Left Termination of the query pattern permute_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 PisEmptyProof (⇔)
        ↳24 YES

(0) Obligation:

Clauses:

permute([], []).
permute(.(X, Y), .(U, V)) :- ','(delete(U, .(X, Y), W), permute(W, V)).
delete(X, .(X, Y), Y).
delete(U, .(X, Y), .(X, Z)) :- delete(U, Y, Z).

Queries:

permute(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

delete15(T59, .(T59, T60), T60).
delete15(T70, .(T68, T69), .(T68, X83)) :- delete15(T70, T69, X83).
permute1([], []).
permute1(.(T21, T22), .(T21, T23)) :- permute1(T22, T23).
permute1(.(T37, T38), .(T39, T40)) :- delete15(T39, T38, X48).
permute1(.(T37, T38), .(T39, T46)) :- ','(delete15(T39, T38, T45), permute1(.(T37, T45), T46)).

Queries:

permute1(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:
permute1_in: (b,f)
delete15_in: (f,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

permute1_in_ga([], []) → permute1_out_ga([], [])
permute1_in_ga(.(T21, T22), .(T21, T23)) → U2_ga(T21, T22, T23, permute1_in_ga(T22, T23))
permute1_in_ga(.(T37, T38), .(T39, T40)) → U3_ga(T37, T38, T39, T40, delete15_in_aga(T39, T38, X48))
delete15_in_aga(T59, .(T59, T60), T60) → delete15_out_aga(T59, .(T59, T60), T60)
delete15_in_aga(T70, .(T68, T69), .(T68, X83)) → U1_aga(T70, T68, T69, X83, delete15_in_aga(T70, T69, X83))
U1_aga(T70, T68, T69, X83, delete15_out_aga(T70, T69, X83)) → delete15_out_aga(T70, .(T68, T69), .(T68, X83))
U3_ga(T37, T38, T39, T40, delete15_out_aga(T39, T38, X48)) → permute1_out_ga(.(T37, T38), .(T39, T40))
permute1_in_ga(.(T37, T38), .(T39, T46)) → U4_ga(T37, T38, T39, T46, delete15_in_aga(T39, T38, T45))
U4_ga(T37, T38, T39, T46, delete15_out_aga(T39, T38, T45)) → U5_ga(T37, T38, T39, T46, permute1_in_ga(.(T37, T45), T46))
U5_ga(T37, T38, T39, T46, permute1_out_ga(.(T37, T45), T46)) → permute1_out_ga(.(T37, T38), .(T39, T46))
U2_ga(T21, T22, T23, permute1_out_ga(T22, T23)) → permute1_out_ga(.(T21, T22), .(T21, T23))

The argument filtering Pi contains the following mapping:
permute1_in_ga(x1, x2)  =  permute1_in_ga(x1)
[]  =  []
permute1_out_ga(x1, x2)  =  permute1_out_ga
.(x1, x2)  =  .(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
delete15_in_aga(x1, x2, x3)  =  delete15_in_aga(x2)
delete15_out_aga(x1, x2, x3)  =  delete15_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x2, x5)
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:

permute1_in_ga([], []) → permute1_out_ga([], [])
permute1_in_ga(.(T21, T22), .(T21, T23)) → U2_ga(T21, T22, T23, permute1_in_ga(T22, T23))
permute1_in_ga(.(T37, T38), .(T39, T40)) → U3_ga(T37, T38, T39, T40, delete15_in_aga(T39, T38, X48))
delete15_in_aga(T59, .(T59, T60), T60) → delete15_out_aga(T59, .(T59, T60), T60)
delete15_in_aga(T70, .(T68, T69), .(T68, X83)) → U1_aga(T70, T68, T69, X83, delete15_in_aga(T70, T69, X83))
U1_aga(T70, T68, T69, X83, delete15_out_aga(T70, T69, X83)) → delete15_out_aga(T70, .(T68, T69), .(T68, X83))
U3_ga(T37, T38, T39, T40, delete15_out_aga(T39, T38, X48)) → permute1_out_ga(.(T37, T38), .(T39, T40))
permute1_in_ga(.(T37, T38), .(T39, T46)) → U4_ga(T37, T38, T39, T46, delete15_in_aga(T39, T38, T45))
U4_ga(T37, T38, T39, T46, delete15_out_aga(T39, T38, T45)) → U5_ga(T37, T38, T39, T46, permute1_in_ga(.(T37, T45), T46))
U5_ga(T37, T38, T39, T46, permute1_out_ga(.(T37, T45), T46)) → permute1_out_ga(.(T37, T38), .(T39, T46))
U2_ga(T21, T22, T23, permute1_out_ga(T22, T23)) → permute1_out_ga(.(T21, T22), .(T21, T23))

