(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).

Query: permute(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

(2) Obligation:

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

permuteA_in_ga([], []) → permuteA_out_ga([], [])
permuteA_in_ga(.(T21, T22), .(T21, T23)) → U1_ga(T21, T22, T23, permuteA_in_ga(T22, T23))
permuteA_in_ga(.(T37, T38), .(T39, T40)) → U2_ga(T37, T38, T39, T40, pB_in_agaga(T39, T38, X42, T37, T40))
pB_in_agaga(T39, T38, T45, T37, T46) → U4_agaga(T39, T38, T45, T37, T46, deleteC_in_aga(T39, T38, T45))
deleteC_in_aga(T59, .(T59, T60), T60) → deleteC_out_aga(T59, .(T59, T60), T60)
deleteC_in_aga(T70, .(T68, T69), .(T68, X75)) → U3_aga(T70, T68, T69, X75, deleteC_in_aga(T70, T69, X75))
U3_aga(T70, T68, T69, X75, deleteC_out_aga(T70, T69, X75)) → deleteC_out_aga(T70, .(T68, T69), .(T68, X75))
U4_agaga(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → U5_agaga(T39, T38, T45, T37, T46, permuteA_in_ga(.(T37, T45), T46))
U5_agaga(T39, T38, T45, T37, T46, permuteA_out_ga(.(T37, T45), T46)) → pB_out_agaga(T39, T38, T45, T37, T46)
U2_ga(T37, T38, T39, T40, pB_out_agaga(T39, T38, X42, T37, T40)) → permuteA_out_ga(.(T37, T38), .(T39, T40))
U1_ga(T21, T22, T23, permuteA_out_ga(T22, T23)) → permuteA_out_ga(.(T21, T22), .(T21, T23))

The argument filtering Pi contains the following mapping:
permuteA_in_ga(x1, x2)  =  permuteA_in_ga(x1)
[]  =  []
permuteA_out_ga(x1, x2)  =  permuteA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x5)
pB_in_agaga(x1, x2, x3, x4, x5)  =  pB_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6)  =  U4_agaga(x2, x4, x6)
deleteC_in_aga(x1, x2, x3)  =  deleteC_in_aga(x2)
deleteC_out_aga(x1, x2, x3)  =  deleteC_out_aga(x1, x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x3, x5)
U5_agaga(x1, x2, x3, x4, x5, x6)  =  U5_agaga(x1, x2, x3, x4, x6)
pB_out_agaga(x1, x2, x3, x4, x5)  =  pB_out_agaga(x1, x2, x3, x4, x5)

(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:

PERMUTEA_IN_GA(.(T21, T22), .(T21, T23)) → U1_GA(T21, T22, T23, permuteA_in_ga(T22, T23))
PERMUTEA_IN_GA(.(T21, T22), .(T21, T23)) → PERMUTEA_IN_GA(T22, T23)
PERMUTEA_IN_GA(.(T37, T38), .(T39, T40)) → U2_GA(T37, T38, T39, T40, pB_in_agaga(T39, T38, X42, T37, T40))
PERMUTEA_IN_GA(.(T37, T38), .(T39, T40)) → PB_IN_AGAGA(T39, T38, X42, T37, T40)
PB_IN_AGAGA(T39, T38, T45, T37, T46) → U4_AGAGA(T39, T38, T45, T37, T46, deleteC_in_aga(T39, T38, T45))
PB_IN_AGAGA(T39, T38, T45, T37, T46) → DELETEC_IN_AGA(T39, T38, T45)
DELETEC_IN_AGA(T70, .(T68, T69), .(T68, X75)) → U3_AGA(T70, T68, T69, X75, deleteC_in_aga(T70, T69, X75))
DELETEC_IN_AGA(T70, .(T68, T69), .(T68, X75)) → DELETEC_IN_AGA(T70, T69, X75)
U4_AGAGA(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → U5_AGAGA(T39, T38, T45, T37, T46, permuteA_in_ga(.(T37, T45), T46))
U4_AGAGA(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → PERMUTEA_IN_GA(.(T37, T45), T46)

