Left Termination of the query pattern flat_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
↳3 PrologToPiTRSProof (⇐)
↳4 PiTRS
↳5 DependencyPairsProof (⇔)
↳6 PiDP
↳7 DependencyGraphProof (⇔)
↳8 PiDP
↳9 UsableRulesProof (⇔)
↳10 PiDP
↳11 PiDPToQDPProof (⇐)
↳12 QDP
↳13 UsableRulesReductionPairsProof (⇔)
↳14 QDP
↳15 MRRProof (⇔)
↳16 QDP
↳17 PisEmptyProof (⇔)
↳18 YES

(0) Obligation:

Clauses:

flat([], []).
flat(.([], T), R) :- flat(T, R).
flat(.(.(H, T), TT), .(H, R)) :- flat(.(T, TT), R).

Queries:

flat(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

flat1([], []).
flat1(.([], []), []).
flat1(.([], .([], T16)), T18) :- flat1(T16, T18).
flat1(.([], .(.(T35, T36), T37)), .(T35, T39)) :- flat1(.(T36, T37), T39).
flat1(.(.(T46, []), T59), .(T46, T61)) :- flat1(T59, T61).
flat1(.(.(T46, .(T70, T71)), T72), .(T46, .(T70, T74))) :- flat1(.(T71, T72), T74).

Queries:

flat1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
flat1_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

flat1_in_ga([], []) → flat1_out_ga([], [])
flat1_in_ga(.([], []), []) → flat1_out_ga(.([], []), [])
flat1_in_ga(.([], .([], T16)), T18) → U1_ga(T16, T18, flat1_in_ga(T16, T18))
flat1_in_ga(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_ga(T35, T36, T37, T39, flat1_in_ga(.(T36, T37), T39))
flat1_in_ga(.(.(T46, []), T59), .(T46, T61)) → U3_ga(T46, T59, T61, flat1_in_ga(T59, T61))
flat1_in_ga(.(.(T46, .(T70, T71)), T72), .(T46, .(T70, T74))) → U4_ga(T46, T70, T71, T72, T74, flat1_in_ga(.(T71, T72), T74))
U4_ga(T46, T70, T71, T72, T74, flat1_out_ga(.(T71, T72), T74)) → flat1_out_ga(.(.(T46, .(T70, T71)), T72), .(T46, .(T70, T74)))
U3_ga(T46, T59, T61, flat1_out_ga(T59, T61)) → flat1_out_ga(.(.(T46, []), T59), .(T46, T61))
U2_ga(T35, T36, T37, T39, flat1_out_ga(.(T36, T37), T39)) → flat1_out_ga(.([], .(.(T35, T36), T37)), .(T35, T39))
U1_ga(T16, T18, flat1_out_ga(T16, T18)) → flat1_out_ga(.([], .([], T16)), T18)

The argument filtering Pi contains the following mapping:
flat1_in_ga(x1, x2)  =  flat1_in_ga(x1)
[]  =  []
flat1_out_ga(x1, x2)  =  flat1_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x2, x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

flat1_in_ga([], []) → flat1_out_ga([], [])
flat1_in_ga(.([], []), []) → flat1_out_ga(.([], []), [])
flat1_in_ga(.([], .([], T16)), T18) → U1_ga(T16, T18, flat1_in_ga(T16, T18))
flat1_in_ga(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_ga(T35, T36, T37, T39, flat1_in_ga(.(T36, T37), T39))
flat1_in_ga(.(.(T46, []), T59), .(T46, T61)) → U3_ga(T46, T59, T61, flat1_in_ga(T59, T61))
flat1_in_ga(.(.(T46, .(T70, T71)), T72), .(T46, .(T70, T74))) → U4_ga(T46, T70, T71, T72, T74, flat1_in_ga(.(T71, T72), T74))
U4_ga(T46, T70, T71, T72, T74, flat1_out_ga(.(T71, T72), T74)) → flat1_out_ga(.(.(T46, .(T70, T71)), T72), .(T46, .(T70, T74)))
U3_ga(T46, T59, T61, flat1_out_ga(T59, T61)) → flat1_out_ga(.(.(T46, []), T59), .(T46, T61))
U2_ga(T35, T36, T37, T39, flat1_out_ga(.(T36, T37), T39)) → flat1_out_ga(.([], .(.(T35, T36), T37)), .(T35, T39))
U1_ga(T16, T18, flat1_out_ga(T16, T18)) → flat1_out_ga(.([], .([], T16)), T18)

