Left Termination of the query pattern in_in_2(g, a) w.r.t. the given Prolog program could not be shown:



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof

Clauses:

in(X, tree(X, X1, X2)).
in(X, tree(Y, Left, X1)) :- ','(less(X, Y), in(X, Left)).
in(X, tree(Y, X1, Right)) :- ','(less(Y, X), in(X, Right)).
less(0, s(X)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

in(g,a).

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

in_in_ga(X, tree(X, X1, X2)) → in_out_ga(X, tree(X, X1, X2))
in_in_ga(X, tree(Y, Left, X1)) → U1_ga(X, Y, Left, X1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U1_ga(X, Y, Left, X1, less_out_ga(X, Y)) → U2_ga(X, Y, Left, X1, in_in_ga(X, Left))
in_in_ga(X, tree(Y, X1, Right)) → U3_ga(X, Y, X1, Right, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U3_ga(X, Y, X1, Right, less_out_ag(Y, X)) → U4_ga(X, Y, X1, Right, in_in_ga(X, Right))
U4_ga(X, Y, X1, Right, in_out_ga(X, Right)) → in_out_ga(X, tree(Y, X1, Right))
U2_ga(X, Y, Left, X1, in_out_ga(X, Left)) → in_out_ga(X, tree(Y, Left, X1))

The argument filtering Pi contains the following mapping:
in_in_ga(x1, x2)  =  in_in_ga(x1)
in_out_ga(x1, x2)  =  in_out_ga(x2)
tree(x1, x2, x3)  =  tree(x1)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x2)
s(x1)  =  s
U5_ga(x1, x2, x3)  =  U5_ga(x3)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x2, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x2, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
  ↳ PrologToPiTRSProof

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

in_in_ga(X, tree(X, X1, X2)) → in_out_ga(X, tree(X, X1, X2))
in_in_ga(X, tree(Y, Left, X1)) → U1_ga(X, Y, Left, X1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U1_ga(X, Y, Left, X1, less_out_ga(X, Y)) → U2_ga(X, Y, Left, X1, in_in_ga(X, Left))
in_in_ga(X, tree(Y, X1, Right)) → U3_ga(X, Y, X1, Right, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U3_ga(X, Y, X1, Right, less_out_ag(Y, X)) → U4_ga(X, Y, X1, Right, in_in_ga(X, Right))
U4_ga(X, Y, X1, Right, in_out_ga(X, Right)) → in_out_ga(X, tree(Y, X1, Right))
U2_ga(X, Y, Left, X1, in_out_ga(X, Left)) → in_out_ga(X, tree(Y, Left, X1))

The argument filtering Pi contains the following mapping:
in_in_ga(x1, x2)  =  in_in_ga(x1)
in_out_ga(x1, x2)  =  in_out_ga(x2)
tree(x1, x2, x3)  =  tree(x1)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x2)
s(x1)  =  s
U5_ga(x1, x2, x3)  =  U5_ga(x3)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x2, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x2, x5)


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

IN_IN_GA(X, tree(Y, Left, X1)) → U1_GA(X, Y, Left, X1, less_in_ga(X, Y))
IN_IN_GA(X, tree(Y, Left, X1)) → LESS_IN_GA(X, Y)
LESS_IN_GA(s(X), s(Y)) → U5_GA(X, Y, less_in_aa(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U5_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U1_GA(X, Y, Left, X1, less_out_ga(X, Y)) → U2_GA(X, Y, Left, X1, in_in_ga(X, Left))
U1_GA(X, Y, Left, X1, less_out_ga(X, Y)) → IN_IN_GA(X, Left)
IN_IN_GA(X, tree(Y, X1, Right)) → U3_GA(X, Y, X1, Right, less_in_ag(Y, X))
IN_IN_GA(X, tree(Y, X1, Right)) → LESS_IN_AG(Y, X)
LESS_IN_AG(s(X), s(Y)) → U5_AG(X, Y, less_in_aa(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AA(X, Y)
U3_GA(X, Y, X1, Right, less_out_ag(Y, X)) → U4_GA(X, Y, X1, Right, in_in_ga(X, Right))
U3_GA(X, Y, X1, Right, less_out_ag(Y, X)) → IN_IN_GA(X, Right)

The TRS R consists of the following rules:

in_in_ga(X, tree(X, X1, X2)) → in_out_ga(X, tree(X, X1, X2))
in_in_ga(X, tree(Y, Left, X1)) → U1_ga(X, Y, Left, X1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U1_ga(X, Y, Left, X1, less_out_ga(X, Y)) → U2_ga(X, Y, Left, X1, in_in_ga(X, Left))
in_in_ga(X, tree(Y, X1, Right)) → U3_ga(X, Y, X1, Right, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U3_ga(X, Y, X1, Right, less_out_ag(Y, X)) → U4_ga(X, Y, X1, Right, in_in_ga(X, Right))
U4_ga(X, Y, X1, Right, in_out_ga(X, Right)) → in_out_ga(X, tree(Y, X1, Right))
U2_ga(X, Y, Left, X1, in_out_ga(X, Left)) → in_out_ga(X, tree(Y, Left, X1))

The argument filtering Pi contains the following mapping:
in_in_ga(x1, x2)  =  in_in_ga(x1)
in_out_ga(x1, x2)  =  in_out_ga(x2)
tree(x1, x2, x3)  =  tree(x1)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x2)
s(x1)  =  s
U5_ga(x1, x2, x3)  =  U5_ga(x3)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x2, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x2, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x2, x5)
U5_AA(x1, x2, x3)  =  U5_AA(x3)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x2, x5)
U5_AG(x1, x2, x3)  =  U5_AG(x3)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x5)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
IN_IN_GA(x1, x2)  =  IN_IN_GA(x1)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)

