(0) Obligation:

Clauses:

conf(X) :- ','(del2(X, Z), ','(del(U, Y, Z), conf(Y))).
del2(X, Y) :- ','(del(U, X, Z), del(V, Z, Y)).
del(X, .(X, T), T).
del(X, .(H, T), .(H, T1)) :- del(X, T, T1).
s2l(s(X), .(Y, Xs)) :- s2l(X, Xs).
s2l(0, []).
goal(X) :- ','(s2l(X, XS), conf(XS)).

Query: conf(g)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

(2) Obligation:

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

confA_in_g(T11) → U1_g(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
pB_in_agaaaaa(T16, T11, T17, X25, X26, X4, X5) → U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
delC_in_aga(T30, .(T30, T31), T31) → delC_out_aga(T30, .(T30, T31), T31)
delC_in_aga(X70, .(T36, T37), .(T36, X71)) → U2_aga(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
U2_aga(X70, T36, T37, X71, delC_out_aga(X70, T37, X71)) → delC_out_aga(X70, .(T36, T37), .(T36, X71))
U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
pK_in_agaaa(T46, T17, T47, X4, X5) → U10_agaaa(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
U10_agaaa(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_agaaa(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
pL_in_aag(T53, T54, T47) → U12_aag(T53, T54, T47, delD_in_aag(T53, T54, T47))
delD_in_aag(X121, .(X121, T60), T60) → delD_out_aag(X121, .(X121, T60), T60)
delD_in_aag(X141, .(T65, X143), .(T65, T66)) → U3_aag(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
U3_aag(X141, T65, X143, T66, delD_out_aag(X141, X143, T66)) → delD_out_aag(X141, .(T65, X143), .(T65, T66))
U12_aag(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_aag(T53, T54, T47, confF_in_g(T54))
confF_in_g(T74) → U5_g(T74, pG_in_gaaa(T74, X164, X165, X166))
pG_in_gaaa(T74, T76, X165, X166) → U14_gaaa(T74, T76, X165, X166, del2I_in_ga(T74, T76))
del2I_in_ga(T86, X199) → U7_ga(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
pJ_in_agaaa(T91, T86, T92, X198, X199) → U18_agaaa(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
delE_in_aga(T109, .(T109, T110), T110) → delE_out_aga(T109, .(T109, T110), T110)
delE_in_aga(X249, .(T115, T117), .(T115, X250)) → U4_aga(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
U4_aga(X249, T115, T117, X250, delE_out_aga(X249, T117, X250)) → delE_out_aga(X249, .(T115, T117), .(T115, X250))
U18_agaaa(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_agaaa(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U19_agaaa(T91, T86, T92, X198, X199, delE_out_aga(X198, T92, X199)) → pJ_out_agaaa(T91, T86, T92, X198, X199)
U7_ga(T86, X199, pJ_out_agaaa(X196, T86, X197, X198, X199)) → del2I_out_ga(T86, X199)
U14_gaaa(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_gaaa(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
pM_in_aag(T123, T124, T76) → U16_aag(T123, T124, T76, delH_in_aag(T123, T124, T76))
delH_in_aag(X288, .(X288, T130), T130) → delH_out_aag(X288, .(X288, T130), T130)
delH_in_aag(X308, .(T135, X310), .(T135, T137)) → U6_aag(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
U6_aag(X308, T135, X310, T137, delH_out_aag(X308, X310, T137)) → delH_out_aag(X308, .(T135, X310), .(T135, T137))
U16_aag(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_aag(T123, T124, T76, confF_in_g(T124))
U17_aag(T123, T124, T76, confF_out_g(T124)) → pM_out_aag(T123, T124, T76)
U15_gaaa(T74, T76, X165, X166, pM_out_aag(X165, X166, T76)) → pG_out_gaaa(T74, T76, X165, X166)
U5_g(T74, pG_out_gaaa(T74, X164, X165, X166)) → confF_out_g(T74)
U13_aag(T53, T54, T47, confF_out_g(T54)) → pL_out_aag(T53, T54, T47)
U11_agaaa(T46, T17, T47, X4, X5, pL_out_aag(X4, X5, T47)) → pK_out_agaaa(T46, T17, T47, X4, X5)
U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_out_agaaa(X25, T17, X26, X4, X5)) → pB_out_agaaaaa(T16, T11, T17, X25, X26, X4, X5)
U1_g(T11, pB_out_agaaaaa(X23, T11, X24, X25, X26, X4, X5)) → confA_out_g(T11)

