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



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

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

Queries:

delete(a,g,a).

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

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
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_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
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_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
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_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
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_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_AGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_GAA(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_GAA(Left, Y, Left1)
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_AGA(X, Y, Left, Right, Left1, less_in_ag(X, Y))
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(X), s(Y)) → U7_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_AGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U7_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GGA(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GA(Y, X)
LESS_IN_GA(s(X), s(Y)) → U7_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_AGA(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AGA(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
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_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
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_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
U4_AGA(x1, x2, x3, x4, x5, x6)  =  U4_AGA(x4, x6)
U6_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GAA(x7)
U2_AGA(x1, x2, x3, x4, x5, x6)  =  U2_AGA(x3, x6)
U7_AG(x1, x2, x3)  =  U7_AG(x3)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
U5_AGA(x1, x2, x3, x4, x5, x6)  =  U5_AGA(x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U1_AGA(x1, x2, x3, x4, x5, x6)  =  U1_AGA(x1, x6)
U3_AGA(x1, x2, x3, x4, x5, x6)  =  U3_AGA(x1, x6)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x3, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x4, x6)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x6)
DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x6)
U7_GG(x1, x2, x3)  =  U7_GG(x3)

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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

DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_AGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_GAA(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_GAA(Left, Y, Left1)
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_AGA(X, Y, Left, Right, Left1, less_in_ag(X, Y))
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(X), s(Y)) → U7_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_AGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U7_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GGA(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GA(Y, X)
LESS_IN_GA(s(X), s(Y)) → U7_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_AGA(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AGA(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
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_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
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_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
U4_AGA(x1, x2, x3, x4, x5, x6)  =  U4_AGA(x4, x6)
U6_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GAA(x7)
U2_AGA(x1, x2, x3, x4, x5, x6)  =  U2_AGA(x3, x6)
U7_AG(x1, x2, x3)  =  U7_AG(x3)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
U5_AGA(x1, x2, x3, x4, x5, x6)  =  U5_AGA(x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U1_AGA(x1, x2, x3, x4, x5, x6)  =  U1_AGA(x1, x6)
U3_AGA(x1, x2, x3, x4, x5, x6)  =  U3_AGA(x1, x6)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x3, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x4, x6)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x6)
DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x6)
U7_GG(x1, x2, x3)  =  U7_GG(x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 6 SCCs with 18 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
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_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
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_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

LESS_IN_GA(s(X)) → LESS_IN_GA(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

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

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
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_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
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_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

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

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

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

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
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_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
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_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

LESS_IN_AG(s(Y)) → LESS_IN_AG(Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP

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

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

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
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_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
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_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP

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

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

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

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP

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

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

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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

U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
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_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
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_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x3, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x4, x6)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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

U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x3, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x4, x6)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP

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

DELETE_IN_GGA(X, tree(Y, Left, Right)) → U2_GGA(X, Left, less_in_gg(X, Y))
U2_GGA(X, Left, less_out_gg) → DELETE_IN_GGA(X, Left)
U4_GGA(X, Right, less_out_gg) → DELETE_IN_GGA(X, Right)
DELETE_IN_GGA(X, tree(Y, Left, Right)) → U4_GGA(X, Right, less_in_gg(Y, X))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

less_in_gg(x0, x1)
U7_gg(x0)

We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof

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

U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AGA(X, Right, Right1)
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
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_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
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_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
U4_AGA(x1, x2, x3, x4, x5, x6)  =  U4_AGA(x4, x6)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof

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

U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AGA(X, Right, Right1)
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))

The TRS R consists of the following rules:

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_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
0  =  0
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U4_AGA(x1, x2, x3, x4, x5, x6)  =  U4_AGA(x4, x6)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof

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

U4_AGA(Right, less_out_ga) → DELETE_IN_AGA(Right)
DELETE_IN_AGA(tree(Y, Left, Right)) → U4_AGA(Right, less_in_ga(Y))

The TRS R consists of the following rules:

less_in_ga(0) → less_out_ga
less_in_ga(s(X)) → U7_ga(less_in_ga(X))
U7_ga(less_out_ga) → less_out_ga

The set Q consists of the following terms:

less_in_ga(x0)
U7_ga(x0)

We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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