The argument filtering Pi contains the following mapping:
flat1_in_ga(x1, x2)  =  flat1_in_ga(x1)
[]  =  []
flat1_out_ga(x1, x2)  =  flat1_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x2, x6)

(5) 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:

FLAT1_IN_GA(.([], .([], T16)), T18) → U1_GA(T16, T18, flat1_in_ga(T16, T18))
FLAT1_IN_GA(.([], .([], T16)), T18) → FLAT1_IN_GA(T16, T18)
FLAT1_IN_GA(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_GA(T35, T36, T37, T39, flat1_in_ga(.(T36, T37), T39))
FLAT1_IN_GA(.([], .(.(T35, T36), T37)), .(T35, T39)) → FLAT1_IN_GA(.(T36, T37), T39)
FLAT1_IN_GA(.(.(T46, []), T59), .(T46, T61)) → U3_GA(T46, T59, T61, flat1_in_ga(T59, T61))
FLAT1_IN_GA(.(.(T46, []), T59), .(T46, T61)) → FLAT1_IN_GA(T59, T61)
FLAT1_IN_GA(.(.(T46, .(T70, T71)), T72), .(T46, .(T70, T74))) → U4_GA(T46, T70, T71, T72, T74, flat1_in_ga(.(T71, T72), T74))
FLAT1_IN_GA(.(.(T46, .(T70, T71)), T72), .(T46, .(T70, T74))) → FLAT1_IN_GA(.(T71, T72), T74)

The TRS R consists of the following rules:

flat1_in_ga([], []) → flat1_out_ga([], [])
flat1_in_ga(.([], []), []) → flat1_out_ga(.([], []), [])
flat1_in_ga(.([], .([], T16)), T18) → U1_ga(T16, T18, flat1_in_ga(T16, T18))
flat1_in_ga(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_ga(T35, T36, T37, T39, flat1_in_ga(.(T36, T37), T39))
flat1_in_ga(.(.(T46, []), T59), .(T46, T61)) → U3_ga(T46, T59, T61, flat1_in_ga(T59, T61))
flat1_in_ga(.(.(T46, .(T70, T71)), T72), .(T46, .(T70, T74))) → U4_ga(T46, T70, T71, T72, T74, flat1_in_ga(.(T71, T72), T74))
U4_ga(T46, T70, T71, T72, T74, flat1_out_ga(.(T71, T72), T74)) → flat1_out_ga(.(.(T46, .(T70, T71)), T72), .(T46, .(T70, T74)))
U3_ga(T46, T59, T61, flat1_out_ga(T59, T61)) → flat1_out_ga(.(.(T46, []), T59), .(T46, T61))
U2_ga(T35, T36, T37, T39, flat1_out_ga(.(T36, T37), T39)) → flat1_out_ga(.([], .(.(T35, T36), T37)), .(T35, T39))
U1_ga(T16, T18, flat1_out_ga(T16, T18)) → flat1_out_ga(.([], .([], T16)), T18)

The argument filtering Pi contains the following mapping:
flat1_in_ga(x1, x2)  =  flat1_in_ga(x1)
[]  =  []
flat1_out_ga(x1, x2)  =  flat1_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x2, x6)
FLAT1_IN_GA(x1, x2)  =  FLAT1_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x5)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x2, x6)

