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

(0) Obligation:

Clauses:

preorder(T, Xs) :- preorder_dl(T, -(Xs, [])).
preorder_dl(nil, -(X, X)).
preorder_dl(tree(L, X, R), -(.(X, Xs), Zs)) :- ','(preorder_dl(L, -(Xs, Ys)), preorder_dl(R, -(Ys, Zs))).

Queries:

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

preorder_in_ga(T, Xs) → U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
preorder_dl_in_ga(nil, -(X, X)) → preorder_dl_out_ga(nil, -(X, X))
preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) → U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) → preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) → preorder_out_ga(T, Xs)

The argument filtering Pi contains the following mapping:
preorder_in_ga(x1, x2)  =  preorder_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
preorder_dl_in_ga(x1, x2)  =  preorder_dl_in_ga(x1)
nil  =  nil
preorder_dl_out_ga(x1, x2)  =  preorder_dl_out_ga
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x3, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x6)
preorder_out_ga(x1, x2)  =  preorder_out_ga

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

preorder_in_ga(T, Xs) → U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
preorder_dl_in_ga(nil, -(X, X)) → preorder_dl_out_ga(nil, -(X, X))
preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) → U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) → preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) → preorder_out_ga(T, Xs)

The argument filtering Pi contains the following mapping:
preorder_in_ga(x1, x2)  =  preorder_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
preorder_dl_in_ga(x1, x2)  =  preorder_dl_in_ga(x1)
nil  =  nil
preorder_dl_out_ga(x1, x2)  =  preorder_dl_out_ga
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x3, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x6)
preorder_out_ga(x1, x2)  =  preorder_out_ga

(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:

PREORDER_IN_GA(T, Xs) → U1_GA(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
PREORDER_IN_GA(T, Xs) → PREORDER_DL_IN_GA(T, -(Xs, []))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN_GA(L, -(Xs, Ys))
U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → PREORDER_DL_IN_GA(R, -(Ys, Zs))

The TRS R consists of the following rules:

preorder_in_ga(T, Xs) → U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
preorder_dl_in_ga(nil, -(X, X)) → preorder_dl_out_ga(nil, -(X, X))
preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) → U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) → preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) → preorder_out_ga(T, Xs)

The argument filtering Pi contains the following mapping:
preorder_in_ga(x1, x2)  =  preorder_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
preorder_dl_in_ga(x1, x2)  =  preorder_dl_in_ga(x1)
nil  =  nil
preorder_dl_out_ga(x1, x2)  =  preorder_dl_out_ga
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x3, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x6)
preorder_out_ga(x1, x2)  =  preorder_out_ga
PREORDER_IN_GA(x1, x2)  =  PREORDER_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
PREORDER_DL_IN_GA(x1, x2)  =  PREORDER_DL_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x3, x6)
U3_GA(x1, x2, x3, x4, x5, x6)  =  U3_GA(x6)

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

(4) Obligation:

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

PREORDER_IN_GA(T, Xs) → U1_GA(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
PREORDER_IN_GA(T, Xs) → PREORDER_DL_IN_GA(T, -(Xs, []))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN_GA(L, -(Xs, Ys))
U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → PREORDER_DL_IN_GA(R, -(Ys, Zs))

The TRS R consists of the following rules:

preorder_in_ga(T, Xs) → U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
preorder_dl_in_ga(nil, -(X, X)) → preorder_dl_out_ga(nil, -(X, X))
preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) → U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) → preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) → preorder_out_ga(T, Xs)

The argument filtering Pi contains the following mapping:
preorder_in_ga(x1, x2)  =  preorder_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
preorder_dl_in_ga(x1, x2)  =  preorder_dl_in_ga(x1)
nil  =  nil
preorder_dl_out_ga(x1, x2)  =  preorder_dl_out_ga
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x3, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x6)
preorder_out_ga(x1, x2)  =  preorder_out_ga
PREORDER_IN_GA(x1, x2)  =  PREORDER_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
PREORDER_DL_IN_GA(x1, x2)  =  PREORDER_DL_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x3, x6)
U3_GA(x1, x2, x3, x4, x5, x6)  =  U3_GA(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 3 less nodes.

