Left Termination proof of /tmp/tmp5VIeLV/delete-bff.pl

Left Termination of the query pattern delete_in_3(g, a, 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(g,a,a).

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,f)
delmin_in: (f,f,f)
less_in: (b,f) (f,f) (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

delete_in_gaa(X, tree(X, void, Right), Right) → delete_out_gaa(X, tree(X, void, Right), Right)
delete_in_gaa(X, tree(X, Left, void), Left) → delete_out_gaa(X, tree(X, Left, void), Left)
delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
delmin_in_aaa(tree(Y, void, Right), Y, Right) → delmin_out_aaa(tree(Y, void, Right), Y, Right)
delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) → delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) → delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U7_aa(X, Y, less_in_aa(X, Y))
U7_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U7_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_aa(X, Y))
U7_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right))

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_gaa(X, tree(X, void, Right), Right) → delete_out_gaa(X, tree(X, void, Right), Right)
delete_in_gaa(X, tree(X, Left, void), Left) → delete_out_gaa(X, tree(X, Left, void), Left)
delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
delmin_in_aaa(tree(Y, void, Right), Y, Right) → delmin_out_aaa(tree(Y, void, Right), Y, Right)
delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) → delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) → delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U7_aa(X, Y, less_in_aa(X, Y))
U7_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U7_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_aa(X, Y))
U7_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right))

DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GAA(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_AAA(Right, Y, Right1)
DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_AAA(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_AAA(Left, Y, Left1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y))
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GA(X, Y)
LESS_IN_GA(s(X), s(Y)) → U7_GA(X, Y, less_in_aa(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U7_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_GAA(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → DELETE_IN_GAA(X, Left, Left1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X))
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_AG(Y, X)
LESS_IN_AG(s(X), s(Y)) → U7_AG(X, Y, less_in_aa(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AA(X, Y)
U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_GAA(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → DELETE_IN_GAA(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_gaa(X, tree(X, void, Right), Right) → delete_out_gaa(X, tree(X, void, Right), Right)
delete_in_gaa(X, tree(X, Left, void), Left) → delete_out_gaa(X, tree(X, Left, void), Left)
delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
delmin_in_aaa(tree(Y, void, Right), Y, Right) → delmin_out_aaa(tree(Y, void, Right), Y, Right)
delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) → delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) → delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U7_aa(X, Y, less_in_aa(X, Y))
U7_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U7_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_aa(X, Y))
U7_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3) = delete_out_gaa(x2)
tree(x1, x2, x3) = tree
U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6)
delmin_in_aaa(x1, x2, x3) = delmin_in_aaa
delmin_out_aaa(x1, x2, x3) = delmin_out_aaa(x1)
U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7)
U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6)
less_in_ga(x1, x2) = less_in_ga(x1)
0 = 0
less_out_ga(x1, x2) = less_out_ga(x2)
s(x1) = s
U7_ga(x1, x2, x3) = U7_ga(x3)
less_in_aa(x1, x2) = less_in_aa
less_out_aa(x1, x2) = less_out_aa(x1, x2)
U7_aa(x1, x2, x3) = U7_aa(x3)
U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x6)
U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6)
less_in_ag(x1, x2) = less_in_ag(x2)
less_out_ag(x1, x2) = less_out_ag(x1)
U7_ag(x1, x2, x3) = U7_ag(x3)
U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x6)
LESS_IN_AA(x1, x2) = LESS_IN_AA
U7_AG(x1, x2, x3) = U7_AG(x3)
U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6)
U7_GA(x1, x2, x3) = U7_GA(x3)
U3_GAA(x1, x2, x3, x4, x5, x6) = U3_GAA(x6)
LESS_IN_GA(x1, x2) = LESS_IN_GA(x1)
LESS_IN_AG(x1, x2) = LESS_IN_AG(x2)
DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1)
U6_AAA(x1, x2, x3, x4, x5, x6, x7) = U6_AAA(x7)
U7_AA(x1, x2, x3) = U7_AA(x3)
U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6)
U5_GAA(x1, x2, x3, x4, x5, x6) = U5_GAA(x6)
U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x6)
DELMIN_IN_AAA(x1, x2, x3) = DELMIN_IN_AAA

