Left Termination proof of /tmp/tmpTfjtSK/insert-bfb.pl

Left Termination of the query pattern insert(b,f,b) w.r.t. the given Prolog program could successfully be proven:

↳ PROLOG

  ↳ PrologToPiTRSProof

insert₃(X, void₀, tree₃(X, void₀, void₀)).
insert₃(X, tree₃(X, Left, Right), tree₃(X, Left, Right)).
insert₃(X, tree₃(Y, Left, Right), tree₃(Y, Left1, Right)) :- less₂(X, Y), insert₃(X, Left, Left1).
insert₃(X, tree₃(Y, Left, Right), tree₃(Y, Left, Right1)) :- less₂(Y, X), insert₃(X, Right, Right1).
less₂(0₀, s₁(underscore)).
less₂(s₁(X), s₁(Y)) :- less₂(X, Y).

With regard to the inferred argument filtering the predicates were used in the following modes:
insert₃: (b,f,b)
less₂: (b,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

insert_3_in_gag₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_gag₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_gag₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_gag₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_gag₆(X, Y, Left, Right, Left1, less_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(underscore)) -> less_2_out_gg₂(0_0, s_1₁(underscore))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_gg₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_gg₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_gag₆(X, Y, Left, Right, Left1, less_2_out_gg₂(X, Y)) -> if_insert_3_in_2_gag₆(X, Y, Left, Right, Left1, insert_3_in_gag₃(X, Left, Left1))
insert_3_in_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_gag₆(X, Y, Left, Right, Right1, less_2_in_gg₂(Y, X))
if_insert_3_in_3_gag₆(X, Y, Left, Right, Right1, less_2_out_gg₂(Y, X)) -> if_insert_3_in_4_gag₆(X, Y, Left, Right, Right1, insert_3_in_gag₃(X, Right, Right1))
if_insert_3_in_4_gag₆(X, Y, Left, Right, Right1, insert_3_out_gag₃(X, Right, Right1)) -> insert_3_out_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))
if_insert_3_in_2_gag₆(X, Y, Left, Right, Left1, insert_3_out_gag₃(X, Left, Left1)) -> insert_3_out_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(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:

insert_3_in_gag₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_gag₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_gag₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_gag₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_gag₆(X, Y, Left, Right, Left1, less_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(underscore)) -> less_2_out_gg₂(0_0, s_1₁(underscore))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_gg₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_gg₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_gag₆(X, Y, Left, Right, Left1, less_2_out_gg₂(X, Y)) -> if_insert_3_in_2_gag₆(X, Y, Left, Right, Left1, insert_3_in_gag₃(X, Left, Left1))
insert_3_in_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_gag₆(X, Y, Left, Right, Right1, less_2_in_gg₂(Y, X))
if_insert_3_in_3_gag₆(X, Y, Left, Right, Right1, less_2_out_gg₂(Y, X)) -> if_insert_3_in_4_gag₆(X, Y, Left, Right, Right1, insert_3_in_gag₃(X, Right, Right1))
if_insert_3_in_4_gag₆(X, Y, Left, Right, Right1, insert_3_out_gag₃(X, Right, Right1)) -> insert_3_out_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))
if_insert_3_in_2_gag₆(X, Y, Left, Right, Left1, insert_3_out_gag₃(X, Left, Left1)) -> insert_3_out_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))

INSERT_3_IN_GAG₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GAG₆(X, Y, Left, Right, Left1, less_2_in_gg₂(X, Y))
INSERT_3_IN_GAG₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> LESS_2_IN_GG₂(X, Y)
LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_LESS_2_IN_1_GG₃(X, Y, less_2_in_gg₂(X, Y))
LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_GG₂(X, Y)
IF_INSERT_3_IN_1_GAG₆(X, Y, Left, Right, Left1, less_2_out_gg₂(X, Y)) -> IF_INSERT_3_IN_2_GAG₆(X, Y, Left, Right, Left1, insert_3_in_gag₃(X, Left, Left1))
IF_INSERT_3_IN_1_GAG₆(X, Y, Left, Right, Left1, less_2_out_gg₂(X, Y)) -> INSERT_3_IN_GAG₃(X, Left, Left1)
INSERT_3_IN_GAG₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GAG₆(X, Y, Left, Right, Right1, less_2_in_gg₂(Y, X))
INSERT_3_IN_GAG₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> LESS_2_IN_GG₂(Y, X)
IF_INSERT_3_IN_3_GAG₆(X, Y, Left, Right, Right1, less_2_out_gg₂(Y, X)) -> IF_INSERT_3_IN_4_GAG₆(X, Y, Left, Right, Right1, insert_3_in_gag₃(X, Right, Right1))
IF_INSERT_3_IN_3_GAG₆(X, Y, Left, Right, Right1, less_2_out_gg₂(Y, X)) -> INSERT_3_IN_GAG₃(X, Right, Right1)