The TRS R consists of the following rules:

permuteA_in_ga([], []) → permuteA_out_ga([], [])
permuteA_in_ga(.(T21, T22), .(T21, T23)) → U1_ga(T21, T22, T23, permuteA_in_ga(T22, T23))
permuteA_in_ga(.(T37, T38), .(T39, T40)) → U2_ga(T37, T38, T39, T40, pB_in_agaga(T39, T38, X42, T37, T40))
pB_in_agaga(T39, T38, T45, T37, T46) → U4_agaga(T39, T38, T45, T37, T46, deleteC_in_aga(T39, T38, T45))
deleteC_in_aga(T59, .(T59, T60), T60) → deleteC_out_aga(T59, .(T59, T60), T60)
deleteC_in_aga(T70, .(T68, T69), .(T68, X75)) → U3_aga(T70, T68, T69, X75, deleteC_in_aga(T70, T69, X75))
U3_aga(T70, T68, T69, X75, deleteC_out_aga(T70, T69, X75)) → deleteC_out_aga(T70, .(T68, T69), .(T68, X75))
U4_agaga(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → U5_agaga(T39, T38, T45, T37, T46, permuteA_in_ga(.(T37, T45), T46))
U5_agaga(T39, T38, T45, T37, T46, permuteA_out_ga(.(T37, T45), T46)) → pB_out_agaga(T39, T38, T45, T37, T46)
U2_ga(T37, T38, T39, T40, pB_out_agaga(T39, T38, X42, T37, T40)) → permuteA_out_ga(.(T37, T38), .(T39, T40))
U1_ga(T21, T22, T23, permuteA_out_ga(T22, T23)) → permuteA_out_ga(.(T21, T22), .(T21, T23))

The argument filtering Pi contains the following mapping:
permuteA_in_ga(x1, x2)  =  permuteA_in_ga(x1)
[]  =  []
permuteA_out_ga(x1, x2)  =  permuteA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x5)
pB_in_agaga(x1, x2, x3, x4, x5)  =  pB_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6)  =  U4_agaga(x2, x4, x6)
deleteC_in_aga(x1, x2, x3)  =  deleteC_in_aga(x2)
deleteC_out_aga(x1, x2, x3)  =  deleteC_out_aga(x1, x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x3, x5)
U5_agaga(x1, x2, x3, x4, x5, x6)  =  U5_agaga(x1, x2, x3, x4, x6)
pB_out_agaga(x1, x2, x3, x4, x5)  =  pB_out_agaga(x1, x2, x3, x4, x5)
PERMUTEA_IN_GA(x1, x2)  =  PERMUTEA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x5)
PB_IN_AGAGA(x1, x2, x3, x4, x5)  =  PB_IN_AGAGA(x2, x4)
U4_AGAGA(x1, x2, x3, x4, x5, x6)  =  U4_AGAGA(x2, x4, x6)
DELETEC_IN_AGA(x1, x2, x3)  =  DELETEC_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x2, x3, x5)
U5_AGAGA(x1, x2, x3, x4, x5, x6)  =  U5_AGAGA(x1, x2, x3, x4, x6)

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

(4) Obligation:

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

PERMUTEA_IN_GA(.(T21, T22), .(T21, T23)) → U1_GA(T21, T22, T23, permuteA_in_ga(T22, T23))
PERMUTEA_IN_GA(.(T21, T22), .(T21, T23)) → PERMUTEA_IN_GA(T22, T23)
PERMUTEA_IN_GA(.(T37, T38), .(T39, T40)) → U2_GA(T37, T38, T39, T40, pB_in_agaga(T39, T38, X42, T37, T40))
PERMUTEA_IN_GA(.(T37, T38), .(T39, T40)) → PB_IN_AGAGA(T39, T38, X42, T37, T40)
PB_IN_AGAGA(T39, T38, T45, T37, T46) → U4_AGAGA(T39, T38, T45, T37, T46, deleteC_in_aga(T39, T38, T45))
PB_IN_AGAGA(T39, T38, T45, T37, T46) → DELETEC_IN_AGA(T39, T38, T45)
DELETEC_IN_AGA(T70, .(T68, T69), .(T68, X75)) → U3_AGA(T70, T68, T69, X75, deleteC_in_aga(T70, T69, X75))
DELETEC_IN_AGA(T70, .(T68, T69), .(T68, X75)) → DELETEC_IN_AGA(T70, T69, X75)
U4_AGAGA(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → U5_AGAGA(T39, T38, T45, T37, T46, permuteA_in_ga(.(T37, T45), T46))
U4_AGAGA(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → PERMUTEA_IN_GA(.(T37, T45), T46)