The argument filtering Pi contains the following mapping:
permute1_in_ga(x1, x2)  =  permute1_in_ga(x1)
[]  =  []
permute1_out_ga(x1, x2)  =  permute1_out_ga
.(x1, x2)  =  .(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
delete15_in_aga(x1, x2, x3)  =  delete15_in_aga(x2)
delete15_out_aga(x1, x2, x3)  =  delete15_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x2, x5)
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:

PERMUTE1_IN_GA(.(T21, T22), .(T21, T23)) → U2_GA(T21, T22, T23, permute1_in_ga(T22, T23))
PERMUTE1_IN_GA(.(T21, T22), .(T21, T23)) → PERMUTE1_IN_GA(T22, T23)
PERMUTE1_IN_GA(.(T37, T38), .(T39, T40)) → U3_GA(T37, T38, T39, T40, delete15_in_aga(T39, T38, X48))
PERMUTE1_IN_GA(.(T37, T38), .(T39, T40)) → DELETE15_IN_AGA(T39, T38, X48)
DELETE15_IN_AGA(T70, .(T68, T69), .(T68, X83)) → U1_AGA(T70, T68, T69, X83, delete15_in_aga(T70, T69, X83))
DELETE15_IN_AGA(T70, .(T68, T69), .(T68, X83)) → DELETE15_IN_AGA(T70, T69, X83)
PERMUTE1_IN_GA(.(T37, T38), .(T39, T46)) → U4_GA(T37, T38, T39, T46, delete15_in_aga(T39, T38, T45))
U4_GA(T37, T38, T39, T46, delete15_out_aga(T39, T38, T45)) → U5_GA(T37, T38, T39, T46, permute1_in_ga(.(T37, T45), T46))
U4_GA(T37, T38, T39, T46, delete15_out_aga(T39, T38, T45)) → PERMUTE1_IN_GA(.(T37, T45), T46)

The TRS R consists of the following rules:

permute1_in_ga([], []) → permute1_out_ga([], [])
permute1_in_ga(.(T21, T22), .(T21, T23)) → U2_ga(T21, T22, T23, permute1_in_ga(T22, T23))
permute1_in_ga(.(T37, T38), .(T39, T40)) → U3_ga(T37, T38, T39, T40, delete15_in_aga(T39, T38, X48))
delete15_in_aga(T59, .(T59, T60), T60) → delete15_out_aga(T59, .(T59, T60), T60)
delete15_in_aga(T70, .(T68, T69), .(T68, X83)) → U1_aga(T70, T68, T69, X83, delete15_in_aga(T70, T69, X83))
U1_aga(T70, T68, T69, X83, delete15_out_aga(T70, T69, X83)) → delete15_out_aga(T70, .(T68, T69), .(T68, X83))
U3_ga(T37, T38, T39, T40, delete15_out_aga(T39, T38, X48)) → permute1_out_ga(.(T37, T38), .(T39, T40))
permute1_in_ga(.(T37, T38), .(T39, T46)) → U4_ga(T37, T38, T39, T46, delete15_in_aga(T39, T38, T45))
U4_ga(T37, T38, T39, T46, delete15_out_aga(T39, T38, T45)) → U5_ga(T37, T38, T39, T46, permute1_in_ga(.(T37, T45), T46))
U5_ga(T37, T38, T39, T46, permute1_out_ga(.(T37, T45), T46)) → permute1_out_ga(.(T37, T38), .(T39, T46))
U2_ga(T21, T22, T23, permute1_out_ga(T22, T23)) → permute1_out_ga(.(T21, T22), .(T21, T23))

