Left Termination of the query pattern delete_in_3(g, a, a) w.r.t. the given Prolog program could not be shown:

(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(g,a,a).

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

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

The argument filtering Pi contains the following mapping:
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5, x6)  =  U1_gaa(x1, x6)
delmin_in_aaa(x1, x2, x3)  =  delmin_in_aaa
delmin_out_aaa(x1, x2, x3)  =  delmin_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_aaa(x7)
U2_gaa(x1, x2, x3, x4, x5, x6)  =  U2_gaa(x1, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x1, x6)
U4_gaa(x1, x2, x3, x4, x5, x6)  =  U4_gaa(x1, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x1, 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_gaa(X, tree(X, void, Right), Right) → delete_out_gaa(X, tree(X, void, Right), Right)
delete_in_gaa(X, tree(X, Left, void), Left) → delete_out_gaa(X, tree(X, Left, void), Left)
delete_in_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gaa(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
delmin_in_aaa(tree(Y, void, Right), Y, Right) → delmin_out_aaa(tree(Y, void, Right), Y, Right)
delmin_in_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_aaa(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
U6_aaa(X, Left, X1, Y, Left1, X2, delmin_out_aaa(Left, Y, Left1)) → delmin_out_aaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gaa(X, Left, Right, Y, Right1, delmin_out_aaa(Right, Y, Right1)) → delete_out_gaa(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gaa(X, Y, Left, Right, Left1, less_in_ga(X, Y))
less_in_ga(0, s(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))
U2_gaa(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_gaa(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
delete_in_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gaa(X, Y, Left, Right, Right1, less_in_ag(Y, X))
less_in_ag(0, s(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))
U4_gaa(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_gaa(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U5_gaa(X, Y, Left, Right, Right1, delete_out_gaa(X, Right, Right1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gaa(X, Y, Left, Right, Left1, delete_out_gaa(X, Left, Left1)) → delete_out_gaa(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5, x6)  =  U1_gaa(x1, x6)
delmin_in_aaa(x1, x2, x3)  =  delmin_in_aaa
delmin_out_aaa(x1, x2, x3)  =  delmin_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_aaa(x7)
U2_gaa(x1, x2, x3, x4, x5, x6)  =  U2_gaa(x1, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x1, x6)
U4_gaa(x1, x2, x3, x4, x5, x6)  =  U4_gaa(x1, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x1, 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_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GAA(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_AAA(Right, Y, Right1)
DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_AAA(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_AAA(Left, Y, Left1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y))
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GA(X, Y)
LESS_IN_GA(s(X), s(Y)) → U7_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_GAA(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → DELETE_IN_GAA(X, Left, Left1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X))
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_AG(Y, X)
LESS_IN_AG(s(X), s(Y)) → U7_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_GAA(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → DELETE_IN_GAA(X, Right, Right1)

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa(x1)
U1_gaa(x1, x2, x3, x4, x5, x6)  =  U1_gaa(x1, x6)
delmin_in_aaa(x1, x2, x3)  =  delmin_in_aaa
delmin_out_aaa(x1, x2, x3)  =  delmin_out_aaa
U6_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_aaa(x7)
U2_gaa(x1, x2, x3, x4, x5, x6)  =  U2_gaa(x1, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
s(x1)  =  s(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x1, x6)
U4_gaa(x1, x2, x3, x4, x5, x6)  =  U4_gaa(x1, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U5_gaa(x1, x2, x3, x4, x5, x6)  =  U5_gaa(x1, x6)
DELETE_IN_GAA(x1, x2, x3)  =  DELETE_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5, x6)  =  U1_GAA(x1, x6)
DELMIN_IN_AAA(x1, x2, x3)  =  DELMIN_IN_AAA
U6_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_AAA(x7)
U2_GAA(x1, x2, x3, x4, x5, x6)  =  U2_GAA(x1, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U3_GAA(x1, x2, x3, x4, x5, x6)  =  U3_GAA(x1, x6)
U4_GAA(x1, x2, x3, x4, x5, x6)  =  U4_GAA(x1, x6)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x2, x3)
U5_GAA(x1, x2, x3, x4, x5, x6)  =  U5_GAA(x1, 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_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GAA(X, Left, Right, Y, Right1, delmin_in_aaa(Right, Y, Right1))
DELETE_IN_GAA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_AAA(Right, Y, Right1)
DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_AAA(X, Left, X1, Y, Left1, X2, delmin_in_aaa(Left, Y, Left1))
DELMIN_IN_AAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_AAA(Left, Y, Left1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GAA(X, Y, Left, Right, Left1, less_in_ga(X, Y))
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GA(X, Y)
LESS_IN_GA(s(X), s(Y)) → U7_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → U3_GAA(X, Y, Left, Right, Left1, delete_in_gaa(X, Left, Left1))
U2_GAA(X, Y, Left, Right, Left1, less_out_ga(X, Y)) → DELETE_IN_GAA(X, Left, Left1)
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GAA(X, Y, Left, Right, Right1, less_in_ag(Y, X))
DELETE_IN_GAA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_AG(Y, X)
LESS_IN_AG(s(X), s(Y)) → U7_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → U5_GAA(X, Y, Left, Right, Right1, delete_in_gaa(X, Right, Right1))
U4_GAA(X, Y, Left, Right, Right1, less_out_ag(Y, X)) → DELETE_IN_GAA(X, Right, Right1)