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

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

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

IN_IN_GA(X, tree(Y, Left, X1)) → U1_GA(X, Y, Left, X1, less_in_ga(X, Y))
IN_IN_GA(X, tree(Y, Left, X1)) → LESS_IN_GA(X, Y)
LESS_IN_GA(s(X), s(Y)) → U5_GA(X, Y, less_in_aa(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U5_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U1_GA(X, Y, Left, X1, less_out_ga(X, Y)) → U2_GA(X, Y, Left, X1, in_in_ga(X, Left))
U1_GA(X, Y, Left, X1, less_out_ga(X, Y)) → IN_IN_GA(X, Left)
IN_IN_GA(X, tree(Y, X1, Right)) → U3_GA(X, Y, X1, Right, less_in_ag(Y, X))
IN_IN_GA(X, tree(Y, X1, Right)) → LESS_IN_AG(Y, X)
LESS_IN_AG(s(X), s(Y)) → U5_AG(X, Y, less_in_aa(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AA(X, Y)
U3_GA(X, Y, X1, Right, less_out_ag(Y, X)) → U4_GA(X, Y, X1, Right, in_in_ga(X, Right))
U3_GA(X, Y, X1, Right, less_out_ag(Y, X)) → IN_IN_GA(X, Right)

The TRS R consists of the following rules:

in_in_ga(X, tree(X, X1, X2)) → in_out_ga(X, tree(X, X1, X2))
in_in_ga(X, tree(Y, Left, X1)) → U1_ga(X, Y, Left, X1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U1_ga(X, Y, Left, X1, less_out_ga(X, Y)) → U2_ga(X, Y, Left, X1, in_in_ga(X, Left))
in_in_ga(X, tree(Y, X1, Right)) → U3_ga(X, Y, X1, Right, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U3_ga(X, Y, X1, Right, less_out_ag(Y, X)) → U4_ga(X, Y, X1, Right, in_in_ga(X, Right))
U4_ga(X, Y, X1, Right, in_out_ga(X, Right)) → in_out_ga(X, tree(Y, X1, Right))
U2_ga(X, Y, Left, X1, in_out_ga(X, Left)) → in_out_ga(X, tree(Y, Left, X1))

The argument filtering Pi contains the following mapping:
in_in_ga(x1, x2)  =  in_in_ga(x1)
in_out_ga(x1, x2)  =  in_out_ga(x2)
tree(x1, x2, x3)  =  tree(x1)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x2)
s(x1)  =  s
U5_ga(x1, x2, x3)  =  U5_ga(x3)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x2, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x2, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x2, x5)
U5_AA(x1, x2, x3)  =  U5_AA(x3)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x2, x5)
U5_AG(x1, x2, x3)  =  U5_AG(x3)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x5)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
IN_IN_GA(x1, x2)  =  IN_IN_GA(x1)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 9 less nodes.

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

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

The TRS R consists of the following rules:

in_in_ga(X, tree(X, X1, X2)) → in_out_ga(X, tree(X, X1, X2))
in_in_ga(X, tree(Y, Left, X1)) → U1_ga(X, Y, Left, X1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U1_ga(X, Y, Left, X1, less_out_ga(X, Y)) → U2_ga(X, Y, Left, X1, in_in_ga(X, Left))
in_in_ga(X, tree(Y, X1, Right)) → U3_ga(X, Y, X1, Right, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U3_ga(X, Y, X1, Right, less_out_ag(Y, X)) → U4_ga(X, Y, X1, Right, in_in_ga(X, Right))
U4_ga(X, Y, X1, Right, in_out_ga(X, Right)) → in_out_ga(X, tree(Y, X1, Right))
U2_ga(X, Y, Left, X1, in_out_ga(X, Left)) → in_out_ga(X, tree(Y, Left, X1))

The argument filtering Pi contains the following mapping:
in_in_ga(x1, x2)  =  in_in_ga(x1)
in_out_ga(x1, x2)  =  in_out_ga(x2)
tree(x1, x2, x3)  =  tree(x1)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x2)
s(x1)  =  s
U5_ga(x1, x2, x3)  =  U5_ga(x3)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x2, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x2, x5)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA

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

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

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

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

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

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

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

LESS_IN_AA → LESS_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

LESS_IN_AA → LESS_IN_AA

The TRS R consists of the following rules:none


s = LESS_IN_AA evaluates to t =LESS_IN_AA

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



Rewriting sequence

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





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

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

U1_GA(X, Y, Left, X1, less_out_ga(X, Y)) → IN_IN_GA(X, Left)
IN_IN_GA(X, tree(Y, Left, X1)) → U1_GA(X, Y, Left, X1, less_in_ga(X, Y))
U3_GA(X, Y, X1, Right, less_out_ag(Y, X)) → IN_IN_GA(X, Right)
IN_IN_GA(X, tree(Y, X1, Right)) → U3_GA(X, Y, X1, Right, less_in_ag(Y, X))

The TRS R consists of the following rules:

in_in_ga(X, tree(X, X1, X2)) → in_out_ga(X, tree(X, X1, X2))
in_in_ga(X, tree(Y, Left, X1)) → U1_ga(X, Y, Left, X1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U1_ga(X, Y, Left, X1, less_out_ga(X, Y)) → U2_ga(X, Y, Left, X1, in_in_ga(X, Left))
in_in_ga(X, tree(Y, X1, Right)) → U3_ga(X, Y, X1, Right, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U3_ga(X, Y, X1, Right, less_out_ag(Y, X)) → U4_ga(X, Y, X1, Right, in_in_ga(X, Right))
U4_ga(X, Y, X1, Right, in_out_ga(X, Right)) → in_out_ga(X, tree(Y, X1, Right))
U2_ga(X, Y, Left, X1, in_out_ga(X, Left)) → in_out_ga(X, tree(Y, Left, X1))

The argument filtering Pi contains the following mapping:
in_in_ga(x1, x2)  =  in_in_ga(x1)
in_out_ga(x1, x2)  =  in_out_ga(x2)
tree(x1, x2, x3)  =  tree(x1)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x2)
s(x1)  =  s
U5_ga(x1, x2, x3)  =  U5_ga(x3)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x2, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x2, x5)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x5)
IN_IN_GA(x1, x2)  =  IN_IN_GA(x1)

