Left Termination of the query pattern normal_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 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 TRUE
      ↳14 PiDP
        ↳15 UsableRulesProof (⇔)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇐)
        ↳18 QDP
        ↳19 MRRProof (⇔)
        ↳20 QDP
        ↳21 DependencyGraphProof (⇔)
        ↳22 TRUE
  ↳23 PrologToPiTRSProof (⇐)
    ↳24 PiTRS
    ↳25 DependencyPairsProof (⇔)
    ↳26 PiDP
    ↳27 DependencyGraphProof (⇔)
    ↳28 AND
      ↳29 PiDP
        ↳30 UsableRulesProof (⇔)
        ↳31 PiDP
        ↳32 PiDPToQDPProof (⇐)
        ↳33 QDP
        ↳34 QDPSizeChangeProof (⇔)
        ↳35 TRUE
      ↳36 PiDP
        ↳37 UsableRulesProof (⇔)
        ↳38 PiDP
        ↳39 PiDPToQDPProof (⇐)
        ↳40 QDP

(0) Obligation:

Clauses:

normal(F, N) :- ','(rewrite(F, F1), normal(F1, N)).
normal(F, F).
rewrite(op(op(A, B), C), op(A, op(B, C))).
rewrite(op(A, op(B, C)), op(A, L)) :- rewrite(op(B, C), L).

Queries:

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

normal_in_ga(F, N) → U1_ga(F, N, rewrite_in_ga(F, F1))
rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) → rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
rewrite_in_ga(op(A, op(B, C)), op(A, L)) → U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) → rewrite_out_ga(op(A, op(B, C)), op(A, L))
U1_ga(F, N, rewrite_out_ga(F, F1)) → U2_ga(F, N, normal_in_ga(F1, N))
normal_in_ga(F, F) → normal_out_ga(F, F)
U2_ga(F, N, normal_out_ga(F1, N)) → normal_out_ga(F, N)

The argument filtering Pi contains the following mapping:
normal_in_ga(x1, x2)  =  normal_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
normal_out_ga(x1, x2)  =  normal_out_ga(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:

normal_in_ga(F, N) → U1_ga(F, N, rewrite_in_ga(F, F1))
rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) → rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
rewrite_in_ga(op(A, op(B, C)), op(A, L)) → U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) → rewrite_out_ga(op(A, op(B, C)), op(A, L))
U1_ga(F, N, rewrite_out_ga(F, F1)) → U2_ga(F, N, normal_in_ga(F1, N))
normal_in_ga(F, F) → normal_out_ga(F, F)
U2_ga(F, N, normal_out_ga(F1, N)) → normal_out_ga(F, N)

The argument filtering Pi contains the following mapping:
normal_in_ga(x1, x2)  =  normal_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
normal_out_ga(x1, x2)  =  normal_out_ga(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:

NORMAL_IN_GA(F, N) → U1_GA(F, N, rewrite_in_ga(F, F1))
NORMAL_IN_GA(F, N) → REWRITE_IN_GA(F, F1)
REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) → U3_GA(A, B, C, L, rewrite_in_ga(op(B, C), L))
REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) → REWRITE_IN_GA(op(B, C), L)
U1_GA(F, N, rewrite_out_ga(F, F1)) → U2_GA(F, N, normal_in_ga(F1, N))
U1_GA(F, N, rewrite_out_ga(F, F1)) → NORMAL_IN_GA(F1, N)

The TRS R consists of the following rules:

normal_in_ga(F, N) → U1_ga(F, N, rewrite_in_ga(F, F1))
rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) → rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
rewrite_in_ga(op(A, op(B, C)), op(A, L)) → U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) → rewrite_out_ga(op(A, op(B, C)), op(A, L))
U1_ga(F, N, rewrite_out_ga(F, F1)) → U2_ga(F, N, normal_in_ga(F1, N))
normal_in_ga(F, F) → normal_out_ga(F, F)
U2_ga(F, N, normal_out_ga(F1, N)) → normal_out_ga(F, N)

