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

(0) Obligation:

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(X3)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

delete(a,a,g).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
delete_in: (f,f,b) (b,f,b)
delmin_in: (f,b,b)
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_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

delete_in_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

DELETE_IN_AAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_AAG(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
DELETE_IN_AAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_AGG(Right, Y, Right1)
DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_AGG(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_AGG(Left, Y, Left1)
DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_AAG(X, Y, Left, Right, Left1, less_in_ag(X, Y))
DELETE_IN_AAG(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_AAG(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_AAG(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
U2_AAG(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → DELETE_IN_GAG(X, Left, Left1)
DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GAG(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_AGG(Right, Y, Right1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U7_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GAG(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GAG(X, Left, Left1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GAG(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GAG(X, Right, Right1)
DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AAG(X, Y, Left, Right, Right1, less_in_ga(Y, X))
DELETE_IN_AAG(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_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_AAG(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U4_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AAG(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)
DELETE_IN_AAG(x1, x2, x3)  =  DELETE_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5, x6)  =  U1_AAG(x6)
DELMIN_IN_AGG(x1, x2, x3)  =  DELMIN_IN_AGG(x2, x3)
U6_AGG(x1, x2, x3, x4, x5, x6, x7)  =  U6_AGG(x7)
U2_AAG(x1, x2, x3, x4, x5, x6)  =  U2_AAG(x5, x6)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x3)
U3_AAG(x1, x2, x3, x4, x5, x6)  =  U3_AAG(x6)
DELETE_IN_GAG(x1, x2, x3)  =  DELETE_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x6)
U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x5, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x3)
U3_GAG(x1, x2, x3, x4, x5, x6)  =  U3_GAG(x6)
U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x1, x5, x6)
U5_GAG(x1, x2, x3, x4, x5, x6)  =  U5_GAG(x6)
U4_AAG(x1, x2, x3, x4, x5, x6)  =  U4_AAG(x5, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
U5_AAG(x1, x2, x3, x4, x5, x6)  =  U5_AAG(x6)

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

(4) Obligation:

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

DELETE_IN_AAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_AAG(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
DELETE_IN_AAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_AGG(Right, Y, Right1)
DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_AGG(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_AGG(Left, Y, Left1)
DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_AAG(X, Y, Left, Right, Left1, less_in_ag(X, Y))
DELETE_IN_AAG(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_AAG(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_AAG(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
U2_AAG(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → DELETE_IN_GAG(X, Left, Left1)
DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GAG(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_AGG(Right, Y, Right1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U7_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GAG(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GAG(X, Left, Left1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GAG(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GAG(X, Right, Right1)
DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AAG(X, Y, Left, Right, Right1, less_in_ga(Y, X))
DELETE_IN_AAG(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_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_AAG(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U4_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AAG(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)
DELETE_IN_AAG(x1, x2, x3)  =  DELETE_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5, x6)  =  U1_AAG(x6)
DELMIN_IN_AGG(x1, x2, x3)  =  DELMIN_IN_AGG(x2, x3)
U6_AGG(x1, x2, x3, x4, x5, x6, x7)  =  U6_AGG(x7)
U2_AAG(x1, x2, x3, x4, x5, x6)  =  U2_AAG(x5, x6)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x3)
U3_AAG(x1, x2, x3, x4, x5, x6)  =  U3_AAG(x6)
DELETE_IN_GAG(x1, x2, x3)  =  DELETE_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x6)
U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x5, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x3)
U3_GAG(x1, x2, x3, x4, x5, x6)  =  U3_GAG(x6)
U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x1, x5, x6)
U5_GAG(x1, x2, x3, x4, x5, x6)  =  U5_GAG(x6)
U4_AAG(x1, x2, x3, x4, x5, x6)  =  U4_AAG(x5, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
U5_AAG(x1, x2, x3, x4, x5, x6)  =  U5_AAG(x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 18 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)

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

(8) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(9) Obligation:

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

(10) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(11) Obligation:

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.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

• LESS_IN_GA(s(X)) → LESS_IN_GA(X)
  The graph contains the following edges 1 > 1

(13) TRUE

(14) Obligation:

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_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(16) Obligation:

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

(17) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(18) Obligation:

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.

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

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

(20) TRUE

(21) Obligation:

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_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)

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

(22) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(23) Obligation:

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

(24) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(25) Obligation:

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.

