1, delmin_out(Right, Y, Right1)) → delete_out(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in(X, tree(X, Left, void), Left) → delete_out(X, tree(X, Left, void), Left)
delete_in(X, tree(X, void, Right), Right) → delete_out(X, tree(X, void, Right), Right)
U3(X, Y, Left, Right, Left1, delete_out(X, Left, Left1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U5(X, Y, Left, Right, Right1, delete_out(X, Right, Right1)) → delete_out(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in(x1, x2, x3) = delete_in(x2)
tree(x1, x2, x3) = tree(x1, x2, x3)
U4(x1, x2, x3, x4, x5, x6) = U4(x2, x3, x4, x6)
less_in(x1, x2) = less_in
s(x1) = s(x1)
U7(x1, x2, x3) = U7(x3)
0 = 0
less_out(x1, x2) = less_out(x1)
U5(x1, x2, x3, x4, x5, x6) = U5(x2, x3, x4, x6)
U2(x1, x2, x3, x4, x5, x6) = U2(x2, x3, x4, x6)
U3(x1, x2, x3, x4, x5, x6) = U3(x1, x2, x3, x4, x6)
U1(x1, x2, x3, x4, x5, x6) = U1(x1, x2, x3, x6)
delmin_in(x1, x2, x3) = delmin_in(x1)
U6(x1, x2, x3, x4, x5, x6, x7) = U6(x1, x2, x3, x7)
void = void
delmin_out(x1, x2, x3) = delmin_out(x1, x2)
delete_out(x1, x2, x3) = delete_out(x1, x2)
DELETE_IN(x1, x2, x3) = DELETE_IN(x2)
U2¹(x1, x2, x3, x4, x5, x6) = U2¹(x2, x3, x4, x6)
U4¹(x1, x2, x3, x4, x5, x6) = U4¹(x2, x3, x4, x6)

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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2¹(X, Y, Left, Right, Left1, less_in(X, Y))
U2¹(X, Y, Left, Right, Left1, less_out(X, Y)) → DELETE_IN(X, Left, Left1)
U4¹(X, Y, Left, Right, Right1, less_out(Y, X)) → DELETE_IN(X, Right, Right1)
DELETE_IN(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4¹(X, Y, Left, Right, Right1, less_in(Y, X))

The TRS R consists of the following rules:

less_in(s(X), s(Y)) → U7(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U7(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3) = tree(x1, x2, x3)
less_in(x1, x2) = less_in
s(x1) = s(x1)
U7(x1, x2, x3) = U7(x3)
0 = 0
less_out(x1, x2) = less_out(x1)
DELETE_IN(x1, x2, x3) = DELETE_IN(x2)
U2¹(x1, x2, x3, x4, x5, x6) = U2¹(x2, x3, x4, x6)
U4¹(x1, x2, x3, x4, x5, x6) = U4¹(x2, x3, x4, x6)

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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

DELETE_IN(tree(Y, Left, Right)) → U2¹(Y, Left, Right, less_in)
DELETE_IN(tree(Y, Left, Right)) → U4¹(Y, Left, Right, less_in)
U4¹(Y, Left, Right, less_out(Y)) → DELETE_IN(Right)
U2¹(Y, Left, Right, less_out(X)) → DELETE_IN(Left)

The TRS R consists of the following rules:

less_in → U7(less_in)
less_in → less_out(0)
U7(less_out(X)) → less_out(s(X))

The set Q consists of the following terms:

less_in
U7(x0)

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:

U2¹(Y, Left, Right, less_out(X)) → DELETE_IN(Left)
The graph contains the following edges 2 >= 1
U4¹(Y, Left, Right, less_out(Y)) → DELETE_IN(Right)
The graph contains the following edges 3 >= 1
DELETE_IN(tree(Y, Left, Right)) → U2¹(Y, Left, Right, less_in)
The graph contains the following edges 1 > 1, 1 > 2, 1 > 3
DELETE_IN(tree(Y, Left, Right)) → U4¹(Y, Left, Right, less_in)
The graph contains the following edges 1 > 1, 1 > 2, 1 > 3