The TRS R consists of the following rules:

insert_3_in_gag₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_gag₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_gag₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_gag₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_gag₆(X, Y, Left, Right, Left1, less_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(underscore)) -> less_2_out_gg₂(0_0, s_1₁(underscore))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_gg₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_gg₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_gag₆(X, Y, Left, Right, Left1, less_2_out_gg₂(X, Y)) -> if_insert_3_in_2_gag₆(X, Y, Left, Right, Left1, insert_3_in_gag₃(X, Left, Left1))
insert_3_in_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_gag₆(X, Y, Left, Right, Right1, less_2_in_gg₂(Y, X))
if_insert_3_in_3_gag₆(X, Y, Left, Right, Right1, less_2_out_gg₂(Y, X)) -> if_insert_3_in_4_gag₆(X, Y, Left, Right, Right1, insert_3_in_gag₃(X, Right, Right1))
if_insert_3_in_4_gag₆(X, Y, Left, Right, Right1, insert_3_out_gag₃(X, Right, Right1)) -> insert_3_out_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))
if_insert_3_in_2_gag₆(X, Y, Left, Right, Left1, insert_3_out_gag₃(X, Left, Left1)) -> insert_3_out_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gag₃(x1, x2, x3) = insert_3_in_gag₂(x1, x3)
void_0 = void_0
tree_3₃(x1, x2, x3) = tree_3₃(x1, x2, x3)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
insert_3_out_gag₃(x1, x2, x3) = insert_3_out_gag₁(x2)
if_insert_3_in_1_gag₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_1_gag₅(x1, x2, x4, x5, x6)
less_2_in_gg₂(x1, x2) = less_2_in_gg₂(x1, x2)
less_2_out_gg₂(x1, x2) = less_2_out_gg
if_less_2_in_1_gg₃(x1, x2, x3) = if_less_2_in_1_gg₁(x3)
if_insert_3_in_2_gag₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_2_gag₃(x2, x4, x6)
if_insert_3_in_3_gag₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_3_gag₅(x1, x2, x3, x5, x6)
if_insert_3_in_4_gag₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_4_gag₃(x2, x3, x6)
IF_INSERT_3_IN_4_GAG₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_4_GAG₃(x2, x3, x6)
IF_LESS_2_IN_1_GG₃(x1, x2, x3) = IF_LESS_2_IN_1_GG₁(x3)
IF_INSERT_3_IN_2_GAG₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_2_GAG₃(x2, x4, x6)
IF_INSERT_3_IN_3_GAG₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_3_GAG₅(x1, x2, x3, x5, x6)
IF_INSERT_3_IN_1_GAG₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_1_GAG₅(x1, x2, x4, x5, x6)
INSERT_3_IN_GAG₃(x1, x2, x3) = INSERT_3_IN_GAG₂(x1, x3)
LESS_2_IN_GG₂(x1, x2) = LESS_2_IN_GG₂(x1, x2)

We have to consider all (P,R,Pi)-chains

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

INSERT_3_IN_GAG₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GAG₆(X, Y, Left, Right, Left1, less_2_in_gg₂(X, Y))
INSERT_3_IN_GAG₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> LESS_2_IN_GG₂(X, Y)
LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_LESS_2_IN_1_GG₃(X, Y, less_2_in_gg₂(X, Y))
LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_GG₂(X, Y)
IF_INSERT_3_IN_1_GAG₆(X, Y, Left, Right, Left1, less_2_out_gg₂(X, Y)) -> IF_INSERT_3_IN_2_GAG₆(X, Y, Left, Right, Left1, insert_3_in_gag₃(X, Left, Left1))
IF_INSERT_3_IN_1_GAG₆(X, Y, Left, Right, Left1, less_2_out_gg₂(X, Y)) -> INSERT_3_IN_GAG₃(X, Left, Left1)
INSERT_3_IN_GAG₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GAG₆(X, Y, Left, Right, Right1, less_2_in_gg₂(Y, X))
INSERT_3_IN_GAG₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> LESS_2_IN_GG₂(Y, X)
IF_INSERT_3_IN_3_GAG₆(X, Y, Left, Right, Right1, less_2_out_gg₂(Y, X)) -> IF_INSERT_3_IN_4_GAG₆(X, Y, Left, Right, Right1, insert_3_in_gag₃(X, Right, Right1))
IF_INSERT_3_IN_3_GAG₆(X, Y, Left, Right, Right1, less_2_out_gg₂(Y, X)) -> INSERT_3_IN_GAG₃(X, Right, Right1)