(26) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

• LESS_IN_AG(s(Y)) → LESS_IN_AG(Y)
  The graph contains the following edges 1 > 1

(27) TRUE

(28) Obligation:

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

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

The TRS R consists of the following rules:

delete_in_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)
DELMIN_IN_AGG(x1, x2, x3)  =  DELMIN_IN_AGG(x2, x3)

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

(29) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(30) Obligation:

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

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

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

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

(31) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(32) Obligation:

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

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

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(33) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

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

(34) TRUE

(35) Obligation:

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

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

The TRS R consists of the following rules:

delete_in_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)
DELETE_IN_GAG(x1, x2, x3)  =  DELETE_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x5, x6)
U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x1, x5, x6)

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

(36) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(37) Obligation:

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

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

The TRS R consists of the following rules:

less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
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_GAG(x1, x2, x3)  =  DELETE_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x5, x6)
U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x1, x5, x6)

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

(38) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(39) Obligation:

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

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

The TRS R consists of the following rules:

less_in_gg(0, s(X3)) → 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.

(40) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

• U2_GAG(X, Left1, less_out_gg) → DELETE_IN_GAG(X, Left1)
  The graph contains the following edges 1 >= 1, 2 >= 2

• U4_GAG(X, Right1, less_out_gg) → DELETE_IN_GAG(X, Right1)
  The graph contains the following edges 1 >= 1, 2 >= 2

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

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

(41) TRUE

(42) Obligation:

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

DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AAG(X, Y, Left, Right, Right1, less_in_ga(Y, X))
U4_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AAG(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, 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_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)
DELETE_IN_AAG(x1, x2, x3)  =  DELETE_IN_AAG(x3)
U4_AAG(x1, x2, x3, x4, x5, x6)  =  U4_AAG(x5, x6)

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

(43) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(44) Obligation:

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

DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AAG(X, Y, Left, Right, Right1, less_in_ga(Y, X))
U4_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AAG(X, Right, Right1)

The TRS R consists of the following rules:

less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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)
DELETE_IN_AAG(x1, x2, x3)  =  DELETE_IN_AAG(x3)
U4_AAG(x1, x2, x3, x4, x5, x6)  =  U4_AAG(x5, x6)

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

(45) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(46) Obligation:

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

DELETE_IN_AAG(tree(Y, Left, Right1)) → U4_AAG(Right1, less_in_ga(Y))
U4_AAG(Right1, less_out_ga) → DELETE_IN_AAG(Right1)

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.

(47) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

• U4_AAG(Right1, less_out_ga) → DELETE_IN_AAG(Right1)
  The graph contains the following edges 1 >= 1

• DELETE_IN_AAG(tree(Y, Left, Right1)) → U4_AAG(Right1, less_in_ga(Y))
  The graph contains the following edges 1 > 1

(48) TRUE

(49) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
delete_in: (f,f,b) (b,f,b)
delmin_in: (f,b,b)
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_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag(x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x2, x4, x5, x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg(x2, x3)
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x1, x4, x5, x6, x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x2, x4, x5, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x2, x4, x5, x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x4, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x2, x3, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x1, x2, x3, x5, x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x2, x3, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x2, x3, x5, x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(50) Obligation:

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

delete_in_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag(x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x2, x4, x5, x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg(x2, x3)
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x1, x4, x5, x6, x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x2, x4, x5, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x2, x4, x5, x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x4, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x2, x3, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x1, x2, x3, x5, x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x2, x3, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x2, x3, x5, x6)

(51) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

DELETE_IN_AAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_AAG(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
DELETE_IN_AAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_AGG(Right, Y, Right1)
DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_AGG(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_AGG(Left, Y, Left1)
DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_AAG(X, Y, Left, Right, Left1, less_in_ag(X, Y))
DELETE_IN_AAG(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_AAG(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_AAG(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
U2_AAG(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → DELETE_IN_GAG(X, Left, Left1)
DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GAG(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_AGG(Right, Y, Right1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U7_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GAG(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GAG(X, Left, Left1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GAG(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GAG(X, Right, Right1)
DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AAG(X, Y, Left, Right, Right1, less_in_ga(Y, X))
DELETE_IN_AAG(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_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_AAG(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U4_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AAG(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag(x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x2, x4, x5, x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg(x2, x3)
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x1, x4, x5, x6, x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x2, x4, x5, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x2, x4, x5, x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x4, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x2, x3, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x1, x2, x3, x5, x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x2, x3, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x2, x3, x5, x6)
DELETE_IN_AAG(x1, x2, x3)  =  DELETE_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5, x6)  =  U1_AAG(x2, x4, x5, x6)
DELMIN_IN_AGG(x1, x2, x3)  =  DELMIN_IN_AGG(x2, x3)
U6_AGG(x1, x2, x3, x4, x5, x6, x7)  =  U6_AGG(x1, x4, x5, x6, x7)
U2_AAG(x1, x2, x3, x4, x5, x6)  =  U2_AAG(x2, x4, x5, x6)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x2, x3)
U3_AAG(x1, x2, x3, x4, x5, x6)  =  U3_AAG(x2, x4, x5, x6)
DELETE_IN_GAG(x1, x2, x3)  =  DELETE_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x1, x2, x4, x5, x6)
U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x2, x4, x5, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)
U3_GAG(x1, x2, x3, x4, x5, x6)  =  U3_GAG(x1, x2, x4, x5, x6)
U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x1, x2, x3, x5, x6)
U5_GAG(x1, x2, x3, x4, x5, x6)  =  U5_GAG(x1, x2, x3, x5, x6)
U4_AAG(x1, x2, x3, x4, x5, x6)  =  U4_AAG(x2, x3, x5, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U5_AAG(x1, x2, x3, x4, x5, x6)  =  U5_AAG(x2, x3, x5, x6)