(6) Obligation:

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

U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → PREORDER_DL_IN_GA(R, -(Ys, Zs))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN_GA(L, -(Xs, Ys))

The TRS R consists of the following rules:

preorder_in_ga(T, Xs) → U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
preorder_dl_in_ga(nil, -(X, X)) → preorder_dl_out_ga(nil, -(X, X))
preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) → U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) → preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) → preorder_out_ga(T, Xs)

The argument filtering Pi contains the following mapping:
preorder_in_ga(x1, x2)  =  preorder_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
preorder_dl_in_ga(x1, x2)  =  preorder_dl_in_ga(x1)
nil  =  nil
preorder_dl_out_ga(x1, x2)  =  preorder_dl_out_ga
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x3, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x6)
preorder_out_ga(x1, x2)  =  preorder_out_ga
PREORDER_DL_IN_GA(x1, x2)  =  PREORDER_DL_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x3, x6)

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:

U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → PREORDER_DL_IN_GA(R, -(Ys, Zs))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN_GA(L, -(Xs, Ys))

The TRS R consists of the following rules:

preorder_dl_in_ga(nil, -(X, X)) → preorder_dl_out_ga(nil, -(X, X))
preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) → U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) → preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))

The argument filtering Pi contains the following mapping:
preorder_dl_in_ga(x1, x2)  =  preorder_dl_in_ga(x1)
nil  =  nil
preorder_dl_out_ga(x1, x2)  =  preorder_dl_out_ga
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x3, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x6)
PREORDER_DL_IN_GA(x1, x2)  =  PREORDER_DL_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x3, x6)

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:

U2_GA(R, preorder_dl_out_ga) → PREORDER_DL_IN_GA(R)
PREORDER_DL_IN_GA(tree(L, X, R)) → U2_GA(R, preorder_dl_in_ga(L))
PREORDER_DL_IN_GA(tree(L, X, R)) → PREORDER_DL_IN_GA(L)

The TRS R consists of the following rules:

preorder_dl_in_ga(nil) → preorder_dl_out_ga
preorder_dl_in_ga(tree(L, X, R)) → U2_ga(R, preorder_dl_in_ga(L))
U2_ga(R, preorder_dl_out_ga) → U3_ga(preorder_dl_in_ga(R))
U3_ga(preorder_dl_out_ga) → preorder_dl_out_ga

The set Q consists of the following terms:

preorder_dl_in_ga(x0)
U2_ga(x0, x1)
U3_ga(x0)

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

(11) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • PREORDER_DL_IN_GA(tree(L, X, R)) → U2_GA(R, preorder_dl_in_ga(L))
    The graph contains the following edges 1 > 1

  • PREORDER_DL_IN_GA(tree(L, X, R)) → PREORDER_DL_IN_GA(L)
    The graph contains the following edges 1 > 1

  • U2_GA(R, preorder_dl_out_ga) → PREORDER_DL_IN_GA(R)
    The graph contains the following edges 1 >= 1

(12) TRUE

(13) PrologToPiTRSProof (SOUND transformation)

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

preorder_in_ga(T, Xs) → U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
preorder_dl_in_ga(nil, -(X, X)) → preorder_dl_out_ga(nil, -(X, X))
preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) → U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) → preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) → preorder_out_ga(T, Xs)

The argument filtering Pi contains the following mapping:
preorder_in_ga(x1, x2)  =  preorder_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
preorder_dl_in_ga(x1, x2)  =  preorder_dl_in_ga(x1)
nil  =  nil
preorder_dl_out_ga(x1, x2)  =  preorder_dl_out_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x1, x2, x3, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x1, x2, x3, x6)
preorder_out_ga(x1, x2)  =  preorder_out_ga(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(14) Obligation:

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

preorder_in_ga(T, Xs) → U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
preorder_dl_in_ga(nil, -(X, X)) → preorder_dl_out_ga(nil, -(X, X))
preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) → U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) → preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) → preorder_out_ga(T, Xs)

The argument filtering Pi contains the following mapping:
preorder_in_ga(x1, x2)  =  preorder_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
preorder_dl_in_ga(x1, x2)  =  preorder_dl_in_ga(x1)
nil  =  nil
preorder_dl_out_ga(x1, x2)  =  preorder_dl_out_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x1, x2, x3, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x1, x2, x3, x6)
preorder_out_ga(x1, x2)  =  preorder_out_ga(x1)

(15) 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:

PREORDER_IN_GA(T, Xs) → U1_GA(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
PREORDER_IN_GA(T, Xs) → PREORDER_DL_IN_GA(T, -(Xs, []))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN_GA(L, -(Xs, Ys))
U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → PREORDER_DL_IN_GA(R, -(Ys, Zs))

The TRS R consists of the following rules:

preorder_in_ga(T, Xs) → U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
preorder_dl_in_ga(nil, -(X, X)) → preorder_dl_out_ga(nil, -(X, X))
preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) → U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) → preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) → preorder_out_ga(T, Xs)