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_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GAA(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_AAA(Right, Y, Right1)
DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_AAA(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_AAA(Left, Y, Left1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y))
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GA(X, Y)
LESS_IN_GA(s(X), s(Y)) → U7_GA(X, Y, less_in_aa(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U7_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_GAA(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → DELETE_IN_GAA(X, Left, Left1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X))
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_AG(Y, X)
LESS_IN_AG(s(X), s(Y)) → U7_AG(X, Y, less_in_aa(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AA(X, Y)
U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_GAA(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → DELETE_IN_GAA(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_gaa(X, tree(X, void, Right), Right) → delete_out_gaa(X, tree(X, void, Right), Right)
delete_in_gaa(X, tree(X, Left, void), Left) → delete_out_gaa(X, tree(X, Left, void), Left)
delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
delmin_in_aaa(tree(Y, void, Right), Y, Right) → delmin_out_aaa(tree(Y, void, Right), Y, Right)
delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) → delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) → delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U7_aa(X, Y, less_in_aa(X, Y))
U7_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U7_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_aa(X, Y))
U7_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

The TRS R consists of the following rules:

delete_in_gaa(X, tree(X, void, Right), Right) → delete_out_gaa(X, tree(X, void, Right), Right)
delete_in_gaa(X, tree(X, Left, void), Left) → delete_out_gaa(X, tree(X, Left, void), Left)
delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
delmin_in_aaa(tree(Y, void, Right), Y, Right) → delmin_out_aaa(tree(Y, void, Right), Y, Right)
delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) → delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) → delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U7_aa(X, Y, less_in_aa(X, Y))
U7_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U7_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_aa(X, Y))
U7_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right))

↳ 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:

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

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

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

                        ↳ NonTerminationProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LESS_IN_AA → LESS_IN_AA

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_AA → LESS_IN_AA

The TRS R consists of the following rules:none

s = LESS_IN_AA evaluates to t =LESS_IN_AA

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_AA to LESS_IN_AA.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

delete_in_gaa(X, tree(X, void, Right), Right) → delete_out_gaa(X, tree(X, void, Right), Right)
delete_in_gaa(X, tree(X, Left, void), Left) → delete_out_gaa(X, tree(X, Left, void), Left)
delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
delmin_in_aaa(tree(Y, void, Right), Y, Right) → delmin_out_aaa(tree(Y, void, Right), Y, Right)
delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) → delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) → delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U7_aa(X, Y, less_in_aa(X, Y))
U7_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U7_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_aa(X, Y))
U7_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right))

↳ 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:

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

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

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:

DELMIN_IN_AAA → DELMIN_IN_AAA

The TRS R consists of the following rules:none

s = DELMIN_IN_AAA evaluates to t =DELMIN_IN_AAA

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 DELMIN_IN_AAA to DELMIN_IN_AAA.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → DELETE_IN_GAA(X, Right, Right1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X))
U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → DELETE_IN_GAA(X, Left, Left1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y))

The TRS R consists of the following rules:

delete_in_gaa(X, tree(X, void, Right), Right) → delete_out_gaa(X, tree(X, void, Right), Right)
delete_in_gaa(X, tree(X, Left, void), Left) → delete_out_gaa(X, tree(X, Left, void), Left)
delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
delmin_in_aaa(tree(Y, void, Right), Y, Right) → delmin_out_aaa(tree(Y, void, Right), Y, Right)
delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) → delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) → delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U7_aa(X, Y, less_in_aa(X, Y))
U7_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U7_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_aa(X, Y))
U7_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3) = delete_out_gaa(x2)
tree(x1, x2, x3) = tree
U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x6)
delmin_in_aaa(x1, x2, x3) = delmin_in_aaa
delmin_out_aaa(x1, x2, x3) = delmin_out_aaa(x1)
U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7)
U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6)
less_in_ga(x1, x2) = less_in_ga(x1)
0 = 0
less_out_ga(x1, x2) = less_out_ga(x2)
s(x1) = s
U7_ga(x1, x2, x3) = U7_ga(x3)
less_in_aa(x1, x2) = less_in_aa
less_out_aa(x1, x2) = less_out_aa(x1, x2)
U7_aa(x1, x2, x3) = U7_aa(x3)
U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x6)
U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6)
less_in_ag(x1, x2) = less_in_ag(x2)
less_out_ag(x1, x2) = less_out_ag(x1)
U7_ag(x1, x2, x3) = U7_ag(x3)
U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x6)
U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6)
DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, 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:

U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → DELETE_IN_GAA(X, Right, Right1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X))
U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → DELETE_IN_GAA(X, Left, Left1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y))

The TRS R consists of the following rules:

less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_aa(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_aa(X, Y))
U7_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U7_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U7_aa(X, Y, less_in_aa(X, Y))
U7_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3) = tree
less_in_ga(x1, x2) = less_in_ga(x1)
0 = 0
less_out_ga(x1, x2) = less_out_ga(x2)
s(x1) = s
U7_ga(x1, x2, x3) = U7_ga(x3)
less_in_aa(x1, x2) = less_in_aa
less_out_aa(x1, x2) = less_out_aa(x1, x2)
U7_aa(x1, x2, x3) = U7_aa(x3)
less_in_ag(x1, x2) = less_in_ag(x2)
less_out_ag(x1, x2) = less_out_ag(x1)
U7_ag(x1, x2, x3) = U7_ag(x3)
U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6)
DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, 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

