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

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇔)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 TRUE
      ↳14 PiDP
        ↳15 UsableRulesProof (⇔)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇐)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔)
        ↳20 TRUE
      ↳21 PiDP
        ↳22 UsableRulesProof (⇔)
        ↳23 PiDP
        ↳24 PiDPToQDPProof (⇐)
        ↳25 QDP
        ↳26 QDPSizeChangeProof (⇔)
        ↳27 TRUE
  ↳28 PrologToPiTRSProof (⇐)
    ↳29 PiTRS
    ↳30 DependencyPairsProof (⇔)
    ↳31 PiDP
    ↳32 DependencyGraphProof (⇔)
    ↳33 AND
      ↳34 PiDP
        ↳35 UsableRulesProof (⇔)
        ↳36 PiDP
        ↳37 PiDPToQDPProof (⇔)
        ↳38 QDP
        ↳39 QDPSizeChangeProof (⇔)
        ↳40 TRUE
      ↳41 PiDP
        ↳42 UsableRulesProof (⇔)
        ↳43 PiDP
        ↳44 PiDPToQDPProof (⇐)
        ↳45 QDP
        ↳46 QDPSizeChangeProof (⇔)
        ↳47 TRUE
      ↳48 PiDP
        ↳49 UsableRulesProof (⇔)
        ↳50 PiDP

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

delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(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_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)

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_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(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_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)

(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_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_GAA(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_GAA(Left, Y, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U7_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GGA(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(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_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x6)
DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)
U6_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GAA(x7)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x3, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x3)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x4, x6)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x6)

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

(4) Obligation:

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

DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_GAA(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_GAA(Left, Y, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U7_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GGA(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(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_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x6)
DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)
U6_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GAA(x7)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x3, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x3)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x4, x6)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 8 less nodes.

(6) Complex Obligation (AND)

(7) 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_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(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_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, 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_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

(10) PiDPToQDPProof (EQUIVALENT 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_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.

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

(13) TRUE

(14) Obligation:

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

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

The TRS R consists of the following rules:

delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(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_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(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:

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

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

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

(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:

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

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

(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:

  • DELMIN_IN_GAA(tree(X, Left, X1)) → DELMIN_IN_GAA(Left)
    The graph contains the following edges 1 > 1

(20) TRUE

(21) Obligation:

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

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

The TRS R consists of the following rules:

delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(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_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x3, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x4, x6)

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

(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:

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

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)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x3, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x4, x6)

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

(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:

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

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.

(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:

  • U2_GGA(X, Left, less_out_gg) → DELETE_IN_GGA(X, Left)
    The graph contains the following edges 1 >= 1, 2 >= 2

  • U4_GGA(X, Right, less_out_gg) → DELETE_IN_GGA(X, Right)
    The graph contains the following edges 1 >= 1, 2 >= 2

  • DELETE_IN_GGA(X, tree(Y, Left, Right)) → U2_GGA(X, Left, less_in_gg(X, Y))
    The graph contains the following edges 1 >= 1, 2 > 2

  • DELETE_IN_GGA(X, tree(Y, Left, Right)) → U4_GGA(X, Right, less_in_gg(Y, X))
    The graph contains the following edges 1 >= 1, 2 > 2

(27) TRUE

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

delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(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_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_gga(x1, x2, x3)  =  delete_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x1, x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x1, x2, x3, x7)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(29) Obligation:

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

delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(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_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_gga(x1, x2, x3)  =  delete_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x1, x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x1, x2, x3, x7)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)

(30) 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_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_GAA(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_GAA(Left, Y, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U7_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GGA(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(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_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_gga(x1, x2, x3)  =  delete_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x1, x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x1, x2, x3, x7)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x1, x2, x3, x6)
DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)
U6_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GAA(x1, x2, x3, x7)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x4, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, x4, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x3, x4, x6)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x4, x6)

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

(31) Obligation:

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

DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_GAA(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_GAA(Left, Y, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U7_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GGA(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(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_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_gga(x1, x2, x3)  =  delete_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x1, x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x1, x2, x3, x7)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x1, x2, x3, x6)
DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)
U6_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GAA(x1, x2, x3, x7)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x4, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, x4, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x3, x4, x6)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x4, x6)

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

(32) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 8 less nodes.

(33) Complex Obligation (AND)

(34) 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_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(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_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_gga(x1, x2, x3)  =  delete_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x1, x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x1, x2, x3, x7)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)

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

(35) UsableRulesProof (EQUIVALENT transformation)

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

(36) 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

(37) PiDPToQDPProof (EQUIVALENT transformation)

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

(38) 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.

(39) 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

(40) TRUE

(41) Obligation:

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

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

The TRS R consists of the following rules:

delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(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_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_gga(x1, x2, x3)  =  delete_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x1, x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x1, x2, x3, x7)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)

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

(42) UsableRulesProof (EQUIVALENT transformation)

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

(43) Obligation:

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

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

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

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

(44) PiDPToQDPProof (SOUND transformation)

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

(45) Obligation:

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

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

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

(46) 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_GAA(tree(X, Left, X1)) → DELMIN_IN_GAA(Left)
    The graph contains the following edges 1 > 1

(47) TRUE

(48) Obligation:

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

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

The TRS R consists of the following rules:

delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(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_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_gga(x1, x2, x3)  =  delete_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x1, x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x1, x2, x3, x7)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x4, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x3, x4, x6)

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

(49) UsableRulesProof (EQUIVALENT transformation)

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

(50) Obligation:

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

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

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)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
s(x1)  =  s(x1)
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x4, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x3, x4, x6)

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