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 PrologToDTProblemTransformerProof (⇐)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇐)
↳4 PiDP
↳5 DependencyGraphProof (⇔)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 YES
    ↳14 PiDP
        ↳15 PiDPToQDPProof (⇐)
        ↳16 QDP
        ↳17 QDPOrderProof (⇔)
        ↳18 QDP
        ↳19 QDPOrderProof (⇔)
        ↳20 QDP
        ↳21 DependencyGraphProof (⇔)
        ↳22 TRUE

(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) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

rewrite13(T76, op(T77, T78), op(T76, X100)) :- rewrite13(T77, T78, X100).
normal1(op(op(T20, T21), T22), T7) :- normal1(op(T20, op(T21, T22)), T7).
normal1(op(T39, op(T40, T41)), T7) :- rewrite13(T40, T41, X51).
normal1(op(T39, op(T40, T41)), T7) :- ','(rewritec13(T40, T41, T48), normal1(op(T39, T48), T7)).

Clauses:

normalc1(op(op(T20, T21), T22), T7) :- normalc1(op(T20, op(T21, T22)), T7).
normalc1(op(T39, op(T40, T41)), T7) :- ','(rewritec13(T40, T41, T48), normalc1(op(T39, T48), T7)).
normalc1(T90, T90).
rewritec13(op(T67, T68), T69, op(T67, op(T68, T69))).
rewritec13(T76, op(T77, T78), op(T76, X100)) :- rewritec13(T77, T78, X100).

Afs:

normal1(x1, x2)  =  normal1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
normal1_in: (b,f)
rewrite13_in: (b,b,f)
rewritec13_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

NORMAL1_IN_GA(op(op(T20, T21), T22), T7) → U2_GA(T20, T21, T22, T7, normal1_in_ga(op(T20, op(T21, T22)), T7))
NORMAL1_IN_GA(op(op(T20, T21), T22), T7) → NORMAL1_IN_GA(op(T20, op(T21, T22)), T7)
NORMAL1_IN_GA(op(T39, op(T40, T41)), T7) → U3_GA(T39, T40, T41, T7, rewrite13_in_gga(T40, T41, X51))
NORMAL1_IN_GA(op(T39, op(T40, T41)), T7) → REWRITE13_IN_GGA(T40, T41, X51)
REWRITE13_IN_GGA(T76, op(T77, T78), op(T76, X100)) → U1_GGA(T76, T77, T78, X100, rewrite13_in_gga(T77, T78, X100))
REWRITE13_IN_GGA(T76, op(T77, T78), op(T76, X100)) → REWRITE13_IN_GGA(T77, T78, X100)
NORMAL1_IN_GA(op(T39, op(T40, T41)), T7) → U4_GA(T39, T40, T41, T7, rewritec13_in_gga(T40, T41, T48))
U4_GA(T39, T40, T41, T7, rewritec13_out_gga(T40, T41, T48)) → U5_GA(T39, T40, T41, T7, normal1_in_ga(op(T39, T48), T7))
U4_GA(T39, T40, T41, T7, rewritec13_out_gga(T40, T41, T48)) → NORMAL1_IN_GA(op(T39, T48), T7)

The TRS R consists of the following rules:

rewritec13_in_gga(op(T67, T68), T69, op(T67, op(T68, T69))) → rewritec13_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewritec13_in_gga(T76, op(T77, T78), op(T76, X100)) → U10_gga(T76, T77, T78, X100, rewritec13_in_gga(T77, T78, X100))
U10_gga(T76, T77, T78, X100, rewritec13_out_gga(T77, T78, X100)) → rewritec13_out_gga(T76, op(T77, T78), op(T76, X100))

The argument filtering Pi contains the following mapping:
normal1_in_ga(x1, x2)  =  normal1_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
rewrite13_in_gga(x1, x2, x3)  =  rewrite13_in_gga(x1, x2)
rewritec13_in_gga(x1, x2, x3)  =  rewritec13_in_gga(x1, x2)
rewritec13_out_gga(x1, x2, x3)  =  rewritec13_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
NORMAL1_IN_GA(x1, x2)  =  NORMAL1_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x3, x5)
REWRITE13_IN_GGA(x1, x2, x3)  =  REWRITE13_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x3, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x3, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