                        ↳ Narrowing

  ↳ PrologToPiTRSProof

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

DELETE_IN_GAA(X) → U2_GAA(X, less_in_ga(X))
U2_GAA(X, less_out_ga(Y)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(X) → U4_GAA(X, less_in_ag(X))
U4_GAA(X, less_out_ag(Y)) → DELETE_IN_GAA(X)

The TRS R consists of the following rules:

less_in_ag(s) → less_out_ag(0)
less_in_ag(s) → U7_ag(less_in_aa)
less_in_ga(0) → less_out_ga(s)
less_in_ga(s) → U7_ga(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s)
less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)

The set Q consists of the following terms:

less_in_ag(x0)
less_in_ga(x0)
U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule DELETE_IN_GAA(X) → U4_GAA(X, less_in_ag(X)) at position [1] we obtained the following new rules:

DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0))
DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
U2_GAA(X, less_out_ga(Y)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(X) → U2_GAA(X, less_in_ga(X))
DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0))
U4_GAA(X, less_out_ag(Y)) → DELETE_IN_GAA(X)

The TRS R consists of the following rules:

less_in_ag(s) → less_out_ag(0)
less_in_ag(s) → U7_ag(less_in_aa)
less_in_ga(0) → less_out_ga(s)
less_in_ga(s) → U7_ga(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s)
less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)

The set Q consists of the following terms:

less_in_ag(x0)
less_in_ga(x0)
U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

  ↳ PrologToPiTRSProof

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

DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
DELETE_IN_GAA(X) → U2_GAA(X, less_in_ga(X))
U2_GAA(X, less_out_ga(Y)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0))
U4_GAA(X, less_out_ag(Y)) → DELETE_IN_GAA(X)

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
less_in_ga(0) → less_out_ga(s)
less_in_ga(s) → U7_ga(less_in_aa)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s)

The set Q consists of the following terms:

less_in_ag(x0)
less_in_ga(x0)
U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

less_in_ag(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

  ↳ PrologToPiTRSProof

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

DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
U2_GAA(X, less_out_ga(Y)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(X) → U2_GAA(X, less_in_ga(X))
DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0))
U4_GAA(X, less_out_ag(Y)) → DELETE_IN_GAA(X)

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
less_in_ga(0) → less_out_ga(s)
less_in_ga(s) → U7_ga(less_in_aa)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s)

The set Q consists of the following terms:

less_in_ga(x0)
U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule DELETE_IN_GAA(X) → U2_GAA(X, less_in_ga(X)) at position [1] we obtained the following new rules:

DELETE_IN_GAA(s) → U2_GAA(s, U7_ga(less_in_aa))
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(s))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
U2_GAA(X, less_out_ga(Y)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(s))
DELETE_IN_GAA(s) → U2_GAA(s, U7_ga(less_in_aa))
DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0))
U4_GAA(X, less_out_ag(Y)) → DELETE_IN_GAA(X)

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
less_in_ga(0) → less_out_ga(s)
less_in_ga(s) → U7_ga(less_in_aa)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s)

The set Q consists of the following terms:

less_in_ga(x0)
U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

  ↳ PrologToPiTRSProof

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

DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
U2_GAA(X, less_out_ga(Y)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(s))
DELETE_IN_GAA(s) → U2_GAA(s, U7_ga(less_in_aa))
DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0))
U4_GAA(X, less_out_ag(Y)) → DELETE_IN_GAA(X)

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s)

The set Q consists of the following terms:

less_in_ga(x0)
U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

less_in_ga(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

  ↳ PrologToPiTRSProof

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

DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
U2_GAA(X, less_out_ga(Y)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(s))
DELETE_IN_GAA(s) → U2_GAA(s, U7_ga(less_in_aa))
DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0))
U4_GAA(X, less_out_ag(Y)) → DELETE_IN_GAA(X)

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s)

The set Q consists of the following terms:

U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GAA(X, less_out_ga(Y)) → DELETE_IN_GAA(X) we obtained the following new rules:

U2_GAA(0, less_out_ga(s)) → DELETE_IN_GAA(0)
U2_GAA(s, less_out_ga(x1)) → DELETE_IN_GAA(s)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

