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 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
↳3 PrologToPiTRSProof (⇐)
↳4 PiTRS
↳5 DependencyPairsProof (⇔)
↳6 PiDP
↳7 DependencyGraphProof (⇔)
↳8 AND
    ↳9 PiDP
        ↳10 UsableRulesProof (⇔)
        ↳11 PiDP
        ↳12 PiDPToQDPProof (⇐)
        ↳13 QDP
        ↳14 QDPSizeChangeProof (⇔)
        ↳15 YES
    ↳16 PiDP
        ↳17 UsableRulesProof (⇔)
        ↳18 PiDP
        ↳19 PiDPToQDPProof (⇐)
        ↳20 QDP
        ↳21 MRRProof (⇔)
        ↳22 QDP
        ↳23 DependencyGraphProof (⇔)
        ↳24 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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

rewrite13(op(T67, T68), T69, op(T67, op(T68, T69))).
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) :- ','(rewrite13(T40, T41, T48), normal1(op(T39, T48), T7)).
normal1(T90, T90).

Queries:

normal1(g,a).

(3) PrologToPiTRSProof (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)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R 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(T39, op(T40, T41)), T7) → U3_ga(T39, T40, T41, T7, rewrite13_in_gga(T40, T41, X51))
rewrite13_in_gga(op(T67, T68), T69, op(T67, op(T68, T69))) → rewrite13_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewrite13_in_gga(T76, op(T77, T78), op(T76, X100)) → U1_gga(T76, T77, T78, X100, rewrite13_in_gga(T77, T78, X100))
U1_gga(T76, T77, T78, X100, rewrite13_out_gga(T77, T78, X100)) → rewrite13_out_gga(T76, op(T77, T78), op(T76, X100))
U3_ga(T39, T40, T41, T7, rewrite13_out_gga(T40, T41, X51)) → normal1_out_ga(op(T39, op(T40, T41)), T7)
normal1_in_ga(op(T39, op(T40, T41)), T7) → U4_ga(T39, T40, T41, T7, rewrite13_in_gga(T40, T41, T48))
U4_ga(T39, T40, T41, T7, rewrite13_out_gga(T40, T41, T48)) → U5_ga(T39, T40, T41, T7, normal1_in_ga(op(T39, T48), T7))
normal1_in_ga(T90, T90) → normal1_out_ga(T90, T90)
U5_ga(T39, T40, T41, T7, normal1_out_ga(op(T39, T48), T7)) → normal1_out_ga(op(T39, op(T40, T41)), T7)
U2_ga(T20, T21, T22, T7, normal1_out_ga(op(T20, op(T21, T22)), T7)) → normal1_out_ga(op(op(T20, T21), T22), T7)

The argument filtering Pi contains the following mapping:
normal1_in_ga(x1, x2)  =  normal1_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
rewrite13_in_gga(x1, x2, x3)  =  rewrite13_in_gga(x1, x2)
rewrite13_out_gga(x1, x2, x3)  =  rewrite13_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
normal1_out_ga(x1, x2)  =  normal1_out_ga
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R 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(T39, op(T40, T41)), T7) → U3_ga(T39, T40, T41, T7, rewrite13_in_gga(T40, T41, X51))
rewrite13_in_gga(op(T67, T68), T69, op(T67, op(T68, T69))) → rewrite13_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewrite13_in_gga(T76, op(T77, T78), op(T76, X100)) → U1_gga(T76, T77, T78, X100, rewrite13_in_gga(T77, T78, X100))
U1_gga(T76, T77, T78, X100, rewrite13_out_gga(T77, T78, X100)) → rewrite13_out_gga(T76, op(T77, T78), op(T76, X100))
U3_ga(T39, T40, T41, T7, rewrite13_out_gga(T40, T41, X51)) → normal1_out_ga(op(T39, op(T40, T41)), T7)
normal1_in_ga(op(T39, op(T40, T41)), T7) → U4_ga(T39, T40, T41, T7, rewrite13_in_gga(T40, T41, T48))
U4_ga(T39, T40, T41, T7, rewrite13_out_gga(T40, T41, T48)) → U5_ga(T39, T40, T41, T7, normal1_in_ga(op(T39, T48), T7))
normal1_in_ga(T90, T90) → normal1_out_ga(T90, T90)
U5_ga(T39, T40, T41, T7, normal1_out_ga(op(T39, T48), T7)) → normal1_out_ga(op(T39, op(T40, T41)), T7)
U2_ga(T20, T21, T22, T7, normal1_out_ga(op(T20, op(T21, T22)), T7)) → normal1_out_ga(op(op(T20, T21), T22), T7)