The TRS R consists of the following rules:

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

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

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 9 less nodes.

(6) Complex Obligation (AND)

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

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

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_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

(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_AG(s(Y)) → LESS_IN_AG(Y)

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_AG(s(Y)) → LESS_IN_AG(Y)
  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_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

The TRS R consists of the following rules:

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

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

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_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

(17) PiDPToQDPProof (SOUND 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_GA(s(X)) → LESS_IN_GA(X)

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

(20) TRUE

(21) Obligation:

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

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

The TRS R consists of the following rules:

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

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

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:

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

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

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:

DELMIN_IN_AAA → DELMIN_IN_AAA

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

(26) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = DELMIN_IN_AAA evaluates to t =DELMIN_IN_AAA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from DELMIN_IN_AAA to DELMIN_IN_AAA.

(27) FALSE

(28) Obligation:

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

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

The TRS R consists of the following rules:

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

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

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:

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

The TRS R consists of the following rules:

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))
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_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))

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

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:

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

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))
less_in_ag(s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(Y)) → U7_ag(Y, less_in_ag(Y))
U7_ga(X, less_out_ga(X)) → less_out_ga(s(X))
U7_ag(Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U7_ga(x0, x1)
U7_ag(x0, x1)

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

(33) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule DELETE_IN_GAA(X) → U2_GAA(X, less_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(0))
DELETE_IN_GAA(s(x0)) → U2_GAA(s(x0), U7_ga(x0, less_in_ga(x0)))

(34) Obligation:

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

U2_GAA(X, less_out_ga(X)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(X) → U4_GAA(X, less_in_ag(X))
U4_GAA(X, less_out_ag(Y, X)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(0))
DELETE_IN_GAA(s(x0)) → U2_GAA(s(x0), U7_ga(x0, less_in_ga(x0)))

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))
less_in_ag(s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(Y)) → U7_ag(Y, less_in_ag(Y))
U7_ga(X, less_out_ga(X)) → less_out_ga(s(X))
U7_ag(Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U7_ga(x0, x1)
U7_ag(x0, x1)

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

(35) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule DELETE_IN_GAA(X) → U4_GAA(X, less_in_ag(X)) at position [1] we obtained the following new rules [LPAR04]:

DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), less_out_ag(0, s(x0)))
DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), U7_ag(x0, less_in_ag(x0)))

(36) Obligation:

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

U2_GAA(X, less_out_ga(X)) → DELETE_IN_GAA(X)
U4_GAA(X, less_out_ag(Y, X)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(0))
DELETE_IN_GAA(s(x0)) → U2_GAA(s(x0), U7_ga(x0, less_in_ga(x0)))
DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), less_out_ag(0, s(x0)))
DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), U7_ag(x0, less_in_ag(x0)))

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))
less_in_ag(s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(Y)) → U7_ag(Y, less_in_ag(Y))
U7_ga(X, less_out_ga(X)) → less_out_ga(s(X))
U7_ag(Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U7_ga(x0, x1)
U7_ag(x0, x1)

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

(37) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GAA(X, less_out_ga(X)) → DELETE_IN_GAA(X) we obtained the following new rules [LPAR04]:

U2_GAA(0, less_out_ga(0)) → DELETE_IN_GAA(0)
U2_GAA(s(z0), less_out_ga(s(z0))) → DELETE_IN_GAA(s(z0))

(38) Obligation:

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

U4_GAA(X, less_out_ag(Y, X)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga(0))
DELETE_IN_GAA(s(x0)) → U2_GAA(s(x0), U7_ga(x0, less_in_ga(x0)))
DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), less_out_ag(0, s(x0)))
DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), U7_ag(x0, less_in_ag(x0)))
U2_GAA(0, less_out_ga(0)) → DELETE_IN_GAA(0)
U2_GAA(s(z0), less_out_ga(s(z0))) → DELETE_IN_GAA(s(z0))

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))
less_in_ag(s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(Y)) → U7_ag(Y, less_in_ag(Y))
U7_ga(X, less_out_ga(X)) → less_out_ga(s(X))
U7_ag(Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U7_ga(x0, x1)
U7_ag(x0, x1)

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

(39) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(40) Complex Obligation (AND)

(41) Obligation:

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

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

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))
less_in_ag(s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(Y)) → U7_ag(Y, less_in_ag(Y))
U7_ga(X, less_out_ga(X)) → less_out_ga(s(X))
U7_ag(Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U7_ga(x0, x1)
U7_ag(x0, x1)

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

(42) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(43) Obligation:

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

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

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

less_in_ga(x0)
less_in_ag(x0)
U7_ga(x0, x1)
U7_ag(x0, x1)

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

(44) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

less_in_ga(x0)
less_in_ag(x0)
U7_ga(x0, x1)
U7_ag(x0, x1)

(45) Obligation:

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

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

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

(46) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

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

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

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

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

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


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

(47) FALSE

(48) Obligation:

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

DELETE_IN_GAA(s(x0)) → U2_GAA(s(x0), U7_ga(x0, less_in_ga(x0)))
U2_GAA(s(z0), less_out_ga(s(z0))) → DELETE_IN_GAA(s(z0))
DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), less_out_ag(0, s(x0)))
U4_GAA(X, less_out_ag(Y, X)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), U7_ag(x0, less_in_ag(x0)))

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))
less_in_ag(s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(Y)) → U7_ag(Y, less_in_ag(Y))
U7_ga(X, less_out_ga(X)) → less_out_ga(s(X))
U7_ag(Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U7_ga(x0, x1)
U7_ag(x0, x1)

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

(49) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U4_GAA(X, less_out_ag(Y, X)) → DELETE_IN_GAA(X) we obtained the following new rules [LPAR04]:

U4_GAA(s(z0), less_out_ag(0, s(z0))) → DELETE_IN_GAA(s(z0))
U4_GAA(s(z0), less_out_ag(x1, s(z0))) → DELETE_IN_GAA(s(z0))

(50) Obligation:

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

DELETE_IN_GAA(s(x0)) → U2_GAA(s(x0), U7_ga(x0, less_in_ga(x0)))
U2_GAA(s(z0), less_out_ga(s(z0))) → DELETE_IN_GAA(s(z0))
DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), less_out_ag(0, s(x0)))
DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), U7_ag(x0, less_in_ag(x0)))
U4_GAA(s(z0), less_out_ag(0, s(z0))) → DELETE_IN_GAA(s(z0))
U4_GAA(s(z0), less_out_ag(x1, s(z0))) → DELETE_IN_GAA(s(z0))

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))
less_in_ag(s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(Y)) → U7_ag(Y, less_in_ag(Y))
U7_ga(X, less_out_ga(X)) → less_out_ga(s(X))
U7_ag(Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U7_ga(x0, x1)
U7_ag(x0, x1)

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

(51) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U4_GAA(s(z0), less_out_ag(0, s(z0))) evaluates to t =U4_GAA(s(z0), less_out_ag(0, s(z0)))

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

U4_GAA(s(z0), less_out_ag(0, s(z0))) → DELETE_IN_GAA(s(z0))
with rule U4_GAA(s(z0'), less_out_ag(0, s(z0'))) → DELETE_IN_GAA(s(z0')) at position [] and matcher [z0' / z0]

DELETE_IN_GAA(s(z0)) → U4_GAA(s(z0), less_out_ag(0, s(z0)))
with rule DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), less_out_ag(0, s(x0)))

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


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

(52) FALSE

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

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

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

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(54) Obligation:

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

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

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

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

The TRS R consists of the following rules:

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

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

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

(56) Obligation:

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

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

The TRS R consists of the following rules:

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

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

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

(57) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 9 less nodes.

