Left Termination of the query pattern insert_in_3(g, a, g) 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 PrologToPiTRSProof (⇐)
    ↳22 PiTRS
    ↳23 DependencyPairsProof (⇔)
    ↳24 PiDP
    ↳25 DependencyGraphProof (⇔)
    ↳26 AND
      ↳27 PiDP
        ↳28 UsableRulesProof (⇔)
        ↳29 PiDP
        ↳30 PiDPToQDPProof (⇔)
        ↳31 QDP
        ↳32 QDPSizeChangeProof (⇔)
        ↳33 TRUE
      ↳34 PiDP
        ↳35 UsableRulesProof (⇔)
        ↳36 PiDP

(0) Obligation:

Clauses:

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

Queries:

insert(g,a,g).

(1) PrologToPiTRSProof (SOUND transformation)

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

insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X1)) → less_out_gg(0, s(X1))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_in_gag(x1, x2, x3)  =  insert_in_gag(x1, x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
insert_out_gag(x1, x2, x3)  =  insert_out_gag(x1, x2, x3)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, 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)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, x6)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x3, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x2, x3, x5, 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:

insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X1)) → less_out_gg(0, s(X1))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_in_gag(x1, x2, x3)  =  insert_in_gag(x1, x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
insert_out_gag(x1, x2, x3)  =  insert_out_gag(x1, x2, x3)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, 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)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, x6)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x3, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x2, x3, x5, 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:

INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U1_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_GAG(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
U1_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → INSERT_IN_GAG(X, Left, Left1)
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U3_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_GAG(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U3_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → INSERT_IN_GAG(X, Right, Right1)

The TRS R consists of the following rules:

insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X1)) → less_out_gg(0, s(X1))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_in_gag(x1, x2, x3)  =  insert_in_gag(x1, x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
insert_out_gag(x1, x2, x3)  =  insert_out_gag(x1, x2, x3)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, 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)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, x6)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x3, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x2, x3, x5, x6)
INSERT_IN_GAG(x1, x2, x3)  =  INSERT_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x1, x2, x4, x5, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)
U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x2, x4, x5, x6)
U3_GAG(x1, x2, x3, x4, x5, x6)  =  U3_GAG(x1, x2, x3, x5, x6)
U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x1, x2, x3, x5, x6)

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

(4) Obligation:

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

INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U1_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_GAG(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
U1_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → INSERT_IN_GAG(X, Left, Left1)
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U3_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_GAG(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U3_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → INSERT_IN_GAG(X, Right, Right1)

The TRS R consists of the following rules:

insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X1)) → less_out_gg(0, s(X1))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_in_gag(x1, x2, x3)  =  insert_in_gag(x1, x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
insert_out_gag(x1, x2, x3)  =  insert_out_gag(x1, x2, x3)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, 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)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, x6)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x3, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x2, x3, x5, x6)
INSERT_IN_GAG(x1, x2, x3)  =  INSERT_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x1, x2, x4, x5, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)
U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x2, x4, x5, x6)
U3_GAG(x1, x2, x3, x4, x5, x6)  =  U3_GAG(x1, x2, x3, x5, x6)
U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x1, x2, x3, x5, x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 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:

insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X1)) → less_out_gg(0, s(X1))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_in_gag(x1, x2, x3)  =  insert_in_gag(x1, x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
insert_out_gag(x1, x2, x3)  =  insert_out_gag(x1, x2, x3)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, 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)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, x6)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x3, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x2, x3, x5, 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:

U1_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → INSERT_IN_GAG(X, Left, Left1)
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → INSERT_IN_GAG(X, Right, Right1)

The TRS R consists of the following rules:

insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X1)) → less_out_gg(0, s(X1))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_in_gag(x1, x2, x3)  =  insert_in_gag(x1, x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
insert_out_gag(x1, x2, x3)  =  insert_out_gag(x1, x2, x3)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, 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)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, x6)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x3, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x2, x3, x5, x6)
INSERT_IN_GAG(x1, x2, x3)  =  INSERT_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x1, x2, x4, x5, x6)
U3_GAG(x1, x2, x3, x4, x5, x6)  =  U3_GAG(x1, x2, x3, x5, x6)

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:

U1_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → INSERT_IN_GAG(X, Left, Left1)
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → INSERT_IN_GAG(X, Right, Right1)

The TRS R consists of the following rules:

less_in_gg(0, s(X1)) → less_out_gg(0, s(X1))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_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)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
INSERT_IN_GAG(x1, x2, x3)  =  INSERT_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x1, x2, x4, x5, x6)
U3_GAG(x1, x2, x3, x4, x5, x6)  =  U3_GAG(x1, x2, x3, x5, x6)

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:

U1_GAG(X, Y, Right, Left1, less_out_gg(X, Y)) → INSERT_IN_GAG(X, Left1)
INSERT_IN_GAG(X, tree(Y, Left1, Right)) → U1_GAG(X, Y, Right, Left1, less_in_gg(X, Y))
INSERT_IN_GAG(X, tree(Y, Left, Right1)) → U3_GAG(X, Y, Left, Right1, less_in_gg(Y, X))
U3_GAG(X, Y, Left, Right1, less_out_gg(Y, X)) → INSERT_IN_GAG(X, Right1)