The TRS R consists of the following rules:

permuteA_in_ga([], []) → permuteA_out_ga([], [])
permuteA_in_ga(.(T21, T22), .(T21, T23)) → U1_ga(T21, T22, T23, permuteA_in_ga(T22, T23))
permuteA_in_ga(.(T37, T38), .(T39, T40)) → U2_ga(T37, T38, T39, T40, pB_in_agaga(T39, T38, X42, T37, T40))
pB_in_agaga(T39, T38, T45, T37, T46) → U4_agaga(T39, T38, T45, T37, T46, deleteC_in_aga(T39, T38, T45))
deleteC_in_aga(T59, .(T59, T60), T60) → deleteC_out_aga(T59, .(T59, T60), T60)
deleteC_in_aga(T70, .(T68, T69), .(T68, X75)) → U3_aga(T70, T68, T69, X75, deleteC_in_aga(T70, T69, X75))
U3_aga(T70, T68, T69, X75, deleteC_out_aga(T70, T69, X75)) → deleteC_out_aga(T70, .(T68, T69), .(T68, X75))
U4_agaga(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → U5_agaga(T39, T38, T45, T37, T46, permuteA_in_ga(.(T37, T45), T46))
U5_agaga(T39, T38, T45, T37, T46, permuteA_out_ga(.(T37, T45), T46)) → pB_out_agaga(T39, T38, T45, T37, T46)
U2_ga(T37, T38, T39, T40, pB_out_agaga(T39, T38, X42, T37, T40)) → permuteA_out_ga(.(T37, T38), .(T39, T40))
U1_ga(T21, T22, T23, permuteA_out_ga(T22, T23)) → permuteA_out_ga(.(T21, T22), .(T21, T23))

The argument filtering Pi contains the following mapping:
permuteA_in_ga(x1, x2)  =  permuteA_in_ga(x1)
[]  =  []
permuteA_out_ga(x1, x2)  =  permuteA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x5)
pB_in_agaga(x1, x2, x3, x4, x5)  =  pB_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6)  =  U4_agaga(x2, x4, x6)
deleteC_in_aga(x1, x2, x3)  =  deleteC_in_aga(x2)
deleteC_out_aga(x1, x2, x3)  =  deleteC_out_aga(x1, x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x3, x5)
U5_agaga(x1, x2, x3, x4, x5, x6)  =  U5_agaga(x1, x2, x3, x4, x6)
pB_out_agaga(x1, x2, x3, x4, x5)  =  pB_out_agaga(x1, x2, x3, x4, x5)
PERMUTEA_IN_GA(x1, x2)  =  PERMUTEA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x5)
PB_IN_AGAGA(x1, x2, x3, x4, x5)  =  PB_IN_AGAGA(x2, x4)
U4_AGAGA(x1, x2, x3, x4, x5, x6)  =  U4_AGAGA(x2, x4, x6)
DELETEC_IN_AGA(x1, x2, x3)  =  DELETEC_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x2, x3, x5)
U5_AGAGA(x1, x2, x3, x4, x5, x6)  =  U5_AGAGA(x1, x2, x3, x4, x6)

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:

DELETEC_IN_AGA(T70, .(T68, T69), .(T68, X75)) → DELETEC_IN_AGA(T70, T69, X75)

The TRS R consists of the following rules:

permuteA_in_ga([], []) → permuteA_out_ga([], [])
permuteA_in_ga(.(T21, T22), .(T21, T23)) → U1_ga(T21, T22, T23, permuteA_in_ga(T22, T23))
permuteA_in_ga(.(T37, T38), .(T39, T40)) → U2_ga(T37, T38, T39, T40, pB_in_agaga(T39, T38, X42, T37, T40))
pB_in_agaga(T39, T38, T45, T37, T46) → U4_agaga(T39, T38, T45, T37, T46, deleteC_in_aga(T39, T38, T45))
deleteC_in_aga(T59, .(T59, T60), T60) → deleteC_out_aga(T59, .(T59, T60), T60)
deleteC_in_aga(T70, .(T68, T69), .(T68, X75)) → U3_aga(T70, T68, T69, X75, deleteC_in_aga(T70, T69, X75))
U3_aga(T70, T68, T69, X75, deleteC_out_aga(T70, T69, X75)) → deleteC_out_aga(T70, .(T68, T69), .(T68, X75))
U4_agaga(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → U5_agaga(T39, T38, T45, T37, T46, permuteA_in_ga(.(T37, T45), T46))
U5_agaga(T39, T38, T45, T37, T46, permuteA_out_ga(.(T37, T45), T46)) → pB_out_agaga(T39, T38, T45, T37, T46)
U2_ga(T37, T38, T39, T40, pB_out_agaga(T39, T38, X42, T37, T40)) → permuteA_out_ga(.(T37, T38), .(T39, T40))
U1_ga(T21, T22, T23, permuteA_out_ga(T22, T23)) → permuteA_out_ga(.(T21, T22), .(T21, T23))