(58) Complex Obligation (AND)

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

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

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

(60) UsableRulesProof (EQUIVALENT transformation)

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

(61) 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

(62) PiDPToQDPProof (SOUND transformation)

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

(63) 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.

(64) 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

(65) TRUE

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

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

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

(67) UsableRulesProof (EQUIVALENT transformation)

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

(68) 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

(69) PiDPToQDPProof (SOUND transformation)

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

(70) 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.

(71) 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

(72) TRUE

(73) Obligation:

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

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

The TRS R consists of the following rules:

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

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

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

(74) UsableRulesProof (EQUIVALENT transformation)

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

(75) Obligation:

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

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

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

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

(76) PiDPToQDPProof (SOUND transformation)

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

(77) Obligation:

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

DELMIN_IN_AAA → DELMIN_IN_AAA

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

(78) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = DELMIN_IN_AAA evaluates to t =DELMIN_IN_AAA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from DELMIN_IN_AAA to DELMIN_IN_AAA.

(79) FALSE

(80) Obligation:

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

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

The TRS R consists of the following rules:

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

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

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

(81) UsableRulesProof (EQUIVALENT transformation)

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

(82) Obligation:

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

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

The TRS R consists of the following rules:

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))
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_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))

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

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

(83) PiDPToQDPProof (SOUND transformation)

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

(84) Obligation:

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

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

The TRS R consists of the following rules:

less_in_ga(0) → less_out_ga
less_in_ga(s(X)) → U7_ga(less_in_ga(X))
less_in_ag(s(X3)) → less_out_ag(0)
less_in_ag(s(Y)) → U7_ag(less_in_ag(Y))
U7_ga(less_out_ga) → less_out_ga
U7_ag(less_out_ag(X)) → less_out_ag(s(X))

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U7_ga(x0)
U7_ag(x0)

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

(85) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule DELETE_IN_GAA(X) → U2_GAA(X, less_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga)
DELETE_IN_GAA(s(x0)) → U2_GAA(s(x0), U7_ga(less_in_ga(x0)))

(86) Obligation:

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

U2_GAA(X, less_out_ga) → DELETE_IN_GAA(X)
DELETE_IN_GAA(X) → U4_GAA(X, less_in_ag(X))
U4_GAA(X, less_out_ag(Y)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga)
DELETE_IN_GAA(s(x0)) → U2_GAA(s(x0), U7_ga(less_in_ga(x0)))

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))
less_in_ag(s(X3)) → less_out_ag(0)
less_in_ag(s(Y)) → U7_ag(less_in_ag(Y))
U7_ga(less_out_ga) → less_out_ga
U7_ag(less_out_ag(X)) → less_out_ag(s(X))

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U7_ga(x0)
U7_ag(x0)

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

(87) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule DELETE_IN_GAA(X) → U4_GAA(X, less_in_ag(X)) at position [1] we obtained the following new rules [LPAR04]:

DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), less_out_ag(0))
DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), U7_ag(less_in_ag(x0)))

(88) Obligation:

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

U2_GAA(X, less_out_ga) → DELETE_IN_GAA(X)
U4_GAA(X, less_out_ag(Y)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga)
DELETE_IN_GAA(s(x0)) → U2_GAA(s(x0), U7_ga(less_in_ga(x0)))
DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), less_out_ag(0))
DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), U7_ag(less_in_ag(x0)))

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))
less_in_ag(s(X3)) → less_out_ag(0)
less_in_ag(s(Y)) → U7_ag(less_in_ag(Y))
U7_ga(less_out_ga) → less_out_ga
U7_ag(less_out_ag(X)) → less_out_ag(s(X))

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U7_ga(x0)
U7_ag(x0)

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

(89) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GAA(X, less_out_ga) → DELETE_IN_GAA(X) we obtained the following new rules [LPAR04]:

U2_GAA(0, less_out_ga) → DELETE_IN_GAA(0)
U2_GAA(s(z0), less_out_ga) → DELETE_IN_GAA(s(z0))