The TRS R consists of the following rules:

less_in_gg(0, s(X1)) → less_out_gg(0, s(X1))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

The set Q consists of the following terms:

less_in_gg(x0, x1)
U5_gg(x0, x1, x2)

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:

  • INSERT_IN_GAG(X, tree(Y, Left1, Right)) → U1_GAG(X, Y, Right, Left1, less_in_gg(X, Y))
    The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4

  • INSERT_IN_GAG(X, tree(Y, Left, Right1)) → U3_GAG(X, Y, Left, Right1, less_in_gg(Y, X))
    The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4

  • U1_GAG(X, Y, Right, Left1, less_out_gg(X, Y)) → INSERT_IN_GAG(X, Left1)
    The graph contains the following edges 1 >= 1, 5 > 1, 4 >= 2

  • U3_GAG(X, Y, Left, Right1, less_out_gg(Y, X)) → INSERT_IN_GAG(X, Right1)
    The graph contains the following edges 1 >= 1, 5 > 1, 4 >= 2

(20) TRUE

(21) PrologToPiTRSProof (SOUND transformation)

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

insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X1)) → less_out_gg(0, s(X1))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_in_gag(x1, x2, x3)  =  insert_in_gag(x1, x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
insert_out_gag(x1, x2, x3)  =  insert_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, 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
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x2, x4, x6)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x3, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x2, x3, x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(22) Obligation:

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

insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X1)) → less_out_gg(0, s(X1))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_in_gag(x1, x2, x3)  =  insert_in_gag(x1, x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
insert_out_gag(x1, x2, x3)  =  insert_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, 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
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x2, x4, x6)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x3, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x2, x3, x6)

(23) 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:

INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U1_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_GAG(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
U1_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → INSERT_IN_GAG(X, Left, Left1)
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U3_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_GAG(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U3_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → INSERT_IN_GAG(X, Right, Right1)

The TRS R consists of the following rules:

insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X1)) → less_out_gg(0, s(X1))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_in_gag(x1, x2, x3)  =  insert_in_gag(x1, x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
insert_out_gag(x1, x2, x3)  =  insert_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, 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
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x2, x4, x6)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x3, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x2, x3, x6)
INSERT_IN_GAG(x1, x2, x3)  =  INSERT_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x1, x2, x4, x5, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x3)
U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x2, x4, x6)
U3_GAG(x1, x2, x3, x4, x5, x6)  =  U3_GAG(x1, x2, x3, x5, x6)
U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x2, x3, x6)

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

(24) Obligation:

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

INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U1_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_GAG(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
U1_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → INSERT_IN_GAG(X, Left, Left1)
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U3_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_GAG(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U3_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → INSERT_IN_GAG(X, Right, Right1)

The TRS R consists of the following rules:

insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X1)) → less_out_gg(0, s(X1))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_in_gag(x1, x2, x3)  =  insert_in_gag(x1, x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
insert_out_gag(x1, x2, x3)  =  insert_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, 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
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x2, x4, x6)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x3, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x2, x3, x6)
INSERT_IN_GAG(x1, x2, x3)  =  INSERT_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x1, x2, x4, x5, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x3)
U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x2, x4, x6)
U3_GAG(x1, x2, x3, x4, x5, x6)  =  U3_GAG(x1, x2, x3, x5, x6)
U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x2, x3, x6)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(26) Complex Obligation (AND)

(27) 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:

insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X1)) → less_out_gg(0, s(X1))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_in_gag(x1, x2, x3)  =  insert_in_gag(x1, x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
insert_out_gag(x1, x2, x3)  =  insert_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, 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
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x2, x4, x6)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x3, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x2, x3, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)

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

(28) UsableRulesProof (EQUIVALENT transformation)

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

(29) 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

(30) PiDPToQDPProof (EQUIVALENT transformation)

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

(31) 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.

(32) 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

(33) TRUE

(34) Obligation:

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

U1_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → INSERT_IN_GAG(X, Left, Left1)
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → INSERT_IN_GAG(X, Right, Right1)

The TRS R consists of the following rules:

insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X1)) → less_out_gg(0, s(X1))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_in_gag(x1, x2, x3)  =  insert_in_gag(x1, x3)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
insert_out_gag(x1, x2, x3)  =  insert_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x1, x2, x4, x5, 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
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x2, x4, x6)
U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x1, x2, x3, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x2, x3, x6)
INSERT_IN_GAG(x1, x2, x3)  =  INSERT_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x1, x2, x4, x5, x6)
U3_GAG(x1, x2, x3, x4, x5, x6)  =  U3_GAG(x1, x2, x3, x5, x6)

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:

U1_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → INSERT_IN_GAG(X, Left, Left1)
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → INSERT_IN_GAG(X, Right, Right1)

The TRS R consists of the following rules:

less_in_gg(0, s(X1)) → less_out_gg(0, s(X1))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_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
U5_gg(x1, x2, x3)  =  U5_gg(x3)
INSERT_IN_GAG(x1, x2, x3)  =  INSERT_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x1, x2, x4, x5, x6)
U3_GAG(x1, x2, x3, x4, x5, x6)  =  U3_GAG(x1, x2, x3, x5, x6)

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