The argument filtering Pi contains the following mapping:
preorder_in_ga(x1, x2)  =  preorder_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
preorder_dl_in_ga(x1, x2)  =  preorder_dl_in_ga(x1)
nil  =  nil
preorder_dl_out_ga(x1, x2)  =  preorder_dl_out_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x1, x2, x3, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x1, x2, x3, x6)
preorder_out_ga(x1, x2)  =  preorder_out_ga(x1)
PREORDER_IN_GA(x1, x2)  =  PREORDER_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
PREORDER_DL_IN_GA(x1, x2)  =  PREORDER_DL_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x1, x2, x3, x6)
U3_GA(x1, x2, x3, x4, x5, x6)  =  U3_GA(x1, x2, x3, x6)

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

(16) Obligation:

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

PREORDER_IN_GA(T, Xs) → U1_GA(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
PREORDER_IN_GA(T, Xs) → PREORDER_DL_IN_GA(T, -(Xs, []))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN_GA(L, -(Xs, Ys))
U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → PREORDER_DL_IN_GA(R, -(Ys, Zs))

The TRS R consists of the following rules:

preorder_in_ga(T, Xs) → U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
preorder_dl_in_ga(nil, -(X, X)) → preorder_dl_out_ga(nil, -(X, X))
preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) → U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) → preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) → preorder_out_ga(T, Xs)

The argument filtering Pi contains the following mapping:
preorder_in_ga(x1, x2)  =  preorder_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
preorder_dl_in_ga(x1, x2)  =  preorder_dl_in_ga(x1)
nil  =  nil
preorder_dl_out_ga(x1, x2)  =  preorder_dl_out_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x1, x2, x3, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x1, x2, x3, x6)
preorder_out_ga(x1, x2)  =  preorder_out_ga(x1)
PREORDER_IN_GA(x1, x2)  =  PREORDER_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
PREORDER_DL_IN_GA(x1, x2)  =  PREORDER_DL_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x1, x2, x3, x6)
U3_GA(x1, x2, x3, x4, x5, x6)  =  U3_GA(x1, x2, x3, x6)

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

(17) DependencyGraphProof (EQUIVALENT transformation)

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

(18) Obligation:

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

U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → PREORDER_DL_IN_GA(R, -(Ys, Zs))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN_GA(L, -(Xs, Ys))

The TRS R consists of the following rules:

preorder_in_ga(T, Xs) → U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
preorder_dl_in_ga(nil, -(X, X)) → preorder_dl_out_ga(nil, -(X, X))
preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) → U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) → preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) → preorder_out_ga(T, Xs)

The argument filtering Pi contains the following mapping:
preorder_in_ga(x1, x2)  =  preorder_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
preorder_dl_in_ga(x1, x2)  =  preorder_dl_in_ga(x1)
nil  =  nil
preorder_dl_out_ga(x1, x2)  =  preorder_dl_out_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x1, x2, x3, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x1, x2, x3, x6)
preorder_out_ga(x1, x2)  =  preorder_out_ga(x1)
PREORDER_DL_IN_GA(x1, x2)  =  PREORDER_DL_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x1, x2, x3, x6)

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

(19) UsableRulesProof (EQUIVALENT transformation)

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

(20) Obligation:

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

U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → PREORDER_DL_IN_GA(R, -(Ys, Zs))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) → PREORDER_DL_IN_GA(L, -(Xs, Ys))

The TRS R consists of the following rules:

preorder_dl_in_ga(nil, -(X, X)) → preorder_dl_out_ga(nil, -(X, X))
preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) → U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) → U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) → preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))

The argument filtering Pi contains the following mapping:
preorder_dl_in_ga(x1, x2)  =  preorder_dl_in_ga(x1)
nil  =  nil
preorder_dl_out_ga(x1, x2)  =  preorder_dl_out_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x1, x2, x3, x6)
U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x1, x2, x3, x6)
PREORDER_DL_IN_GA(x1, x2)  =  PREORDER_DL_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x1, x2, x3, x6)

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

(21) PiDPToQDPProof (SOUND transformation)

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

(22) Obligation:

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

U2_GA(L, X, R, preorder_dl_out_ga(L)) → PREORDER_DL_IN_GA(R)
PREORDER_DL_IN_GA(tree(L, X, R)) → U2_GA(L, X, R, preorder_dl_in_ga(L))
PREORDER_DL_IN_GA(tree(L, X, R)) → PREORDER_DL_IN_GA(L)

The TRS R consists of the following rules:

preorder_dl_in_ga(nil) → preorder_dl_out_ga(nil)
preorder_dl_in_ga(tree(L, X, R)) → U2_ga(L, X, R, preorder_dl_in_ga(L))
U2_ga(L, X, R, preorder_dl_out_ga(L)) → U3_ga(L, X, R, preorder_dl_in_ga(R))
U3_ga(L, X, R, preorder_dl_out_ga(R)) → preorder_dl_out_ga(tree(L, X, R))

The set Q consists of the following terms:

preorder_dl_in_ga(x0)
U2_ga(x0, x1, x2, x3)
U3_ga(x0, x1, x2, x3)

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