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

(6) Obligation:

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

FLAT1_IN_GA(.([], .([], T16)), T18) → U1_GA(T16, T18, flat1_in_ga(T16, T18))
FLAT1_IN_GA(.([], .([], T16)), T18) → FLAT1_IN_GA(T16, T18)
FLAT1_IN_GA(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_GA(T35, T36, T37, T39, flat1_in_ga(.(T36, T37), T39))
FLAT1_IN_GA(.([], .(.(T35, T36), T37)), .(T35, T39)) → FLAT1_IN_GA(.(T36, T37), T39)
FLAT1_IN_GA(.(.(T46, []), T59), .(T46, T61)) → U3_GA(T46, T59, T61, flat1_in_ga(T59, T61))
FLAT1_IN_GA(.(.(T46, []), T59), .(T46, T61)) → FLAT1_IN_GA(T59, T61)
FLAT1_IN_GA(.(.(T46, .(T70, T71)), T72), .(T46, .(T70, T74))) → U4_GA(T46, T70, T71, T72, T74, flat1_in_ga(.(T71, T72), T74))
FLAT1_IN_GA(.(.(T46, .(T70, T71)), T72), .(T46, .(T70, T74))) → FLAT1_IN_GA(.(T71, T72), T74)

The TRS R consists of the following rules:

flat1_in_ga([], []) → flat1_out_ga([], [])
flat1_in_ga(.([], []), []) → flat1_out_ga(.([], []), [])
flat1_in_ga(.([], .([], T16)), T18) → U1_ga(T16, T18, flat1_in_ga(T16, T18))
flat1_in_ga(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_ga(T35, T36, T37, T39, flat1_in_ga(.(T36, T37), T39))
flat1_in_ga(.(.(T46, []), T59), .(T46, T61)) → U3_ga(T46, T59, T61, flat1_in_ga(T59, T61))
flat1_in_ga(.(.(T46, .(T70, T71)), T72), .(T46, .(T70, T74))) → U4_ga(T46, T70, T71, T72, T74, flat1_in_ga(.(T71, T72), T74))
U4_ga(T46, T70, T71, T72, T74, flat1_out_ga(.(T71, T72), T74)) → flat1_out_ga(.(.(T46, .(T70, T71)), T72), .(T46, .(T70, T74)))
U3_ga(T46, T59, T61, flat1_out_ga(T59, T61)) → flat1_out_ga(.(.(T46, []), T59), .(T46, T61))
U2_ga(T35, T36, T37, T39, flat1_out_ga(.(T36, T37), T39)) → flat1_out_ga(.([], .(.(T35, T36), T37)), .(T35, T39))
U1_ga(T16, T18, flat1_out_ga(T16, T18)) → flat1_out_ga(.([], .([], T16)), T18)

The argument filtering Pi contains the following mapping:
flat1_in_ga(x1, x2)  =  flat1_in_ga(x1)
[]  =  []
flat1_out_ga(x1, x2)  =  flat1_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x2, x6)
FLAT1_IN_GA(x1, x2)  =  FLAT1_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x5)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x2, x6)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Obligation:

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

FLAT1_IN_GA(.([], .(.(T35, T36), T37)), .(T35, T39)) → FLAT1_IN_GA(.(T36, T37), T39)
FLAT1_IN_GA(.([], .([], T16)), T18) → FLAT1_IN_GA(T16, T18)
FLAT1_IN_GA(.(.(T46, []), T59), .(T46, T61)) → FLAT1_IN_GA(T59, T61)
FLAT1_IN_GA(.(.(T46, .(T70, T71)), T72), .(T46, .(T70, T74))) → FLAT1_IN_GA(.(T71, T72), T74)