NORMAL1_IN_GA(op(op(T20, T21), T22), T7) → U2_GA(T20, T21, T22, T7, normal1_in_ga(op(T20, op(T21, T22)), T7))
NORMAL1_IN_GA(op(op(T20, T21), T22), T7) → NORMAL1_IN_GA(op(T20, op(T21, T22)), T7)
NORMAL1_IN_GA(op(T39, op(T40, T41)), T7) → U3_GA(T39, T40, T41, T7, rewrite13_in_gga(T40, T41, X51))
NORMAL1_IN_GA(op(T39, op(T40, T41)), T7) → REWRITE13_IN_GGA(T40, T41, X51)
REWRITE13_IN_GGA(T76, op(T77, T78), op(T76, X100)) → U1_GGA(T76, T77, T78, X100, rewrite13_in_gga(T77, T78, X100))
REWRITE13_IN_GGA(T76, op(T77, T78), op(T76, X100)) → REWRITE13_IN_GGA(T77, T78, X100)
NORMAL1_IN_GA(op(T39, op(T40, T41)), T7) → U4_GA(T39, T40, T41, T7, rewritec13_in_gga(T40, T41, T48))
U4_GA(T39, T40, T41, T7, rewritec13_out_gga(T40, T41, T48)) → U5_GA(T39, T40, T41, T7, normal1_in_ga(op(T39, T48), T7))
U4_GA(T39, T40, T41, T7, rewritec13_out_gga(T40, T41, T48)) → NORMAL1_IN_GA(op(T39, T48), T7)

The TRS R consists of the following rules:

rewritec13_in_gga(op(T67, T68), T69, op(T67, op(T68, T69))) → rewritec13_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewritec13_in_gga(T76, op(T77, T78), op(T76, X100)) → U10_gga(T76, T77, T78, X100, rewritec13_in_gga(T77, T78, X100))
U10_gga(T76, T77, T78, X100, rewritec13_out_gga(T77, T78, X100)) → rewritec13_out_gga(T76, op(T77, T78), op(T76, X100))

The argument filtering Pi contains the following mapping:
normal1_in_ga(x1, x2)  =  normal1_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
rewrite13_in_gga(x1, x2, x3)  =  rewrite13_in_gga(x1, x2)
rewritec13_in_gga(x1, x2, x3)  =  rewritec13_in_gga(x1, x2)
rewritec13_out_gga(x1, x2, x3)  =  rewritec13_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
NORMAL1_IN_GA(x1, x2)  =  NORMAL1_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x3, x5)
REWRITE13_IN_GGA(x1, x2, x3)  =  REWRITE13_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x3, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

REWRITE13_IN_GGA(T76, op(T77, T78), op(T76, X100)) → REWRITE13_IN_GGA(T77, T78, X100)

The TRS R consists of the following rules:

rewritec13_in_gga(op(T67, T68), T69, op(T67, op(T68, T69))) → rewritec13_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewritec13_in_gga(T76, op(T77, T78), op(T76, X100)) → U10_gga(T76, T77, T78, X100, rewritec13_in_gga(T77, T78, X100))
U10_gga(T76, T77, T78, X100, rewritec13_out_gga(T77, T78, X100)) → rewritec13_out_gga(T76, op(T77, T78), op(T76, X100))

The argument filtering Pi contains the following mapping:
op(x1, x2)  =  op(x1, x2)
rewritec13_in_gga(x1, x2, x3)  =  rewritec13_in_gga(x1, x2)
rewritec13_out_gga(x1, x2, x3)  =  rewritec13_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
REWRITE13_IN_GGA(x1, x2, x3)  =  REWRITE13_IN_GGA(x1, x2)

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:

REWRITE13_IN_GGA(T76, op(T77, T78), op(T76, X100)) → REWRITE13_IN_GGA(T77, T78, X100)

R is empty.
The argument filtering Pi contains the following mapping:
op(x1, x2)  =  op(x1, x2)
REWRITE13_IN_GGA(x1, x2, x3)  =  REWRITE13_IN_GGA(x1, x2)

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:

REWRITE13_IN_GGA(T76, op(T77, T78)) → REWRITE13_IN_GGA(T77, T78)

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:

  • REWRITE13_IN_GGA(T76, op(T77, T78)) → REWRITE13_IN_GGA(T77, T78)
    The graph contains the following edges 2 > 1, 2 > 2

(13) YES

(14) Obligation:

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

NORMAL1_IN_GA(op(T39, op(T40, T41)), T7) → U4_GA(T39, T40, T41, T7, rewritec13_in_gga(T40, T41, T48))
U4_GA(T39, T40, T41, T7, rewritec13_out_gga(T40, T41, T48)) → NORMAL1_IN_GA(op(T39, T48), T7)
NORMAL1_IN_GA(op(op(T20, T21), T22), T7) → NORMAL1_IN_GA(op(T20, op(T21, T22)), T7)

The TRS R consists of the following rules:

rewritec13_in_gga(op(T67, T68), T69, op(T67, op(T68, T69))) → rewritec13_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewritec13_in_gga(T76, op(T77, T78), op(T76, X100)) → U10_gga(T76, T77, T78, X100, rewritec13_in_gga(T77, T78, X100))
U10_gga(T76, T77, T78, X100, rewritec13_out_gga(T77, T78, X100)) → rewritec13_out_gga(T76, op(T77, T78), op(T76, X100))