The TRS R consists of the following rules:

insert_3_in_gag₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_gag₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_gag₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_gag₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_gag₆(X, Y, Left, Right, Left1, less_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(underscore)) -> less_2_out_gg₂(0_0, s_1₁(underscore))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_gg₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_gg₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_gag₆(X, Y, Left, Right, Left1, less_2_out_gg₂(X, Y)) -> if_insert_3_in_2_gag₆(X, Y, Left, Right, Left1, insert_3_in_gag₃(X, Left, Left1))
insert_3_in_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_gag₆(X, Y, Left, Right, Right1, less_2_in_gg₂(Y, X))
if_insert_3_in_3_gag₆(X, Y, Left, Right, Right1, less_2_out_gg₂(Y, X)) -> if_insert_3_in_4_gag₆(X, Y, Left, Right, Right1, insert_3_in_gag₃(X, Right, Right1))
if_insert_3_in_4_gag₆(X, Y, Left, Right, Right1, insert_3_out_gag₃(X, Right, Right1)) -> insert_3_out_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))
if_insert_3_in_2_gag₆(X, Y, Left, Right, Left1, insert_3_out_gag₃(X, Left, Left1)) -> insert_3_out_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_GG₂(X, Y)

The TRS R consists of the following rules:

insert_3_in_gag₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_gag₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_gag₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_gag₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_gag₆(X, Y, Left, Right, Left1, less_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(underscore)) -> less_2_out_gg₂(0_0, s_1₁(underscore))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_gg₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_gg₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_gag₆(X, Y, Left, Right, Left1, less_2_out_gg₂(X, Y)) -> if_insert_3_in_2_gag₆(X, Y, Left, Right, Left1, insert_3_in_gag₃(X, Left, Left1))
insert_3_in_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_gag₆(X, Y, Left, Right, Right1, less_2_in_gg₂(Y, X))
if_insert_3_in_3_gag₆(X, Y, Left, Right, Right1, less_2_out_gg₂(Y, X)) -> if_insert_3_in_4_gag₆(X, Y, Left, Right, Right1, insert_3_in_gag₃(X, Right, Right1))
if_insert_3_in_4_gag₆(X, Y, Left, Right, Right1, insert_3_out_gag₃(X, Right, Right1)) -> insert_3_out_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))
if_insert_3_in_2_gag₆(X, Y, Left, Right, Left1, insert_3_out_gag₃(X, Left, Left1)) -> insert_3_out_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_GG₂(X, Y)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem 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:

LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_GG₂(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {LESS_2_IN_GG₂}.
By using the subterm criterion together with the size-change analysis 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_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_GG₂(X, Y)
The graph contains the following edges 1 > 1, 2 > 2

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

INSERT_3_IN_GAG₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GAG₆(X, Y, Left, Right, Right1, less_2_in_gg₂(Y, X))
IF_INSERT_3_IN_3_GAG₆(X, Y, Left, Right, Right1, less_2_out_gg₂(Y, X)) -> INSERT_3_IN_GAG₃(X, Right, Right1)
IF_INSERT_3_IN_1_GAG₆(X, Y, Left, Right, Left1, less_2_out_gg₂(X, Y)) -> INSERT_3_IN_GAG₃(X, Left, Left1)
INSERT_3_IN_GAG₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GAG₆(X, Y, Left, Right, Left1, less_2_in_gg₂(X, Y))

The TRS R consists of the following rules:

insert_3_in_gag₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_gag₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_gag₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_gag₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_gag₆(X, Y, Left, Right, Left1, less_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(underscore)) -> less_2_out_gg₂(0_0, s_1₁(underscore))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_gg₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_gg₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_gag₆(X, Y, Left, Right, Left1, less_2_out_gg₂(X, Y)) -> if_insert_3_in_2_gag₆(X, Y, Left, Right, Left1, insert_3_in_gag₃(X, Left, Left1))
insert_3_in_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_gag₆(X, Y, Left, Right, Right1, less_2_in_gg₂(Y, X))
if_insert_3_in_3_gag₆(X, Y, Left, Right, Right1, less_2_out_gg₂(Y, X)) -> if_insert_3_in_4_gag₆(X, Y, Left, Right, Right1, insert_3_in_gag₃(X, Right, Right1))
if_insert_3_in_4_gag₆(X, Y, Left, Right, Right1, insert_3_out_gag₃(X, Right, Right1)) -> insert_3_out_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))
if_insert_3_in_2_gag₆(X, Y, Left, Right, Left1, insert_3_out_gag₃(X, Left, Left1)) -> insert_3_out_gag₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gag₃(x1, x2, x3) = insert_3_in_gag₂(x1, x3)
void_0 = void_0
tree_3₃(x1, x2, x3) = tree_3₃(x1, x2, x3)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
insert_3_out_gag₃(x1, x2, x3) = insert_3_out_gag₁(x2)
if_insert_3_in_1_gag₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_1_gag₅(x1, x2, x4, x5, x6)
less_2_in_gg₂(x1, x2) = less_2_in_gg₂(x1, x2)
less_2_out_gg₂(x1, x2) = less_2_out_gg
if_less_2_in_1_gg₃(x1, x2, x3) = if_less_2_in_1_gg₁(x3)
if_insert_3_in_2_gag₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_2_gag₃(x2, x4, x6)
if_insert_3_in_3_gag₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_3_gag₅(x1, x2, x3, x5, x6)
if_insert_3_in_4_gag₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_4_gag₃(x2, x3, x6)
IF_INSERT_3_IN_3_GAG₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_3_GAG₅(x1, x2, x3, x5, x6)
IF_INSERT_3_IN_1_GAG₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_1_GAG₅(x1, x2, x4, x5, x6)
INSERT_3_IN_GAG₃(x1, x2, x3) = INSERT_3_IN_GAG₂(x1, x3)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting 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:

INSERT_3_IN_GAG₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GAG₆(X, Y, Left, Right, Right1, less_2_in_gg₂(Y, X))
IF_INSERT_3_IN_3_GAG₆(X, Y, Left, Right, Right1, less_2_out_gg₂(Y, X)) -> INSERT_3_IN_GAG₃(X, Right, Right1)
IF_INSERT_3_IN_1_GAG₆(X, Y, Left, Right, Left1, less_2_out_gg₂(X, Y)) -> INSERT_3_IN_GAG₃(X, Left, Left1)
INSERT_3_IN_GAG₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GAG₆(X, Y, Left, Right, Left1, less_2_in_gg₂(X, Y))

The TRS R consists of the following rules:

less_2_in_gg₂(0_0, s_1₁(underscore)) -> less_2_out_gg₂(0_0, s_1₁(underscore))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_gg₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_gg₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))

The argument filtering Pi contains the following mapping:
tree_3₃(x1, x2, x3) = tree_3₃(x1, x2, x3)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
less_2_in_gg₂(x1, x2) = less_2_in_gg₂(x1, x2)
less_2_out_gg₂(x1, x2) = less_2_out_gg
if_less_2_in_1_gg₃(x1, x2, x3) = if_less_2_in_1_gg₁(x3)
IF_INSERT_3_IN_3_GAG₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_3_GAG₅(x1, x2, x3, x5, x6)
IF_INSERT_3_IN_1_GAG₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_1_GAG₅(x1, x2, x4, x5, x6)
INSERT_3_IN_GAG₃(x1, x2, x3) = INSERT_3_IN_GAG₂(x1, x3)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem 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:

INSERT_3_IN_GAG₂(X, tree_3₃(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GAG₅(X, Y, Left, Right1, less_2_in_gg₂(Y, X))
IF_INSERT_3_IN_3_GAG₅(X, Y, Left, Right1, less_2_out_gg) -> INSERT_3_IN_GAG₂(X, Right1)
IF_INSERT_3_IN_1_GAG₅(X, Y, Right, Left1, less_2_out_gg) -> INSERT_3_IN_GAG₂(X, Left1)
INSERT_3_IN_GAG₂(X, tree_3₃(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GAG₅(X, Y, Right, Left1, less_2_in_gg₂(X, Y))

The TRS R consists of the following rules:

less_2_in_gg₂(0_0, s_1₁(underscore)) -> less_2_out_gg
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₁(less_2_in_gg₂(X, Y))
if_less_2_in_1_gg₁(less_2_out_gg) -> less_2_out_gg

The set Q consists of the following terms:

less_2_in_gg₂(x0, x1)
if_less_2_in_1_gg₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_INSERT_3_IN_3_GAG₅, INSERT_3_IN_GAG₂, IF_INSERT_3_IN_1_GAG₅}.
By using the subterm criterion together with the size-change analysis 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:

IF_INSERT_3_IN_3_GAG₅(X, Y, Left, Right1, less_2_out_gg) -> INSERT_3_IN_GAG₂(X, Right1)
The graph contains the following edges 1 >= 1, 4 >= 2
IF_INSERT_3_IN_1_GAG₅(X, Y, Right, Left1, less_2_out_gg) -> INSERT_3_IN_GAG₂(X, Left1)
The graph contains the following edges 1 >= 1, 4 >= 2
INSERT_3_IN_GAG₂(X, tree_3₃(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GAG₅(X, Y, Left, Right1, less_2_in_gg₂(Y, X))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4
INSERT_3_IN_GAG₂(X, tree_3₃(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GAG₅(X, Y, Right, Left1, less_2_in_gg₂(X, Y))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4