The argument filtering Pi contains the following mapping:
permute1_in_ga(x1, x2)  =  permute1_in_ga(x1)
[]  =  []
permute1_out_ga(x1, x2)  =  permute1_out_ga
.(x1, x2)  =  .(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
delete15_in_aga(x1, x2, x3)  =  delete15_in_aga(x2)
delete15_out_aga(x1, x2, x3)  =  delete15_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x2, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
PERMUTE1_IN_GA(x1, x2)  =  PERMUTE1_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
DELETE15_IN_AGA(x1, x2, x3)  =  DELETE15_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x2, 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:

PERMUTE1_IN_GA(.(T21, T22), .(T21, T23)) → U2_GA(T21, T22, T23, permute1_in_ga(T22, T23))
PERMUTE1_IN_GA(.(T21, T22), .(T21, T23)) → PERMUTE1_IN_GA(T22, T23)
PERMUTE1_IN_GA(.(T37, T38), .(T39, T40)) → U3_GA(T37, T38, T39, T40, delete15_in_aga(T39, T38, X48))
PERMUTE1_IN_GA(.(T37, T38), .(T39, T40)) → DELETE15_IN_AGA(T39, T38, X48)
DELETE15_IN_AGA(T70, .(T68, T69), .(T68, X83)) → U1_AGA(T70, T68, T69, X83, delete15_in_aga(T70, T69, X83))
DELETE15_IN_AGA(T70, .(T68, T69), .(T68, X83)) → DELETE15_IN_AGA(T70, T69, X83)
PERMUTE1_IN_GA(.(T37, T38), .(T39, T46)) → U4_GA(T37, T38, T39, T46, delete15_in_aga(T39, T38, T45))
U4_GA(T37, T38, T39, T46, delete15_out_aga(T39, T38, T45)) → U5_GA(T37, T38, T39, T46, permute1_in_ga(.(T37, T45), T46))
U4_GA(T37, T38, T39, T46, delete15_out_aga(T39, T38, T45)) → PERMUTE1_IN_GA(.(T37, T45), T46)

The TRS R consists of the following rules:

permute1_in_ga([], []) → permute1_out_ga([], [])
permute1_in_ga(.(T21, T22), .(T21, T23)) → U2_ga(T21, T22, T23, permute1_in_ga(T22, T23))
permute1_in_ga(.(T37, T38), .(T39, T40)) → U3_ga(T37, T38, T39, T40, delete15_in_aga(T39, T38, X48))
delete15_in_aga(T59, .(T59, T60), T60) → delete15_out_aga(T59, .(T59, T60), T60)
delete15_in_aga(T70, .(T68, T69), .(T68, X83)) → U1_aga(T70, T68, T69, X83, delete15_in_aga(T70, T69, X83))
U1_aga(T70, T68, T69, X83, delete15_out_aga(T70, T69, X83)) → delete15_out_aga(T70, .(T68, T69), .(T68, X83))
U3_ga(T37, T38, T39, T40, delete15_out_aga(T39, T38, X48)) → permute1_out_ga(.(T37, T38), .(T39, T40))
permute1_in_ga(.(T37, T38), .(T39, T46)) → U4_ga(T37, T38, T39, T46, delete15_in_aga(T39, T38, T45))
U4_ga(T37, T38, T39, T46, delete15_out_aga(T39, T38, T45)) → U5_ga(T37, T38, T39, T46, permute1_in_ga(.(T37, T45), T46))
U5_ga(T37, T38, T39, T46, permute1_out_ga(.(T37, T45), T46)) → permute1_out_ga(.(T37, T38), .(T39, T46))
U2_ga(T21, T22, T23, permute1_out_ga(T22, T23)) → permute1_out_ga(.(T21, T22), .(T21, T23))

The argument filtering Pi contains the following mapping:
permute1_in_ga(x1, x2)  =  permute1_in_ga(x1)
[]  =  []
permute1_out_ga(x1, x2)  =  permute1_out_ga
.(x1, x2)  =  .(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
delete15_in_aga(x1, x2, x3)  =  delete15_in_aga(x2)
delete15_out_aga(x1, x2, x3)  =  delete15_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x2, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
PERMUTE1_IN_GA(x1, x2)  =  PERMUTE1_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
DELETE15_IN_AGA(x1, x2, x3)  =  DELETE15_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x2, 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:

DELETE15_IN_AGA(T70, .(T68, T69), .(T68, X83)) → DELETE15_IN_AGA(T70, T69, X83)