The argument filtering Pi contains the following mapping:
normal_in_ga(x1, x2)  =  normal_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
normal_out_ga(x1, x2)  =  normal_out_ga(x2)
NORMAL_IN_GA(x1, x2)  =  NORMAL_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
REWRITE_IN_GA(x1, x2)  =  REWRITE_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x5)
U2_GA(x1, x2, x3)  =  U2_GA(x3)

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

(4) Obligation:

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

NORMAL_IN_GA(F, N) → U1_GA(F, N, rewrite_in_ga(F, F1))
NORMAL_IN_GA(F, N) → REWRITE_IN_GA(F, F1)
REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) → U3_GA(A, B, C, L, rewrite_in_ga(op(B, C), L))
REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) → REWRITE_IN_GA(op(B, C), L)
U1_GA(F, N, rewrite_out_ga(F, F1)) → U2_GA(F, N, normal_in_ga(F1, N))
U1_GA(F, N, rewrite_out_ga(F, F1)) → NORMAL_IN_GA(F1, N)

The TRS R consists of the following rules:

normal_in_ga(F, N) → U1_ga(F, N, rewrite_in_ga(F, F1))
rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) → rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
rewrite_in_ga(op(A, op(B, C)), op(A, L)) → U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) → rewrite_out_ga(op(A, op(B, C)), op(A, L))
U1_ga(F, N, rewrite_out_ga(F, F1)) → U2_ga(F, N, normal_in_ga(F1, N))
normal_in_ga(F, F) → normal_out_ga(F, F)
U2_ga(F, N, normal_out_ga(F1, N)) → normal_out_ga(F, N)

The argument filtering Pi contains the following mapping:
normal_in_ga(x1, x2)  =  normal_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
normal_out_ga(x1, x2)  =  normal_out_ga(x2)
NORMAL_IN_GA(x1, x2)  =  NORMAL_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
REWRITE_IN_GA(x1, x2)  =  REWRITE_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x5)
U2_GA(x1, x2, x3)  =  U2_GA(x3)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) → REWRITE_IN_GA(op(B, C), L)

The TRS R consists of the following rules:

normal_in_ga(F, N) → U1_ga(F, N, rewrite_in_ga(F, F1))
rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) → rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
rewrite_in_ga(op(A, op(B, C)), op(A, L)) → U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) → rewrite_out_ga(op(A, op(B, C)), op(A, L))
U1_ga(F, N, rewrite_out_ga(F, F1)) → U2_ga(F, N, normal_in_ga(F1, N))
normal_in_ga(F, F) → normal_out_ga(F, F)
U2_ga(F, N, normal_out_ga(F1, N)) → normal_out_ga(F, N)

The argument filtering Pi contains the following mapping:
normal_in_ga(x1, x2)  =  normal_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
normal_out_ga(x1, x2)  =  normal_out_ga(x2)
REWRITE_IN_GA(x1, x2)  =  REWRITE_IN_GA(x1)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) → REWRITE_IN_GA(op(B, C), L)

R is empty.
The argument filtering Pi contains the following mapping:
op(x1, x2)  =  op(x1, x2)
REWRITE_IN_GA(x1, x2)  =  REWRITE_IN_GA(x1)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

REWRITE_IN_GA(op(A, op(B, C))) → REWRITE_IN_GA(op(B, C))

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

(12) 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:

  • REWRITE_IN_GA(op(A, op(B, C))) → REWRITE_IN_GA(op(B, C))
    The graph contains the following edges 1 > 1

(13) TRUE

(14) Obligation:

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

U1_GA(F, N, rewrite_out_ga(F, F1)) → NORMAL_IN_GA(F1, N)
NORMAL_IN_GA(F, N) → U1_GA(F, N, rewrite_in_ga(F, F1))

