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

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

(0) Obligation:

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).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
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

(2) Obligation:

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)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

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)
SAMEFRINGE_IN_GG(x1, x2)  =  SAMEFRINGE_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x3, x4, x5)
GOPHER_IN_GA(x1, x2)  =  GOPHER_IN_GA(x1)
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)

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

(4) Obligation:

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)
SAMEFRINGE_IN_GG(x1, x2)  =  SAMEFRINGE_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x3, x4, x5)
GOPHER_IN_GA(x1, x2)  =  GOPHER_IN_GA(x1)
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)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

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.

(12) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, 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: Polynomial interpretation [POLO]:

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

(13) Obligation:

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

(14) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(15) TRUE

(16) Obligation:

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

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))) → 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)))

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)
SAMEFRINGE_IN_GG(x1, x2)  =  SAMEFRINGE_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x3, x4, x5)
U3_GG(x1, x2, x3, x4, x5, x6, x7)  =  U3_GG(x6, x7)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

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))) → 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)))

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)
SAMEFRINGE_IN_GG(x1, x2)  =  SAMEFRINGE_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x3, x4, x5)
U3_GG(x1, x2, x3, x4, x5, x6, x7)  =  U3_GG(x6, x7)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

U2_GG(X, Y, gopher_out_ga(cons(U1, V1))) → U3_GG(V1, gopher_in_ga(cons(X, Y)))
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)))

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.

(21) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

U2_GG(X, Y, gopher_out_ga(cons(U1, V1))) → U3_GG(V1, gopher_in_ga(cons(X, Y)))
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)))

Strictly oriented rules of the TRS R:

gopher_in_ga(cons(nil, Y)) → gopher_out_ga(cons(nil, Y))

Used ordering: Polynomial interpretation [POLO]:

POL(SAMEFRINGE_IN_GG(x1, x2)) = x1 + x2   
POL(U1_ga(x1)) = x1   
POL(U2_GG(x1, x2, x3)) = 2 + x1 + x2 + x3   
POL(U3_GG(x1, x2)) = x1 + x2   
POL(cons(x1, x2)) = 4 + x1 + x2   
POL(gopher_in_ga(x1)) = 1 + x1   
POL(gopher_out_ga(x1)) = x1   
POL(nil) = 0   

(22) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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.

(23) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(24) TRUE

(25) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. 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(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x1, x2, x3, x4, x5)
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
gopher_out_ga(x1, x2)  =  gopher_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
U3_gg(x1, x2, x3, x4, x5, x6, x7)  =  U3_gg(x1, x2, x3, x4, x6, x7)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x1, x2, x3, x4, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(26) Obligation:

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(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x1, x2, x3, x4, x5)
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
gopher_out_ga(x1, x2)  =  gopher_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
U3_gg(x1, x2, x3, x4, x5, x6, x7)  =  U3_gg(x1, x2, x3, x4, x6, x7)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x1, x2, x3, x4, x5)

(27) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

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(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x1, x2, x3, x4, x5)
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
gopher_out_ga(x1, x2)  =  gopher_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
U3_gg(x1, x2, x3, x4, x5, x6, x7)  =  U3_gg(x1, x2, x3, x4, x6, x7)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x1, x2, x3, x4, x5)
SAMEFRINGE_IN_GG(x1, x2)  =  SAMEFRINGE_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x1, x2, x3, x4, x5)
GOPHER_IN_GA(x1, x2)  =  GOPHER_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
U3_GG(x1, x2, x3, x4, x5, x6, x7)  =  U3_GG(x1, x2, x3, x4, x6, x7)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x1, x2, x3, x4, x5)

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

(28) Obligation:

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(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x1, x2, x3, x4, x5)
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
gopher_out_ga(x1, x2)  =  gopher_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
U3_gg(x1, x2, x3, x4, x5, x6, x7)  =  U3_gg(x1, x2, x3, x4, x6, x7)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x1, x2, x3, x4, x5)
SAMEFRINGE_IN_GG(x1, x2)  =  SAMEFRINGE_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x1, x2, x3, x4, x5)
GOPHER_IN_GA(x1, x2)  =  GOPHER_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
U3_GG(x1, x2, x3, x4, x5, x6, x7)  =  U3_GG(x1, x2, x3, x4, x6, x7)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x1, x2, x3, x4, x5)

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

(29) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.

(30) Complex Obligation (AND)

(31) Obligation:

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(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x1, x2, x3, x4, x5)
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
gopher_out_ga(x1, x2)  =  gopher_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
U3_gg(x1, x2, x3, x4, x5, x6, x7)  =  U3_gg(x1, x2, x3, x4, x6, x7)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x1, x2, x3, x4, x5)
GOPHER_IN_GA(x1, x2)  =  GOPHER_IN_GA(x1)

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

(32) UsableRulesProof (EQUIVALENT transformation)

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

(33) Obligation:

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

(34) PiDPToQDPProof (SOUND transformation)

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

(35) Obligation:

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.

(36) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, 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: Polynomial interpretation [POLO]:

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

(37) Obligation:

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

(38) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(39) TRUE

(40) Obligation:

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

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))) → 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)))

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(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x1, x2, x3, x4, x5)
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
gopher_out_ga(x1, x2)  =  gopher_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
U3_gg(x1, x2, x3, x4, x5, x6, x7)  =  U3_gg(x1, x2, x3, x4, x6, x7)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x1, x2, x3, x4, x5)
SAMEFRINGE_IN_GG(x1, x2)  =  SAMEFRINGE_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x1, x2, x3, x4, x5)
U3_GG(x1, x2, x3, x4, x5, x6, x7)  =  U3_GG(x1, x2, x3, x4, x6, x7)

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

(41) UsableRulesProof (EQUIVALENT transformation)

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

(42) Obligation:

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

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))) → 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)))

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(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
SAMEFRINGE_IN_GG(x1, x2)  =  SAMEFRINGE_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x1, x2, x3, x4, x5)
U3_GG(x1, x2, x3, x4, x5, x6, x7)  =  U3_GG(x1, x2, x3, x4, x6, x7)

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

(43) PiDPToQDPProof (SOUND transformation)

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

(44) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gopher_in_ga(x0)
U1_ga(x0, x1, x2, x3)

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