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 PrologToDTProblemTransformerProof (⇐)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇐)
↳4 PiDP
↳5 DependencyGraphProof (⇔)
↳6 PiDP
↳7 PiDPToQDPProof (⇐)
↳8 QDP
↳9 UsableRulesReductionPairsProof (⇔)
↳10 QDP
↳11 MRRProof (⇔)
↳12 QDP
↳13 PisEmptyProof (⇔)
↳14 YES

(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) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

flatten1(cons(atom(T8), cons(atom(T29), T30)), .(T8, .(T29, T32))) :- flatten1(T30, T32).
flatten1(cons(atom(T8), cons(cons(T51, T52), T53)), .(T8, T55)) :- flatten1(cons(T51, cons(T52, T53)), T55).
flatten1(cons(cons(atom(T85), T86), T87), .(T85, T89)) :- flatten1(cons(T86, T87), T89).
flatten1(cons(cons(cons(T103, T104), T105), T106), T108) :- flatten1(cons(T103, cons(T104, cons(T105, T106))), T108).

Clauses:

flattenc1(atom(T4), .(T4, [])).
flattenc1(cons(atom(T8), atom(T16)), .(T8, .(T16, []))).
flattenc1(cons(atom(T8), cons(atom(T29), T30)), .(T8, .(T29, T32))) :- flattenc1(T30, T32).
flattenc1(cons(atom(T8), cons(cons(T51, T52), T53)), .(T8, T55)) :- flattenc1(cons(T51, cons(T52, T53)), T55).
flattenc1(cons(cons(atom(T85), T86), T87), .(T85, T89)) :- flattenc1(cons(T86, T87), T89).
flattenc1(cons(cons(cons(T103, T104), T105), T106), T108) :- flattenc1(cons(T103, cons(T104, cons(T105, T106))), T108).

Afs:

flatten1(x1, x2)  =  flatten1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
flatten1_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

FLATTEN1_IN_GA(cons(atom(T8), cons(atom(T29), T30)), .(T8, .(T29, T32))) → U1_GA(T8, T29, T30, T32, flatten1_in_ga(T30, T32))
FLATTEN1_IN_GA(cons(atom(T8), cons(atom(T29), T30)), .(T8, .(T29, T32))) → FLATTEN1_IN_GA(T30, T32)
FLATTEN1_IN_GA(cons(atom(T8), cons(cons(T51, T52), T53)), .(T8, T55)) → U2_GA(T8, T51, T52, T53, T55, flatten1_in_ga(cons(T51, cons(T52, T53)), T55))
FLATTEN1_IN_GA(cons(atom(T8), cons(cons(T51, T52), T53)), .(T8, T55)) → FLATTEN1_IN_GA(cons(T51, cons(T52, T53)), T55)
FLATTEN1_IN_GA(cons(cons(atom(T85), T86), T87), .(T85, T89)) → U3_GA(T85, T86, T87, T89, flatten1_in_ga(cons(T86, T87), T89))
FLATTEN1_IN_GA(cons(cons(atom(T85), T86), T87), .(T85, T89)) → FLATTEN1_IN_GA(cons(T86, T87), T89)
FLATTEN1_IN_GA(cons(cons(cons(T103, T104), T105), T106), T108) → U4_GA(T103, T104, T105, T106, T108, flatten1_in_ga(cons(T103, cons(T104, cons(T105, T106))), T108))
FLATTEN1_IN_GA(cons(cons(cons(T103, T104), T105), T106), T108) → FLATTEN1_IN_GA(cons(T103, cons(T104, cons(T105, T106))), T108)

R is empty.
The argument filtering Pi contains the following mapping:
flatten1_in_ga(x1, x2)  =  flatten1_in_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
atom(x1)  =  atom(x1)
.(x1, x2)  =  .(x1, x2)
FLATTEN1_IN_GA(x1, x2)  =  FLATTEN1_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x1, x2, x3, x4, x6)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x3, x5)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x2, x3, x4, x6)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