The argument filtering Pi contains the following mapping:
confA_in_g(x1)  =  confA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
pB_in_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_in_agaaaaa(x2)
U8_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_agaaaaa(x2, x8)
delC_in_aga(x1, x2, x3)  =  delC_in_aga(x2)
.(x1, x2)  =  .(x2)
delC_out_aga(x1, x2, x3)  =  delC_out_aga(x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U9_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_agaaaaa(x2, x3, x8)
pK_in_agaaa(x1, x2, x3, x4, x5)  =  pK_in_agaaa(x2)
U10_agaaa(x1, x2, x3, x4, x5, x6)  =  U10_agaaa(x2, x6)
U11_agaaa(x1, x2, x3, x4, x5, x6)  =  U11_agaaa(x2, x3, x6)
pL_in_aag(x1, x2, x3)  =  pL_in_aag(x3)
U12_aag(x1, x2, x3, x4)  =  U12_aag(x3, x4)
delD_in_aag(x1, x2, x3)  =  delD_in_aag(x3)
delD_out_aag(x1, x2, x3)  =  delD_out_aag(x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x4, x5)
U13_aag(x1, x2, x3, x4)  =  U13_aag(x2, x3, x4)
confF_in_g(x1)  =  confF_in_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
pG_in_gaaa(x1, x2, x3, x4)  =  pG_in_gaaa(x1)
U14_gaaa(x1, x2, x3, x4, x5)  =  U14_gaaa(x1, x5)
del2I_in_ga(x1, x2)  =  del2I_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
pJ_in_agaaa(x1, x2, x3, x4, x5)  =  pJ_in_agaaa(x2)
U18_agaaa(x1, x2, x3, x4, x5, x6)  =  U18_agaaa(x2, x6)
delE_in_aga(x1, x2, x3)  =  delE_in_aga(x2)
delE_out_aga(x1, x2, x3)  =  delE_out_aga(x2, x3)
U4_aga(x1, x2, x3, x4, x5)  =  U4_aga(x3, x5)
U19_agaaa(x1, x2, x3, x4, x5, x6)  =  U19_agaaa(x2, x3, x6)
pJ_out_agaaa(x1, x2, x3, x4, x5)  =  pJ_out_agaaa(x2, x3, x5)
del2I_out_ga(x1, x2)  =  del2I_out_ga(x1, x2)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x2, x5)
pM_in_aag(x1, x2, x3)  =  pM_in_aag(x3)
U16_aag(x1, x2, x3, x4)  =  U16_aag(x3, x4)
delH_in_aag(x1, x2, x3)  =  delH_in_aag(x3)
delH_out_aag(x1, x2, x3)  =  delH_out_aag(x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x4, x5)
U17_aag(x1, x2, x3, x4)  =  U17_aag(x2, x3, x4)
confF_out_g(x1)  =  confF_out_g(x1)
pM_out_aag(x1, x2, x3)  =  pM_out_aag(x2, x3)
pG_out_gaaa(x1, x2, x3, x4)  =  pG_out_gaaa(x1, x2, x4)
pL_out_aag(x1, x2, x3)  =  pL_out_aag(x2, x3)
pK_out_agaaa(x1, x2, x3, x4, x5)  =  pK_out_agaaa(x2, x3, x5)
pB_out_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_out_agaaaaa(x2, x3, x5, x7)
confA_out_g(x1)  =  confA_out_g(x1)

(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:

CONFA_IN_G(T11) → U1_G(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
CONFA_IN_G(T11) → PB_IN_AGAAAAA(X23, T11, X24, X25, X26, X4, X5)
PB_IN_AGAAAAA(T16, T11, T17, X25, X26, X4, X5) → U8_AGAAAAA(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
PB_IN_AGAAAAA(T16, T11, T17, X25, X26, X4, X5) → DELC_IN_AGA(T16, T11, T17)
DELC_IN_AGA(X70, .(T36, T37), .(T36, X71)) → U2_AGA(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
DELC_IN_AGA(X70, .(T36, T37), .(T36, X71)) → DELC_IN_AGA(X70, T37, X71)
U8_AGAAAAA(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_AGAAAAA(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
U8_AGAAAAA(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → PK_IN_AGAAA(X25, T17, X26, X4, X5)
PK_IN_AGAAA(T46, T17, T47, X4, X5) → U10_AGAAA(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
PK_IN_AGAAA(T46, T17, T47, X4, X5) → DELC_IN_AGA(T46, T17, T47)
U10_AGAAA(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_AGAAA(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
U10_AGAAA(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → PL_IN_AAG(X4, X5, T47)
PL_IN_AAG(T53, T54, T47) → U12_AAG(T53, T54, T47, delD_in_aag(T53, T54, T47))
PL_IN_AAG(T53, T54, T47) → DELD_IN_AAG(T53, T54, T47)
DELD_IN_AAG(X141, .(T65, X143), .(T65, T66)) → U3_AAG(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
DELD_IN_AAG(X141, .(T65, X143), .(T65, T66)) → DELD_IN_AAG(X141, X143, T66)
U12_AAG(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_AAG(T53, T54, T47, confF_in_g(T54))
U12_AAG(T53, T54, T47, delD_out_aag(T53, T54, T47)) → CONFF_IN_G(T54)
CONFF_IN_G(T74) → U5_G(T74, pG_in_gaaa(T74, X164, X165, X166))
CONFF_IN_G(T74) → PG_IN_GAAA(T74, X164, X165, X166)
PG_IN_GAAA(T74, T76, X165, X166) → U14_GAAA(T74, T76, X165, X166, del2I_in_ga(T74, T76))
PG_IN_GAAA(T74, T76, X165, X166) → DEL2I_IN_GA(T74, T76)
DEL2I_IN_GA(T86, X199) → U7_GA(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
DEL2I_IN_GA(T86, X199) → PJ_IN_AGAAA(X196, T86, X197, X198, X199)
PJ_IN_AGAAA(T91, T86, T92, X198, X199) → U18_AGAAA(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
PJ_IN_AGAAA(T91, T86, T92, X198, X199) → DELE_IN_AGA(T91, T86, T92)
DELE_IN_AGA(X249, .(T115, T117), .(T115, X250)) → U4_AGA(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
DELE_IN_AGA(X249, .(T115, T117), .(T115, X250)) → DELE_IN_AGA(X249, T117, X250)
U18_AGAAA(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_AGAAA(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U18_AGAAA(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → DELE_IN_AGA(X198, T92, X199)
U14_GAAA(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_GAAA(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
U14_GAAA(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → PM_IN_AAG(X165, X166, T76)
PM_IN_AAG(T123, T124, T76) → U16_AAG(T123, T124, T76, delH_in_aag(T123, T124, T76))
PM_IN_AAG(T123, T124, T76) → DELH_IN_AAG(T123, T124, T76)
DELH_IN_AAG(X308, .(T135, X310), .(T135, T137)) → U6_AAG(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
DELH_IN_AAG(X308, .(T135, X310), .(T135, T137)) → DELH_IN_AAG(X308, X310, T137)
U16_AAG(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_AAG(T123, T124, T76, confF_in_g(T124))
U16_AAG(T123, T124, T76, delH_out_aag(T123, T124, T76)) → CONFF_IN_G(T124)

The TRS R consists of the following rules:

confA_in_g(T11) → U1_g(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
pB_in_agaaaaa(T16, T11, T17, X25, X26, X4, X5) → U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
delC_in_aga(T30, .(T30, T31), T31) → delC_out_aga(T30, .(T30, T31), T31)
delC_in_aga(X70, .(T36, T37), .(T36, X71)) → U2_aga(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
U2_aga(X70, T36, T37, X71, delC_out_aga(X70, T37, X71)) → delC_out_aga(X70, .(T36, T37), .(T36, X71))
U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
pK_in_agaaa(T46, T17, T47, X4, X5) → U10_agaaa(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
U10_agaaa(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_agaaa(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
pL_in_aag(T53, T54, T47) → U12_aag(T53, T54, T47, delD_in_aag(T53, T54, T47))
delD_in_aag(X121, .(X121, T60), T60) → delD_out_aag(X121, .(X121, T60), T60)
delD_in_aag(X141, .(T65, X143), .(T65, T66)) → U3_aag(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
U3_aag(X141, T65, X143, T66, delD_out_aag(X141, X143, T66)) → delD_out_aag(X141, .(T65, X143), .(T65, T66))
U12_aag(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_aag(T53, T54, T47, confF_in_g(T54))
confF_in_g(T74) → U5_g(T74, pG_in_gaaa(T74, X164, X165, X166))
pG_in_gaaa(T74, T76, X165, X166) → U14_gaaa(T74, T76, X165, X166, del2I_in_ga(T74, T76))
del2I_in_ga(T86, X199) → U7_ga(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
pJ_in_agaaa(T91, T86, T92, X198, X199) → U18_agaaa(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
delE_in_aga(T109, .(T109, T110), T110) → delE_out_aga(T109, .(T109, T110), T110)
delE_in_aga(X249, .(T115, T117), .(T115, X250)) → U4_aga(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
U4_aga(X249, T115, T117, X250, delE_out_aga(X249, T117, X250)) → delE_out_aga(X249, .(T115, T117), .(T115, X250))
U18_agaaa(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_agaaa(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U19_agaaa(T91, T86, T92, X198, X199, delE_out_aga(X198, T92, X199)) → pJ_out_agaaa(T91, T86, T92, X198, X199)
U7_ga(T86, X199, pJ_out_agaaa(X196, T86, X197, X198, X199)) → del2I_out_ga(T86, X199)
U14_gaaa(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_gaaa(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
pM_in_aag(T123, T124, T76) → U16_aag(T123, T124, T76, delH_in_aag(T123, T124, T76))
delH_in_aag(X288, .(X288, T130), T130) → delH_out_aag(X288, .(X288, T130), T130)
delH_in_aag(X308, .(T135, X310), .(T135, T137)) → U6_aag(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
U6_aag(X308, T135, X310, T137, delH_out_aag(X308, X310, T137)) → delH_out_aag(X308, .(T135, X310), .(T135, T137))
U16_aag(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_aag(T123, T124, T76, confF_in_g(T124))
U17_aag(T123, T124, T76, confF_out_g(T124)) → pM_out_aag(T123, T124, T76)
U15_gaaa(T74, T76, X165, X166, pM_out_aag(X165, X166, T76)) → pG_out_gaaa(T74, T76, X165, X166)
U5_g(T74, pG_out_gaaa(T74, X164, X165, X166)) → confF_out_g(T74)
U13_aag(T53, T54, T47, confF_out_g(T54)) → pL_out_aag(T53, T54, T47)
U11_agaaa(T46, T17, T47, X4, X5, pL_out_aag(X4, X5, T47)) → pK_out_agaaa(T46, T17, T47, X4, X5)
U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_out_agaaa(X25, T17, X26, X4, X5)) → pB_out_agaaaaa(T16, T11, T17, X25, X26, X4, X5)
U1_g(T11, pB_out_agaaaaa(X23, T11, X24, X25, X26, X4, X5)) → confA_out_g(T11)