The TRS R consists of the following rules:

permute1_in_ga([], []) → permute1_out_ga([], [])
permute1_in_ga(.(T21, T22), .(T21, T23)) → U2_ga(T21, T22, T23, permute1_in_ga(T22, T23))
permute1_in_ga(.(T37, T38), .(T39, T40)) → U3_ga(T37, T38, T39, T40, delete15_in_aga(T39, T38, X48))
delete15_in_aga(T59, .(T59, T60), T60) → delete15_out_aga(T59, .(T59, T60), T60)
delete15_in_aga(T70, .(T68, T69), .(T68, X83)) → U1_aga(T70, T68, T69, X83, delete15_in_aga(T70, T69, X83))
U1_aga(T70, T68, T69, X83, delete15_out_aga(T70, T69, X83)) → delete15_out_aga(T70, .(T68, T69), .(T68, X83))
U3_ga(T37, T38, T39, T40, delete15_out_aga(T39, T38, X48)) → permute1_out_ga(.(T37, T38), .(T39, T40))
permute1_in_ga(.(T37, T38), .(T39, T46)) → U4_ga(T37, T38, T39, T46, delete15_in_aga(T39, T38, T45))
U4_ga(T37, T38, T39, T46, delete15_out_aga(T39, T38, T45)) → U5_ga(T37, T38, T39, T46, permute1_in_ga(.(T37, T45), T46))
U5_ga(T37, T38, T39, T46, permute1_out_ga(.(T37, T45), T46)) → permute1_out_ga(.(T37, T38), .(T39, T46))
U2_ga(T21, T22, T23, permute1_out_ga(T22, T23)) → permute1_out_ga(.(T21, T22), .(T21, T23))

The argument filtering Pi contains the following mapping:
permute1_in_ga(x1, x2)  =  permute1_in_ga(x1)
[]  =  []
permute1_out_ga(x1, x2)  =  permute1_out_ga
.(x1, x2)  =  .(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
delete15_in_aga(x1, x2, x3)  =  delete15_in_aga(x2)
delete15_out_aga(x1, x2, x3)  =  delete15_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x2, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
DELETE15_IN_AGA(x1, x2, x3)  =  DELETE15_IN_AGA(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:

DELETE15_IN_AGA(T70, .(T68, T69), .(T68, X83)) → DELETE15_IN_AGA(T70, T69, X83)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
DELETE15_IN_AGA(x1, x2, x3)  =  DELETE15_IN_AGA(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:

DELETE15_IN_AGA(.(T68, T69)) → DELETE15_IN_AGA(T69)

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:

  • DELETE15_IN_AGA(.(T68, T69)) → DELETE15_IN_AGA(T69)
    The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

PERMUTE1_IN_GA(.(T37, T38), .(T39, T46)) → U4_GA(T37, T38, T39, T46, delete15_in_aga(T39, T38, T45))
U4_GA(T37, T38, T39, T46, delete15_out_aga(T39, T38, T45)) → PERMUTE1_IN_GA(.(T37, T45), T46)
PERMUTE1_IN_GA(.(T21, T22), .(T21, T23)) → PERMUTE1_IN_GA(T22, T23)

The TRS R consists of the following rules:

permute1_in_ga([], []) → permute1_out_ga([], [])
permute1_in_ga(.(T21, T22), .(T21, T23)) → U2_ga(T21, T22, T23, permute1_in_ga(T22, T23))
permute1_in_ga(.(T37, T38), .(T39, T40)) → U3_ga(T37, T38, T39, T40, delete15_in_aga(T39, T38, X48))
delete15_in_aga(T59, .(T59, T60), T60) → delete15_out_aga(T59, .(T59, T60), T60)
delete15_in_aga(T70, .(T68, T69), .(T68, X83)) → U1_aga(T70, T68, T69, X83, delete15_in_aga(T70, T69, X83))
U1_aga(T70, T68, T69, X83, delete15_out_aga(T70, T69, X83)) → delete15_out_aga(T70, .(T68, T69), .(T68, X83))
U3_ga(T37, T38, T39, T40, delete15_out_aga(T39, T38, X48)) → permute1_out_ga(.(T37, T38), .(T39, T40))
permute1_in_ga(.(T37, T38), .(T39, T46)) → U4_ga(T37, T38, T39, T46, delete15_in_aga(T39, T38, T45))
U4_ga(T37, T38, T39, T46, delete15_out_aga(T39, T38, T45)) → U5_ga(T37, T38, T39, T46, permute1_in_ga(.(T37, T45), T46))
U5_ga(T37, T38, T39, T46, permute1_out_ga(.(T37, T45), T46)) → permute1_out_ga(.(T37, T38), .(T39, T46))
U2_ga(T21, T22, T23, permute1_out_ga(T22, T23)) → permute1_out_ga(.(T21, T22), .(T21, T23))