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

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

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

U1_GA(X, Y, Left, X1, less_out_ga(X, Y)) → IN_IN_GA(X, Left)
IN_IN_GA(X, tree(Y, Left, X1)) → U1_GA(X, Y, Left, X1, less_in_ga(X, Y))
U3_GA(X, Y, X1, Right, less_out_ag(Y, X)) → IN_IN_GA(X, Right)
IN_IN_GA(X, tree(Y, X1, Right)) → U3_GA(X, Y, X1, Right, less_in_ag(Y, X))

The TRS R consists of the following rules:

less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x2)
s(x1)  =  s
U5_ga(x1, x2, x3)  =  U5_ga(x3)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
less_in_ag(x1, x2)  =  less_in_ag(x2)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x5)
IN_IN_GA(x1, x2)  =  IN_IN_GA(x1)

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

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

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

U3_GA(X, less_out_ag(Y)) → IN_IN_GA(X)
U1_GA(X, less_out_ga(Y)) → IN_IN_GA(X)
IN_IN_GA(X) → U3_GA(X, less_in_ag(X))
IN_IN_GA(X) → U1_GA(X, less_in_ga(X))

The TRS R consists of the following rules:

less_in_ga(0) → less_out_ga(s)
less_in_ga(s) → U5_ga(less_in_aa)
less_in_ag(s) → less_out_ag(0)
less_in_ag(s) → U5_ag(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s)
less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule IN_IN_GA(X) → U1_GA(X, less_in_ga(X)) at position [1] we obtained the following new rules:

IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))
IN_IN_GA(0) → U1_GA(0, less_out_ga(s))



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

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

U3_GA(X, less_out_ag(Y)) → IN_IN_GA(X)
IN_IN_GA(0) → U1_GA(0, less_out_ga(s))
U1_GA(X, less_out_ga(Y)) → IN_IN_GA(X)
IN_IN_GA(X) → U3_GA(X, less_in_ag(X))
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))

The TRS R consists of the following rules:

less_in_ga(0) → less_out_ga(s)
less_in_ga(s) → U5_ga(less_in_aa)
less_in_ag(s) → less_out_ag(0)
less_in_ag(s) → U5_ag(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s)
less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
  ↳ PrologToPiTRSProof

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

U3_GA(X, less_out_ag(Y)) → IN_IN_GA(X)
IN_IN_GA(0) → U1_GA(0, less_out_ga(s))
U1_GA(X, less_out_ga(Y)) → IN_IN_GA(X)
IN_IN_GA(X) → U3_GA(X, less_in_ag(X))
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
less_in_ag(s) → less_out_ag(0)
less_in_ag(s) → U5_ag(less_in_aa)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s)

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

less_in_ga(x0)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
  ↳ PrologToPiTRSProof

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

U3_GA(X, less_out_ag(Y)) → IN_IN_GA(X)
IN_IN_GA(0) → U1_GA(0, less_out_ga(s))
U1_GA(X, less_out_ga(Y)) → IN_IN_GA(X)
IN_IN_GA(X) → U3_GA(X, less_in_ag(X))
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
less_in_ag(s) → less_out_ag(0)
less_in_ag(s) → U5_ag(less_in_aa)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s)

The set Q consists of the following terms:

less_in_ag(x0)
U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule IN_IN_GA(X) → U3_GA(X, less_in_ag(X)) at position [1] we obtained the following new rules:

IN_IN_GA(s) → U3_GA(s, U5_ag(less_in_aa))
IN_IN_GA(s) → U3_GA(s, less_out_ag(0))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ UsableRulesProof
  ↳ PrologToPiTRSProof

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

U3_GA(X, less_out_ag(Y)) → IN_IN_GA(X)
IN_IN_GA(0) → U1_GA(0, less_out_ga(s))
U1_GA(X, less_out_ga(Y)) → IN_IN_GA(X)
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))
IN_IN_GA(s) → U3_GA(s, U5_ag(less_in_aa))
IN_IN_GA(s) → U3_GA(s, less_out_ag(0))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
less_in_ag(s) → less_out_ag(0)
less_in_ag(s) → U5_ag(less_in_aa)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s)

The set Q consists of the following terms:

less_in_ag(x0)
U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ UsableRulesProof
                                          ↳ QDP
                                            ↳ QReductionProof
  ↳ PrologToPiTRSProof

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

U3_GA(X, less_out_ag(Y)) → IN_IN_GA(X)
IN_IN_GA(0) → U1_GA(0, less_out_ga(s))
U1_GA(X, less_out_ga(Y)) → IN_IN_GA(X)
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))
IN_IN_GA(s) → U3_GA(s, U5_ag(less_in_aa))
IN_IN_GA(s) → U3_GA(s, less_out_ag(0))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s)

The set Q consists of the following terms:

less_in_ag(x0)
U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

less_in_ag(x0)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ UsableRulesProof
                                          ↳ QDP
                                            ↳ QReductionProof
                                              ↳ QDP
                                                ↳ Instantiation
  ↳ PrologToPiTRSProof

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

U3_GA(X, less_out_ag(Y)) → IN_IN_GA(X)
IN_IN_GA(0) → U1_GA(0, less_out_ga(s))
U1_GA(X, less_out_ga(Y)) → IN_IN_GA(X)
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))
IN_IN_GA(s) → U3_GA(s, U5_ag(less_in_aa))
IN_IN_GA(s) → U3_GA(s, less_out_ag(0))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s)

The set Q consists of the following terms:

U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U3_GA(X, less_out_ag(Y)) → IN_IN_GA(X) we obtained the following new rules:

U3_GA(s, less_out_ag(0)) → IN_IN_GA(s)
U3_GA(s, less_out_ag(x1)) → IN_IN_GA(s)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ UsableRulesProof
                                          ↳ QDP
                                            ↳ QReductionProof
                                              ↳ QDP
                                                ↳ Instantiation
                                                  ↳ QDP
                                                    ↳ Instantiation
  ↳ PrologToPiTRSProof

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

U3_GA(s, less_out_ag(0)) → IN_IN_GA(s)
IN_IN_GA(0) → U1_GA(0, less_out_ga(s))
U1_GA(X, less_out_ga(Y)) → IN_IN_GA(X)
U3_GA(s, less_out_ag(x1)) → IN_IN_GA(s)
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))
IN_IN_GA(s) → U3_GA(s, U5_ag(less_in_aa))
IN_IN_GA(s) → U3_GA(s, less_out_ag(0))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s)

The set Q consists of the following terms:

U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U1_GA(X, less_out_ga(Y)) → IN_IN_GA(X) we obtained the following new rules:

U1_GA(s, less_out_ga(x1)) → IN_IN_GA(s)
U1_GA(0, less_out_ga(s)) → IN_IN_GA(0)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ UsableRulesProof
                                          ↳ QDP
                                            ↳ QReductionProof
                                              ↳ QDP
                                                ↳ Instantiation
                                                  ↳ QDP
                                                    ↳ Instantiation
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
  ↳ PrologToPiTRSProof

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

U1_GA(s, less_out_ga(x1)) → IN_IN_GA(s)
U1_GA(0, less_out_ga(s)) → IN_IN_GA(0)
U3_GA(s, less_out_ag(0)) → IN_IN_GA(s)
IN_IN_GA(0) → U1_GA(0, less_out_ga(s))
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))
U3_GA(s, less_out_ag(x1)) → IN_IN_GA(s)
IN_IN_GA(s) → U3_GA(s, U5_ag(less_in_aa))
IN_IN_GA(s) → U3_GA(s, less_out_ag(0))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s)

The set Q consists of the following terms:

U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ UsableRulesProof
                                          ↳ QDP
                                            ↳ QReductionProof
                                              ↳ QDP
                                                ↳ Instantiation
                                                  ↳ QDP
                                                    ↳ Instantiation
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ AND
                                                            ↳ QDP
                                                              ↳ NonTerminationProof
                                                            ↳ QDP
  ↳ PrologToPiTRSProof

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

U1_GA(s, less_out_ga(x1)) → IN_IN_GA(s)
U3_GA(s, less_out_ag(0)) → IN_IN_GA(s)
U3_GA(s, less_out_ag(x1)) → IN_IN_GA(s)
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))
IN_IN_GA(s) → U3_GA(s, U5_ag(less_in_aa))
IN_IN_GA(s) → U3_GA(s, less_out_ag(0))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s)

The set Q consists of the following terms:

U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

U1_GA(s, less_out_ga(x1)) → IN_IN_GA(s)
U3_GA(s, less_out_ag(0)) → IN_IN_GA(s)
U3_GA(s, less_out_ag(x1)) → IN_IN_GA(s)
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))
IN_IN_GA(s) → U3_GA(s, U5_ag(less_in_aa))
IN_IN_GA(s) → U3_GA(s, less_out_ag(0))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s)


s = IN_IN_GA(s) evaluates to t =IN_IN_GA(s)

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



Rewriting sequence

IN_IN_GA(s) → U3_GA(s, less_out_ag(0))
with rule IN_IN_GA(s) → U3_GA(s, less_out_ag(0)) at position [] and matcher [ ]

U3_GA(s, less_out_ag(0)) → IN_IN_GA(s)
with rule U3_GA(s, less_out_ag(0)) → IN_IN_GA(s)

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


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





↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ UsableRulesProof
                                          ↳ QDP
                                            ↳ QReductionProof
                                              ↳ QDP
                                                ↳ Instantiation
                                                  ↳ QDP
                                                    ↳ Instantiation
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ AND
                                                            ↳ QDP
                                                            ↳ QDP
                                                              ↳ UsableRulesProof
  ↳ PrologToPiTRSProof

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

U1_GA(0, less_out_ga(s)) → IN_IN_GA(0)
IN_IN_GA(0) → U1_GA(0, less_out_ga(s))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s)

The set Q consists of the following terms:

U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ UsableRulesProof
                                          ↳ QDP
                                            ↳ QReductionProof
                                              ↳ QDP
                                                ↳ Instantiation
                                                  ↳ QDP
                                                    ↳ Instantiation
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ AND
                                                            ↳ QDP
                                                            ↳ QDP
                                                              ↳ UsableRulesProof
                                                                ↳ QDP
                                                                  ↳ QReductionProof
  ↳ PrologToPiTRSProof

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

U1_GA(0, less_out_ga(s)) → IN_IN_GA(0)
IN_IN_GA(0) → U1_GA(0, less_out_ga(s))

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

U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ UsableRulesProof
                                          ↳ QDP
                                            ↳ QReductionProof
                                              ↳ QDP
                                                ↳ Instantiation
                                                  ↳ QDP
                                                    ↳ Instantiation
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ AND
                                                            ↳ QDP
                                                            ↳ QDP
                                                              ↳ UsableRulesProof
                                                                ↳ QDP
                                                                  ↳ QReductionProof
                                                                    ↳ QDP
                                                                      ↳ NonTerminationProof
  ↳ PrologToPiTRSProof

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

U1_GA(0, less_out_ga(s)) → IN_IN_GA(0)
IN_IN_GA(0) → U1_GA(0, less_out_ga(s))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

U1_GA(0, less_out_ga(s)) → IN_IN_GA(0)
IN_IN_GA(0) → U1_GA(0, less_out_ga(s))

The TRS R consists of the following rules:none


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

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



Rewriting sequence

IN_IN_GA(0) → U1_GA(0, less_out_ga(s))
with rule IN_IN_GA(0) → U1_GA(0, less_out_ga(s)) at position [] and matcher [ ]