The argument filtering Pi contains the following mapping:
confA_in_g(x1)  =  confA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
pB_in_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_in_agaaaaa(x2)
U8_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_agaaaaa(x2, x8)
delC_in_aga(x1, x2, x3)  =  delC_in_aga(x2)
.(x1, x2)  =  .(x2)
delC_out_aga(x1, x2, x3)  =  delC_out_aga(x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U9_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_agaaaaa(x2, x3, x8)
pK_in_agaaa(x1, x2, x3, x4, x5)  =  pK_in_agaaa(x2)
U10_agaaa(x1, x2, x3, x4, x5, x6)  =  U10_agaaa(x2, x6)
U11_agaaa(x1, x2, x3, x4, x5, x6)  =  U11_agaaa(x2, x3, x6)
pL_in_aag(x1, x2, x3)  =  pL_in_aag(x3)
U12_aag(x1, x2, x3, x4)  =  U12_aag(x3, x4)
delD_in_aag(x1, x2, x3)  =  delD_in_aag(x3)
delD_out_aag(x1, x2, x3)  =  delD_out_aag(x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x4, x5)
U13_aag(x1, x2, x3, x4)  =  U13_aag(x2, x3, x4)
confF_in_g(x1)  =  confF_in_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
pG_in_gaaa(x1, x2, x3, x4)  =  pG_in_gaaa(x1)
U14_gaaa(x1, x2, x3, x4, x5)  =  U14_gaaa(x1, x5)
del2I_in_ga(x1, x2)  =  del2I_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
pJ_in_agaaa(x1, x2, x3, x4, x5)  =  pJ_in_agaaa(x2)
U18_agaaa(x1, x2, x3, x4, x5, x6)  =  U18_agaaa(x2, x6)
delE_in_aga(x1, x2, x3)  =  delE_in_aga(x2)
delE_out_aga(x1, x2, x3)  =  delE_out_aga(x2, x3)
U4_aga(x1, x2, x3, x4, x5)  =  U4_aga(x3, x5)
U19_agaaa(x1, x2, x3, x4, x5, x6)  =  U19_agaaa(x2, x3, x6)
pJ_out_agaaa(x1, x2, x3, x4, x5)  =  pJ_out_agaaa(x2, x3, x5)
del2I_out_ga(x1, x2)  =  del2I_out_ga(x1, x2)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x2, x5)
pM_in_aag(x1, x2, x3)  =  pM_in_aag(x3)
U16_aag(x1, x2, x3, x4)  =  U16_aag(x3, x4)
delH_in_aag(x1, x2, x3)  =  delH_in_aag(x3)
delH_out_aag(x1, x2, x3)  =  delH_out_aag(x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x4, x5)
U17_aag(x1, x2, x3, x4)  =  U17_aag(x2, x3, x4)
confF_out_g(x1)  =  confF_out_g(x1)
pM_out_aag(x1, x2, x3)  =  pM_out_aag(x2, x3)
pG_out_gaaa(x1, x2, x3, x4)  =  pG_out_gaaa(x1, x2, x4)
pL_out_aag(x1, x2, x3)  =  pL_out_aag(x2, x3)
pK_out_agaaa(x1, x2, x3, x4, x5)  =  pK_out_agaaa(x2, x3, x5)
pB_out_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_out_agaaaaa(x2, x3, x5, x7)
confA_out_g(x1)  =  confA_out_g(x1)
CONFA_IN_G(x1)  =  CONFA_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
PB_IN_AGAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  PB_IN_AGAAAAA(x2)
U8_AGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_AGAAAAA(x2, x8)
DELC_IN_AGA(x1, x2, x3)  =  DELC_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U9_AGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_AGAAAAA(x2, x3, x8)
PK_IN_AGAAA(x1, x2, x3, x4, x5)  =  PK_IN_AGAAA(x2)
U10_AGAAA(x1, x2, x3, x4, x5, x6)  =  U10_AGAAA(x2, x6)
U11_AGAAA(x1, x2, x3, x4, x5, x6)  =  U11_AGAAA(x2, x3, x6)
PL_IN_AAG(x1, x2, x3)  =  PL_IN_AAG(x3)
U12_AAG(x1, x2, x3, x4)  =  U12_AAG(x3, x4)
DELD_IN_AAG(x1, x2, x3)  =  DELD_IN_AAG(x3)
U3_AAG(x1, x2, x3, x4, x5)  =  U3_AAG(x4, x5)
U13_AAG(x1, x2, x3, x4)  =  U13_AAG(x2, x3, x4)
CONFF_IN_G(x1)  =  CONFF_IN_G(x1)
U5_G(x1, x2)  =  U5_G(x1, x2)
PG_IN_GAAA(x1, x2, x3, x4)  =  PG_IN_GAAA(x1)
U14_GAAA(x1, x2, x3, x4, x5)  =  U14_GAAA(x1, x5)
DEL2I_IN_GA(x1, x2)  =  DEL2I_IN_GA(x1)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
PJ_IN_AGAAA(x1, x2, x3, x4, x5)  =  PJ_IN_AGAAA(x2)
U18_AGAAA(x1, x2, x3, x4, x5, x6)  =  U18_AGAAA(x2, x6)
DELE_IN_AGA(x1, x2, x3)  =  DELE_IN_AGA(x2)
U4_AGA(x1, x2, x3, x4, x5)  =  U4_AGA(x3, x5)
U19_AGAAA(x1, x2, x3, x4, x5, x6)  =  U19_AGAAA(x2, x3, x6)
U15_GAAA(x1, x2, x3, x4, x5)  =  U15_GAAA(x1, x2, x5)
PM_IN_AAG(x1, x2, x3)  =  PM_IN_AAG(x3)
U16_AAG(x1, x2, x3, x4)  =  U16_AAG(x3, x4)
DELH_IN_AAG(x1, x2, x3)  =  DELH_IN_AAG(x3)
U6_AAG(x1, x2, x3, x4, x5)  =  U6_AAG(x4, x5)
U17_AAG(x1, x2, x3, x4)  =  U17_AAG(x2, x3, x4)

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

(4) Obligation:

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

CONFA_IN_G(T11) → U1_G(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
CONFA_IN_G(T11) → PB_IN_AGAAAAA(X23, T11, X24, X25, X26, X4, X5)
PB_IN_AGAAAAA(T16, T11, T17, X25, X26, X4, X5) → U8_AGAAAAA(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
PB_IN_AGAAAAA(T16, T11, T17, X25, X26, X4, X5) → DELC_IN_AGA(T16, T11, T17)
DELC_IN_AGA(X70, .(T36, T37), .(T36, X71)) → U2_AGA(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
DELC_IN_AGA(X70, .(T36, T37), .(T36, X71)) → DELC_IN_AGA(X70, T37, X71)
U8_AGAAAAA(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_AGAAAAA(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
U8_AGAAAAA(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → PK_IN_AGAAA(X25, T17, X26, X4, X5)
PK_IN_AGAAA(T46, T17, T47, X4, X5) → U10_AGAAA(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
PK_IN_AGAAA(T46, T17, T47, X4, X5) → DELC_IN_AGA(T46, T17, T47)
U10_AGAAA(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_AGAAA(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
U10_AGAAA(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → PL_IN_AAG(X4, X5, T47)
PL_IN_AAG(T53, T54, T47) → U12_AAG(T53, T54, T47, delD_in_aag(T53, T54, T47))
PL_IN_AAG(T53, T54, T47) → DELD_IN_AAG(T53, T54, T47)
DELD_IN_AAG(X141, .(T65, X143), .(T65, T66)) → U3_AAG(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
DELD_IN_AAG(X141, .(T65, X143), .(T65, T66)) → DELD_IN_AAG(X141, X143, T66)
U12_AAG(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_AAG(T53, T54, T47, confF_in_g(T54))
U12_AAG(T53, T54, T47, delD_out_aag(T53, T54, T47)) → CONFF_IN_G(T54)
CONFF_IN_G(T74) → U5_G(T74, pG_in_gaaa(T74, X164, X165, X166))
CONFF_IN_G(T74) → PG_IN_GAAA(T74, X164, X165, X166)
PG_IN_GAAA(T74, T76, X165, X166) → U14_GAAA(T74, T76, X165, X166, del2I_in_ga(T74, T76))
PG_IN_GAAA(T74, T76, X165, X166) → DEL2I_IN_GA(T74, T76)
DEL2I_IN_GA(T86, X199) → U7_GA(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
DEL2I_IN_GA(T86, X199) → PJ_IN_AGAAA(X196, T86, X197, X198, X199)
PJ_IN_AGAAA(T91, T86, T92, X198, X199) → U18_AGAAA(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
PJ_IN_AGAAA(T91, T86, T92, X198, X199) → DELE_IN_AGA(T91, T86, T92)
DELE_IN_AGA(X249, .(T115, T117), .(T115, X250)) → U4_AGA(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
DELE_IN_AGA(X249, .(T115, T117), .(T115, X250)) → DELE_IN_AGA(X249, T117, X250)
U18_AGAAA(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_AGAAA(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U18_AGAAA(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → DELE_IN_AGA(X198, T92, X199)
U14_GAAA(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_GAAA(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
U14_GAAA(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → PM_IN_AAG(X165, X166, T76)
PM_IN_AAG(T123, T124, T76) → U16_AAG(T123, T124, T76, delH_in_aag(T123, T124, T76))
PM_IN_AAG(T123, T124, T76) → DELH_IN_AAG(T123, T124, T76)
DELH_IN_AAG(X308, .(T135, X310), .(T135, T137)) → U6_AAG(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
DELH_IN_AAG(X308, .(T135, X310), .(T135, T137)) → DELH_IN_AAG(X308, X310, T137)
U16_AAG(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_AAG(T123, T124, T76, confF_in_g(T124))
U16_AAG(T123, T124, T76, delH_out_aag(T123, T124, T76)) → CONFF_IN_G(T124)

The TRS R consists of the following rules:

confA_in_g(T11) → U1_g(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
pB_in_agaaaaa(T16, T11, T17, X25, X26, X4, X5) → U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
delC_in_aga(T30, .(T30, T31), T31) → delC_out_aga(T30, .(T30, T31), T31)
delC_in_aga(X70, .(T36, T37), .(T36, X71)) → U2_aga(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
U2_aga(X70, T36, T37, X71, delC_out_aga(X70, T37, X71)) → delC_out_aga(X70, .(T36, T37), .(T36, X71))
U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
pK_in_agaaa(T46, T17, T47, X4, X5) → U10_agaaa(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
U10_agaaa(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_agaaa(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
pL_in_aag(T53, T54, T47) → U12_aag(T53, T54, T47, delD_in_aag(T53, T54, T47))
delD_in_aag(X121, .(X121, T60), T60) → delD_out_aag(X121, .(X121, T60), T60)
delD_in_aag(X141, .(T65, X143), .(T65, T66)) → U3_aag(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
U3_aag(X141, T65, X143, T66, delD_out_aag(X141, X143, T66)) → delD_out_aag(X141, .(T65, X143), .(T65, T66))
U12_aag(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_aag(T53, T54, T47, confF_in_g(T54))
confF_in_g(T74) → U5_g(T74, pG_in_gaaa(T74, X164, X165, X166))
pG_in_gaaa(T74, T76, X165, X166) → U14_gaaa(T74, T76, X165, X166, del2I_in_ga(T74, T76))
del2I_in_ga(T86, X199) → U7_ga(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
pJ_in_agaaa(T91, T86, T92, X198, X199) → U18_agaaa(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
delE_in_aga(T109, .(T109, T110), T110) → delE_out_aga(T109, .(T109, T110), T110)
delE_in_aga(X249, .(T115, T117), .(T115, X250)) → U4_aga(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
U4_aga(X249, T115, T117, X250, delE_out_aga(X249, T117, X250)) → delE_out_aga(X249, .(T115, T117), .(T115, X250))
U18_agaaa(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_agaaa(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U19_agaaa(T91, T86, T92, X198, X199, delE_out_aga(X198, T92, X199)) → pJ_out_agaaa(T91, T86, T92, X198, X199)
U7_ga(T86, X199, pJ_out_agaaa(X196, T86, X197, X198, X199)) → del2I_out_ga(T86, X199)
U14_gaaa(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_gaaa(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
pM_in_aag(T123, T124, T76) → U16_aag(T123, T124, T76, delH_in_aag(T123, T124, T76))
delH_in_aag(X288, .(X288, T130), T130) → delH_out_aag(X288, .(X288, T130), T130)
delH_in_aag(X308, .(T135, X310), .(T135, T137)) → U6_aag(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
U6_aag(X308, T135, X310, T137, delH_out_aag(X308, X310, T137)) → delH_out_aag(X308, .(T135, X310), .(T135, T137))
U16_aag(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_aag(T123, T124, T76, confF_in_g(T124))
U17_aag(T123, T124, T76, confF_out_g(T124)) → pM_out_aag(T123, T124, T76)
U15_gaaa(T74, T76, X165, X166, pM_out_aag(X165, X166, T76)) → pG_out_gaaa(T74, T76, X165, X166)
U5_g(T74, pG_out_gaaa(T74, X164, X165, X166)) → confF_out_g(T74)
U13_aag(T53, T54, T47, confF_out_g(T54)) → pL_out_aag(T53, T54, T47)
U11_agaaa(T46, T17, T47, X4, X5, pL_out_aag(X4, X5, T47)) → pK_out_agaaa(T46, T17, T47, X4, X5)
U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_out_agaaa(X25, T17, X26, X4, X5)) → pB_out_agaaaaa(T16, T11, T17, X25, X26, X4, X5)
U1_g(T11, pB_out_agaaaaa(X23, T11, X24, X25, X26, X4, X5)) → confA_out_g(T11)