The argument filtering Pi contains the following mapping:
permute1_in_ga(x1, x2)  =  permute1_in_ga(x1)
[]  =  []
permute1_out_ga(x1, x2)  =  permute1_out_ga
.(x1, x2)  =  .(x1, x2)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
delete15_in_aga(x1, x2, x3)  =  delete15_in_aga(x2)
delete15_out_aga(x1, x2, x3)  =  delete15_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x2, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
PERMUTE1_IN_GA(x1, x2)  =  PERMUTE1_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:

PERMUTE1_IN_GA(.(T37, T38), .(T39, T46)) → U4_GA(T37, T38, T39, T46, delete15_in_aga(T39, T38, T45))
U4_GA(T37, T38, T39, T46, delete15_out_aga(T39, T38, T45)) → PERMUTE1_IN_GA(.(T37, T45), T46)
PERMUTE1_IN_GA(.(T21, T22), .(T21, T23)) → PERMUTE1_IN_GA(T22, T23)

The TRS R consists of the following rules:

delete15_in_aga(T59, .(T59, T60), T60) → delete15_out_aga(T59, .(T59, T60), T60)
delete15_in_aga(T70, .(T68, T69), .(T68, X83)) → U1_aga(T70, T68, T69, X83, delete15_in_aga(T70, T69, X83))
U1_aga(T70, T68, T69, X83, delete15_out_aga(T70, T69, X83)) → delete15_out_aga(T70, .(T68, T69), .(T68, X83))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
delete15_in_aga(x1, x2, x3)  =  delete15_in_aga(x2)
delete15_out_aga(x1, x2, x3)  =  delete15_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x2, x5)
PERMUTE1_IN_GA(x1, x2)  =  PERMUTE1_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:

PERMUTE1_IN_GA(.(T37, T38)) → U4_GA(T37, delete15_in_aga(T38))
U4_GA(T37, delete15_out_aga(T39, T45)) → PERMUTE1_IN_GA(.(T37, T45))
PERMUTE1_IN_GA(.(T21, T22)) → PERMUTE1_IN_GA(T22)

The TRS R consists of the following rules:

delete15_in_aga(.(T59, T60)) → delete15_out_aga(T59, T60)
delete15_in_aga(.(T68, T69)) → U1_aga(T68, delete15_in_aga(T69))
U1_aga(T68, delete15_out_aga(T70, X83)) → delete15_out_aga(T70, .(T68, X83))

The set Q consists of the following terms:

delete15_in_aga(x0)
U1_aga(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:

PERMUTE1_IN_GA(.(T37, T38)) → U4_GA(T37, delete15_in_aga(T38))
U4_GA(T37, delete15_out_aga(T39, T45)) → PERMUTE1_IN_GA(.(T37, T45))
PERMUTE1_IN_GA(.(T21, T22)) → PERMUTE1_IN_GA(T22)

Strictly oriented rules of the TRS R:

delete15_in_aga(.(T59, T60)) → delete15_out_aga(T59, T60)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 3 + x1 + x2   
POL(PERMUTE1_IN_GA(x1)) = x1   
POL(U1_aga(x1, x2)) = 3 + x1 + x2   
POL(U4_GA(x1, x2)) = x1 + x2   
POL(delete15_in_aga(x1)) = 2 + x1   
POL(delete15_out_aga(x1, x2)) = 4 + x1 + x2   

(22) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

delete15_in_aga(.(T68, T69)) → U1_aga(T68, delete15_in_aga(T69))
U1_aga(T68, delete15_out_aga(T70, X83)) → delete15_out_aga(T70, .(T68, X83))

The set Q consists of the following terms:

delete15_in_aga(x0)
U1_aga(x0, x1)

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

(23) PisEmptyProof (EQUIVALENT transformation)

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

(24) YES