FLATTEN1_IN_GA(cons(atom(T8), cons(atom(T29), T30)), .(T8, .(T29, T32))) → U1_GA(T8, T29, T30, T32, flatten1_in_ga(T30, T32))
FLATTEN1_IN_GA(cons(atom(T8), cons(atom(T29), T30)), .(T8, .(T29, T32))) → FLATTEN1_IN_GA(T30, T32)
FLATTEN1_IN_GA(cons(atom(T8), cons(cons(T51, T52), T53)), .(T8, T55)) → U2_GA(T8, T51, T52, T53, T55, flatten1_in_ga(cons(T51, cons(T52, T53)), T55))
FLATTEN1_IN_GA(cons(atom(T8), cons(cons(T51, T52), T53)), .(T8, T55)) → FLATTEN1_IN_GA(cons(T51, cons(T52, T53)), T55)
FLATTEN1_IN_GA(cons(cons(atom(T85), T86), T87), .(T85, T89)) → U3_GA(T85, T86, T87, T89, flatten1_in_ga(cons(T86, T87), T89))
FLATTEN1_IN_GA(cons(cons(atom(T85), T86), T87), .(T85, T89)) → FLATTEN1_IN_GA(cons(T86, T87), T89)
FLATTEN1_IN_GA(cons(cons(cons(T103, T104), T105), T106), T108) → U4_GA(T103, T104, T105, T106, T108, flatten1_in_ga(cons(T103, cons(T104, cons(T105, T106))), T108))
FLATTEN1_IN_GA(cons(cons(cons(T103, T104), T105), T106), T108) → FLATTEN1_IN_GA(cons(T103, cons(T104, cons(T105, T106))), T108)

R is empty.
The argument filtering Pi contains the following mapping:
flatten1_in_ga(x1, x2)  =  flatten1_in_ga(x1)
cons(x1, x2)  =  cons(x1, x2)
atom(x1)  =  atom(x1)
.(x1, x2)  =  .(x1, x2)
FLATTEN1_IN_GA(x1, x2)  =  FLATTEN1_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x1, x2, x3, x4, x6)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x3, x5)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x2, x3, x4, x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes.

(6) Obligation:

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

FLATTEN1_IN_GA(cons(atom(T8), cons(cons(T51, T52), T53)), .(T8, T55)) → FLATTEN1_IN_GA(cons(T51, cons(T52, T53)), T55)
FLATTEN1_IN_GA(cons(atom(T8), cons(atom(T29), T30)), .(T8, .(T29, T32))) → FLATTEN1_IN_GA(T30, T32)
FLATTEN1_IN_GA(cons(cons(atom(T85), T86), T87), .(T85, T89)) → FLATTEN1_IN_GA(cons(T86, T87), T89)
FLATTEN1_IN_GA(cons(cons(cons(T103, T104), T105), T106), T108) → FLATTEN1_IN_GA(cons(T103, cons(T104, cons(T105, T106))), T108)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
atom(x1)  =  atom(x1)
.(x1, x2)  =  .(x1, x2)
FLATTEN1_IN_GA(x1, x2)  =  FLATTEN1_IN_GA(x1)

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

(7) PiDPToQDPProof (SOUND transformation)

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

(8) Obligation:

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

FLATTEN1_IN_GA(cons(atom(T8), cons(cons(T51, T52), T53))) → FLATTEN1_IN_GA(cons(T51, cons(T52, T53)))
FLATTEN1_IN_GA(cons(atom(T8), cons(atom(T29), T30))) → FLATTEN1_IN_GA(T30)
FLATTEN1_IN_GA(cons(cons(atom(T85), T86), T87)) → FLATTEN1_IN_GA(cons(T86, T87))
FLATTEN1_IN_GA(cons(cons(cons(T103, T104), T105), T106)) → FLATTEN1_IN_GA(cons(T103, cons(T104, cons(T105, T106))))

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

(9) 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:

FLATTEN1_IN_GA(cons(atom(T8), cons(cons(T51, T52), T53))) → FLATTEN1_IN_GA(cons(T51, cons(T52, T53)))
FLATTEN1_IN_GA(cons(atom(T8), cons(atom(T29), T30))) → FLATTEN1_IN_GA(T30)
FLATTEN1_IN_GA(cons(cons(atom(T85), T86), T87)) → FLATTEN1_IN_GA(cons(T86, T87))
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(FLATTEN1_IN_GA(x1)) = 2·x1   
POL(atom(x1)) = x1   
POL(cons(x1, x2)) = x1 + x2   

(10) Obligation:

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

FLATTEN1_IN_GA(cons(cons(cons(T103, T104), T105), T106)) → FLATTEN1_IN_GA(cons(T103, cons(T104, cons(T105, T106))))

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:

FLATTEN1_IN_GA(cons(cons(cons(T103, T104), T105), T106)) → FLATTEN1_IN_GA(cons(T103, cons(T104, cons(T105, T106))))


Used ordering: Polynomial interpretation [POLO]:

POL(FLATTEN1_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) YES