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

(52) Obligation:

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

DELETE_IN_AAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_AAG(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
DELETE_IN_AAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_AGG(Right, Y, Right1)
DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_AGG(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_AGG(Left, Y, Left1)
DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_AAG(X, Y, Left, Right, Left1, less_in_ag(X, Y))
DELETE_IN_AAG(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_AAG(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_AAG(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
U2_AAG(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → DELETE_IN_GAG(X, Left, Left1)
DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GAG(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_AGG(Right, Y, Right1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U7_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GAG(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GAG(X, Left, Left1)
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GAG(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GAG(X, Right, Right1)
DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AAG(X, Y, Left, Right, Right1, less_in_ga(Y, X))
DELETE_IN_AAG(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_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_AAG(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U4_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AAG(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag(x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x2, x4, x5, x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg(x2, x3)
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x1, x4, x5, x6, x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x2, x4, x5, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x2, x4, x5, x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x4, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x2, x3, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x1, x2, x3, x5, x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x2, x3, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x2, x3, x5, x6)
DELETE_IN_AAG(x1, x2, x3)  =  DELETE_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5, x6)  =  U1_AAG(x2, x4, x5, x6)
DELMIN_IN_AGG(x1, x2, x3)  =  DELMIN_IN_AGG(x2, x3)
U6_AGG(x1, x2, x3, x4, x5, x6, x7)  =  U6_AGG(x1, x4, x5, x6, x7)
U2_AAG(x1, x2, x3, x4, x5, x6)  =  U2_AAG(x2, x4, x5, x6)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x2, x3)
U3_AAG(x1, x2, x3, x4, x5, x6)  =  U3_AAG(x2, x4, x5, x6)
DELETE_IN_GAG(x1, x2, x3)  =  DELETE_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x1, x2, x4, x5, x6)
U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x2, x4, x5, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)
U3_GAG(x1, x2, x3, x4, x5, x6)  =  U3_GAG(x1, x2, x4, x5, x6)
U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x1, x2, x3, x5, x6)
U5_GAG(x1, x2, x3, x4, x5, x6)  =  U5_GAG(x1, x2, x3, x5, x6)
U4_AAG(x1, x2, x3, x4, x5, x6)  =  U4_AAG(x2, x3, x5, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U5_AAG(x1, x2, x3, x4, x5, x6)  =  U5_AAG(x2, x3, x5, x6)

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

(53) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 18 less nodes.

(54) Complex Obligation (AND)

(55) Obligation:

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_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag(x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x2, x4, x5, x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg(x2, x3)
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x1, x4, x5, x6, x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x2, x4, x5, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x2, x4, x5, x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x4, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x2, x3, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x1, x2, x3, x5, x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x2, x3, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x2, x3, x5, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)

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

(56) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(57) Obligation:

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

(58) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(59) Obligation:

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.

(60) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

• LESS_IN_GA(s(X)) → LESS_IN_GA(X)
  The graph contains the following edges 1 > 1

(61) TRUE

(62) Obligation:

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_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag(x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x2, x4, x5, x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg(x2, x3)
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x1, x4, x5, x6, x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x2, x4, x5, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x2, x4, x5, x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x4, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x2, x3, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x1, x2, x3, x5, x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x2, x3, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x2, x3, x5, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)

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

(63) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(64) Obligation:

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

(65) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(66) Obligation:

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.

(67) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

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

(68) TRUE

(69) Obligation:

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_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag(x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x2, x4, x5, x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg(x2, x3)
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x1, x4, x5, x6, x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x2, x4, x5, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x2, x4, x5, x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x4, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x2, x3, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x1, x2, x3, x5, x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x2, x3, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x2, x3, x5, x6)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)

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

(70) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(71) Obligation:

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

(72) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(73) Obligation:

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.

