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 QDPOrderProof (⇔)
    ↳12 QDP
    ↳13 DependencyGraphProof (⇔)
    ↳14 QDP
    ↳15 UsableRulesProof (⇔)
    ↳16 QDP
    ↳17 QReductionProof (⇔)
    ↳18 QDP
    ↳19 MRRProof (⇔)
    ↳20 QDP
    ↳21 PisEmptyProof (⇔)
    ↳22 TRUE
  ↳23 PrologToPiTRSProof (⇐)
    ↳24 PiTRS
    ↳25 DependencyPairsProof (⇔)
    ↳26 PiDP
    ↳27 DependencyGraphProof (⇔)
    ↳28 PiDP
    ↳29 UsableRulesProof (⇔)
    ↳30 PiDP
    ↳31 PiDPToQDPProof (⇐)
    ↳32 QDP
    ↳33 QDPOrderProof (⇔)
    ↳34 QDP
    ↳35 DependencyGraphProof (⇔)
    ↳36 QDP
    ↳37 UsableRulesProof (⇔)
    ↳38 QDP
    ↳39 QReductionProof (⇔)
    ↳40 QDP

(0) Obligation:

Clauses:

right(tree(X, XS1, XS2), XS2).
flat(niltree, nil).
flat(tree(X, niltree, XS), cons(X, YS)) :- ','(right(tree(X, niltree, XS), ZS), flat(ZS, YS)).
flat(tree(X, tree(Y, YS1, YS2), XS), ZS) :- flat(tree(Y, YS1, tree(X, YS2, XS)), ZS).

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(niltree, nil) → flat_out_ga(niltree, nil)
flat_in_ga(tree(X, niltree, XS), cons(X, YS)) → U1_ga(X, XS, YS, right_in_ga(tree(X, niltree, XS), ZS))
right_in_ga(tree(X, XS1, XS2), XS2) → right_out_ga(tree(X, XS1, XS2), XS2)
U1_ga(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → U2_ga(X, XS, YS, flat_in_ga(ZS, YS))
flat_in_ga(tree(X, tree(Y, YS1, YS2), XS), ZS) → U3_ga(X, Y, YS1, YS2, XS, ZS, flat_in_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS))
U3_ga(X, Y, YS1, YS2, XS, ZS, flat_out_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS)) → flat_out_ga(tree(X, tree(Y, YS1, YS2), XS), ZS)
U2_ga(X, XS, YS, flat_out_ga(ZS, YS)) → flat_out_ga(tree(X, niltree, XS), cons(X, YS))

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
niltree  =  niltree
flat_out_ga(x1, x2)  =  flat_out_ga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
right_in_ga(x1, x2)  =  right_in_ga(x1)
right_out_ga(x1, x2)  =  right_out_ga(x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
U3_ga(x1, x2, x3, x4, x5, x6, x7)  =  U3_ga(x7)
cons(x1, x2)  =  cons(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:

flat_in_ga(niltree, nil) → flat_out_ga(niltree, nil)
flat_in_ga(tree(X, niltree, XS), cons(X, YS)) → U1_ga(X, XS, YS, right_in_ga(tree(X, niltree, XS), ZS))
right_in_ga(tree(X, XS1, XS2), XS2) → right_out_ga(tree(X, XS1, XS2), XS2)
U1_ga(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → U2_ga(X, XS, YS, flat_in_ga(ZS, YS))
flat_in_ga(tree(X, tree(Y, YS1, YS2), XS), ZS) → U3_ga(X, Y, YS1, YS2, XS, ZS, flat_in_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS))
U3_ga(X, Y, YS1, YS2, XS, ZS, flat_out_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS)) → flat_out_ga(tree(X, tree(Y, YS1, YS2), XS), ZS)
U2_ga(X, XS, YS, flat_out_ga(ZS, YS)) → flat_out_ga(tree(X, niltree, XS), cons(X, YS))

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
niltree  =  niltree
flat_out_ga(x1, x2)  =  flat_out_ga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
right_in_ga(x1, x2)  =  right_in_ga(x1)
right_out_ga(x1, x2)  =  right_out_ga(x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
U3_ga(x1, x2, x3, x4, x5, x6, x7)  =  U3_ga(x7)
cons(x1, x2)  =  cons(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:

FLAT_IN_GA(tree(X, niltree, XS), cons(X, YS)) → U1_GA(X, XS, YS, right_in_ga(tree(X, niltree, XS), ZS))
FLAT_IN_GA(tree(X, niltree, XS), cons(X, YS)) → RIGHT_IN_GA(tree(X, niltree, XS), ZS)
U1_GA(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → U2_GA(X, XS, YS, flat_in_ga(ZS, YS))
U1_GA(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → FLAT_IN_GA(ZS, YS)
FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS), ZS) → U3_GA(X, Y, YS1, YS2, XS, ZS, flat_in_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS))
FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS), ZS) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)), ZS)

The TRS R consists of the following rules:

flat_in_ga(niltree, nil) → flat_out_ga(niltree, nil)
flat_in_ga(tree(X, niltree, XS), cons(X, YS)) → U1_ga(X, XS, YS, right_in_ga(tree(X, niltree, XS), ZS))
right_in_ga(tree(X, XS1, XS2), XS2) → right_out_ga(tree(X, XS1, XS2), XS2)
U1_ga(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → U2_ga(X, XS, YS, flat_in_ga(ZS, YS))
flat_in_ga(tree(X, tree(Y, YS1, YS2), XS), ZS) → U3_ga(X, Y, YS1, YS2, XS, ZS, flat_in_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS))
U3_ga(X, Y, YS1, YS2, XS, ZS, flat_out_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS)) → flat_out_ga(tree(X, tree(Y, YS1, YS2), XS), ZS)
U2_ga(X, XS, YS, flat_out_ga(ZS, YS)) → flat_out_ga(tree(X, niltree, XS), cons(X, YS))

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
niltree  =  niltree
flat_out_ga(x1, x2)  =  flat_out_ga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
right_in_ga(x1, x2)  =  right_in_ga(x1)
right_out_ga(x1, x2)  =  right_out_ga(x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
U3_ga(x1, x2, x3, x4, x5, x6, x7)  =  U3_ga(x7)
cons(x1, x2)  =  cons(x1, x2)
FLAT_IN_GA(x1, x2)  =  FLAT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
RIGHT_IN_GA(x1, x2)  =  RIGHT_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
U3_GA(x1, x2, x3, x4, x5, x6, x7)  =  U3_GA(x7)

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(tree(X, niltree, XS), cons(X, YS)) → U1_GA(X, XS, YS, right_in_ga(tree(X, niltree, XS), ZS))
FLAT_IN_GA(tree(X, niltree, XS), cons(X, YS)) → RIGHT_IN_GA(tree(X, niltree, XS), ZS)
U1_GA(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → U2_GA(X, XS, YS, flat_in_ga(ZS, YS))
U1_GA(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → FLAT_IN_GA(ZS, YS)
FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS), ZS) → U3_GA(X, Y, YS1, YS2, XS, ZS, flat_in_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS))
FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS), ZS) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)), ZS)

The TRS R consists of the following rules:

flat_in_ga(niltree, nil) → flat_out_ga(niltree, nil)
flat_in_ga(tree(X, niltree, XS), cons(X, YS)) → U1_ga(X, XS, YS, right_in_ga(tree(X, niltree, XS), ZS))
right_in_ga(tree(X, XS1, XS2), XS2) → right_out_ga(tree(X, XS1, XS2), XS2)
U1_ga(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → U2_ga(X, XS, YS, flat_in_ga(ZS, YS))
flat_in_ga(tree(X, tree(Y, YS1, YS2), XS), ZS) → U3_ga(X, Y, YS1, YS2, XS, ZS, flat_in_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS))
U3_ga(X, Y, YS1, YS2, XS, ZS, flat_out_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS)) → flat_out_ga(tree(X, tree(Y, YS1, YS2), XS), ZS)
U2_ga(X, XS, YS, flat_out_ga(ZS, YS)) → flat_out_ga(tree(X, niltree, XS), cons(X, YS))

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
niltree  =  niltree
flat_out_ga(x1, x2)  =  flat_out_ga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
right_in_ga(x1, x2)  =  right_in_ga(x1)
right_out_ga(x1, x2)  =  right_out_ga(x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
U3_ga(x1, x2, x3, x4, x5, x6, x7)  =  U3_ga(x7)
cons(x1, x2)  =  cons(x1, x2)
FLAT_IN_GA(x1, x2)  =  FLAT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
RIGHT_IN_GA(x1, x2)  =  RIGHT_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
U3_GA(x1, x2, x3, x4, x5, x6, x7)  =  U3_GA(x7)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(6) Obligation:

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

U1_GA(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → FLAT_IN_GA(ZS, YS)
FLAT_IN_GA(tree(X, niltree, XS), cons(X, YS)) → U1_GA(X, XS, YS, right_in_ga(tree(X, niltree, XS), ZS))
FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS), ZS) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)), ZS)

The TRS R consists of the following rules:

flat_in_ga(niltree, nil) → flat_out_ga(niltree, nil)
flat_in_ga(tree(X, niltree, XS), cons(X, YS)) → U1_ga(X, XS, YS, right_in_ga(tree(X, niltree, XS), ZS))
right_in_ga(tree(X, XS1, XS2), XS2) → right_out_ga(tree(X, XS1, XS2), XS2)
U1_ga(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → U2_ga(X, XS, YS, flat_in_ga(ZS, YS))
flat_in_ga(tree(X, tree(Y, YS1, YS2), XS), ZS) → U3_ga(X, Y, YS1, YS2, XS, ZS, flat_in_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS))
U3_ga(X, Y, YS1, YS2, XS, ZS, flat_out_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS)) → flat_out_ga(tree(X, tree(Y, YS1, YS2), XS), ZS)
U2_ga(X, XS, YS, flat_out_ga(ZS, YS)) → flat_out_ga(tree(X, niltree, XS), cons(X, YS))

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
niltree  =  niltree
flat_out_ga(x1, x2)  =  flat_out_ga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
right_in_ga(x1, x2)  =  right_in_ga(x1)
right_out_ga(x1, x2)  =  right_out_ga(x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
U3_ga(x1, x2, x3, x4, x5, x6, x7)  =  U3_ga(x7)
cons(x1, x2)  =  cons(x1, x2)
FLAT_IN_GA(x1, x2)  =  FLAT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)

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:

U1_GA(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → FLAT_IN_GA(ZS, YS)
FLAT_IN_GA(tree(X, niltree, XS), cons(X, YS)) → U1_GA(X, XS, YS, right_in_ga(tree(X, niltree, XS), ZS))
FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS), ZS) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)), ZS)

The TRS R consists of the following rules:

right_in_ga(tree(X, XS1, XS2), XS2) → right_out_ga(tree(X, XS1, XS2), XS2)

The argument filtering Pi contains the following mapping:
niltree  =  niltree
tree(x1, x2, x3)  =  tree(x1, x2, x3)
right_in_ga(x1, x2)  =  right_in_ga(x1)
right_out_ga(x1, x2)  =  right_out_ga(x2)
cons(x1, x2)  =  cons(x1, x2)
FLAT_IN_GA(x1, x2)  =  FLAT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)

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:

U1_GA(X, right_out_ga(ZS)) → FLAT_IN_GA(ZS)
FLAT_IN_GA(tree(X, niltree, XS)) → U1_GA(X, right_in_ga(tree(X, niltree, XS)))
FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS)) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)))

The TRS R consists of the following rules:

right_in_ga(tree(X, XS1, XS2)) → right_out_ga(XS2)

The set Q consists of the following terms:

right_in_ga(x0)

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1_GA(X, right_out_ga(ZS)) → FLAT_IN_GA(ZS)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(FLAT_IN_GA(x1)) = x1   
POL(U1_GA(x1, x2)) = x2   
POL(niltree) = 0   
POL(right_in_ga(x1)) = x1   
POL(right_out_ga(x1)) = 1 + x1   
POL(tree(x1, x2, x3)) = 1 + x2 + x3   

The following usable rules [FROCOS05] were oriented:

right_in_ga(tree(X, XS1, XS2)) → right_out_ga(XS2)

(12) Obligation:

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

FLAT_IN_GA(tree(X, niltree, XS)) → U1_GA(X, right_in_ga(tree(X, niltree, XS)))
FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS)) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)))

The TRS R consists of the following rules:

right_in_ga(tree(X, XS1, XS2)) → right_out_ga(XS2)

The set Q consists of the following terms:

right_in_ga(x0)

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

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(14) Obligation:

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

FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS)) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)))

The TRS R consists of the following rules:

right_in_ga(tree(X, XS1, XS2)) → right_out_ga(XS2)

The set Q consists of the following terms:

right_in_ga(x0)

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

(15) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(16) Obligation:

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

FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS)) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)))

R is empty.
The set Q consists of the following terms:

right_in_ga(x0)

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

(17) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

right_in_ga(x0)

