Left Termination of the query pattern flatten_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 PrologToPiTRSProof (⇐)
    ↳12 PiTRS
    ↳13 DependencyPairsProof (⇔)
    ↳14 PiDP
    ↳15 DependencyGraphProof (⇔)
    ↳16 PiDP
    ↳17 UsableRulesProof (⇔)
    ↳18 PiDP
    ↳19 PiDPToQDPProof (⇐)
    ↳20 QDP
    ↳21 UsableRulesReductionPairsProof (⇔)
    ↳22 QDP
    ↳23 PisEmptyProof (⇔)
    ↳24 TRUE

(0) Obligation:

Clauses:

flatten(atom(X), .(X, [])).
flatten(cons(atom(X), U), .(X, Y)) :- flatten(U, Y).
flatten(cons(cons(U, V), W), X) :- flatten(cons(U, cons(V, W)), X).

Queries:

flatten(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:
flatten_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))

The argument filtering Pi contains the following mapping:
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
atom(x1)  =  atom(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
.(x1, x2)  =  .(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))

The argument filtering Pi contains the following mapping:
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
atom(x1)  =  atom(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
.(x1, x2)  =  .(x1, x2)

(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:

FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → U1_GA(X, U, Y, flatten_in_ga(U, Y))
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)
FLATTEN_IN_GA(cons(cons(U, V), W), X) → U2_GA(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)

The TRS R consists of the following rules:

flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))

The argument filtering Pi contains the following mapping:
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
atom(x1)  =  atom(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
.(x1, x2)  =  .(x1, x2)
FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_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:

FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → U1_GA(X, U, Y, flatten_in_ga(U, Y))
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)
FLATTEN_IN_GA(cons(cons(U, V), W), X) → U2_GA(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)

The TRS R consists of the following rules:

flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))

The argument filtering Pi contains the following mapping:
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
atom(x1)  =  atom(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
.(x1, x2)  =  .(x1, x2)
FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_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 2 less nodes.

(6) Obligation:

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

FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)

The TRS R consists of the following rules:

flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))

The argument filtering Pi contains the following mapping:
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
atom(x1)  =  atom(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
.(x1, x2)  =  .(x1, x2)
FLATTEN_IN_GA(x1, x2)  =  FLATTEN_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:

FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)

R is empty.
The argument filtering Pi contains the following mapping:
atom(x1)  =  atom(x1)
cons(x1, x2)  =  cons(x1, x2)
.(x1, x2)  =  .(x1, x2)
FLATTEN_IN_GA(x1, x2)  =  FLATTEN_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:

FLATTEN_IN_GA(cons(cons(U, V), W)) → FLATTEN_IN_GA(cons(U, cons(V, W)))
FLATTEN_IN_GA(cons(atom(X), U)) → FLATTEN_IN_GA(U)

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

(11) PrologToPiTRSProof (SOUND transformation)

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

flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))

The argument filtering Pi contains the following mapping:
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
atom(x1)  =  atom(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
.(x1, x2)  =  .(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(12) Obligation:

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

flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))

The argument filtering Pi contains the following mapping:
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
atom(x1)  =  atom(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
.(x1, x2)  =  .(x1, x2)

(13) 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:

FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → U1_GA(X, U, Y, flatten_in_ga(U, Y))
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)
FLATTEN_IN_GA(cons(cons(U, V), W), X) → U2_GA(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)

The TRS R consists of the following rules:

flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))

The argument filtering Pi contains the following mapping:
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
atom(x1)  =  atom(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
.(x1, x2)  =  .(x1, x2)
FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x5)

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

(14) Obligation:

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

FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → U1_GA(X, U, Y, flatten_in_ga(U, Y))
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)
FLATTEN_IN_GA(cons(cons(U, V), W), X) → U2_GA(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)

The TRS R consists of the following rules:

flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))

The argument filtering Pi contains the following mapping:
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
atom(x1)  =  atom(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
.(x1, x2)  =  .(x1, x2)
FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x5)

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

(15) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(16) Obligation:

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

FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)

The TRS R consists of the following rules:

flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))

The argument filtering Pi contains the following mapping:
flatten_in_ga(x1, x2)  =  flatten_in_ga(x1)
atom(x1)  =  atom(x1)
flatten_out_ga(x1, x2)  =  flatten_out_ga(x2)
cons(x1, x2)  =  cons(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
.(x1, x2)  =  .(x1, x2)
FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)

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:

FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)

R is empty.
The argument filtering Pi contains the following mapping:
atom(x1)  =  atom(x1)
cons(x1, x2)  =  cons(x1, x2)
.(x1, x2)  =  .(x1, x2)
FLATTEN_IN_GA(x1, x2)  =  FLATTEN_IN_GA(x1)

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:

FLATTEN_IN_GA(cons(cons(U, V), W)) → FLATTEN_IN_GA(cons(U, cons(V, W)))
FLATTEN_IN_GA(cons(atom(X), U)) → FLATTEN_IN_GA(U)

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

(21) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

FLATTEN_IN_GA(cons(cons(U, V), W)) → FLATTEN_IN_GA(cons(U, cons(V, W)))
FLATTEN_IN_GA(cons(atom(X), U)) → FLATTEN_IN_GA(U)
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(FLATTEN_IN_GA(x1)) = 2·x1   
POL(atom(x1)) = x1   
POL(cons(x1, x2)) = 2 + 2·x1 + x2   

(22) Obligation:

Q DP problem:
P is empty.
R is empty.
Q is empty.
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