The TRS R consists of the following rules:

normal_in_ga(F, N) → U1_ga(F, N, rewrite_in_ga(F, F1))
rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) → rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
rewrite_in_ga(op(A, op(B, C)), op(A, L)) → U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) → rewrite_out_ga(op(A, op(B, C)), op(A, L))
U1_ga(F, N, rewrite_out_ga(F, F1)) → U2_ga(F, N, normal_in_ga(F1, N))
normal_in_ga(F, F) → normal_out_ga(F, F)
U2_ga(F, N, normal_out_ga(F1, N)) → normal_out_ga(F, N)

The argument filtering Pi contains the following mapping:
normal_in_ga(x1, x2)  =  normal_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
normal_out_ga(x1, x2)  =  normal_out_ga(x2)
NORMAL_IN_GA(x1, x2)  =  NORMAL_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

U1_GA(F, N, rewrite_out_ga(F, F1)) → NORMAL_IN_GA(F1, N)
NORMAL_IN_GA(F, N) → U1_GA(F, N, rewrite_in_ga(F, F1))

The TRS R consists of the following rules:

rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) → rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
rewrite_in_ga(op(A, op(B, C)), op(A, L)) → U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) → rewrite_out_ga(op(A, op(B, C)), op(A, L))

The argument filtering Pi contains the following mapping:
rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
NORMAL_IN_GA(x1, x2)  =  NORMAL_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

U1_GA(rewrite_out_ga(F1)) → NORMAL_IN_GA(F1)
NORMAL_IN_GA(F) → U1_GA(rewrite_in_ga(F))

The TRS R consists of the following rules:

rewrite_in_ga(op(op(A, B), C)) → rewrite_out_ga(op(A, op(B, C)))
rewrite_in_ga(op(A, op(B, C))) → U3_ga(A, rewrite_in_ga(op(B, C)))
U3_ga(A, rewrite_out_ga(L)) → rewrite_out_ga(op(A, L))

The set Q consists of the following terms:

rewrite_in_ga(x0)
U3_ga(x0, x1)

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:

U1_GA(rewrite_out_ga(F1)) → NORMAL_IN_GA(F1)


Used ordering: Polynomial interpretation [POLO]:

POL(NORMAL_IN_GA(x1)) = 2 + 2·x1   
POL(U1_GA(x1)) = 2·x1   
POL(U3_ga(x1, x2)) = 1 + 2·x1 + x2   
POL(op(x1, x2)) = 1 + 2·x1 + x2   
POL(rewrite_in_ga(x1)) = 1 + x1   
POL(rewrite_out_ga(x1)) = 2 + x1   

(20) Obligation:

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

NORMAL_IN_GA(F) → U1_GA(rewrite_in_ga(F))

The TRS R consists of the following rules:

rewrite_in_ga(op(op(A, B), C)) → rewrite_out_ga(op(A, op(B, C)))
rewrite_in_ga(op(A, op(B, C))) → U3_ga(A, rewrite_in_ga(op(B, C)))
U3_ga(A, rewrite_out_ga(L)) → rewrite_out_ga(op(A, L))

The set Q consists of the following terms:

rewrite_in_ga(x0)
U3_ga(x0, x1)

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

(21) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

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

normal_in_ga(F, N) → U1_ga(F, N, rewrite_in_ga(F, F1))
rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) → rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
rewrite_in_ga(op(A, op(B, C)), op(A, L)) → U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) → rewrite_out_ga(op(A, op(B, C)), op(A, L))
U1_ga(F, N, rewrite_out_ga(F, F1)) → U2_ga(F, N, normal_in_ga(F1, N))
normal_in_ga(F, F) → normal_out_ga(F, F)
U2_ga(F, N, normal_out_ga(F1, N)) → normal_out_ga(F, N)

The argument filtering Pi contains the following mapping:
normal_in_ga(x1, x2)  =  normal_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
normal_out_ga(x1, x2)  =  normal_out_ga(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:

normal_in_ga(F, N) → U1_ga(F, N, rewrite_in_ga(F, F1))
rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) → rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
rewrite_in_ga(op(A, op(B, C)), op(A, L)) → U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) → rewrite_out_ga(op(A, op(B, C)), op(A, L))
U1_ga(F, N, rewrite_out_ga(F, F1)) → U2_ga(F, N, normal_in_ga(F1, N))
normal_in_ga(F, F) → normal_out_ga(F, F)
U2_ga(F, N, normal_out_ga(F1, N)) → normal_out_ga(F, N)

