Left Termination of the query pattern gopher_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 PiDP
    ↳7 UsableRulesProof (⇔)
    ↳8 PiDP
    ↳9 PiDPToQDPProof (⇐)
    ↳10 QDP
    ↳11 MRRProof (⇔)
    ↳12 QDP
    ↳13 PisEmptyProof (⇔)
    ↳14 TRUE
  ↳15 PrologToPiTRSProof (⇐)
    ↳16 PiTRS
    ↳17 DependencyPairsProof (⇔)
    ↳18 PiDP
    ↳19 DependencyGraphProof (⇔)
    ↳20 PiDP
    ↳21 UsableRulesProof (⇔)
    ↳22 PiDP
    ↳23 PiDPToQDPProof (⇐)
    ↳24 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).

Queries:

gopher(g,a).

(1) PrologToPiTRSProof (SOUND transformation)

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

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)

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

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)

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

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)

The TRS R consists of the following rules:

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)

The argument filtering Pi contains the following mapping:
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
nil  =  nil
gopher_out_ga(x1, x2)  =  gopher_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
GOPHER_IN_GA(x1, x2)  =  GOPHER_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)

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

(4) Obligation:

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

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)

The TRS R consists of the following rules:

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)

The argument filtering Pi contains the following mapping:
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
nil  =  nil
gopher_out_ga(x1, x2)  =  gopher_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
GOPHER_IN_GA(x1, x2)  =  GOPHER_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.

(6) 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:

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)

The argument filtering Pi contains the following mapping:
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
nil  =  nil
gopher_out_ga(x1, x2)  =  gopher_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
GOPHER_IN_GA(x1, x2)  =  GOPHER_IN_GA(x1)

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) 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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) 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.

(11) 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   

(12) Obligation:

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) PrologToPiTRSProof (SOUND transformation)

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

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)

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

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(16) Obligation:

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

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)

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

(17) 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:

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)

The TRS R consists of the following rules:

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)

The argument filtering Pi contains the following mapping:
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
nil  =  nil
gopher_out_ga(x1, x2)  =  gopher_out_ga(x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
GOPHER_IN_GA(x1, x2)  =  GOPHER_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)

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

(18) Obligation:

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

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)

The TRS R consists of the following rules:

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)

The argument filtering Pi contains the following mapping:
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
nil  =  nil
gopher_out_ga(x1, x2)  =  gopher_out_ga(x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
GOPHER_IN_GA(x1, x2)  =  GOPHER_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)

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

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.

(20) 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:

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)

The argument filtering Pi contains the following mapping:
gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
nil  =  nil
gopher_out_ga(x1, x2)  =  gopher_out_ga(x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
GOPHER_IN_GA(x1, x2)  =  GOPHER_IN_GA(x1)

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

(21) UsableRulesProof (EQUIVALENT transformation)

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

(22) 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

(23) PiDPToQDPProof (SOUND transformation)

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

(24) 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.