(18) Obligation:

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

FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS)) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)))

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

(19) 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(tree(X, tree(Y, YS1, YS2), XS)) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)))


Used ordering: Polynomial interpretation [POLO]:

POL(FLAT_IN_GA(x1)) = 2·x1   
POL(tree(x1, x2, x3)) = 2 + 2·x1 + 2·x2 + x3   

(20) Obligation:

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

(21) PisEmptyProof (EQUIVALENT transformation)

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

(22) TRUE

(23) 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(niltree, nil) → flat_out_ga(niltree, nil)
flat_in_ga(tree(X, niltree, XS), cons(X, YS)) → U1_ga(X, XS, YS, right_in_ga(tree(X, niltree, XS), ZS))
right_in_ga(tree(X, XS1, XS2), XS2) → right_out_ga(tree(X, XS1, XS2), XS2)
U1_ga(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → U2_ga(X, XS, YS, flat_in_ga(ZS, YS))
flat_in_ga(tree(X, tree(Y, YS1, YS2), XS), ZS) → U3_ga(X, Y, YS1, YS2, XS, ZS, flat_in_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS))
U3_ga(X, Y, YS1, YS2, XS, ZS, flat_out_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS)) → flat_out_ga(tree(X, tree(Y, YS1, YS2), XS), ZS)
U2_ga(X, XS, YS, flat_out_ga(ZS, YS)) → flat_out_ga(tree(X, niltree, XS), cons(X, YS))

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
niltree  =  niltree
flat_out_ga(x1, x2)  =  flat_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
right_in_ga(x1, x2)  =  right_in_ga(x1)
right_out_ga(x1, x2)  =  right_out_ga(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
U3_ga(x1, x2, x3, x4, x5, x6, x7)  =  U3_ga(x1, x2, x3, x4, x5, x7)
cons(x1, x2)  =  cons(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(24) Obligation:

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

flat_in_ga(niltree, nil) → flat_out_ga(niltree, nil)
flat_in_ga(tree(X, niltree, XS), cons(X, YS)) → U1_ga(X, XS, YS, right_in_ga(tree(X, niltree, XS), ZS))
right_in_ga(tree(X, XS1, XS2), XS2) → right_out_ga(tree(X, XS1, XS2), XS2)
U1_ga(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → U2_ga(X, XS, YS, flat_in_ga(ZS, YS))
flat_in_ga(tree(X, tree(Y, YS1, YS2), XS), ZS) → U3_ga(X, Y, YS1, YS2, XS, ZS, flat_in_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS))
U3_ga(X, Y, YS1, YS2, XS, ZS, flat_out_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS)) → flat_out_ga(tree(X, tree(Y, YS1, YS2), XS), ZS)
U2_ga(X, XS, YS, flat_out_ga(ZS, YS)) → flat_out_ga(tree(X, niltree, XS), cons(X, YS))

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
niltree  =  niltree
flat_out_ga(x1, x2)  =  flat_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
right_in_ga(x1, x2)  =  right_in_ga(x1)
right_out_ga(x1, x2)  =  right_out_ga(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
U3_ga(x1, x2, x3, x4, x5, x6, x7)  =  U3_ga(x1, x2, x3, x4, x5, x7)
cons(x1, x2)  =  cons(x1, x2)

(25) 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(tree(X, niltree, XS), cons(X, YS)) → U1_GA(X, XS, YS, right_in_ga(tree(X, niltree, XS), ZS))
FLAT_IN_GA(tree(X, niltree, XS), cons(X, YS)) → RIGHT_IN_GA(tree(X, niltree, XS), ZS)
U1_GA(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → U2_GA(X, XS, YS, flat_in_ga(ZS, YS))
U1_GA(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → FLAT_IN_GA(ZS, YS)
FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS), ZS) → U3_GA(X, Y, YS1, YS2, XS, ZS, flat_in_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS))
FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS), ZS) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)), ZS)