U2_GAA(0, less_out_ga(s)) → DELETE_IN_GAA(0)
DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(s))
DELETE_IN_GAA(s) → U2_GAA(s, U7_ga(less_in_aa))
U2_GAA(s, less_out_ga(x1)) → DELETE_IN_GAA(s)
DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0))
U4_GAA(X, less_out_ag(Y)) → DELETE_IN_GAA(X)

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s)

The set Q consists of the following terms:

U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ AND

                                                        ↳ QDP

                                                          ↳ UsableRulesProof

                                                        ↳ QDP

  ↳ PrologToPiTRSProof

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

U2_GAA(0, less_out_ga(s)) → DELETE_IN_GAA(0)
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(s))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s)

The set Q consists of the following terms:

U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ AND

                                                        ↳ QDP

                                                          ↳ UsableRulesProof

                                                            ↳ QDP

                                                              ↳ QReductionProof

                                                        ↳ QDP

  ↳ PrologToPiTRSProof

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

U2_GAA(0, less_out_ga(s)) → DELETE_IN_GAA(0)
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(s))

R is empty.
The set Q consists of the following terms:

U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ AND

                                                        ↳ QDP

                                                          ↳ UsableRulesProof

                                                            ↳ QDP

                                                              ↳ QReductionProof

                                                                ↳ QDP

                                                                  ↳ NonTerminationProof

                                                        ↳ QDP

  ↳ PrologToPiTRSProof

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

U2_GAA(0, less_out_ga(s)) → DELETE_IN_GAA(0)
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(s))

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 narrowing to the left:

The TRS P consists of the following rules:

U2_GAA(0, less_out_ga(s)) → DELETE_IN_GAA(0)
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(s))

The TRS R consists of the following rules:none

s = DELETE_IN_GAA(0) evaluates to t =DELETE_IN_GAA(0)

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(s))
with rule DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(s)) at position [] and matcher [ ]

U2_GAA(0, less_out_ga(s)) → DELETE_IN_GAA(0)
with rule U2_GAA(0, less_out_ga(s)) → DELETE_IN_GAA(0)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ AND

                                                        ↳ QDP

                                                        ↳ QDP

                                                          ↳ Instantiation

  ↳ PrologToPiTRSProof

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

DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
DELETE_IN_GAA(s) → U2_GAA(s, U7_ga(less_in_aa))
U2_GAA(s, less_out_ga(x1)) → DELETE_IN_GAA(s)
DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0))
U4_GAA(X, less_out_ag(Y)) → DELETE_IN_GAA(X)

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s)

The set Q consists of the following terms:

U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U4_GAA(X, less_out_ag(Y)) → DELETE_IN_GAA(X) we obtained the following new rules:

U4_GAA(s, less_out_ag(0)) → DELETE_IN_GAA(s)
U4_GAA(s, less_out_ag(x1)) → DELETE_IN_GAA(s)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ AND

                                                        ↳ QDP

                                                        ↳ QDP

                                                          ↳ Instantiation

                                                            ↳ QDP

                                                              ↳ NonTerminationProof

  ↳ PrologToPiTRSProof

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

DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
U4_GAA(s, less_out_ag(0)) → DELETE_IN_GAA(s)
DELETE_IN_GAA(s) → U2_GAA(s, U7_ga(less_in_aa))
DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0))
U2_GAA(s, less_out_ga(x1)) → DELETE_IN_GAA(s)
U4_GAA(s, less_out_ag(x1)) → DELETE_IN_GAA(s)

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s)

The set Q consists of the following terms:

U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

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 narrowing to the left:

The TRS P consists of the following rules:

DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
U4_GAA(s, less_out_ag(0)) → DELETE_IN_GAA(s)
DELETE_IN_GAA(s) → U2_GAA(s, U7_ga(less_in_aa))
DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0))
U2_GAA(s, less_out_ga(x1)) → DELETE_IN_GAA(s)
U4_GAA(s, less_out_ag(x1)) → DELETE_IN_GAA(s)

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s)

s = DELETE_IN_GAA(s) evaluates to t =DELETE_IN_GAA(s)

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0))
with rule DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0)) at position [] and matcher [ ]

U4_GAA(s, less_out_ag(0)) → DELETE_IN_GAA(s)
with rule U4_GAA(s, less_out_ag(0)) → DELETE_IN_GAA(s)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

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,f)
delmin_in: (f,f,f)
less_in: (b,f) (f,f) (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

delete_in_gaa(X, tree(X, void, Right), Right) → delete_out_gaa(X, tree(X, void, Right), Right)
delete_in_gaa(X, tree(X, Left, void), Left) → delete_out_gaa(X, tree(X, Left, void), Left)
delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
delmin_in_aaa(tree(Y, void, Right), Y, Right) → delmin_out_aaa(tree(Y, void, Right), Y, Right)
delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) → delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) → delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U7_aa(X, Y, less_in_aa(X, Y))
U7_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U7_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_aa(X, Y))
U7_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right))