The argument filtering Pi contains the following mapping:
normal1_in_ga(x1, x2)  =  normal1_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
rewrite13_in_gga(x1, x2, x3)  =  rewrite13_in_gga(x1, x2)
rewrite13_out_gga(x1, x2, x3)  =  rewrite13_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
normal1_out_ga(x1, x2)  =  normal1_out_ga
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)

(5) 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:

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, rewrite13_in_gga(T40, T41, T48))
U4_GA(T39, T40, T41, T7, rewrite13_out_gga(T40, T41, T48)) → U5_GA(T39, T40, T41, T7, normal1_in_ga(op(T39, T48), T7))
U4_GA(T39, T40, T41, T7, rewrite13_out_gga(T40, T41, T48)) → NORMAL1_IN_GA(op(T39, T48), T7)

The TRS R 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(T39, op(T40, T41)), T7) → U3_ga(T39, T40, T41, T7, rewrite13_in_gga(T40, T41, X51))
rewrite13_in_gga(op(T67, T68), T69, op(T67, op(T68, T69))) → rewrite13_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewrite13_in_gga(T76, op(T77, T78), op(T76, X100)) → U1_gga(T76, T77, T78, X100, rewrite13_in_gga(T77, T78, X100))
U1_gga(T76, T77, T78, X100, rewrite13_out_gga(T77, T78, X100)) → rewrite13_out_gga(T76, op(T77, T78), op(T76, X100))
U3_ga(T39, T40, T41, T7, rewrite13_out_gga(T40, T41, X51)) → normal1_out_ga(op(T39, op(T40, T41)), T7)
normal1_in_ga(op(T39, op(T40, T41)), T7) → U4_ga(T39, T40, T41, T7, rewrite13_in_gga(T40, T41, T48))
U4_ga(T39, T40, T41, T7, rewrite13_out_gga(T40, T41, T48)) → U5_ga(T39, T40, T41, T7, normal1_in_ga(op(T39, T48), T7))
normal1_in_ga(T90, T90) → normal1_out_ga(T90, T90)
U5_ga(T39, T40, T41, T7, normal1_out_ga(op(T39, T48), T7)) → normal1_out_ga(op(T39, op(T40, T41)), T7)
U2_ga(T20, T21, T22, T7, normal1_out_ga(op(T20, op(T21, T22)), T7)) → normal1_out_ga(op(op(T20, T21), T22), T7)

The argument filtering Pi contains the following mapping:
normal1_in_ga(x1, x2)  =  normal1_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
rewrite13_in_gga(x1, x2, x3)  =  rewrite13_in_gga(x1, x2)
rewrite13_out_gga(x1, x2, x3)  =  rewrite13_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
normal1_out_ga(x1, x2)  =  normal1_out_ga
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
NORMAL1_IN_GA(x1, x2)  =  NORMAL1_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
REWRITE13_IN_GGA(x1, x2, x3)  =  REWRITE13_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x5)

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

(6) 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, rewrite13_in_gga(T40, T41, T48))
U4_GA(T39, T40, T41, T7, rewrite13_out_gga(T40, T41, T48)) → U5_GA(T39, T40, T41, T7, normal1_in_ga(op(T39, T48), T7))
U4_GA(T39, T40, T41, T7, rewrite13_out_gga(T40, T41, T48)) → NORMAL1_IN_GA(op(T39, T48), T7)