The argument filtering Pi contains the following mapping:
permuteA_in_ga(x1, x2)  =  permuteA_in_ga(x1)
[]  =  []
permuteA_out_ga(x1, x2)  =  permuteA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x5)
pB_in_agaga(x1, x2, x3, x4, x5)  =  pB_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6)  =  U4_agaga(x2, x4, x6)
deleteC_in_aga(x1, x2, x3)  =  deleteC_in_aga(x2)
deleteC_out_aga(x1, x2, x3)  =  deleteC_out_aga(x1, x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x3, x5)
U5_agaga(x1, x2, x3, x4, x5, x6)  =  U5_agaga(x1, x2, x3, x4, x6)
pB_out_agaga(x1, x2, x3, x4, x5)  =  pB_out_agaga(x1, x2, x3, x4, x5)
DELETEC_IN_AGA(x1, x2, x3)  =  DELETEC_IN_AGA(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:

DELETEC_IN_AGA(T70, .(T68, T69), .(T68, X75)) → DELETEC_IN_AGA(T70, T69, X75)

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

DELETEC_IN_AGA(.(T68, T69)) → DELETEC_IN_AGA(T69)

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:

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

(13) YES

(14) Obligation:

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

PERMUTEA_IN_GA(.(T37, T38), .(T39, T40)) → PB_IN_AGAGA(T39, T38, X42, T37, T40)
PB_IN_AGAGA(T39, T38, T45, T37, T46) → U4_AGAGA(T39, T38, T45, T37, T46, deleteC_in_aga(T39, T38, T45))
U4_AGAGA(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → PERMUTEA_IN_GA(.(T37, T45), T46)
PERMUTEA_IN_GA(.(T21, T22), .(T21, T23)) → PERMUTEA_IN_GA(T22, T23)

The TRS R consists of the following rules:

permuteA_in_ga([], []) → permuteA_out_ga([], [])
permuteA_in_ga(.(T21, T22), .(T21, T23)) → U1_ga(T21, T22, T23, permuteA_in_ga(T22, T23))
permuteA_in_ga(.(T37, T38), .(T39, T40)) → U2_ga(T37, T38, T39, T40, pB_in_agaga(T39, T38, X42, T37, T40))
pB_in_agaga(T39, T38, T45, T37, T46) → U4_agaga(T39, T38, T45, T37, T46, deleteC_in_aga(T39, T38, T45))
deleteC_in_aga(T59, .(T59, T60), T60) → deleteC_out_aga(T59, .(T59, T60), T60)
deleteC_in_aga(T70, .(T68, T69), .(T68, X75)) → U3_aga(T70, T68, T69, X75, deleteC_in_aga(T70, T69, X75))
U3_aga(T70, T68, T69, X75, deleteC_out_aga(T70, T69, X75)) → deleteC_out_aga(T70, .(T68, T69), .(T68, X75))
U4_agaga(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → U5_agaga(T39, T38, T45, T37, T46, permuteA_in_ga(.(T37, T45), T46))
U5_agaga(T39, T38, T45, T37, T46, permuteA_out_ga(.(T37, T45), T46)) → pB_out_agaga(T39, T38, T45, T37, T46)
U2_ga(T37, T38, T39, T40, pB_out_agaga(T39, T38, X42, T37, T40)) → permuteA_out_ga(.(T37, T38), .(T39, T40))
U1_ga(T21, T22, T23, permuteA_out_ga(T22, T23)) → permuteA_out_ga(.(T21, T22), .(T21, T23))

The argument filtering Pi contains the following mapping:
permuteA_in_ga(x1, x2)  =  permuteA_in_ga(x1)
[]  =  []
permuteA_out_ga(x1, x2)  =  permuteA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x5)
pB_in_agaga(x1, x2, x3, x4, x5)  =  pB_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6)  =  U4_agaga(x2, x4, x6)
deleteC_in_aga(x1, x2, x3)  =  deleteC_in_aga(x2)
deleteC_out_aga(x1, x2, x3)  =  deleteC_out_aga(x1, x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x3, x5)
U5_agaga(x1, x2, x3, x4, x5, x6)  =  U5_agaga(x1, x2, x3, x4, x6)
pB_out_agaga(x1, x2, x3, x4, x5)  =  pB_out_agaga(x1, x2, x3, x4, x5)
PERMUTEA_IN_GA(x1, x2)  =  PERMUTEA_IN_GA(x1)
PB_IN_AGAGA(x1, x2, x3, x4, x5)  =  PB_IN_AGAGA(x2, x4)
U4_AGAGA(x1, x2, x3, x4, x5, x6)  =  U4_AGAGA(x2, x4, x6)

