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 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 PiDP
    ↳7 UsableRulesProof (⇔)
    ↳8 PiDP
    ↳9 PiDPToQDPProof (⇐)
    ↳10 QDP
    ↳11 UsableRulesReductionPairsProof (⇔)
    ↳12 QDP
    ↳13 MRRProof (⇔)
    ↳14 QDP
    ↳15 PisEmptyProof (⇔)
    ↳16 TRUE
  ↳17 PrologToPiTRSProof (⇐)
    ↳18 PiTRS
    ↳19 DependencyPairsProof (⇔)
    ↳20 PiDP
    ↳21 DependencyGraphProof (⇔)
    ↳22 PiDP
    ↳23 UsableRulesProof (⇔)
    ↳24 PiDP
    ↳25 PiDPToQDPProof (⇐)
    ↳26 QDP
    ↳27 UsableRulesReductionPairsProof (⇔)
    ↳28 QDP

(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) PrologToPiTRSProof (SOUND transformation)

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

flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_ga(T, R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U1_ga(T, R, flat_out_ga(T, R)) → flat_out_ga(.([], T), R)

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
[]  =  []
flat_out_ga(x1, x2)  =  flat_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)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_ga(T, R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U1_ga(T, R, flat_out_ga(T, R)) → flat_out_ga(.([], T), R)

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
[]  =  []
flat_out_ga(x1, x2)  =  flat_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)

(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:

FLAT_IN_GA(.([], T), R) → U1_GA(T, R, flat_in_ga(T, R))
FLAT_IN_GA(.([], T), R) → FLAT_IN_GA(T, R)
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → U2_GA(H, T, TT, R, flat_in_ga(.(T, TT), R))
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)

The TRS R consists of the following rules:

flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_ga(T, R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U1_ga(T, R, flat_out_ga(T, R)) → flat_out_ga(.([], T), R)

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
[]  =  []
flat_out_ga(x1, x2)  =  flat_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)
FLAT_IN_GA(x1, x2)  =  FLAT_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x5)

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

(4) Obligation:

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

FLAT_IN_GA(.([], T), R) → U1_GA(T, R, flat_in_ga(T, R))
FLAT_IN_GA(.([], T), R) → FLAT_IN_GA(T, R)
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → U2_GA(H, T, TT, R, flat_in_ga(.(T, TT), R))
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)

The TRS R consists of the following rules:

flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_ga(T, R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U1_ga(T, R, flat_out_ga(T, R)) → flat_out_ga(.([], T), R)

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
[]  =  []
flat_out_ga(x1, x2)  =  flat_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)
FLAT_IN_GA(x1, x2)  =  FLAT_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, 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:

FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_GA(.([], T), R) → FLAT_IN_GA(T, R)

The TRS R consists of the following rules:

flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_ga(T, R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U1_ga(T, R, flat_out_ga(T, R)) → flat_out_ga(.([], T), R)

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
[]  =  []
flat_out_ga(x1, x2)  =  flat_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)
FLAT_IN_GA(x1, x2)  =  FLAT_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:

FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_GA(.([], T), R) → FLAT_IN_GA(T, R)

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

FLAT_IN_GA(.(.(H, T), TT)) → FLAT_IN_GA(.(T, TT))
FLAT_IN_GA(.([], T)) → FLAT_IN_GA(T)

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

(11) 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:

FLAT_IN_GA(.([], T)) → FLAT_IN_GA(T)
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

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

(12) Obligation:

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

FLAT_IN_GA(.(.(H, T), TT)) → FLAT_IN_GA(.(T, TT))

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

(13) 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:

FLAT_IN_GA(.(.(H, T), TT)) → FLAT_IN_GA(.(T, TT))


Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(FLAT_IN_GA(x1)) = 2·x1   

(14) Obligation:

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

(15) PisEmptyProof (EQUIVALENT transformation)

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

(16) TRUE

(17) PrologToPiTRSProof (SOUND transformation)

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

flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_ga(T, R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U1_ga(T, R, flat_out_ga(T, R)) → flat_out_ga(.([], T), R)

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
[]  =  []
flat_out_ga(x1, x2)  =  flat_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(18) Obligation:

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

flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_ga(T, R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U1_ga(T, R, flat_out_ga(T, R)) → flat_out_ga(.([], T), R)

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
[]  =  []
flat_out_ga(x1, x2)  =  flat_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)

(19) 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:

FLAT_IN_GA(.([], T), R) → U1_GA(T, R, flat_in_ga(T, R))
FLAT_IN_GA(.([], T), R) → FLAT_IN_GA(T, R)
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → U2_GA(H, T, TT, R, flat_in_ga(.(T, TT), R))
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)

The TRS R consists of the following rules:

flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_ga(T, R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U1_ga(T, R, flat_out_ga(T, R)) → flat_out_ga(.([], T), R)

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
[]  =  []
flat_out_ga(x1, x2)  =  flat_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
FLAT_IN_GA(x1, x2)  =  FLAT_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)

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

(20) Obligation:

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

FLAT_IN_GA(.([], T), R) → U1_GA(T, R, flat_in_ga(T, R))
FLAT_IN_GA(.([], T), R) → FLAT_IN_GA(T, R)
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → U2_GA(H, T, TT, R, flat_in_ga(.(T, TT), R))
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)

The TRS R consists of the following rules:

flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_ga(T, R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U1_ga(T, R, flat_out_ga(T, R)) → flat_out_ga(.([], T), R)

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
[]  =  []
flat_out_ga(x1, x2)  =  flat_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
FLAT_IN_GA(x1, x2)  =  FLAT_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)

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

(21) DependencyGraphProof (EQUIVALENT transformation)

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

(22) Obligation:

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

FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_GA(.([], T), R) → FLAT_IN_GA(T, R)

The TRS R consists of the following rules:

flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_ga(T, R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U1_ga(T, R, flat_out_ga(T, R)) → flat_out_ga(.([], T), R)

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
[]  =  []
flat_out_ga(x1, x2)  =  flat_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
FLAT_IN_GA(x1, x2)  =  FLAT_IN_GA(x1)

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

(23) UsableRulesProof (EQUIVALENT transformation)

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

(24) Obligation:

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

FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_GA(.([], T), R) → FLAT_IN_GA(T, R)

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

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

(25) PiDPToQDPProof (SOUND transformation)

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

(26) Obligation:

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

FLAT_IN_GA(.(.(H, T), TT)) → FLAT_IN_GA(.(T, TT))
FLAT_IN_GA(.([], T)) → FLAT_IN_GA(T)

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

(27) 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:

FLAT_IN_GA(.([], T)) → FLAT_IN_GA(T)
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

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

(28) Obligation:

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

FLAT_IN_GA(.(.(H, T), TT)) → FLAT_IN_GA(.(T, TT))

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