The argument filtering Pi contains the following mapping:
normal_in_ga(x1, x2)  =  normal_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
normal_out_ga(x1, x2)  =  normal_out_ga(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:

NORMAL_IN_GA(F, N) → U1_GA(F, N, rewrite_in_ga(F, F1))
NORMAL_IN_GA(F, N) → REWRITE_IN_GA(F, F1)
REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) → U3_GA(A, B, C, L, rewrite_in_ga(op(B, C), L))
REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) → REWRITE_IN_GA(op(B, C), L)
U1_GA(F, N, rewrite_out_ga(F, F1)) → U2_GA(F, N, normal_in_ga(F1, N))
U1_GA(F, N, rewrite_out_ga(F, F1)) → NORMAL_IN_GA(F1, N)

The TRS R consists of the following rules:

normal_in_ga(F, N) → U1_ga(F, N, rewrite_in_ga(F, F1))
rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) → rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
rewrite_in_ga(op(A, op(B, C)), op(A, L)) → U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) → rewrite_out_ga(op(A, op(B, C)), op(A, L))
U1_ga(F, N, rewrite_out_ga(F, F1)) → U2_ga(F, N, normal_in_ga(F1, N))
normal_in_ga(F, F) → normal_out_ga(F, F)
U2_ga(F, N, normal_out_ga(F1, N)) → normal_out_ga(F, N)

The argument filtering Pi contains the following mapping:
normal_in_ga(x1, x2)  =  normal_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
normal_out_ga(x1, x2)  =  normal_out_ga(x1, x2)
NORMAL_IN_GA(x1, x2)  =  NORMAL_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
REWRITE_IN_GA(x1, x2)  =  REWRITE_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x3, x5)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)

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

(26) Obligation:

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

NORMAL_IN_GA(F, N) → U1_GA(F, N, rewrite_in_ga(F, F1))
NORMAL_IN_GA(F, N) → REWRITE_IN_GA(F, F1)
REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) → U3_GA(A, B, C, L, rewrite_in_ga(op(B, C), L))
REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) → REWRITE_IN_GA(op(B, C), L)
U1_GA(F, N, rewrite_out_ga(F, F1)) → U2_GA(F, N, normal_in_ga(F1, N))
U1_GA(F, N, rewrite_out_ga(F, F1)) → NORMAL_IN_GA(F1, N)

The TRS R consists of the following rules:

normal_in_ga(F, N) → U1_ga(F, N, rewrite_in_ga(F, F1))
rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) → rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
rewrite_in_ga(op(A, op(B, C)), op(A, L)) → U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) → rewrite_out_ga(op(A, op(B, C)), op(A, L))
U1_ga(F, N, rewrite_out_ga(F, F1)) → U2_ga(F, N, normal_in_ga(F1, N))
normal_in_ga(F, F) → normal_out_ga(F, F)
U2_ga(F, N, normal_out_ga(F1, N)) → normal_out_ga(F, N)

The argument filtering Pi contains the following mapping:
normal_in_ga(x1, x2)  =  normal_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
normal_out_ga(x1, x2)  =  normal_out_ga(x1, x2)
NORMAL_IN_GA(x1, x2)  =  NORMAL_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
REWRITE_IN_GA(x1, x2)  =  REWRITE_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x3, x5)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)

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

(27) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes.

