Left Termination of the query pattern samefringe_in_2(g, g) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

gopher(nil, nil).
gopher(cons(nil, Y), cons(nil, Y)).
gopher(cons(cons(U, V), W), X) :- gopher(cons(U, cons(V, W)), X).
samefringe(nil, nil).
samefringe(cons(U, V), cons(X, Y)) :- ','(gopher(cons(U, V), cons(U1, V1)), ','(gopher(cons(X, Y), cons(X1, Y1)), samefringe(V1, Y1))).

Queries:

samefringe(g,g).

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

samefringe_in_gg(nil, nil) → samefringe_out_gg(nil, nil)
samefringe_in_gg(cons(U, V), cons(X, Y)) → U2_gg(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1)))
gopher_in_ga(nil, nil) → gopher_out_ga(nil, nil)
gopher_in_ga(cons(nil, Y), cons(nil, Y)) → gopher_out_ga(cons(nil, Y), cons(nil, Y))
gopher_in_ga(cons(cons(U, V), W), X) → U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X))
U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) → gopher_out_ga(cons(cons(U, V), W), X)
U2_gg(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) → U3_gg(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1)))
U3_gg(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) → U4_gg(U, V, X, Y, samefringe_in_gg(V1, Y1))
U4_gg(U, V, X, Y, samefringe_out_gg(V1, Y1)) → samefringe_out_gg(cons(U, V), cons(X, Y))

The argument filtering Pi contains the following mapping:
samefringe_in_gg(x1, x2)  =  samefringe_in_gg(x1, x2)
nil  =  nil
samefringe_out_gg(x1, x2)  =  samefringe_out_gg
cons(x1, x2)  =  cons(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x3, x4, x5)
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
gopher_out_ga(x1, x2)  =  gopher_out_ga(x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U3_gg(x1, x2, x3, x4, x5, x6, x7)  =  U3_gg(x6, x7)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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

samefringe_in_gg(nil, nil) → samefringe_out_gg(nil, nil)
samefringe_in_gg(cons(U, V), cons(X, Y)) → U2_gg(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1)))
gopher_in_ga(nil, nil) → gopher_out_ga(nil, nil)
gopher_in_ga(cons(nil, Y), cons(nil, Y)) → gopher_out_ga(cons(nil, Y), cons(nil, Y))
gopher_in_ga(cons(cons(U, V), W), X) → U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X))
U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) → gopher_out_ga(cons(cons(U, V), W), X)
U2_gg(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) → U3_gg(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1)))
U3_gg(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) → U4_gg(U, V, X, Y, samefringe_in_gg(V1, Y1))
U4_gg(U, V, X, Y, samefringe_out_gg(V1, Y1)) → samefringe_out_gg(cons(U, V), cons(X, Y))

The argument filtering Pi contains the following mapping:
samefringe_in_gg(x1, x2)  =  samefringe_in_gg(x1, x2)
nil  =  nil
samefringe_out_gg(x1, x2)  =  samefringe_out_gg
cons(x1, x2)  =  cons(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x3, x4, x5)
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
gopher_out_ga(x1, x2)  =  gopher_out_ga(x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U3_gg(x1, x2, x3, x4, x5, x6, x7)  =  U3_gg(x6, x7)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(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:

SAMEFRINGE_IN_GG(cons(U, V), cons(X, Y)) → U2_GG(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1)))
SAMEFRINGE_IN_GG(cons(U, V), cons(X, Y)) → GOPHER_IN_GA(cons(U, V), cons(U1, V1))
GOPHER_IN_GA(cons(cons(U, V), W), X) → U1_GA(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X))
GOPHER_IN_GA(cons(cons(U, V), W), X) → GOPHER_IN_GA(cons(U, cons(V, W)), X)
U2_GG(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) → U3_GG(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1)))
U2_GG(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) → GOPHER_IN_GA(cons(X, Y), cons(X1, Y1))
U3_GG(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) → U4_GG(U, V, X, Y, samefringe_in_gg(V1, Y1))
U3_GG(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) → SAMEFRINGE_IN_GG(V1, Y1)

