Left Termination proof of /tmp/tmpl3gL6A/delete-bfb.pl

Left Termination of the query pattern delete_in_3(g, a, g) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ 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(g,a,g).

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

delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

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:

delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GAG(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_AGG(Right, Y, Right1)
DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_AGG(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_AGG(Left, Y, Left1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U7_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GAG(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GAG(X, Left, Left1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GAG(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GAG(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gag(x1, x2, x3) = delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3) = delete_out_gag
tree(x1, x2, x3) = tree(x1, x2, x3)
U1_gag(x1, x2, x3, x4, x5, x6) = U1_gag(x6)
delmin_in_agg(x1, x2, x3) = delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3) = delmin_out_agg
U6_agg(x1, x2, x3, x4, x5, x6, x7) = U6_agg(x7)
U2_gag(x1, x2, x3, x4, x5, x6) = U2_gag(x1, x5, x6)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg
U7_gg(x1, x2, x3) = U7_gg(x3)
U3_gag(x1, x2, x3, x4, x5, x6) = U3_gag(x6)
U4_gag(x1, x2, x3, x4, x5, x6) = U4_gag(x1, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6) = U5_gag(x6)
U5_GAG(x1, x2, x3, x4, x5, x6) = U5_GAG(x6)
LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2)
U3_GAG(x1, x2, x3, x4, x5, x6) = U3_GAG(x6)
U4_GAG(x1, x2, x3, x4, x5, x6) = U4_GAG(x1, x5, x6)
U6_AGG(x1, x2, x3, x4, x5, x6, x7) = U6_AGG(x7)
U2_GAG(x1, x2, x3, x4, x5, x6) = U2_GAG(x1, x5, x6)
DELMIN_IN_AGG(x1, x2, x3) = DELMIN_IN_AGG(x2, x3)
U7_GG(x1, x2, x3) = U7_GG(x3)
U1_GAG(x1, x2, x3, x4, x5, x6) = U1_GAG(x6)
DELETE_IN_GAG(x1, x2, x3) = DELETE_IN_GAG(x1, x3)

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:

DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GAG(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_AGG(Right, Y, Right1)
DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_AGG(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_AGG(Left, Y, Left1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U7_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GAG(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GAG(X, Left, Left1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GAG(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GAG(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

The TRS R consists of the following rules:

delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

R is empty.
Pi is empty.
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

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

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:

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

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

The TRS R consists of the following rules:

delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

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

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

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

              ↳ PiDP

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

DELMIN_IN_AGG(Y, tree(X, Left1, X2)) → DELMIN_IN_AGG(Y, Left1)

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

DELMIN_IN_AGG(Y, tree(X, Left1, X2)) → DELMIN_IN_AGG(Y, Left1)
The graph contains the following edges 1 >= 1, 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GAG(X, Right, Right1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GAG(X, Left, Left1)

The TRS R consists of the following rules:

delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gag(x1, x2, x3) = delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3) = delete_out_gag
tree(x1, x2, x3) = tree(x1, x2, x3)
U1_gag(x1, x2, x3, x4, x5, x6) = U1_gag(x6)
delmin_in_agg(x1, x2, x3) = delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3) = delmin_out_agg
U6_agg(x1, x2, x3, x4, x5, x6, x7) = U6_agg(x7)
U2_gag(x1, x2, x3, x4, x5, x6) = U2_gag(x1, x5, x6)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg
U7_gg(x1, x2, x3) = U7_gg(x3)
U3_gag(x1, x2, x3, x4, x5, x6) = U3_gag(x6)
U4_gag(x1, x2, x3, x4, x5, x6) = U4_gag(x1, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6) = U5_gag(x6)
U4_GAG(x1, x2, x3, x4, x5, x6) = U4_GAG(x1, x5, x6)
U2_GAG(x1, x2, x3, x4, x5, x6) = U2_GAG(x1, x5, x6)
DELETE_IN_GAG(x1, x2, x3) = DELETE_IN_GAG(x1, x3)

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:

U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GAG(X, Right, Right1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GAG(X, Left, Left1)

The TRS R consists of the following rules:

less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3) = tree(x1, x2, x3)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg
U7_gg(x1, x2, x3) = U7_gg(x3)
U4_GAG(x1, x2, x3, x4, x5, x6) = U4_GAG(x1, x5, x6)
U2_GAG(x1, x2, x3, x4, x5, x6) = U2_GAG(x1, x5, x6)
DELETE_IN_GAG(x1, x2, x3) = DELETE_IN_GAG(x1, x3)

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

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

DELETE_IN_GAG(X, tree(Y, Left1, Right)) → U2_GAG(X, Left1, less_in_gg(X, Y))
U2_GAG(X, Left1, less_out_gg) → DELETE_IN_GAG(X, Left1)
DELETE_IN_GAG(X, tree(Y, Left, Right1)) → U4_GAG(X, Right1, less_in_gg(Y, X))
U4_GAG(X, Right1, less_out_gg) → DELETE_IN_GAG(X, Right1)

The TRS R consists of the following rules:

less_in_gg(0, s(X)) → less_out_gg
less_in_gg(s(X), s(Y)) → U7_gg(less_in_gg(X, Y))
U7_gg(less_out_gg) → less_out_gg

The set Q consists of the following terms:

less_in_gg(x0, x1)
U7_gg(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:

DELETE_IN_GAG(X, tree(Y, Left, Right1)) → U4_GAG(X, Right1, less_in_gg(Y, X))
The graph contains the following edges 1 >= 1, 2 > 2
DELETE_IN_GAG(X, tree(Y, Left1, Right)) → U2_GAG(X, Left1, less_in_gg(X, Y))
The graph contains the following edges 1 >= 1, 2 > 2
U2_GAG(X, Left1, less_out_gg) → DELETE_IN_GAG(X, Left1)
The graph contains the following edges 1 >= 1, 2 >= 2
U4_GAG(X, Right1, less_out_gg) → DELETE_IN_GAG(X, Right1)
The graph contains the following edges 1 >= 1, 2 >= 2