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



↳ PROLOG
  ↳ PrologToPiTRSProof

gopher2(nil0, nil0).
gopher2(cons2(nil0, Y), cons2(nil0, Y)).
gopher2(cons2(cons2(U, V), W), X) :- gopher2(cons2(U, cons2(V, W)), X).


With regard to the inferred argument filtering the predicates were used in the following modes:
gopher2: (b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


gopher_2_in_ga2(nil_0, nil_0) -> gopher_2_out_ga2(nil_0, nil_0)
gopher_2_in_ga2(cons_22(nil_0, Y), cons_22(nil_0, Y)) -> gopher_2_out_ga2(cons_22(nil_0, Y), cons_22(nil_0, Y))
gopher_2_in_ga2(cons_22(cons_22(U, V), W), X) -> if_gopher_2_in_1_ga5(U, V, W, X, gopher_2_in_ga2(cons_22(U, cons_22(V, W)), X))
if_gopher_2_in_1_ga5(U, V, W, X, gopher_2_out_ga2(cons_22(U, cons_22(V, W)), X)) -> gopher_2_out_ga2(cons_22(cons_22(U, V), W), X)

The argument filtering Pi contains the following mapping:
gopher_2_in_ga2(x1, x2)  =  gopher_2_in_ga1(x1)
nil_0  =  nil_0
cons_22(x1, x2)  =  cons_22(x1, x2)
gopher_2_out_ga2(x1, x2)  =  gopher_2_out_ga1(x2)
if_gopher_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_gopher_2_in_1_ga1(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:

gopher_2_in_ga2(nil_0, nil_0) -> gopher_2_out_ga2(nil_0, nil_0)
gopher_2_in_ga2(cons_22(nil_0, Y), cons_22(nil_0, Y)) -> gopher_2_out_ga2(cons_22(nil_0, Y), cons_22(nil_0, Y))
gopher_2_in_ga2(cons_22(cons_22(U, V), W), X) -> if_gopher_2_in_1_ga5(U, V, W, X, gopher_2_in_ga2(cons_22(U, cons_22(V, W)), X))
if_gopher_2_in_1_ga5(U, V, W, X, gopher_2_out_ga2(cons_22(U, cons_22(V, W)), X)) -> gopher_2_out_ga2(cons_22(cons_22(U, V), W), X)

The argument filtering Pi contains the following mapping:
gopher_2_in_ga2(x1, x2)  =  gopher_2_in_ga1(x1)
nil_0  =  nil_0
cons_22(x1, x2)  =  cons_22(x1, x2)
gopher_2_out_ga2(x1, x2)  =  gopher_2_out_ga1(x2)
if_gopher_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_gopher_2_in_1_ga1(x5)


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

GOPHER_2_IN_GA2(cons_22(cons_22(U, V), W), X) -> IF_GOPHER_2_IN_1_GA5(U, V, W, X, gopher_2_in_ga2(cons_22(U, cons_22(V, W)), X))
GOPHER_2_IN_GA2(cons_22(cons_22(U, V), W), X) -> GOPHER_2_IN_GA2(cons_22(U, cons_22(V, W)), X)

The TRS R consists of the following rules:

gopher_2_in_ga2(nil_0, nil_0) -> gopher_2_out_ga2(nil_0, nil_0)
gopher_2_in_ga2(cons_22(nil_0, Y), cons_22(nil_0, Y)) -> gopher_2_out_ga2(cons_22(nil_0, Y), cons_22(nil_0, Y))
gopher_2_in_ga2(cons_22(cons_22(U, V), W), X) -> if_gopher_2_in_1_ga5(U, V, W, X, gopher_2_in_ga2(cons_22(U, cons_22(V, W)), X))
if_gopher_2_in_1_ga5(U, V, W, X, gopher_2_out_ga2(cons_22(U, cons_22(V, W)), X)) -> gopher_2_out_ga2(cons_22(cons_22(U, V), W), X)

The argument filtering Pi contains the following mapping:
gopher_2_in_ga2(x1, x2)  =  gopher_2_in_ga1(x1)
nil_0  =  nil_0
cons_22(x1, x2)  =  cons_22(x1, x2)
gopher_2_out_ga2(x1, x2)  =  gopher_2_out_ga1(x2)
if_gopher_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_gopher_2_in_1_ga1(x5)
GOPHER_2_IN_GA2(x1, x2)  =  GOPHER_2_IN_GA1(x1)
IF_GOPHER_2_IN_1_GA5(x1, x2, x3, x4, x5)  =  IF_GOPHER_2_IN_1_GA1(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:

GOPHER_2_IN_GA2(cons_22(cons_22(U, V), W), X) -> IF_GOPHER_2_IN_1_GA5(U, V, W, X, gopher_2_in_ga2(cons_22(U, cons_22(V, W)), X))
GOPHER_2_IN_GA2(cons_22(cons_22(U, V), W), X) -> GOPHER_2_IN_GA2(cons_22(U, cons_22(V, W)), X)

The TRS R consists of the following rules:

gopher_2_in_ga2(nil_0, nil_0) -> gopher_2_out_ga2(nil_0, nil_0)
gopher_2_in_ga2(cons_22(nil_0, Y), cons_22(nil_0, Y)) -> gopher_2_out_ga2(cons_22(nil_0, Y), cons_22(nil_0, Y))
gopher_2_in_ga2(cons_22(cons_22(U, V), W), X) -> if_gopher_2_in_1_ga5(U, V, W, X, gopher_2_in_ga2(cons_22(U, cons_22(V, W)), X))
if_gopher_2_in_1_ga5(U, V, W, X, gopher_2_out_ga2(cons_22(U, cons_22(V, W)), X)) -> gopher_2_out_ga2(cons_22(cons_22(U, V), W), X)

The argument filtering Pi contains the following mapping:
gopher_2_in_ga2(x1, x2)  =  gopher_2_in_ga1(x1)
nil_0  =  nil_0
cons_22(x1, x2)  =  cons_22(x1, x2)
gopher_2_out_ga2(x1, x2)  =  gopher_2_out_ga1(x2)
if_gopher_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_gopher_2_in_1_ga1(x5)
GOPHER_2_IN_GA2(x1, x2)  =  GOPHER_2_IN_GA1(x1)
IF_GOPHER_2_IN_1_GA5(x1, x2, x3, x4, x5)  =  IF_GOPHER_2_IN_1_GA1(x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 1 SCC with 1 less node.

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof

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

GOPHER_2_IN_GA2(cons_22(cons_22(U, V), W), X) -> GOPHER_2_IN_GA2(cons_22(U, cons_22(V, W)), X)

The TRS R consists of the following rules:

gopher_2_in_ga2(nil_0, nil_0) -> gopher_2_out_ga2(nil_0, nil_0)
gopher_2_in_ga2(cons_22(nil_0, Y), cons_22(nil_0, Y)) -> gopher_2_out_ga2(cons_22(nil_0, Y), cons_22(nil_0, Y))
gopher_2_in_ga2(cons_22(cons_22(U, V), W), X) -> if_gopher_2_in_1_ga5(U, V, W, X, gopher_2_in_ga2(cons_22(U, cons_22(V, W)), X))
if_gopher_2_in_1_ga5(U, V, W, X, gopher_2_out_ga2(cons_22(U, cons_22(V, W)), X)) -> gopher_2_out_ga2(cons_22(cons_22(U, V), W), X)

The argument filtering Pi contains the following mapping:
gopher_2_in_ga2(x1, x2)  =  gopher_2_in_ga1(x1)
nil_0  =  nil_0
cons_22(x1, x2)  =  cons_22(x1, x2)
gopher_2_out_ga2(x1, x2)  =  gopher_2_out_ga1(x2)
if_gopher_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_gopher_2_in_1_ga1(x5)
GOPHER_2_IN_GA2(x1, x2)  =  GOPHER_2_IN_GA1(x1)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof

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

GOPHER_2_IN_GA2(cons_22(cons_22(U, V), W), X) -> GOPHER_2_IN_GA2(cons_22(U, cons_22(V, W)), X)

R is empty.
The argument filtering Pi contains the following mapping:
cons_22(x1, x2)  =  cons_22(x1, x2)
GOPHER_2_IN_GA2(x1, x2)  =  GOPHER_2_IN_GA1(x1)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ QDPSizeChangeProof

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

GOPHER_2_IN_GA1(cons_22(cons_22(U, V), W)) -> GOPHER_2_IN_GA1(cons_22(U, cons_22(V, W)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {GOPHER_2_IN_GA1}.
We used the following order and afs together with the size-change analysis to show that there are no infinite chains for this DP problem.

Order:Homeomorphic Embedding Order

AFS:
cons_22(x1, x2)  =  cons_21(x1)

From the DPs we obtained the following set of size-change graphs:

We oriented the following set of usable rules. none