The TRS R consists of the following rules:

flat_in_ga(niltree, nil) → flat_out_ga(niltree, nil)
flat_in_ga(tree(X, niltree, XS), cons(X, YS)) → U1_ga(X, XS, YS, right_in_ga(tree(X, niltree, XS), ZS))
right_in_ga(tree(X, XS1, XS2), XS2) → right_out_ga(tree(X, XS1, XS2), XS2)
U1_ga(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → U2_ga(X, XS, YS, flat_in_ga(ZS, YS))
flat_in_ga(tree(X, tree(Y, YS1, YS2), XS), ZS) → U3_ga(X, Y, YS1, YS2, XS, ZS, flat_in_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS))
U3_ga(X, Y, YS1, YS2, XS, ZS, flat_out_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS)) → flat_out_ga(tree(X, tree(Y, YS1, YS2), XS), ZS)
U2_ga(X, XS, YS, flat_out_ga(ZS, YS)) → flat_out_ga(tree(X, niltree, XS), cons(X, YS))

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
niltree  =  niltree
flat_out_ga(x1, x2)  =  flat_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
right_in_ga(x1, x2)  =  right_in_ga(x1)
right_out_ga(x1, x2)  =  right_out_ga(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
U3_ga(x1, x2, x3, x4, x5, x6, x7)  =  U3_ga(x1, x2, x3, x4, x5, x7)
cons(x1, x2)  =  cons(x1, x2)
FLAT_IN_GA(x1, x2)  =  FLAT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
RIGHT_IN_GA(x1, x2)  =  RIGHT_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
U3_GA(x1, x2, x3, x4, x5, x6, x7)  =  U3_GA(x1, x2, x3, x4, x5, x7)

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

(26) Obligation:

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

FLAT_IN_GA(tree(X, niltree, XS), cons(X, YS)) → U1_GA(X, XS, YS, right_in_ga(tree(X, niltree, XS), ZS))
FLAT_IN_GA(tree(X, niltree, XS), cons(X, YS)) → RIGHT_IN_GA(tree(X, niltree, XS), ZS)
U1_GA(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → U2_GA(X, XS, YS, flat_in_ga(ZS, YS))
U1_GA(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → FLAT_IN_GA(ZS, YS)
FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS), ZS) → U3_GA(X, Y, YS1, YS2, XS, ZS, flat_in_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS))
FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS), ZS) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)), ZS)

The TRS R consists of the following rules:

flat_in_ga(niltree, nil) → flat_out_ga(niltree, nil)
flat_in_ga(tree(X, niltree, XS), cons(X, YS)) → U1_ga(X, XS, YS, right_in_ga(tree(X, niltree, XS), ZS))
right_in_ga(tree(X, XS1, XS2), XS2) → right_out_ga(tree(X, XS1, XS2), XS2)
U1_ga(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → U2_ga(X, XS, YS, flat_in_ga(ZS, YS))
flat_in_ga(tree(X, tree(Y, YS1, YS2), XS), ZS) → U3_ga(X, Y, YS1, YS2, XS, ZS, flat_in_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS))
U3_ga(X, Y, YS1, YS2, XS, ZS, flat_out_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS)) → flat_out_ga(tree(X, tree(Y, YS1, YS2), XS), ZS)
U2_ga(X, XS, YS, flat_out_ga(ZS, YS)) → flat_out_ga(tree(X, niltree, XS), cons(X, YS))

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
niltree  =  niltree
flat_out_ga(x1, x2)  =  flat_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
right_in_ga(x1, x2)  =  right_in_ga(x1)
right_out_ga(x1, x2)  =  right_out_ga(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
U3_ga(x1, x2, x3, x4, x5, x6, x7)  =  U3_ga(x1, x2, x3, x4, x5, x7)
cons(x1, x2)  =  cons(x1, x2)
FLAT_IN_GA(x1, x2)  =  FLAT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
RIGHT_IN_GA(x1, x2)  =  RIGHT_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
U3_GA(x1, x2, x3, x4, x5, x6, x7)  =  U3_GA(x1, x2, x3, x4, x5, x7)

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

(27) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(28) Obligation:

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

U1_GA(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → FLAT_IN_GA(ZS, YS)
FLAT_IN_GA(tree(X, niltree, XS), cons(X, YS)) → U1_GA(X, XS, YS, right_in_ga(tree(X, niltree, XS), ZS))
FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS), ZS) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)), ZS)