(28) Complex Obligation (AND)

(29) Obligation:

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

REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) → REWRITE_IN_GA(op(B, C), L)

The TRS R consists of the following rules:

normal_in_ga(F, N) → U1_ga(F, N, rewrite_in_ga(F, F1))
rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) → rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
rewrite_in_ga(op(A, op(B, C)), op(A, L)) → U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) → rewrite_out_ga(op(A, op(B, C)), op(A, L))
U1_ga(F, N, rewrite_out_ga(F, F1)) → U2_ga(F, N, normal_in_ga(F1, N))
normal_in_ga(F, F) → normal_out_ga(F, F)
U2_ga(F, N, normal_out_ga(F1, N)) → normal_out_ga(F, N)

The argument filtering Pi contains the following mapping:
normal_in_ga(x1, x2)  =  normal_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
normal_out_ga(x1, x2)  =  normal_out_ga(x1, x2)
REWRITE_IN_GA(x1, x2)  =  REWRITE_IN_GA(x1)

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

(30) UsableRulesProof (EQUIVALENT transformation)

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

(31) Obligation:

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

REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) → REWRITE_IN_GA(op(B, C), L)

R is empty.
The argument filtering Pi contains the following mapping:
op(x1, x2)  =  op(x1, x2)
REWRITE_IN_GA(x1, x2)  =  REWRITE_IN_GA(x1)

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

(32) PiDPToQDPProof (SOUND transformation)

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

(33) Obligation:

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

REWRITE_IN_GA(op(A, op(B, C))) → REWRITE_IN_GA(op(B, C))

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

(34) 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:

  • REWRITE_IN_GA(op(A, op(B, C))) → REWRITE_IN_GA(op(B, C))
    The graph contains the following edges 1 > 1

(35) TRUE

(36) Obligation:

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

U1_GA(F, N, rewrite_out_ga(F, F1)) → NORMAL_IN_GA(F1, N)
NORMAL_IN_GA(F, N) → U1_GA(F, N, rewrite_in_ga(F, F1))

The TRS R consists of the following rules:

normal_in_ga(F, N) → U1_ga(F, N, rewrite_in_ga(F, F1))
rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) → rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
rewrite_in_ga(op(A, op(B, C)), op(A, L)) → U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) → rewrite_out_ga(op(A, op(B, C)), op(A, L))
U1_ga(F, N, rewrite_out_ga(F, F1)) → U2_ga(F, N, normal_in_ga(F1, N))
normal_in_ga(F, F) → normal_out_ga(F, F)
U2_ga(F, N, normal_out_ga(F1, N)) → normal_out_ga(F, N)

The argument filtering Pi contains the following mapping:
normal_in_ga(x1, x2)  =  normal_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
normal_out_ga(x1, x2)  =  normal_out_ga(x1, x2)
NORMAL_IN_GA(x1, x2)  =  NORMAL_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)

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

(37) UsableRulesProof (EQUIVALENT transformation)

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

(38) Obligation:

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

U1_GA(F, N, rewrite_out_ga(F, F1)) → NORMAL_IN_GA(F1, N)
NORMAL_IN_GA(F, N) → U1_GA(F, N, rewrite_in_ga(F, F1))

The TRS R consists of the following rules:

rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) → rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
rewrite_in_ga(op(A, op(B, C)), op(A, L)) → U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) → rewrite_out_ga(op(A, op(B, C)), op(A, L))

The argument filtering Pi contains the following mapping:
rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x3, x5)
NORMAL_IN_GA(x1, x2)  =  NORMAL_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)

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

(39) PiDPToQDPProof (SOUND transformation)

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

(40) Obligation:

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

U1_GA(F, rewrite_out_ga(F, F1)) → NORMAL_IN_GA(F1)
NORMAL_IN_GA(F) → U1_GA(F, rewrite_in_ga(F))

The TRS R consists of the following rules:

rewrite_in_ga(op(op(A, B), C)) → rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
rewrite_in_ga(op(A, op(B, C))) → U3_ga(A, B, C, rewrite_in_ga(op(B, C)))
U3_ga(A, B, C, rewrite_out_ga(op(B, C), L)) → rewrite_out_ga(op(A, op(B, C)), op(A, L))

The set Q consists of the following terms:

rewrite_in_ga(x0)
U3_ga(x0, x1, x2, x3)

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