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_gaa(X, tree(X, void, Right), Right) → delete_out_gaa(X, tree(X, void, Right), Right)
delete_in_gaa(X, tree(X, Left, void), Left) → delete_out_gaa(X, tree(X, Left, void), Left)
delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
delmin_in_aaa(tree(Y, void, Right), Y, Right) → delmin_out_aaa(tree(Y, void, Right), Y, Right)
delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) → delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) → delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U7_aa(X, Y, less_in_aa(X, Y))
U7_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U7_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_aa(X, Y))
U7_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right))

DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GAA(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_AAA(Right, Y, Right1)
DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_AAA(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_AAA(Left, Y, Left1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y))
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GA(X, Y)
LESS_IN_GA(s(X), s(Y)) → U7_GA(X, Y, less_in_aa(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U7_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_GAA(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → DELETE_IN_GAA(X, Left, Left1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X))
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_AG(Y, X)
LESS_IN_AG(s(X), s(Y)) → U7_AG(X, Y, less_in_aa(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AA(X, Y)
U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_GAA(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → DELETE_IN_GAA(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_gaa(X, tree(X, void, Right), Right) → delete_out_gaa(X, tree(X, void, Right), Right)
delete_in_gaa(X, tree(X, Left, void), Left) → delete_out_gaa(X, tree(X, Left, void), Left)
delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
delmin_in_aaa(tree(Y, void, Right), Y, Right) → delmin_out_aaa(tree(Y, void, Right), Y, Right)
delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) → delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) → delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U7_aa(X, Y, less_in_aa(X, Y))
U7_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U7_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_aa(X, Y))
U7_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3) = delete_out_gaa(x1, x2)
tree(x1, x2, x3) = tree
U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6)
delmin_in_aaa(x1, x2, x3) = delmin_in_aaa
delmin_out_aaa(x1, x2, x3) = delmin_out_aaa(x1)
U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7)
U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6)
less_in_ga(x1, x2) = less_in_ga(x1)
0 = 0
less_out_ga(x1, x2) = less_out_ga(x1, x2)
s(x1) = s
U7_ga(x1, x2, x3) = U7_ga(x3)
less_in_aa(x1, x2) = less_in_aa
less_out_aa(x1, x2) = less_out_aa(x1, x2)
U7_aa(x1, x2, x3) = U7_aa(x3)
U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6)
U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6)
less_in_ag(x1, x2) = less_in_ag(x2)
less_out_ag(x1, x2) = less_out_ag(x1, x2)
U7_ag(x1, x2, x3) = U7_ag(x3)
U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x1, x6)
LESS_IN_AA(x1, x2) = LESS_IN_AA
U7_AG(x1, x2, x3) = U7_AG(x3)
U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6)
U7_GA(x1, x2, x3) = U7_GA(x3)
U3_GAA(x1, x2, x3, x4, x5, x6) = U3_GAA(x1, x6)
LESS_IN_GA(x1, x2) = LESS_IN_GA(x1)
LESS_IN_AG(x1, x2) = LESS_IN_AG(x2)
DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1)
U6_AAA(x1, x2, x3, x4, x5, x6, x7) = U6_AAA(x7)
U7_AA(x1, x2, x3) = U7_AA(x3)
U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, x6)
U5_GAA(x1, x2, x3, x4, x5, x6) = U5_GAA(x1, x6)
U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x6)
DELMIN_IN_AAA(x1, x2, x3) = DELMIN_IN_AAA

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_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GAA(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_AAA(Right, Y, Right1)
DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_AAA(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_AAA(Left, Y, Left1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y))
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GA(X, Y)
LESS_IN_GA(s(X), s(Y)) → U7_GA(X, Y, less_in_aa(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U7_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_GAA(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → DELETE_IN_GAA(X, Left, Left1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X))
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_AG(Y, X)
LESS_IN_AG(s(X), s(Y)) → U7_AG(X, Y, less_in_aa(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AA(X, Y)
U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_GAA(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → DELETE_IN_GAA(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_gaa(X, tree(X, void, Right), Right) → delete_out_gaa(X, tree(X, void, Right), Right)
delete_in_gaa(X, tree(X, Left, void), Left) → delete_out_gaa(X, tree(X, Left, void), Left)
delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
delmin_in_aaa(tree(Y, void, Right), Y, Right) → delmin_out_aaa(tree(Y, void, Right), Y, Right)
delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) → delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) → delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U7_aa(X, Y, less_in_aa(X, Y))
U7_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U7_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_aa(X, Y))
U7_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