The TRS R consists of the following rules:

flat_in_ga(niltree, nil) → flat_out_ga(niltree, nil)
flat_in_ga(tree(X, niltree, XS), cons(X, YS)) → U1_ga(X, XS, YS, right_in_ga(tree(X, niltree, XS), ZS))
right_in_ga(tree(X, XS1, XS2), XS2) → right_out_ga(tree(X, XS1, XS2), XS2)
U1_ga(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → U2_ga(X, XS, YS, flat_in_ga(ZS, YS))
flat_in_ga(tree(X, tree(Y, YS1, YS2), XS), ZS) → U3_ga(X, Y, YS1, YS2, XS, ZS, flat_in_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS))
U3_ga(X, Y, YS1, YS2, XS, ZS, flat_out_ga(tree(Y, YS1, tree(X, YS2, XS)), ZS)) → flat_out_ga(tree(X, tree(Y, YS1, YS2), XS), ZS)
U2_ga(X, XS, YS, flat_out_ga(ZS, YS)) → flat_out_ga(tree(X, niltree, XS), cons(X, YS))

The argument filtering Pi contains the following mapping:
flat_in_ga(x1, x2)  =  flat_in_ga(x1)
niltree  =  niltree
flat_out_ga(x1, x2)  =  flat_out_ga(x1, x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
right_in_ga(x1, x2)  =  right_in_ga(x1)
right_out_ga(x1, x2)  =  right_out_ga(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
U3_ga(x1, x2, x3, x4, x5, x6, x7)  =  U3_ga(x1, x2, x3, x4, x5, x7)
cons(x1, x2)  =  cons(x1, x2)
FLAT_IN_GA(x1, x2)  =  FLAT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

U1_GA(X, XS, YS, right_out_ga(tree(X, niltree, XS), ZS)) → FLAT_IN_GA(ZS, YS)
FLAT_IN_GA(tree(X, niltree, XS), cons(X, YS)) → U1_GA(X, XS, YS, right_in_ga(tree(X, niltree, XS), ZS))
FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS), ZS) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)), ZS)

The TRS R consists of the following rules:

right_in_ga(tree(X, XS1, XS2), XS2) → right_out_ga(tree(X, XS1, XS2), XS2)

The argument filtering Pi contains the following mapping:
niltree  =  niltree
tree(x1, x2, x3)  =  tree(x1, x2, x3)
right_in_ga(x1, x2)  =  right_in_ga(x1)
right_out_ga(x1, x2)  =  right_out_ga(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
FLAT_IN_GA(x1, x2)  =  FLAT_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)

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

(31) PiDPToQDPProof (SOUND transformation)

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

(32) Obligation:

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

U1_GA(X, XS, right_out_ga(tree(X, niltree, XS), ZS)) → FLAT_IN_GA(ZS)
FLAT_IN_GA(tree(X, niltree, XS)) → U1_GA(X, XS, right_in_ga(tree(X, niltree, XS)))
FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS)) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)))

The TRS R consists of the following rules:

right_in_ga(tree(X, XS1, XS2)) → right_out_ga(tree(X, XS1, XS2), XS2)

The set Q consists of the following terms:

right_in_ga(x0)

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

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1_GA(X, XS, right_out_ga(tree(X, niltree, XS), ZS)) → FLAT_IN_GA(ZS)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(FLAT_IN_GA(x1)) = x1   
POL(U1_GA(x1, x2, x3)) = x3   
POL(niltree) = 0   
POL(right_in_ga(x1)) = x1   
POL(right_out_ga(x1, x2)) = 1 + x2   
POL(tree(x1, x2, x3)) = 1 + x2 + x3   

The following usable rules [FROCOS05] were oriented:

right_in_ga(tree(X, XS1, XS2)) → right_out_ga(tree(X, XS1, XS2), XS2)

(34) Obligation:

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

FLAT_IN_GA(tree(X, niltree, XS)) → U1_GA(X, XS, right_in_ga(tree(X, niltree, XS)))
FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS)) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)))

The TRS R consists of the following rules:

right_in_ga(tree(X, XS1, XS2)) → right_out_ga(tree(X, XS1, XS2), XS2)

The set Q consists of the following terms:

right_in_ga(x0)

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

(35) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(36) Obligation:

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

FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS)) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)))

The TRS R consists of the following rules:

right_in_ga(tree(X, XS1, XS2)) → right_out_ga(tree(X, XS1, XS2), XS2)

The set Q consists of the following terms:

right_in_ga(x0)

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

(37) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(38) Obligation:

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

FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS)) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)))

R is empty.
The set Q consists of the following terms:

right_in_ga(x0)

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

(39) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

right_in_ga(x0)

(40) Obligation:

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

FLAT_IN_GA(tree(X, tree(Y, YS1, YS2), XS)) → FLAT_IN_GA(tree(Y, YS1, tree(X, YS2, XS)))

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