(74) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

• LESS_IN_AG(s(Y)) → LESS_IN_AG(Y)
  The graph contains the following edges 1 > 1

(75) TRUE

(76) Obligation:

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

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

The TRS R consists of the following rules:

delete_in_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag(x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x2, x4, x5, x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg(x2, x3)
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x1, x4, x5, x6, x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x2, x4, x5, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x2, x4, x5, x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x4, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x2, x3, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x1, x2, x3, x5, x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x2, x3, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x2, x3, x5, x6)
DELMIN_IN_AGG(x1, x2, x3)  =  DELMIN_IN_AGG(x2, x3)

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

(77) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(78) Obligation:

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

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

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

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

(79) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(80) Obligation:

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

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

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(81) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

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

(82) TRUE

(83) Obligation:

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

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

The TRS R consists of the following rules:

delete_in_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag(x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x2, x4, x5, x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg(x2, x3)
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x1, x4, x5, x6, x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x2, x4, x5, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x2, x4, x5, x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x4, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x2, x3, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x1, x2, x3, x5, x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x2, x3, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x2, x3, x5, x6)
DELETE_IN_GAG(x1, x2, x3)  =  DELETE_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x2, x4, x5, x6)
U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x1, x2, x3, x5, x6)

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

(84) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(85) Obligation:

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

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

The TRS R consists of the following rules:

less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
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(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
DELETE_IN_GAG(x1, x2, x3)  =  DELETE_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x2, x4, x5, x6)
U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x1, x2, x3, x5, x6)

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

(86) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(87) Obligation:

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

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

The TRS R consists of the following rules:

less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
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 set Q consists of the following terms:

less_in_gg(x0, x1)
U7_gg(x0, x1, x2)

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

(88) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

• U2_GAG(X, Y, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GAG(X, Left1)
  The graph contains the following edges 1 >= 1, 5 > 1, 4 >= 2

• U4_GAG(X, Y, Left, Right1, less_out_gg(Y, X)) → DELETE_IN_GAG(X, Right1)
  The graph contains the following edges 1 >= 1, 5 > 1, 4 >= 2

• DELETE_IN_GAG(X, tree(Y, Left1, Right)) → U2_GAG(X, Y, Right, Left1, less_in_gg(X, Y))
  The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4

• DELETE_IN_GAG(X, tree(Y, Left, Right1)) → U4_GAG(X, Y, Left, Right1, less_in_gg(Y, X))
  The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4

(89) TRUE

(90) Obligation:

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

DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AAG(X, Y, Left, Right, Right1, less_in_ga(Y, X))
U4_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AAG(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_aag(X, tree(X, void, Right), Right) → delete_out_aag(X, tree(X, void, Right), Right)
delete_in_aag(X, tree(X, Left, void), Left) → delete_out_aag(X, tree(X, Left, void), Left)
delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
delmin_in_agg(tree(Y, void, Right), Y, Right) → delmin_out_agg(tree(Y, void, Right), Y, Right)
delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) → delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(X, void, Right), Right) → delete_out_gag(X, tree(X, void, Right), Right)
delete_in_gag(X, tree(X, Left, void), Left) → delete_out_gag(X, tree(X, Left, void), Left)
delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) → delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) → delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
delete_out_aag(x1, x2, x3)  =  delete_out_aag(x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x2, x4, x5, x6)
delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
delmin_out_agg(x1, x2, x3)  =  delmin_out_agg(x2, x3)
U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x1, x4, x5, x6, x7)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x2, x4, x5, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x2, x4, x5, x6)
delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
delete_out_gag(x1, x2, x3)  =  delete_out_gag(x1, x3)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, x6)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x4, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x2, x3, x5, x6)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x1, x2, x3, x5, x6)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x2, x3, x5, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x2, x3, x5, x6)
DELETE_IN_AAG(x1, x2, x3)  =  DELETE_IN_AAG(x3)
U4_AAG(x1, x2, x3, x4, x5, x6)  =  U4_AAG(x2, x3, x5, x6)

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

(91) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(92) Obligation:

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

DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AAG(X, Y, Left, Right, Right1, less_in_ga(Y, X))
U4_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AAG(X, Right, Right1)

The TRS R consists of the following rules:

less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
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(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
DELETE_IN_AAG(x1, x2, x3)  =  DELETE_IN_AAG(x3)
U4_AAG(x1, x2, x3, x4, x5, x6)  =  U4_AAG(x2, x3, x5, x6)

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

(93) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(94) Obligation:

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

DELETE_IN_AAG(tree(Y, Left, Right1)) → U4_AAG(Y, Left, Right1, less_in_ga(Y))
U4_AAG(Y, Left, Right1, less_out_ga(Y)) → DELETE_IN_AAG(Right1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

less_in_ga(x0)
U7_ga(x0, x1)

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