The TRS R consists of the following rules:

delete_in_gaa(X, tree(X, void, Right), Right) → delete_out_gaa(X, tree(X, void, Right), Right)
delete_in_gaa(X, tree(X, Left, void), Left) → delete_out_gaa(X, tree(X, Left, void), Left)
delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
delmin_in_aaa(tree(Y, void, Right), Y, Right) → delmin_out_aaa(tree(Y, void, Right), Y, Right)
delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) → delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) → delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U7_aa(X, Y, less_in_aa(X, Y))
U7_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U7_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_aa(X, Y))
U7_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right))

↳ 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:

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

              ↳ PiDP

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

LESS_IN_AA → LESS_IN_AA

The TRS R consists of the following rules:none

s = LESS_IN_AA evaluates to t =LESS_IN_AA

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_AA to LESS_IN_AA.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

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

The TRS R consists of the following rules:

delete_in_gaa(X, tree(X, void, Right), Right) → delete_out_gaa(X, tree(X, void, Right), Right)
delete_in_gaa(X, tree(X, Left, void), Left) → delete_out_gaa(X, tree(X, Left, void), Left)
delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
delmin_in_aaa(tree(Y, void, Right), Y, Right) → delmin_out_aaa(tree(Y, void, Right), Y, Right)
delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) → delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) → delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U7_aa(X, Y, less_in_aa(X, Y))
U7_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U7_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_aa(X, Y))
U7_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right))

↳ 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:

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

↳ 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:

DELMIN_IN_AAA → DELMIN_IN_AAA

The TRS R consists of the following rules:none

s = DELMIN_IN_AAA evaluates to t =DELMIN_IN_AAA

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 DELMIN_IN_AAA to DELMIN_IN_AAA.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → DELETE_IN_GAA(X, Right, Right1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X))
U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → DELETE_IN_GAA(X, Left, Left1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y))

The TRS R consists of the following rules:

delete_in_gaa(X, tree(X, void, Right), Right) → delete_out_gaa(X, tree(X, void, Right), Right)
delete_in_gaa(X, tree(X, Left, void), Left) → delete_out_gaa(X, tree(X, Left, void), Left)
delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
delmin_in_aaa(tree(Y, void, Right), Y, Right) → delmin_out_aaa(tree(Y, void, Right), Y, Right)
delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) → delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) → delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U7_aa(X, Y, less_in_aa(X, Y))
U7_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U7_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_aa(X, Y))
U7_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gaa(x1, x2, x3) = delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3) = delete_out_gaa(x1, x2)
tree(x1, x2, x3) = tree
U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x6)
delmin_in_aaa(x1, x2, x3) = delmin_in_aaa
delmin_out_aaa(x1, x2, x3) = delmin_out_aaa(x1)
U6_aaa(x1, x2, x3, x4, x5, x6, x7) = U6_aaa(x7)
U2_gaa(x1, x2, x3, x4, x5, x6) = U2_gaa(x1, x6)
less_in_ga(x1, x2) = less_in_ga(x1)
0 = 0
less_out_ga(x1, x2) = less_out_ga(x1, x2)
s(x1) = s
U7_ga(x1, x2, x3) = U7_ga(x3)
less_in_aa(x1, x2) = less_in_aa
less_out_aa(x1, x2) = less_out_aa(x1, x2)
U7_aa(x1, x2, x3) = U7_aa(x3)
U3_gaa(x1, x2, x3, x4, x5, x6) = U3_gaa(x1, x6)
U4_gaa(x1, x2, x3, x4, x5, x6) = U4_gaa(x1, x6)
less_in_ag(x1, x2) = less_in_ag(x2)
less_out_ag(x1, x2) = less_out_ag(x1, x2)
U7_ag(x1, x2, x3) = U7_ag(x3)
U5_gaa(x1, x2, x3, x4, x5, x6) = U5_gaa(x1, x6)
U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6)
DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, 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:

U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → DELETE_IN_GAA(X, Right, Right1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X))
U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → DELETE_IN_GAA(X, Left, Left1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y))