(90) Obligation:

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

U4_GAA(X, less_out_ag(Y)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(0) → U2_GAA(0, less_out_ga)
DELETE_IN_GAA(s(x0)) → U2_GAA(s(x0), U7_ga(less_in_ga(x0)))
DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), less_out_ag(0))
DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), U7_ag(less_in_ag(x0)))
U2_GAA(0, less_out_ga) → DELETE_IN_GAA(0)
U2_GAA(s(z0), less_out_ga) → DELETE_IN_GAA(s(z0))

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))
less_in_ag(s(X3)) → less_out_ag(0)
less_in_ag(s(Y)) → U7_ag(less_in_ag(Y))
U7_ga(less_out_ga) → less_out_ga
U7_ag(less_out_ag(X)) → less_out_ag(s(X))

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U7_ga(x0)
U7_ag(x0)

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

(91) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(92) Complex Obligation (AND)

(93) Obligation:

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

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

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))
less_in_ag(s(X3)) → less_out_ag(0)
less_in_ag(s(Y)) → U7_ag(less_in_ag(Y))
U7_ga(less_out_ga) → less_out_ga
U7_ag(less_out_ag(X)) → less_out_ag(s(X))

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U7_ga(x0)
U7_ag(x0)

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

(94) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(95) Obligation:

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

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

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

less_in_ga(x0)
less_in_ag(x0)
U7_ga(x0)
U7_ag(x0)

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

(96) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

less_in_ga(x0)
less_in_ag(x0)
U7_ga(x0)
U7_ag(x0)

(97) Obligation:

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

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

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

(98) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

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

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

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

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

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


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

(99) FALSE

(100) Obligation:

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

DELETE_IN_GAA(s(x0)) → U2_GAA(s(x0), U7_ga(less_in_ga(x0)))
U2_GAA(s(z0), less_out_ga) → DELETE_IN_GAA(s(z0))
DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), less_out_ag(0))
U4_GAA(X, less_out_ag(Y)) → DELETE_IN_GAA(X)
DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), U7_ag(less_in_ag(x0)))

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))
less_in_ag(s(X3)) → less_out_ag(0)
less_in_ag(s(Y)) → U7_ag(less_in_ag(Y))
U7_ga(less_out_ga) → less_out_ga
U7_ag(less_out_ag(X)) → less_out_ag(s(X))

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U7_ga(x0)
U7_ag(x0)

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

(101) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U4_GAA(X, less_out_ag(Y)) → DELETE_IN_GAA(X) we obtained the following new rules [LPAR04]:

U4_GAA(s(z0), less_out_ag(0)) → DELETE_IN_GAA(s(z0))
U4_GAA(s(z0), less_out_ag(x1)) → DELETE_IN_GAA(s(z0))

(102) Obligation:

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

DELETE_IN_GAA(s(x0)) → U2_GAA(s(x0), U7_ga(less_in_ga(x0)))
U2_GAA(s(z0), less_out_ga) → DELETE_IN_GAA(s(z0))
DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), less_out_ag(0))
DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), U7_ag(less_in_ag(x0)))
U4_GAA(s(z0), less_out_ag(0)) → DELETE_IN_GAA(s(z0))
U4_GAA(s(z0), less_out_ag(x1)) → DELETE_IN_GAA(s(z0))

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))
less_in_ag(s(X3)) → less_out_ag(0)
less_in_ag(s(Y)) → U7_ag(less_in_ag(Y))
U7_ga(less_out_ga) → less_out_ga
U7_ag(less_out_ag(X)) → less_out_ag(s(X))

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U7_ga(x0)
U7_ag(x0)

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

(103) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U4_GAA(s(z0), less_out_ag(0)) evaluates to t =U4_GAA(s(z0), less_out_ag(0))

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

U4_GAA(s(z0), less_out_ag(0)) → DELETE_IN_GAA(s(z0))
with rule U4_GAA(s(z0'), less_out_ag(0)) → DELETE_IN_GAA(s(z0')) at position [] and matcher [z0' / z0]

DELETE_IN_GAA(s(z0)) → U4_GAA(s(z0), less_out_ag(0))
with rule DELETE_IN_GAA(s(x0)) → U4_GAA(s(x0), less_out_ag(0))

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


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

(104) FALSE