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:

PERMUTEA_IN_GA(.(T37, T38), .(T39, T40)) → PB_IN_AGAGA(T39, T38, X42, T37, T40)
PB_IN_AGAGA(T39, T38, T45, T37, T46) → U4_AGAGA(T39, T38, T45, T37, T46, deleteC_in_aga(T39, T38, T45))
U4_AGAGA(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → PERMUTEA_IN_GA(.(T37, T45), T46)
PERMUTEA_IN_GA(.(T21, T22), .(T21, T23)) → PERMUTEA_IN_GA(T22, T23)

The TRS R consists of the following rules:

deleteC_in_aga(T59, .(T59, T60), T60) → deleteC_out_aga(T59, .(T59, T60), T60)
deleteC_in_aga(T70, .(T68, T69), .(T68, X75)) → U3_aga(T70, T68, T69, X75, deleteC_in_aga(T70, T69, X75))
U3_aga(T70, T68, T69, X75, deleteC_out_aga(T70, T69, X75)) → deleteC_out_aga(T70, .(T68, T69), .(T68, X75))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
deleteC_in_aga(x1, x2, x3)  =  deleteC_in_aga(x2)
deleteC_out_aga(x1, x2, x3)  =  deleteC_out_aga(x1, x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x3, x5)
PERMUTEA_IN_GA(x1, x2)  =  PERMUTEA_IN_GA(x1)
PB_IN_AGAGA(x1, x2, x3, x4, x5)  =  PB_IN_AGAGA(x2, x4)
U4_AGAGA(x1, x2, x3, x4, x5, x6)  =  U4_AGAGA(x2, x4, x6)

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:

PERMUTEA_IN_GA(.(T37, T38)) → PB_IN_AGAGA(T38, T37)
PB_IN_AGAGA(T38, T37) → U4_AGAGA(T38, T37, deleteC_in_aga(T38))
U4_AGAGA(T38, T37, deleteC_out_aga(T39, T38, T45)) → PERMUTEA_IN_GA(.(T37, T45))
PERMUTEA_IN_GA(.(T21, T22)) → PERMUTEA_IN_GA(T22)

The TRS R consists of the following rules:

deleteC_in_aga(.(T59, T60)) → deleteC_out_aga(T59, .(T59, T60), T60)
deleteC_in_aga(.(T68, T69)) → U3_aga(T68, T69, deleteC_in_aga(T69))
U3_aga(T68, T69, deleteC_out_aga(T70, T69, X75)) → deleteC_out_aga(T70, .(T68, T69), .(T68, X75))

The set Q consists of the following terms:

deleteC_in_aga(x0)
U3_aga(x0, x1, x2)

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


PB_IN_AGAGA(T38, T37) → U4_AGAGA(T38, T37, deleteC_in_aga(T38))
PERMUTEA_IN_GA(.(T21, T22)) → PERMUTEA_IN_GA(T22)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x2   
POL(PB_IN_AGAGA(x1, x2)) = 1 + x1   
POL(PERMUTEA_IN_GA(x1)) = x1   
POL(U3_aga(x1, x2, x3)) = 1 + x3   
POL(U4_AGAGA(x1, x2, x3)) = x3   
POL(deleteC_in_aga(x1)) = x1   
POL(deleteC_out_aga(x1, x2, x3)) = 1 + x3   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

deleteC_in_aga(.(T59, T60)) → deleteC_out_aga(T59, .(T59, T60), T60)
deleteC_in_aga(.(T68, T69)) → U3_aga(T68, T69, deleteC_in_aga(T69))
U3_aga(T68, T69, deleteC_out_aga(T70, T69, X75)) → deleteC_out_aga(T70, .(T68, T69), .(T68, X75))

(20) Obligation:

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

PERMUTEA_IN_GA(.(T37, T38)) → PB_IN_AGAGA(T38, T37)
U4_AGAGA(T38, T37, deleteC_out_aga(T39, T38, T45)) → PERMUTEA_IN_GA(.(T37, T45))

The TRS R consists of the following rules:

deleteC_in_aga(.(T59, T60)) → deleteC_out_aga(T59, .(T59, T60), T60)
deleteC_in_aga(.(T68, T69)) → U3_aga(T68, T69, deleteC_in_aga(T69))
U3_aga(T68, T69, deleteC_out_aga(T70, T69, X75)) → deleteC_out_aga(T70, .(T68, T69), .(T68, X75))

The set Q consists of the following terms:

deleteC_in_aga(x0)
U3_aga(x0, x1, x2)

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 2 less nodes.

(22) TRUE