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

0 Prolog
↳1 PrologToPiTRSProof (⇒, 0 ms)
↳2 PiTRS
↳3 DependencyPairsProof (⇔, 22 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇒, 0 ms)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔, 0 ms)
        ↳13 YES
    ↳14 PiDP
        ↳15 UsableRulesProof (⇔, 0 ms)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇒, 0 ms)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔, 0 ms)
        ↳20 YES

(0) Obligation:

Clauses:

f(A, [], RES) :- g(A, [], RES).
f(.(A, As), .(B, Bs), RES) :- f(.(B, .(A, As)), Bs, RES).
g([], RES, RES).
g(.(C, Cs), D, RES) :- g(Cs, .(C, D), RES).

Query: f(g,g,a)

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
f_in: (b,b,f)
g_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

f_in_gga(A, [], RES) → U1_gga(A, RES, g_in_gga(A, [], RES))
g_in_gga([], RES, RES) → g_out_gga([], RES, RES)
g_in_gga(.(C, Cs), D, RES) → U3_gga(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES))
U3_gga(C, Cs, D, RES, g_out_gga(Cs, .(C, D), RES)) → g_out_gga(.(C, Cs), D, RES)
U1_gga(A, RES, g_out_gga(A, [], RES)) → f_out_gga(A, [], RES)
f_in_gga(.(A, As), .(B, Bs), RES) → U2_gga(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES))
U2_gga(A, As, B, Bs, RES, f_out_gga(.(B, .(A, As)), Bs, RES)) → f_out_gga(.(A, As), .(B, Bs), RES)

The argument filtering Pi contains the following mapping:
f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x3)
g_in_gga(x1, x2, x3)  =  g_in_gga(x1, x2)
g_out_gga(x1, x2, x3)  =  g_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

f_in_gga(A, [], RES) → U1_gga(A, RES, g_in_gga(A, [], RES))
g_in_gga([], RES, RES) → g_out_gga([], RES, RES)
g_in_gga(.(C, Cs), D, RES) → U3_gga(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES))
U3_gga(C, Cs, D, RES, g_out_gga(Cs, .(C, D), RES)) → g_out_gga(.(C, Cs), D, RES)
U1_gga(A, RES, g_out_gga(A, [], RES)) → f_out_gga(A, [], RES)
f_in_gga(.(A, As), .(B, Bs), RES) → U2_gga(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES))
U2_gga(A, As, B, Bs, RES, f_out_gga(.(B, .(A, As)), Bs, RES)) → f_out_gga(.(A, As), .(B, Bs), RES)

The argument filtering Pi contains the following mapping:
f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x3)
g_in_gga(x1, x2, x3)  =  g_in_gga(x1, x2)
g_out_gga(x1, x2, x3)  =  g_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x6)

(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:

F_IN_GGA(A, [], RES) → U1_GGA(A, RES, g_in_gga(A, [], RES))
F_IN_GGA(A, [], RES) → G_IN_GGA(A, [], RES)
G_IN_GGA(.(C, Cs), D, RES) → U3_GGA(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES))
G_IN_GGA(.(C, Cs), D, RES) → G_IN_GGA(Cs, .(C, D), RES)
F_IN_GGA(.(A, As), .(B, Bs), RES) → U2_GGA(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES))
F_IN_GGA(.(A, As), .(B, Bs), RES) → F_IN_GGA(.(B, .(A, As)), Bs, RES)

The TRS R consists of the following rules:

f_in_gga(A, [], RES) → U1_gga(A, RES, g_in_gga(A, [], RES))
g_in_gga([], RES, RES) → g_out_gga([], RES, RES)
g_in_gga(.(C, Cs), D, RES) → U3_gga(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES))
U3_gga(C, Cs, D, RES, g_out_gga(Cs, .(C, D), RES)) → g_out_gga(.(C, Cs), D, RES)
U1_gga(A, RES, g_out_gga(A, [], RES)) → f_out_gga(A, [], RES)
f_in_gga(.(A, As), .(B, Bs), RES) → U2_gga(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES))
U2_gga(A, As, B, Bs, RES, f_out_gga(.(B, .(A, As)), Bs, RES)) → f_out_gga(.(A, As), .(B, Bs), RES)

The argument filtering Pi contains the following mapping:
f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x3)
g_in_gga(x1, x2, x3)  =  g_in_gga(x1, x2)
g_out_gga(x1, x2, x3)  =  g_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x6)
F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x3)
G_IN_GGA(x1, x2, x3)  =  G_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x5)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x6)

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

(4) Obligation:

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

F_IN_GGA(A, [], RES) → U1_GGA(A, RES, g_in_gga(A, [], RES))
F_IN_GGA(A, [], RES) → G_IN_GGA(A, [], RES)
G_IN_GGA(.(C, Cs), D, RES) → U3_GGA(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES))
G_IN_GGA(.(C, Cs), D, RES) → G_IN_GGA(Cs, .(C, D), RES)
F_IN_GGA(.(A, As), .(B, Bs), RES) → U2_GGA(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES))
F_IN_GGA(.(A, As), .(B, Bs), RES) → F_IN_GGA(.(B, .(A, As)), Bs, RES)

The TRS R consists of the following rules:

f_in_gga(A, [], RES) → U1_gga(A, RES, g_in_gga(A, [], RES))
g_in_gga([], RES, RES) → g_out_gga([], RES, RES)
g_in_gga(.(C, Cs), D, RES) → U3_gga(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES))
U3_gga(C, Cs, D, RES, g_out_gga(Cs, .(C, D), RES)) → g_out_gga(.(C, Cs), D, RES)
U1_gga(A, RES, g_out_gga(A, [], RES)) → f_out_gga(A, [], RES)
f_in_gga(.(A, As), .(B, Bs), RES) → U2_gga(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES))
U2_gga(A, As, B, Bs, RES, f_out_gga(.(B, .(A, As)), Bs, RES)) → f_out_gga(.(A, As), .(B, Bs), RES)

The argument filtering Pi contains the following mapping:
f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x3)
g_in_gga(x1, x2, x3)  =  g_in_gga(x1, x2)
g_out_gga(x1, x2, x3)  =  g_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x6)
F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x3)
G_IN_GGA(x1, x2, x3)  =  G_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x5)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x6)

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:

G_IN_GGA(.(C, Cs), D, RES) → G_IN_GGA(Cs, .(C, D), RES)

The TRS R consists of the following rules:

f_in_gga(A, [], RES) → U1_gga(A, RES, g_in_gga(A, [], RES))
g_in_gga([], RES, RES) → g_out_gga([], RES, RES)
g_in_gga(.(C, Cs), D, RES) → U3_gga(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES))
U3_gga(C, Cs, D, RES, g_out_gga(Cs, .(C, D), RES)) → g_out_gga(.(C, Cs), D, RES)
U1_gga(A, RES, g_out_gga(A, [], RES)) → f_out_gga(A, [], RES)
f_in_gga(.(A, As), .(B, Bs), RES) → U2_gga(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES))
U2_gga(A, As, B, Bs, RES, f_out_gga(.(B, .(A, As)), Bs, RES)) → f_out_gga(.(A, As), .(B, Bs), RES)

The argument filtering Pi contains the following mapping:
f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x3)
g_in_gga(x1, x2, x3)  =  g_in_gga(x1, x2)
g_out_gga(x1, x2, x3)  =  g_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x6)
G_IN_GGA(x1, x2, x3)  =  G_IN_GGA(x1, x2)

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:

G_IN_GGA(.(C, Cs), D, RES) → G_IN_GGA(Cs, .(C, D), RES)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
G_IN_GGA(x1, x2, x3)  =  G_IN_GGA(x1, x2)

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:

G_IN_GGA(.(C, Cs), D) → G_IN_GGA(Cs, .(C, D))

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

  • G_IN_GGA(.(C, Cs), D) → G_IN_GGA(Cs, .(C, D))
    The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

F_IN_GGA(.(A, As), .(B, Bs), RES) → F_IN_GGA(.(B, .(A, As)), Bs, RES)

The TRS R consists of the following rules:

f_in_gga(A, [], RES) → U1_gga(A, RES, g_in_gga(A, [], RES))
g_in_gga([], RES, RES) → g_out_gga([], RES, RES)
g_in_gga(.(C, Cs), D, RES) → U3_gga(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES))
U3_gga(C, Cs, D, RES, g_out_gga(Cs, .(C, D), RES)) → g_out_gga(.(C, Cs), D, RES)
U1_gga(A, RES, g_out_gga(A, [], RES)) → f_out_gga(A, [], RES)
f_in_gga(.(A, As), .(B, Bs), RES) → U2_gga(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES))
U2_gga(A, As, B, Bs, RES, f_out_gga(.(B, .(A, As)), Bs, RES)) → f_out_gga(.(A, As), .(B, Bs), RES)

The argument filtering Pi contains the following mapping:
f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
[]  =  []
U1_gga(x1, x2, x3)  =  U1_gga(x3)
g_in_gga(x1, x2, x3)  =  g_in_gga(x1, x2)
g_out_gga(x1, x2, x3)  =  g_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x6)
F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

F_IN_GGA(.(A, As), .(B, Bs), RES) → F_IN_GGA(.(B, .(A, As)), Bs, RES)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

F_IN_GGA(.(A, As), .(B, Bs)) → F_IN_GGA(.(B, .(A, As)), Bs)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

  • F_IN_GGA(.(A, As), .(B, Bs)) → F_IN_GGA(.(B, .(A, As)), Bs)
    The graph contains the following edges 2 > 2

(20) YES