U1_GA(0, less_out_ga(s)) → IN_IN_GA(0)
with rule U1_GA(0, less_out_ga(s)) → IN_IN_GA(0)

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


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




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

in_in_ga(X, tree(X, X1, X2)) → in_out_ga(X, tree(X, X1, X2))
in_in_ga(X, tree(Y, Left, X1)) → U1_ga(X, Y, Left, X1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U1_ga(X, Y, Left, X1, less_out_ga(X, Y)) → U2_ga(X, Y, Left, X1, in_in_ga(X, Left))
in_in_ga(X, tree(Y, X1, Right)) → U3_ga(X, Y, X1, Right, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U3_ga(X, Y, X1, Right, less_out_ag(Y, X)) → U4_ga(X, Y, X1, Right, in_in_ga(X, Right))
U4_ga(X, Y, X1, Right, in_out_ga(X, Right)) → in_out_ga(X, tree(Y, X1, Right))
U2_ga(X, Y, Left, X1, in_out_ga(X, Left)) → in_out_ga(X, tree(Y, Left, X1))

The argument filtering Pi contains the following mapping:
in_in_ga(x1, x2)  =  in_in_ga(x1)
in_out_ga(x1, x2)  =  in_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1, x2)
s(x1)  =  s
U5_ga(x1, x2, x3)  =  U5_ga(x3)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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

in_in_ga(X, tree(X, X1, X2)) → in_out_ga(X, tree(X, X1, X2))
in_in_ga(X, tree(Y, Left, X1)) → U1_ga(X, Y, Left, X1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U1_ga(X, Y, Left, X1, less_out_ga(X, Y)) → U2_ga(X, Y, Left, X1, in_in_ga(X, Left))
in_in_ga(X, tree(Y, X1, Right)) → U3_ga(X, Y, X1, Right, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U3_ga(X, Y, X1, Right, less_out_ag(Y, X)) → U4_ga(X, Y, X1, Right, in_in_ga(X, Right))
U4_ga(X, Y, X1, Right, in_out_ga(X, Right)) → in_out_ga(X, tree(Y, X1, Right))
U2_ga(X, Y, Left, X1, in_out_ga(X, Left)) → in_out_ga(X, tree(Y, Left, X1))

The argument filtering Pi contains the following mapping:
in_in_ga(x1, x2)  =  in_in_ga(x1)
in_out_ga(x1, x2)  =  in_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1, x2)
s(x1)  =  s
U5_ga(x1, x2, x3)  =  U5_ga(x3)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x5)


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

IN_IN_GA(X, tree(Y, Left, X1)) → U1_GA(X, Y, Left, X1, less_in_ga(X, Y))
IN_IN_GA(X, tree(Y, Left, X1)) → LESS_IN_GA(X, Y)
LESS_IN_GA(s(X), s(Y)) → U5_GA(X, Y, less_in_aa(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U5_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U1_GA(X, Y, Left, X1, less_out_ga(X, Y)) → U2_GA(X, Y, Left, X1, in_in_ga(X, Left))
U1_GA(X, Y, Left, X1, less_out_ga(X, Y)) → IN_IN_GA(X, Left)
IN_IN_GA(X, tree(Y, X1, Right)) → U3_GA(X, Y, X1, Right, less_in_ag(Y, X))
IN_IN_GA(X, tree(Y, X1, Right)) → LESS_IN_AG(Y, X)
LESS_IN_AG(s(X), s(Y)) → U5_AG(X, Y, less_in_aa(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AA(X, Y)
U3_GA(X, Y, X1, Right, less_out_ag(Y, X)) → U4_GA(X, Y, X1, Right, in_in_ga(X, Right))
U3_GA(X, Y, X1, Right, less_out_ag(Y, X)) → IN_IN_GA(X, Right)

The TRS R consists of the following rules:

in_in_ga(X, tree(X, X1, X2)) → in_out_ga(X, tree(X, X1, X2))
in_in_ga(X, tree(Y, Left, X1)) → U1_ga(X, Y, Left, X1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U1_ga(X, Y, Left, X1, less_out_ga(X, Y)) → U2_ga(X, Y, Left, X1, in_in_ga(X, Left))
in_in_ga(X, tree(Y, X1, Right)) → U3_ga(X, Y, X1, Right, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U3_ga(X, Y, X1, Right, less_out_ag(Y, X)) → U4_ga(X, Y, X1, Right, in_in_ga(X, Right))
U4_ga(X, Y, X1, Right, in_out_ga(X, Right)) → in_out_ga(X, tree(Y, X1, Right))
U2_ga(X, Y, Left, X1, in_out_ga(X, Left)) → in_out_ga(X, tree(Y, Left, X1))

The argument filtering Pi contains the following mapping:
in_in_ga(x1, x2)  =  in_in_ga(x1)
in_out_ga(x1, x2)  =  in_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1, x2)
s(x1)  =  s
U5_ga(x1, x2, x3)  =  U5_ga(x3)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x5)
U5_AA(x1, x2, x3)  =  U5_AA(x3)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x5)
U5_AG(x1, x2, x3)  =  U5_AG(x3)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x5)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
IN_IN_GA(x1, x2)  =  IN_IN_GA(x1)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)

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

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

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

IN_IN_GA(X, tree(Y, Left, X1)) → U1_GA(X, Y, Left, X1, less_in_ga(X, Y))
IN_IN_GA(X, tree(Y, Left, X1)) → LESS_IN_GA(X, Y)
LESS_IN_GA(s(X), s(Y)) → U5_GA(X, Y, less_in_aa(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U5_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U1_GA(X, Y, Left, X1, less_out_ga(X, Y)) → U2_GA(X, Y, Left, X1, in_in_ga(X, Left))
U1_GA(X, Y, Left, X1, less_out_ga(X, Y)) → IN_IN_GA(X, Left)
IN_IN_GA(X, tree(Y, X1, Right)) → U3_GA(X, Y, X1, Right, less_in_ag(Y, X))
IN_IN_GA(X, tree(Y, X1, Right)) → LESS_IN_AG(Y, X)
LESS_IN_AG(s(X), s(Y)) → U5_AG(X, Y, less_in_aa(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AA(X, Y)
U3_GA(X, Y, X1, Right, less_out_ag(Y, X)) → U4_GA(X, Y, X1, Right, in_in_ga(X, Right))
U3_GA(X, Y, X1, Right, less_out_ag(Y, X)) → IN_IN_GA(X, Right)