The TRS R 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(T39, op(T40, T41)), T7) → U3_ga(T39, T40, T41, T7, rewrite13_in_gga(T40, T41, X51))
rewrite13_in_gga(op(T67, T68), T69, op(T67, op(T68, T69))) → rewrite13_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewrite13_in_gga(T76, op(T77, T78), op(T76, X100)) → U1_gga(T76, T77, T78, X100, rewrite13_in_gga(T77, T78, X100))
U1_gga(T76, T77, T78, X100, rewrite13_out_gga(T77, T78, X100)) → rewrite13_out_gga(T76, op(T77, T78), op(T76, X100))
U3_ga(T39, T40, T41, T7, rewrite13_out_gga(T40, T41, X51)) → normal1_out_ga(op(T39, op(T40, T41)), T7)
normal1_in_ga(op(T39, op(T40, T41)), T7) → U4_ga(T39, T40, T41, T7, rewrite13_in_gga(T40, T41, T48))
U4_ga(T39, T40, T41, T7, rewrite13_out_gga(T40, T41, T48)) → U5_ga(T39, T40, T41, T7, normal1_in_ga(op(T39, T48), T7))
normal1_in_ga(T90, T90) → normal1_out_ga(T90, T90)
U5_ga(T39, T40, T41, T7, normal1_out_ga(op(T39, T48), T7)) → normal1_out_ga(op(T39, op(T40, T41)), T7)
U2_ga(T20, T21, T22, T7, normal1_out_ga(op(T20, op(T21, T22)), T7)) → normal1_out_ga(op(op(T20, T21), T22), T7)

The argument filtering Pi contains the following mapping:
normal1_in_ga(x1, x2)  =  normal1_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
rewrite13_in_gga(x1, x2, x3)  =  rewrite13_in_gga(x1, x2)
rewrite13_out_gga(x1, x2, x3)  =  rewrite13_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
normal1_out_ga(x1, x2)  =  normal1_out_ga
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
NORMAL1_IN_GA(x1, x2)  =  NORMAL1_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
REWRITE13_IN_GGA(x1, x2, x3)  =  REWRITE13_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Complex Obligation (AND)

(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)

The TRS R 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(T39, op(T40, T41)), T7) → U3_ga(T39, T40, T41, T7, rewrite13_in_gga(T40, T41, X51))
rewrite13_in_gga(op(T67, T68), T69, op(T67, op(T68, T69))) → rewrite13_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewrite13_in_gga(T76, op(T77, T78), op(T76, X100)) → U1_gga(T76, T77, T78, X100, rewrite13_in_gga(T77, T78, X100))
U1_gga(T76, T77, T78, X100, rewrite13_out_gga(T77, T78, X100)) → rewrite13_out_gga(T76, op(T77, T78), op(T76, X100))
U3_ga(T39, T40, T41, T7, rewrite13_out_gga(T40, T41, X51)) → normal1_out_ga(op(T39, op(T40, T41)), T7)
normal1_in_ga(op(T39, op(T40, T41)), T7) → U4_ga(T39, T40, T41, T7, rewrite13_in_gga(T40, T41, T48))
U4_ga(T39, T40, T41, T7, rewrite13_out_gga(T40, T41, T48)) → U5_ga(T39, T40, T41, T7, normal1_in_ga(op(T39, T48), T7))
normal1_in_ga(T90, T90) → normal1_out_ga(T90, T90)
U5_ga(T39, T40, T41, T7, normal1_out_ga(op(T39, T48), T7)) → normal1_out_ga(op(T39, op(T40, T41)), T7)
U2_ga(T20, T21, T22, T7, normal1_out_ga(op(T20, op(T21, T22)), T7)) → normal1_out_ga(op(op(T20, T21), T22), T7)

The argument filtering Pi contains the following mapping:
normal1_in_ga(x1, x2)  =  normal1_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
rewrite13_in_gga(x1, x2, x3)  =  rewrite13_in_gga(x1, x2)
rewrite13_out_gga(x1, x2, x3)  =  rewrite13_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
normal1_out_ga(x1, x2)  =  normal1_out_ga
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
REWRITE13_IN_GGA(x1, x2, x3)  =  REWRITE13_IN_GGA(x1, x2)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) 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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) 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.

(14) 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

(15) YES

(16) 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, rewrite13_in_gga(T40, T41, T48))
U4_GA(T39, T40, T41, T7, rewrite13_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:

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(T39, op(T40, T41)), T7) → U3_ga(T39, T40, T41, T7, rewrite13_in_gga(T40, T41, X51))
rewrite13_in_gga(op(T67, T68), T69, op(T67, op(T68, T69))) → rewrite13_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewrite13_in_gga(T76, op(T77, T78), op(T76, X100)) → U1_gga(T76, T77, T78, X100, rewrite13_in_gga(T77, T78, X100))
U1_gga(T76, T77, T78, X100, rewrite13_out_gga(T77, T78, X100)) → rewrite13_out_gga(T76, op(T77, T78), op(T76, X100))
U3_ga(T39, T40, T41, T7, rewrite13_out_gga(T40, T41, X51)) → normal1_out_ga(op(T39, op(T40, T41)), T7)
normal1_in_ga(op(T39, op(T40, T41)), T7) → U4_ga(T39, T40, T41, T7, rewrite13_in_gga(T40, T41, T48))
U4_ga(T39, T40, T41, T7, rewrite13_out_gga(T40, T41, T48)) → U5_ga(T39, T40, T41, T7, normal1_in_ga(op(T39, T48), T7))
normal1_in_ga(T90, T90) → normal1_out_ga(T90, T90)
U5_ga(T39, T40, T41, T7, normal1_out_ga(op(T39, T48), T7)) → normal1_out_ga(op(T39, op(T40, T41)), T7)
U2_ga(T20, T21, T22, T7, normal1_out_ga(op(T20, op(T21, T22)), T7)) → normal1_out_ga(op(op(T20, T21), T22), T7)

The argument filtering Pi contains the following mapping:
normal1_in_ga(x1, x2)  =  normal1_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
rewrite13_in_gga(x1, x2, x3)  =  rewrite13_in_gga(x1, x2)
rewrite13_out_gga(x1, x2, x3)  =  rewrite13_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
normal1_out_ga(x1, x2)  =  normal1_out_ga
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
NORMAL1_IN_GA(x1, x2)  =  NORMAL1_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x5)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) 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, rewrite13_in_gga(T40, T41, T48))
U4_GA(T39, T40, T41, T7, rewrite13_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:

rewrite13_in_gga(op(T67, T68), T69, op(T67, op(T68, T69))) → rewrite13_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewrite13_in_gga(T76, op(T77, T78), op(T76, X100)) → U1_gga(T76, T77, T78, X100, rewrite13_in_gga(T77, T78, X100))
U1_gga(T76, T77, T78, X100, rewrite13_out_gga(T77, T78, X100)) → rewrite13_out_gga(T76, op(T77, T78), op(T76, X100))

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)
rewrite13_out_gga(x1, x2, x3)  =  rewrite13_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
NORMAL1_IN_GA(x1, x2)  =  NORMAL1_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x5)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

NORMAL1_IN_GA(op(T39, op(T40, T41))) → U4_GA(T39, rewrite13_in_gga(T40, T41))
U4_GA(T39, rewrite13_out_gga(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:

rewrite13_in_gga(op(T67, T68), T69) → rewrite13_out_gga(op(T67, op(T68, T69)))
rewrite13_in_gga(T76, op(T77, T78)) → U1_gga(T76, rewrite13_in_gga(T77, T78))
U1_gga(T76, rewrite13_out_gga(X100)) → rewrite13_out_gga(op(T76, X100))

The set Q consists of the following terms:

rewrite13_in_gga(x0, x1)
U1_gga(x0, x1)

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

(21) 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:

NORMAL1_IN_GA(op(T39, op(T40, T41))) → U4_GA(T39, rewrite13_in_gga(T40, T41))
NORMAL1_IN_GA(op(op(T20, T21), T22)) → NORMAL1_IN_GA(op(T20, op(T21, T22)))


Used ordering: Polynomial interpretation [POLO]:

POL(NORMAL1_IN_GA(x1)) = x1   
POL(U1_gga(x1, x2)) = 2 + 2·x1 + x2   
POL(U4_GA(x1, x2)) = 2 + 2·x1 + x2   
POL(op(x1, x2)) = 2 + 2·x1 + x2   
POL(rewrite13_in_gga(x1, x2)) = 2·x1 + x2   
POL(rewrite13_out_gga(x1)) = x1   

(22) Obligation:

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

U4_GA(T39, rewrite13_out_gga(T48)) → NORMAL1_IN_GA(op(T39, T48))

The TRS R consists of the following rules:

rewrite13_in_gga(op(T67, T68), T69) → rewrite13_out_gga(op(T67, op(T68, T69)))
rewrite13_in_gga(T76, op(T77, T78)) → U1_gga(T76, rewrite13_in_gga(T77, T78))
U1_gga(T76, rewrite13_out_gga(X100)) → rewrite13_out_gga(op(T76, X100))

The set Q consists of the following terms:

rewrite13_in_gga(x0, x1)
U1_gga(x0, x1)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

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

(24) TRUE