The TRS R consists of the following rules:

less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_aa(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_aa(X, Y))
U7_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U7_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U7_aa(X, Y, less_in_aa(X, Y))
U7_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3) = tree
less_in_ga(x1, x2) = less_in_ga(x1)
0 = 0
less_out_ga(x1, x2) = less_out_ga(x1, x2)
s(x1) = s
U7_ga(x1, x2, x3) = U7_ga(x3)
less_in_aa(x1, x2) = less_in_aa
less_out_aa(x1, x2) = less_out_aa(x1, x2)
U7_aa(x1, x2, x3) = U7_aa(x3)
less_in_ag(x1, x2) = less_in_ag(x2)
less_out_ag(x1, x2) = less_out_ag(x1, x2)
U7_ag(x1, x2, x3) = U7_ag(x3)
U4_GAA(x1, x2, x3, x4, x5, x6) = U4_GAA(x1, x6)
DELETE_IN_GAA(x1, x2, x3) = DELETE_IN_GAA(x1)
U2_GAA(x1, x2, x3, x4, x5, x6) = U2_GAA(x1, 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

                        ↳ Narrowing

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

U4_GAA(X, less_out_ag(Y, X)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(X) → U2_GAA(X, less_in_ga(X))
U2_GAA(X, less_out_ga(X, Y)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(X) → U4_GAA(X, less_in_ag(X))

The TRS R consists of the following rules:

less_in_ag(s) → less_out_ag(0, s)
less_in_ag(s) → U7_ag(less_in_aa)
less_in_ga(0) → less_out_ga(0, s)
less_in_ga(s) → U7_ga(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s, s)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s, s)
less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)

The set Q consists of the following terms:

less_in_ag(x0)
less_in_ga(x0)
U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule DELETE_IN_GAA(X) → U4_GAA(X, less_in_ag(X)) at position [1] we obtained the following new rules:

DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0, s))
DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

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

U4_GAA(X, less_out_ag(Y, X)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0, s))
DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
DELETE_IN_GAA(X) → U2_GAA(X, less_in_ga(X))
U2_GAA(X, less_out_ga(X, Y)) → DELETE_IN_GAA(X)

The TRS R consists of the following rules:

less_in_ag(s) → less_out_ag(0, s)
less_in_ag(s) → U7_ag(less_in_aa)
less_in_ga(0) → less_out_ga(0, s)
less_in_ga(s) → U7_ga(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s, s)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s, s)
less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)

The set Q consists of the following terms:

less_in_ag(x0)
less_in_ga(x0)
U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

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

U4_GAA(X, less_out_ag(Y, X)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0, s))
DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
DELETE_IN_GAA(X) → U2_GAA(X, less_in_ga(X))
U2_GAA(X, less_out_ga(X, Y)) → DELETE_IN_GAA(X)

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s, s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
less_in_ga(0) → less_out_ga(0, s)
less_in_ga(s) → U7_ga(less_in_aa)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s, s)

The set Q consists of the following terms:

less_in_ag(x0)
less_in_ga(x0)
U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

less_in_ag(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

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

U4_GAA(X, less_out_ag(Y, X)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0, s))
DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
DELETE_IN_GAA(X) → U2_GAA(X, less_in_ga(X))
U2_GAA(X, less_out_ga(X, Y)) → DELETE_IN_GAA(X)

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s, s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
less_in_ga(0) → less_out_ga(0, s)
less_in_ga(s) → U7_ga(less_in_aa)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s, s)

The set Q consists of the following terms:

less_in_ga(x0)
U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule DELETE_IN_GAA(X) → U2_GAA(X, less_in_ga(X)) at position [1] we obtained the following new rules:

DELETE_IN_GAA(s) → U2_GAA(s, U7_ga(less_in_aa))
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(0, s))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

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

U4_GAA(X, less_out_ag(Y, X)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0, s))
DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
U2_GAA(X, less_out_ga(X, Y)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(s) → U2_GAA(s, U7_ga(less_in_aa))
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(0, s))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s, s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
less_in_ga(0) → less_out_ga(0, s)
less_in_ga(s) → U7_ga(less_in_aa)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s, s)

The set Q consists of the following terms:

less_in_ga(x0)
U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

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

U4_GAA(X, less_out_ag(Y, X)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0, s))
DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
U2_GAA(X, less_out_ga(X, Y)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(s) → U2_GAA(s, U7_ga(less_in_aa))
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(0, s))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s, s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s, s)

The set Q consists of the following terms:

less_in_ga(x0)
U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

less_in_ga(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

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

U4_GAA(X, less_out_ag(Y, X)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0, s))
DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
U2_GAA(X, less_out_ga(X, Y)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(s) → U2_GAA(s, U7_ga(less_in_aa))
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(0, s))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s, s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s, s)

The set Q consists of the following terms:

U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U4_GAA(X, less_out_ag(Y, X)) → DELETE_IN_GAA(X) we obtained the following new rules:

U4_GAA(s, less_out_ag(0, s)) → DELETE_IN_GAA(s)
U4_GAA(s, less_out_ag(x1, s)) → DELETE_IN_GAA(s)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