The TRS R consists of the following rules:

in_in_ga(X, tree(X, X1, X2)) → in_out_ga(X, tree(X, X1, X2))
in_in_ga(X, tree(Y, Left, X1)) → U1_ga(X, Y, Left, X1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U1_ga(X, Y, Left, X1, less_out_ga(X, Y)) → U2_ga(X, Y, Left, X1, in_in_ga(X, Left))
in_in_ga(X, tree(Y, X1, Right)) → U3_ga(X, Y, X1, Right, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U3_ga(X, Y, X1, Right, less_out_ag(Y, X)) → U4_ga(X, Y, X1, Right, in_in_ga(X, Right))
U4_ga(X, Y, X1, Right, in_out_ga(X, Right)) → in_out_ga(X, tree(Y, X1, Right))
U2_ga(X, Y, Left, X1, in_out_ga(X, Left)) → in_out_ga(X, tree(Y, Left, X1))

The argument filtering Pi contains the following mapping:
in_in_ga(x1, x2)  =  in_in_ga(x1)
in_out_ga(x1, x2)  =  in_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1, x2)
s(x1)  =  s
U5_ga(x1, x2, x3)  =  U5_ga(x3)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x5)
U5_AA(x1, x2, x3)  =  U5_AA(x3)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x5)
U5_AG(x1, x2, x3)  =  U5_AG(x3)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x5)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
IN_IN_GA(x1, x2)  =  IN_IN_GA(x1)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 9 less nodes.

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

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

The TRS R consists of the following rules:

in_in_ga(X, tree(X, X1, X2)) → in_out_ga(X, tree(X, X1, X2))
in_in_ga(X, tree(Y, Left, X1)) → U1_ga(X, Y, Left, X1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U1_ga(X, Y, Left, X1, less_out_ga(X, Y)) → U2_ga(X, Y, Left, X1, in_in_ga(X, Left))
in_in_ga(X, tree(Y, X1, Right)) → U3_ga(X, Y, X1, Right, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U3_ga(X, Y, X1, Right, less_out_ag(Y, X)) → U4_ga(X, Y, X1, Right, in_in_ga(X, Right))
U4_ga(X, Y, X1, Right, in_out_ga(X, Right)) → in_out_ga(X, tree(Y, X1, Right))
U2_ga(X, Y, Left, X1, in_out_ga(X, Left)) → in_out_ga(X, tree(Y, Left, X1))

The argument filtering Pi contains the following mapping:
in_in_ga(x1, x2)  =  in_in_ga(x1)
in_out_ga(x1, x2)  =  in_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1, x2)
s(x1)  =  s
U5_ga(x1, x2, x3)  =  U5_ga(x3)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x5)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA

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

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

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

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

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

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

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

LESS_IN_AA → LESS_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

LESS_IN_AA → LESS_IN_AA

The TRS R consists of the following rules:none


s = LESS_IN_AA evaluates to t =LESS_IN_AA

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



Rewriting sequence

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





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

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

U1_GA(X, Y, Left, X1, less_out_ga(X, Y)) → IN_IN_GA(X, Left)
IN_IN_GA(X, tree(Y, Left, X1)) → U1_GA(X, Y, Left, X1, less_in_ga(X, Y))
U3_GA(X, Y, X1, Right, less_out_ag(Y, X)) → IN_IN_GA(X, Right)
IN_IN_GA(X, tree(Y, X1, Right)) → U3_GA(X, Y, X1, Right, less_in_ag(Y, X))

The TRS R consists of the following rules:

in_in_ga(X, tree(X, X1, X2)) → in_out_ga(X, tree(X, X1, X2))
in_in_ga(X, tree(Y, Left, X1)) → U1_ga(X, Y, Left, X1, less_in_ga(X, Y))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U1_ga(X, Y, Left, X1, less_out_ga(X, Y)) → U2_ga(X, Y, Left, X1, in_in_ga(X, Left))
in_in_ga(X, tree(Y, X1, Right)) → U3_ga(X, Y, X1, Right, less_in_ag(Y, X))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U3_ga(X, Y, X1, Right, less_out_ag(Y, X)) → U4_ga(X, Y, X1, Right, in_in_ga(X, Right))
U4_ga(X, Y, X1, Right, in_out_ga(X, Right)) → in_out_ga(X, tree(Y, X1, Right))
U2_ga(X, Y, Left, X1, in_out_ga(X, Left)) → in_out_ga(X, tree(Y, Left, X1))

The argument filtering Pi contains the following mapping:
in_in_ga(x1, x2)  =  in_in_ga(x1)
in_out_ga(x1, x2)  =  in_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1, x2)
s(x1)  =  s
U5_ga(x1, x2, x3)  =  U5_ga(x3)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x5)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x5)
IN_IN_GA(x1, x2)  =  IN_IN_GA(x1)

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

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

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

U1_GA(X, Y, Left, X1, less_out_ga(X, Y)) → IN_IN_GA(X, Left)
IN_IN_GA(X, tree(Y, Left, X1)) → U1_GA(X, Y, Left, X1, less_in_ga(X, Y))
U3_GA(X, Y, X1, Right, less_out_ag(Y, X)) → IN_IN_GA(X, Right)
IN_IN_GA(X, tree(Y, X1, Right)) → U3_GA(X, Y, X1, Right, less_in_ag(Y, X))