The argument filtering Pi contains the following mapping:
op(x1, x2)  =  op(x1, x2)
rewritec13_in_gga(x1, x2, x3)  =  rewritec13_in_gga(x1, x2)
rewritec13_out_gga(x1, x2, x3)  =  rewritec13_out_gga(x1, x2, x3)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x2, x3, x5)
NORMAL1_IN_GA(x1, x2)  =  NORMAL1_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x3, x5)

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

(15) PiDPToQDPProof (SOUND transformation)

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

(16) Obligation:

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

NORMAL1_IN_GA(op(T39, op(T40, T41))) → U4_GA(T39, T40, T41, rewritec13_in_gga(T40, T41))
U4_GA(T39, T40, T41, rewritec13_out_gga(T40, T41, T48)) → NORMAL1_IN_GA(op(T39, T48))
NORMAL1_IN_GA(op(op(T20, T21), T22)) → NORMAL1_IN_GA(op(T20, op(T21, T22)))

The TRS R consists of the following rules:

rewritec13_in_gga(op(T67, T68), T69) → rewritec13_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewritec13_in_gga(T76, op(T77, T78)) → U10_gga(T76, T77, T78, rewritec13_in_gga(T77, T78))
U10_gga(T76, T77, T78, rewritec13_out_gga(T77, T78, X100)) → rewritec13_out_gga(T76, op(T77, T78), op(T76, X100))

The set Q consists of the following terms:

rewritec13_in_gga(x0, x1)
U10_gga(x0, x1, x2, x3)

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


NORMAL1_IN_GA(op(op(T20, T21), T22)) → NORMAL1_IN_GA(op(T20, op(T21, T22)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(NORMAL1_IN_GA(x1)) = x1   
POL(U10_gga(x1, x2, x3, x4)) = 0   
POL(U4_GA(x1, x2, x3, x4)) = 1 + x1   
POL(op(x1, x2)) = 1 + x1   
POL(rewritec13_in_gga(x1, x2)) = 0   
POL(rewritec13_out_gga(x1, x2, x3)) = 0   

The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

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

NORMAL1_IN_GA(op(T39, op(T40, T41))) → U4_GA(T39, T40, T41, rewritec13_in_gga(T40, T41))
U4_GA(T39, T40, T41, rewritec13_out_gga(T40, T41, T48)) → NORMAL1_IN_GA(op(T39, T48))

The TRS R consists of the following rules:

rewritec13_in_gga(op(T67, T68), T69) → rewritec13_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewritec13_in_gga(T76, op(T77, T78)) → U10_gga(T76, T77, T78, rewritec13_in_gga(T77, T78))
U10_gga(T76, T77, T78, rewritec13_out_gga(T77, T78, X100)) → rewritec13_out_gga(T76, op(T77, T78), op(T76, X100))

The set Q consists of the following terms:

rewritec13_in_gga(x0, x1)
U10_gga(x0, x1, x2, x3)

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


NORMAL1_IN_GA(op(T39, op(T40, T41))) → U4_GA(T39, T40, T41, rewritec13_in_gga(T40, T41))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U4_GA(x1, ..., x4) ) = 2x1 + x4 + 2


POL( rewritec13_in_gga(x1, x2) ) = 2x1 + x2


POL( op(x1, x2) ) = 2x1 + x2 + 2


POL( rewritec13_out_gga(x1, ..., x3) ) = x3


POL( U10_gga(x1, ..., x4) ) = 2x1 + x4 + 2


POL( NORMAL1_IN_GA(x1) ) = x1



The following usable rules [FROCOS05] were oriented:

rewritec13_in_gga(op(T67, T68), T69) → rewritec13_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewritec13_in_gga(T76, op(T77, T78)) → U10_gga(T76, T77, T78, rewritec13_in_gga(T77, T78))
U10_gga(T76, T77, T78, rewritec13_out_gga(T77, T78, X100)) → rewritec13_out_gga(T76, op(T77, T78), op(T76, X100))

(20) Obligation:

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

U4_GA(T39, T40, T41, rewritec13_out_gga(T40, T41, T48)) → NORMAL1_IN_GA(op(T39, T48))

The TRS R consists of the following rules:

rewritec13_in_gga(op(T67, T68), T69) → rewritec13_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewritec13_in_gga(T76, op(T77, T78)) → U10_gga(T76, T77, T78, rewritec13_in_gga(T77, T78))
U10_gga(T76, T77, T78, rewritec13_out_gga(T77, T78, X100)) → rewritec13_out_gga(T76, op(T77, T78), op(T76, X100))

The set Q consists of the following terms:

rewritec13_in_gga(x0, x1)
U10_gga(x0, x1, x2, x3)

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