The TRS R consists of the following rules:

samefringe_in_gg(nil, nil) → samefringe_out_gg(nil, nil)
samefringe_in_gg(cons(U, V), cons(X, Y)) → U2_gg(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1)))
gopher_in_ga(nil, nil) → gopher_out_ga(nil, nil)
gopher_in_ga(cons(nil, Y), cons(nil, Y)) → gopher_out_ga(cons(nil, Y), cons(nil, Y))
gopher_in_ga(cons(cons(U, V), W), X) → U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X))
U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) → gopher_out_ga(cons(cons(U, V), W), X)
U2_gg(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) → U3_gg(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1)))
U3_gg(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) → U4_gg(U, V, X, Y, samefringe_in_gg(V1, Y1))
U4_gg(U, V, X, Y, samefringe_out_gg(V1, Y1)) → samefringe_out_gg(cons(U, V), cons(X, Y))

The argument filtering Pi contains the following mapping:
samefringe_in_gg(x1, x2)  =  samefringe_in_gg(x1, x2)
nil  =  nil
samefringe_out_gg(x1, x2)  =  samefringe_out_gg
cons(x1, x2)  =  cons(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x3, x4, x5)
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
gopher_out_ga(x1, x2)  =  gopher_out_ga(x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U3_gg(x1, x2, x3, x4, x5, x6, x7)  =  U3_gg(x6, x7)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U3_GG(x1, x2, x3, x4, x5, x6, x7)  =  U3_GG(x6, x7)
GOPHER_IN_GA(x1, x2)  =  GOPHER_IN_GA(x1)
SAMEFRINGE_IN_GG(x1, x2)  =  SAMEFRINGE_IN_GG(x1, x2)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x3, x4, x5)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x5)

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

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

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

SAMEFRINGE_IN_GG(cons(U, V), cons(X, Y)) → U2_GG(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1)))
SAMEFRINGE_IN_GG(cons(U, V), cons(X, Y)) → GOPHER_IN_GA(cons(U, V), cons(U1, V1))
GOPHER_IN_GA(cons(cons(U, V), W), X) → U1_GA(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X))
GOPHER_IN_GA(cons(cons(U, V), W), X) → GOPHER_IN_GA(cons(U, cons(V, W)), X)
U2_GG(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) → U3_GG(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1)))
U2_GG(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) → GOPHER_IN_GA(cons(X, Y), cons(X1, Y1))
U3_GG(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) → U4_GG(U, V, X, Y, samefringe_in_gg(V1, Y1))
U3_GG(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) → SAMEFRINGE_IN_GG(V1, Y1)

The TRS R consists of the following rules:

samefringe_in_gg(nil, nil) → samefringe_out_gg(nil, nil)
samefringe_in_gg(cons(U, V), cons(X, Y)) → U2_gg(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1)))
gopher_in_ga(nil, nil) → gopher_out_ga(nil, nil)
gopher_in_ga(cons(nil, Y), cons(nil, Y)) → gopher_out_ga(cons(nil, Y), cons(nil, Y))
gopher_in_ga(cons(cons(U, V), W), X) → U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X))
U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) → gopher_out_ga(cons(cons(U, V), W), X)
U2_gg(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) → U3_gg(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1)))
U3_gg(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) → U4_gg(U, V, X, Y, samefringe_in_gg(V1, Y1))
U4_gg(U, V, X, Y, samefringe_out_gg(V1, Y1)) → samefringe_out_gg(cons(U, V), cons(X, Y))

The argument filtering Pi contains the following mapping:
samefringe_in_gg(x1, x2)  =  samefringe_in_gg(x1, x2)
nil  =  nil
samefringe_out_gg(x1, x2)  =  samefringe_out_gg
cons(x1, x2)  =  cons(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x3, x4, x5)
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
gopher_out_ga(x1, x2)  =  gopher_out_ga(x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U3_gg(x1, x2, x3, x4, x5, x6, x7)  =  U3_gg(x6, x7)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U3_GG(x1, x2, x3, x4, x5, x6, x7)  =  U3_GG(x6, x7)
GOPHER_IN_GA(x1, x2)  =  GOPHER_IN_GA(x1)
SAMEFRINGE_IN_GG(x1, x2)  =  SAMEFRINGE_IN_GG(x1, x2)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x3, x4, x5)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 4 less nodes.