The TRS R consists of the following rules:

less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1, x2)
s(x1)  =  s
U5_ga(x1, x2, x3)  =  U5_ga(x3)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
less_in_ag(x1, x2)  =  less_in_ag(x2)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x5)
IN_IN_GA(x1, x2)  =  IN_IN_GA(x1)

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

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

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

U3_GA(X, less_out_ag(Y, X)) → IN_IN_GA(X)
U1_GA(X, less_out_ga(X, Y)) → IN_IN_GA(X)
IN_IN_GA(X) → U3_GA(X, less_in_ag(X))
IN_IN_GA(X) → U1_GA(X, less_in_ga(X))

The TRS R consists of the following rules:

less_in_ga(0) → less_out_ga(0, s)
less_in_ga(s) → U5_ga(less_in_aa)
less_in_ag(s) → less_out_ag(0, s)
less_in_ag(s) → U5_ag(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s, s)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s, s)
less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule IN_IN_GA(X) → U1_GA(X, less_in_ga(X)) at position [1] we obtained the following new rules:

IN_IN_GA(0) → U1_GA(0, less_out_ga(0, s))
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))



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

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

U3_GA(X, less_out_ag(Y, X)) → IN_IN_GA(X)
U1_GA(X, less_out_ga(X, Y)) → IN_IN_GA(X)
IN_IN_GA(X) → U3_GA(X, less_in_ag(X))
IN_IN_GA(0) → U1_GA(0, less_out_ga(0, s))
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))

The TRS R consists of the following rules:

less_in_ga(0) → less_out_ga(0, s)
less_in_ga(s) → U5_ga(less_in_aa)
less_in_ag(s) → less_out_ag(0, s)
less_in_ag(s) → U5_ag(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s, s)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s, s)
less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof

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

U3_GA(X, less_out_ag(Y, X)) → IN_IN_GA(X)
U1_GA(X, less_out_ga(X, Y)) → IN_IN_GA(X)
IN_IN_GA(X) → U3_GA(X, less_in_ag(X))
IN_IN_GA(0) → U1_GA(0, less_out_ga(0, s))
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))

The TRS R consists of the following rules:

less_in_ag(s) → less_out_ag(0, s)
less_in_ag(s) → U5_ag(less_in_aa)
less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s, s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s, s)

The set Q consists of the following terms:

less_in_ga(x0)
less_in_ag(x0)
U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

less_in_ga(x0)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing

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

U3_GA(X, less_out_ag(Y, X)) → IN_IN_GA(X)
U1_GA(X, less_out_ga(X, Y)) → IN_IN_GA(X)
IN_IN_GA(X) → U3_GA(X, less_in_ag(X))
IN_IN_GA(0) → U1_GA(0, less_out_ga(0, s))
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))

The TRS R consists of the following rules:

less_in_ag(s) → less_out_ag(0, s)
less_in_ag(s) → U5_ag(less_in_aa)
less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s, s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s, s)

The set Q consists of the following terms:

less_in_ag(x0)
U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule IN_IN_GA(X) → U3_GA(X, less_in_ag(X)) at position [1] we obtained the following new rules:

IN_IN_GA(s) → U3_GA(s, less_out_ag(0, s))
IN_IN_GA(s) → U3_GA(s, U5_ag(less_in_aa))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ UsableRulesProof

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

U3_GA(X, less_out_ag(Y, X)) → IN_IN_GA(X)
IN_IN_GA(s) → U3_GA(s, less_out_ag(0, s))
U1_GA(X, less_out_ga(X, Y)) → IN_IN_GA(X)
IN_IN_GA(0) → U1_GA(0, less_out_ga(0, s))
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))
IN_IN_GA(s) → U3_GA(s, U5_ag(less_in_aa))

The TRS R consists of the following rules:

less_in_ag(s) → less_out_ag(0, s)
less_in_ag(s) → U5_ag(less_in_aa)
less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s, s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s, s)

The set Q consists of the following terms:

less_in_ag(x0)
U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ UsableRulesProof
                                          ↳ QDP
                                            ↳ QReductionProof

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

U3_GA(X, less_out_ag(Y, X)) → IN_IN_GA(X)
IN_IN_GA(s) → U3_GA(s, less_out_ag(0, s))
U1_GA(X, less_out_ga(X, Y)) → IN_IN_GA(X)
IN_IN_GA(0) → U1_GA(0, less_out_ga(0, s))
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))
IN_IN_GA(s) → U3_GA(s, U5_ag(less_in_aa))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s, s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s, s)

The set Q consists of the following terms:

less_in_ag(x0)
U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

less_in_ag(x0)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ UsableRulesProof
                                          ↳ QDP
                                            ↳ QReductionProof
                                              ↳ QDP
                                                ↳ Instantiation

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

U3_GA(X, less_out_ag(Y, X)) → IN_IN_GA(X)
IN_IN_GA(s) → U3_GA(s, less_out_ag(0, s))
U1_GA(X, less_out_ga(X, Y)) → IN_IN_GA(X)
IN_IN_GA(0) → U1_GA(0, less_out_ga(0, s))
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))
IN_IN_GA(s) → U3_GA(s, U5_ag(less_in_aa))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s, s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s, s)

The set Q consists of the following terms:

U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U3_GA(X, less_out_ag(Y, X)) → IN_IN_GA(X) we obtained the following new rules:

U3_GA(s, less_out_ag(x1, s)) → IN_IN_GA(s)
U3_GA(s, less_out_ag(0, s)) → IN_IN_GA(s)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ UsableRulesProof
                                          ↳ QDP
                                            ↳ QReductionProof
                                              ↳ QDP
                                                ↳ Instantiation
                                                  ↳ QDP
                                                    ↳ Instantiation