The TRS R consists of the following rules:

flat1_in_ga([], []) → flat1_out_ga([], [])
flat1_in_ga(.([], []), []) → flat1_out_ga(.([], []), [])
flat1_in_ga(.([], .([], T16)), T18) → U1_ga(T16, T18, flat1_in_ga(T16, T18))
flat1_in_ga(.([], .(.(T35, T36), T37)), .(T35, T39)) → U2_ga(T35, T36, T37, T39, flat1_in_ga(.(T36, T37), T39))
flat1_in_ga(.(.(T46, []), T59), .(T46, T61)) → U3_ga(T46, T59, T61, flat1_in_ga(T59, T61))
flat1_in_ga(.(.(T46, .(T70, T71)), T72), .(T46, .(T70, T74))) → U4_ga(T46, T70, T71, T72, T74, flat1_in_ga(.(T71, T72), T74))
U4_ga(T46, T70, T71, T72, T74, flat1_out_ga(.(T71, T72), T74)) → flat1_out_ga(.(.(T46, .(T70, T71)), T72), .(T46, .(T70, T74)))
U3_ga(T46, T59, T61, flat1_out_ga(T59, T61)) → flat1_out_ga(.(.(T46, []), T59), .(T46, T61))
U2_ga(T35, T36, T37, T39, flat1_out_ga(.(T36, T37), T39)) → flat1_out_ga(.([], .(.(T35, T36), T37)), .(T35, T39))
U1_ga(T16, T18, flat1_out_ga(T16, T18)) → flat1_out_ga(.([], .([], T16)), T18)

The argument filtering Pi contains the following mapping:
flat1_in_ga(x1, x2)  =  flat1_in_ga(x1)
[]  =  []
flat1_out_ga(x1, x2)  =  flat1_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x5)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x2, x6)
FLAT1_IN_GA(x1, x2)  =  FLAT1_IN_GA(x1)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

FLAT1_IN_GA(.([], .(.(T35, T36), T37)), .(T35, T39)) → FLAT1_IN_GA(.(T36, T37), T39)
FLAT1_IN_GA(.([], .([], T16)), T18) → FLAT1_IN_GA(T16, T18)
FLAT1_IN_GA(.(.(T46, []), T59), .(T46, T61)) → FLAT1_IN_GA(T59, T61)
FLAT1_IN_GA(.(.(T46, .(T70, T71)), T72), .(T46, .(T70, T74))) → FLAT1_IN_GA(.(T71, T72), T74)

R is empty.
The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
FLAT1_IN_GA(x1, x2)  =  FLAT1_IN_GA(x1)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

FLAT1_IN_GA(.([], .(.(T35, T36), T37))) → FLAT1_IN_GA(.(T36, T37))
FLAT1_IN_GA(.([], .([], T16))) → FLAT1_IN_GA(T16)
FLAT1_IN_GA(.(.(T46, []), T59)) → FLAT1_IN_GA(T59)
FLAT1_IN_GA(.(.(T46, .(T70, T71)), T72)) → FLAT1_IN_GA(.(T71, T72))

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

(13) 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:

FLAT1_IN_GA(.([], .(.(T35, T36), T37))) → FLAT1_IN_GA(.(T36, T37))
FLAT1_IN_GA(.([], .([], T16))) → FLAT1_IN_GA(T16)
FLAT1_IN_GA(.(.(T46, []), T59)) → FLAT1_IN_GA(T59)
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 2·x1 + 2·x2   
POL(FLAT1_IN_GA(x1)) = 2·x1   
POL([]) = 0   

(14) Obligation:

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

FLAT1_IN_GA(.(.(T46, .(T70, T71)), T72)) → FLAT1_IN_GA(.(T71, T72))

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

(15) 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:

FLAT1_IN_GA(.(.(T46, .(T70, T71)), T72)) → FLAT1_IN_GA(.(T71, T72))


Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x1 + x2   
POL(FLAT1_IN_GA(x1)) = x1   

(16) Obligation:

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

(17) PisEmptyProof (EQUIVALENT transformation)

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

(18) YES