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

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

GOPHER_IN_GA(cons(cons(U, V), W), X) → GOPHER_IN_GA(cons(U, cons(V, W)), X)

The TRS R consists of the following rules:

samefringe_in_gg(nil, nil) → samefringe_out_gg(nil, nil)
samefringe_in_gg(cons(U, V), cons(X, Y)) → U2_gg(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1)))
gopher_in_ga(nil, nil) → gopher_out_ga(nil, nil)
gopher_in_ga(cons(nil, Y), cons(nil, Y)) → gopher_out_ga(cons(nil, Y), cons(nil, Y))
gopher_in_ga(cons(cons(U, V), W), X) → U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X))
U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) → gopher_out_ga(cons(cons(U, V), W), X)
U2_gg(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) → U3_gg(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1)))
U3_gg(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) → U4_gg(U, V, X, Y, samefringe_in_gg(V1, Y1))
U4_gg(U, V, X, Y, samefringe_out_gg(V1, Y1)) → samefringe_out_gg(cons(U, V), cons(X, Y))

The argument filtering Pi contains the following mapping:
samefringe_in_gg(x1, x2)  =  samefringe_in_gg(x1, x2)
nil  =  nil
samefringe_out_gg(x1, x2)  =  samefringe_out_gg
cons(x1, x2)  =  cons(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x3, x4, x5)
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
gopher_out_ga(x1, x2)  =  gopher_out_ga(x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U3_gg(x1, x2, x3, x4, x5, x6, x7)  =  U3_gg(x6, x7)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
GOPHER_IN_GA(x1, x2)  =  GOPHER_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
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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

GOPHER_IN_GA(cons(cons(U, V), W), X) → GOPHER_IN_GA(cons(U, cons(V, W)), X)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
GOPHER_IN_GA(x1, x2)  =  GOPHER_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
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ RuleRemovalProof
              ↳ PiDP

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

GOPHER_IN_GA(cons(cons(U, V), W)) → GOPHER_IN_GA(cons(U, cons(V, W)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

GOPHER_IN_GA(cons(cons(U, V), W)) → GOPHER_IN_GA(cons(U, cons(V, W)))


Used ordering: POLO with Polynomial interpretation [25]:

POL(GOPHER_IN_GA(x1)) = 2·x1   
POL(cons(x1, x2)) = 2 + 2·x1 + x2   



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ RuleRemovalProof
                          ↳ QDP
                            ↳ PisEmptyProof
              ↳ PiDP

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

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

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

U3_GG(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) → SAMEFRINGE_IN_GG(V1, Y1)
SAMEFRINGE_IN_GG(cons(U, V), cons(X, Y)) → U2_GG(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1)))
U2_GG(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) → U3_GG(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1)))

The TRS R consists of the following rules:

samefringe_in_gg(nil, nil) → samefringe_out_gg(nil, nil)
samefringe_in_gg(cons(U, V), cons(X, Y)) → U2_gg(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1)))
gopher_in_ga(nil, nil) → gopher_out_ga(nil, nil)
gopher_in_ga(cons(nil, Y), cons(nil, Y)) → gopher_out_ga(cons(nil, Y), cons(nil, Y))
gopher_in_ga(cons(cons(U, V), W), X) → U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X))
U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) → gopher_out_ga(cons(cons(U, V), W), X)
U2_gg(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) → U3_gg(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1)))
U3_gg(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) → U4_gg(U, V, X, Y, samefringe_in_gg(V1, Y1))
U4_gg(U, V, X, Y, samefringe_out_gg(V1, Y1)) → samefringe_out_gg(cons(U, V), cons(X, Y))