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

U3_GA(s, less_out_ag(x1, s)) → IN_IN_GA(s)
U3_GA(s, less_out_ag(0, s)) → IN_IN_GA(s)
IN_IN_GA(s) → U3_GA(s, less_out_ag(0, s))
U1_GA(X, less_out_ga(X, Y)) → IN_IN_GA(X)
IN_IN_GA(0) → U1_GA(0, less_out_ga(0, s))
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))
IN_IN_GA(s) → U3_GA(s, U5_ag(less_in_aa))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s, s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s, s)

The set Q consists of the following terms:

U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U1_GA(X, less_out_ga(X, Y)) → IN_IN_GA(X) we obtained the following new rules:

U1_GA(0, less_out_ga(0, s)) → IN_IN_GA(0)
U1_GA(s, less_out_ga(s, x1)) → IN_IN_GA(s)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ UsableRulesProof
                                          ↳ QDP
                                            ↳ QReductionProof
                                              ↳ QDP
                                                ↳ Instantiation
                                                  ↳ QDP
                                                    ↳ Instantiation
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof

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

U1_GA(0, less_out_ga(0, s)) → IN_IN_GA(0)
U3_GA(s, less_out_ag(x1, s)) → IN_IN_GA(s)
U3_GA(s, less_out_ag(0, s)) → IN_IN_GA(s)
IN_IN_GA(s) → U3_GA(s, less_out_ag(0, s))
U1_GA(s, less_out_ga(s, x1)) → IN_IN_GA(s)
IN_IN_GA(0) → U1_GA(0, less_out_ga(0, s))
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))
IN_IN_GA(s) → U3_GA(s, U5_ag(less_in_aa))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s, s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s, s)

The set Q consists of the following terms:

U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ UsableRulesProof
                                          ↳ QDP
                                            ↳ QReductionProof
                                              ↳ QDP
                                                ↳ Instantiation
                                                  ↳ QDP
                                                    ↳ Instantiation
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ AND
                                                            ↳ QDP
                                                              ↳ UsableRulesProof
                                                            ↳ QDP

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

U1_GA(0, less_out_ga(0, s)) → IN_IN_GA(0)
IN_IN_GA(0) → U1_GA(0, less_out_ga(0, s))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s, s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s, s)

The set Q consists of the following terms:

U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ UsableRulesProof
                                          ↳ QDP
                                            ↳ QReductionProof
                                              ↳ QDP
                                                ↳ Instantiation
                                                  ↳ QDP
                                                    ↳ Instantiation
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ AND
                                                            ↳ QDP
                                                              ↳ UsableRulesProof
                                                                ↳ QDP
                                                                  ↳ QReductionProof
                                                            ↳ QDP

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

U1_GA(0, less_out_ga(0, s)) → IN_IN_GA(0)
IN_IN_GA(0) → U1_GA(0, less_out_ga(0, s))

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

U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ UsableRulesProof
                                          ↳ QDP
                                            ↳ QReductionProof
                                              ↳ QDP
                                                ↳ Instantiation
                                                  ↳ QDP
                                                    ↳ Instantiation
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ AND
                                                            ↳ QDP
                                                              ↳ UsableRulesProof
                                                                ↳ QDP
                                                                  ↳ QReductionProof
                                                                    ↳ QDP
                                                                      ↳ NonTerminationProof
                                                            ↳ QDP

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

U1_GA(0, less_out_ga(0, s)) → IN_IN_GA(0)
IN_IN_GA(0) → U1_GA(0, less_out_ga(0, s))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

U1_GA(0, less_out_ga(0, s)) → IN_IN_GA(0)
IN_IN_GA(0) → U1_GA(0, less_out_ga(0, s))

The TRS R consists of the following rules:none


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

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



Rewriting sequence

IN_IN_GA(0) → U1_GA(0, less_out_ga(0, s))
with rule IN_IN_GA(0) → U1_GA(0, less_out_ga(0, s)) at position [] and matcher [ ]

U1_GA(0, less_out_ga(0, s)) → IN_IN_GA(0)
with rule U1_GA(0, less_out_ga(0, s)) → IN_IN_GA(0)

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


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





↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ UsableRulesProof
                                          ↳ QDP
                                            ↳ QReductionProof
                                              ↳ QDP
                                                ↳ Instantiation
                                                  ↳ QDP
                                                    ↳ Instantiation
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ AND
                                                            ↳ QDP
                                                            ↳ QDP
                                                              ↳ NonTerminationProof

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

U3_GA(s, less_out_ag(x1, s)) → IN_IN_GA(s)
U3_GA(s, less_out_ag(0, s)) → IN_IN_GA(s)
IN_IN_GA(s) → U3_GA(s, less_out_ag(0, s))
U1_GA(s, less_out_ga(s, x1)) → IN_IN_GA(s)
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))
IN_IN_GA(s) → U3_GA(s, U5_ag(less_in_aa))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s, s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s, s)

The set Q consists of the following terms:

U5_ga(x0)
U5_ag(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

U3_GA(s, less_out_ag(x1, s)) → IN_IN_GA(s)
U3_GA(s, less_out_ag(0, s)) → IN_IN_GA(s)
IN_IN_GA(s) → U3_GA(s, less_out_ag(0, s))
U1_GA(s, less_out_ga(s, x1)) → IN_IN_GA(s)
IN_IN_GA(s) → U1_GA(s, U5_ga(less_in_aa))
IN_IN_GA(s) → U3_GA(s, U5_ag(less_in_aa))

The TRS R consists of the following rules:

less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_ga(less_out_aa(X, Y)) → less_out_ga(s, s)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)
U5_ag(less_out_aa(X, Y)) → less_out_ag(s, s)


s = IN_IN_GA(s) evaluates to t =IN_IN_GA(s)

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



Rewriting sequence

IN_IN_GA(s) → U3_GA(s, less_out_ag(0, s))
with rule IN_IN_GA(s) → U3_GA(s, less_out_ag(0, s)) at position [] and matcher [ ]

U3_GA(s, less_out_ag(0, s)) → IN_IN_GA(s)
with rule U3_GA(s, less_out_ag(x1, s)) → IN_IN_GA(s)

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


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