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

DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0, s))
DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
U2_GAA(X, less_out_ga(X, Y)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(s) → U2_GAA(s, U7_ga(less_in_aa))
U4_GAA(s, less_out_ag(0, s)) → DELETE_IN_GAA(s)
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(0, s))
U4_GAA(s, less_out_ag(x1, s)) → DELETE_IN_GAA(s)

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s, s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s, s)

The set Q consists of the following terms:

U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GAA(X, less_out_ga(X, Y)) → DELETE_IN_GAA(X) we obtained the following new rules:

U2_GAA(s, less_out_ga(s, x1)) → DELETE_IN_GAA(s)
U2_GAA(0, less_out_ga(0, s)) → DELETE_IN_GAA(0)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

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

DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0, s))
DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
U2_GAA(s, less_out_ga(s, x1)) → DELETE_IN_GAA(s)
DELETE_IN_GAA(s) → U2_GAA(s, U7_ga(less_in_aa))
U4_GAA(s, less_out_ag(0, s)) → DELETE_IN_GAA(s)
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(0, s))
U2_GAA(0, less_out_ga(0, s)) → DELETE_IN_GAA(0)
U4_GAA(s, less_out_ag(x1, s)) → DELETE_IN_GAA(s)

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s, s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s, s)

The set Q consists of the following terms:

U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ AND

                                                            ↳ QDP

                                                              ↳ UsableRulesProof

                                                            ↳ QDP

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

DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(0, s))
U2_GAA(0, less_out_ga(0, s)) → DELETE_IN_GAA(0)

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s, s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s, s)

The set Q consists of the following terms:

U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ AND

                                                            ↳ QDP

                                                              ↳ UsableRulesProof

                                                                ↳ QDP

                                                                  ↳ QReductionProof

                                                            ↳ QDP

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

DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(0, s))
U2_GAA(0, less_out_ga(0, s)) → DELETE_IN_GAA(0)

R is empty.
The set Q consists of the following terms:

U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ AND

                                                            ↳ QDP

                                                              ↳ UsableRulesProof

                                                                ↳ QDP

                                                                  ↳ QReductionProof

                                                                    ↳ QDP

                                                                      ↳ NonTerminationProof

                                                            ↳ QDP

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

DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(0, s))
U2_GAA(0, less_out_ga(0, s)) → DELETE_IN_GAA(0)

The TRS R consists of the following rules:none

s = U2_GAA(0, less_out_ga(0, s)) evaluates to t =U2_GAA(0, less_out_ga(0, s))

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

U2_GAA(0, less_out_ga(0, s)) → DELETE_IN_GAA(0)
with rule U2_GAA(0, less_out_ga(0, s)) → DELETE_IN_GAA(0) at position [] and matcher [ ]

DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(0, s))
with rule DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(0, s))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ AND

                                                            ↳ QDP

                                                            ↳ QDP

                                                              ↳ NonTerminationProof

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

DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0, s))
DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
U2_GAA(s, less_out_ga(s, x1)) → DELETE_IN_GAA(s)
DELETE_IN_GAA(s) → U2_GAA(s, U7_ga(less_in_aa))
U4_GAA(s, less_out_ag(0, s)) → DELETE_IN_GAA(s)
U4_GAA(s, less_out_ag(x1, s)) → DELETE_IN_GAA(s)

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s, s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s, s)

The set Q consists of the following terms:

U7_ag(x0)
U7_ga(x0)
less_in_aa
U7_aa(x0)

DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0, s))
DELETE_IN_GAA(s) → U4_GAA(s, U7_ag(less_in_aa))
U2_GAA(s, less_out_ga(s, x1)) → DELETE_IN_GAA(s)
DELETE_IN_GAA(s) → U2_GAA(s, U7_ga(less_in_aa))
U4_GAA(s, less_out_ag(0, s)) → DELETE_IN_GAA(s)
U4_GAA(s, less_out_ag(x1, s)) → DELETE_IN_GAA(s)

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U7_aa(less_in_aa)
U7_ag(less_out_aa(X, Y)) → less_out_ag(s, s)
U7_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U7_ga(less_out_aa(X, Y)) → less_out_ga(s, s)

s = U4_GAA(s, less_out_ag(0, s)) evaluates to t =U4_GAA(s, less_out_ag(0, s))

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

U4_GAA(s, less_out_ag(0, s)) → DELETE_IN_GAA(s)
with rule U4_GAA(s, less_out_ag(0, s)) → DELETE_IN_GAA(s) at position [] and matcher [ ]

DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0, s))
with rule DELETE_IN_GAA(s) → U4_GAA(s, less_out_ag(0, s))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.