The argument filtering Pi contains the following mapping:
samefringe_in_gg(x1, x2)  =  samefringe_in_gg(x1, x2)
nil  =  nil
samefringe_out_gg(x1, x2)  =  samefringe_out_gg
cons(x1, x2)  =  cons(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x3, x4, x5)
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
gopher_out_ga(x1, x2)  =  gopher_out_ga(x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U3_gg(x1, x2, x3, x4, x5, x6, x7)  =  U3_gg(x6, x7)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U3_GG(x1, x2, x3, x4, x5, x6, x7)  =  U3_GG(x6, x7)
SAMEFRINGE_IN_GG(x1, x2)  =  SAMEFRINGE_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x3, x4, x5)

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

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

U3_GG(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) → SAMEFRINGE_IN_GG(V1, Y1)
SAMEFRINGE_IN_GG(cons(U, V), cons(X, Y)) → U2_GG(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1)))
U2_GG(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) → U3_GG(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1)))

The TRS R consists of the following rules:

gopher_in_ga(cons(nil, Y), cons(nil, Y)) → gopher_out_ga(cons(nil, Y), cons(nil, Y))
gopher_in_ga(cons(cons(U, V), W), X) → U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X))
U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) → gopher_out_ga(cons(cons(U, V), W), X)

The argument filtering Pi contains the following mapping:
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
gopher_out_ga(x1, x2)  =  gopher_out_ga(x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U3_GG(x1, x2, x3, x4, x5, x6, x7)  =  U3_GG(x6, x7)
SAMEFRINGE_IN_GG(x1, x2)  =  SAMEFRINGE_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x3, x4, x5)

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
                        ↳ RuleRemovalProof

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

U3_GG(V1, gopher_out_ga(cons(X1, Y1))) → SAMEFRINGE_IN_GG(V1, Y1)
SAMEFRINGE_IN_GG(cons(U, V), cons(X, Y)) → U2_GG(X, Y, gopher_in_ga(cons(U, V)))
U2_GG(X, Y, gopher_out_ga(cons(U1, V1))) → U3_GG(V1, gopher_in_ga(cons(X, Y)))

The TRS R consists of the following rules:

gopher_in_ga(cons(nil, Y)) → gopher_out_ga(cons(nil, Y))
gopher_in_ga(cons(cons(U, V), W)) → U1_ga(gopher_in_ga(cons(U, cons(V, W))))
U1_ga(gopher_out_ga(X)) → gopher_out_ga(X)

The set Q consists of the following terms:

gopher_in_ga(x0)
U1_ga(x0)

We have to consider all (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

gopher_in_ga(cons(nil, Y)) → gopher_out_ga(cons(nil, Y))
gopher_in_ga(cons(cons(U, V), W)) → U1_ga(gopher_in_ga(cons(U, cons(V, W))))

Used ordering: POLO with Polynomial interpretation [25]:

POL(SAMEFRINGE_IN_GG(x1, x2)) = 1 + x1 + x2   
POL(U1_ga(x1)) = x1   
POL(U2_GG(x1, x2, x3)) = 1 + 2·x1 + x2 + x3   
POL(U3_GG(x1, x2)) = x1 + x2   
POL(cons(x1, x2)) = 1 + 2·x1 + x2   
POL(gopher_in_ga(x1)) = 1 + x1   
POL(gopher_out_ga(x1)) = x1   
POL(nil) = 0   



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

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

U3_GG(V1, gopher_out_ga(cons(X1, Y1))) → SAMEFRINGE_IN_GG(V1, Y1)
SAMEFRINGE_IN_GG(cons(U, V), cons(X, Y)) → U2_GG(X, Y, gopher_in_ga(cons(U, V)))
U2_GG(X, Y, gopher_out_ga(cons(U1, V1))) → U3_GG(V1, gopher_in_ga(cons(X, Y)))

The TRS R consists of the following rules:

U1_ga(gopher_out_ga(X)) → gopher_out_ga(X)

The set Q consists of the following terms:

gopher_in_ga(x0)
U1_ga(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 3 less nodes.