The argument filtering Pi contains the following mapping:
confA_in_g(x1)  =  confA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
pB_in_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_in_agaaaaa(x2)
U8_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_agaaaaa(x2, x8)
delC_in_aga(x1, x2, x3)  =  delC_in_aga(x2)
.(x1, x2)  =  .(x2)
delC_out_aga(x1, x2, x3)  =  delC_out_aga(x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U9_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_agaaaaa(x2, x3, x8)
pK_in_agaaa(x1, x2, x3, x4, x5)  =  pK_in_agaaa(x2)
U10_agaaa(x1, x2, x3, x4, x5, x6)  =  U10_agaaa(x2, x6)
U11_agaaa(x1, x2, x3, x4, x5, x6)  =  U11_agaaa(x2, x3, x6)
pL_in_aag(x1, x2, x3)  =  pL_in_aag(x3)
U12_aag(x1, x2, x3, x4)  =  U12_aag(x3, x4)
delD_in_aag(x1, x2, x3)  =  delD_in_aag(x3)
delD_out_aag(x1, x2, x3)  =  delD_out_aag(x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x4, x5)
U13_aag(x1, x2, x3, x4)  =  U13_aag(x2, x3, x4)
confF_in_g(x1)  =  confF_in_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
pG_in_gaaa(x1, x2, x3, x4)  =  pG_in_gaaa(x1)
U14_gaaa(x1, x2, x3, x4, x5)  =  U14_gaaa(x1, x5)
del2I_in_ga(x1, x2)  =  del2I_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
pJ_in_agaaa(x1, x2, x3, x4, x5)  =  pJ_in_agaaa(x2)
U18_agaaa(x1, x2, x3, x4, x5, x6)  =  U18_agaaa(x2, x6)
delE_in_aga(x1, x2, x3)  =  delE_in_aga(x2)
delE_out_aga(x1, x2, x3)  =  delE_out_aga(x2, x3)
U4_aga(x1, x2, x3, x4, x5)  =  U4_aga(x3, x5)
U19_agaaa(x1, x2, x3, x4, x5, x6)  =  U19_agaaa(x2, x3, x6)
pJ_out_agaaa(x1, x2, x3, x4, x5)  =  pJ_out_agaaa(x2, x3, x5)
del2I_out_ga(x1, x2)  =  del2I_out_ga(x1, x2)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x2, x5)
pM_in_aag(x1, x2, x3)  =  pM_in_aag(x3)
U16_aag(x1, x2, x3, x4)  =  U16_aag(x3, x4)
delH_in_aag(x1, x2, x3)  =  delH_in_aag(x3)
delH_out_aag(x1, x2, x3)  =  delH_out_aag(x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x4, x5)
U17_aag(x1, x2, x3, x4)  =  U17_aag(x2, x3, x4)
confF_out_g(x1)  =  confF_out_g(x1)
pM_out_aag(x1, x2, x3)  =  pM_out_aag(x2, x3)
pG_out_gaaa(x1, x2, x3, x4)  =  pG_out_gaaa(x1, x2, x4)
pL_out_aag(x1, x2, x3)  =  pL_out_aag(x2, x3)
pK_out_agaaa(x1, x2, x3, x4, x5)  =  pK_out_agaaa(x2, x3, x5)
pB_out_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_out_agaaaaa(x2, x3, x5, x7)
confA_out_g(x1)  =  confA_out_g(x1)
CONFA_IN_G(x1)  =  CONFA_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
PB_IN_AGAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  PB_IN_AGAAAAA(x2)
U8_AGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_AGAAAAA(x2, x8)
DELC_IN_AGA(x1, x2, x3)  =  DELC_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U9_AGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_AGAAAAA(x2, x3, x8)
PK_IN_AGAAA(x1, x2, x3, x4, x5)  =  PK_IN_AGAAA(x2)
U10_AGAAA(x1, x2, x3, x4, x5, x6)  =  U10_AGAAA(x2, x6)
U11_AGAAA(x1, x2, x3, x4, x5, x6)  =  U11_AGAAA(x2, x3, x6)
PL_IN_AAG(x1, x2, x3)  =  PL_IN_AAG(x3)
U12_AAG(x1, x2, x3, x4)  =  U12_AAG(x3, x4)
DELD_IN_AAG(x1, x2, x3)  =  DELD_IN_AAG(x3)
U3_AAG(x1, x2, x3, x4, x5)  =  U3_AAG(x4, x5)
U13_AAG(x1, x2, x3, x4)  =  U13_AAG(x2, x3, x4)
CONFF_IN_G(x1)  =  CONFF_IN_G(x1)
U5_G(x1, x2)  =  U5_G(x1, x2)
PG_IN_GAAA(x1, x2, x3, x4)  =  PG_IN_GAAA(x1)
U14_GAAA(x1, x2, x3, x4, x5)  =  U14_GAAA(x1, x5)
DEL2I_IN_GA(x1, x2)  =  DEL2I_IN_GA(x1)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
PJ_IN_AGAAA(x1, x2, x3, x4, x5)  =  PJ_IN_AGAAA(x2)
U18_AGAAA(x1, x2, x3, x4, x5, x6)  =  U18_AGAAA(x2, x6)
DELE_IN_AGA(x1, x2, x3)  =  DELE_IN_AGA(x2)
U4_AGA(x1, x2, x3, x4, x5)  =  U4_AGA(x3, x5)
U19_AGAAA(x1, x2, x3, x4, x5, x6)  =  U19_AGAAA(x2, x3, x6)
U15_GAAA(x1, x2, x3, x4, x5)  =  U15_GAAA(x1, x2, x5)
PM_IN_AAG(x1, x2, x3)  =  PM_IN_AAG(x3)
U16_AAG(x1, x2, x3, x4)  =  U16_AAG(x3, x4)
DELH_IN_AAG(x1, x2, x3)  =  DELH_IN_AAG(x3)
U6_AAG(x1, x2, x3, x4, x5)  =  U6_AAG(x4, x5)
U17_AAG(x1, x2, x3, x4)  =  U17_AAG(x2, x3, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 29 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

DELH_IN_AAG(X308, .(T135, X310), .(T135, T137)) → DELH_IN_AAG(X308, X310, T137)

The TRS R consists of the following rules:

confA_in_g(T11) → U1_g(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
pB_in_agaaaaa(T16, T11, T17, X25, X26, X4, X5) → U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
delC_in_aga(T30, .(T30, T31), T31) → delC_out_aga(T30, .(T30, T31), T31)
delC_in_aga(X70, .(T36, T37), .(T36, X71)) → U2_aga(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
U2_aga(X70, T36, T37, X71, delC_out_aga(X70, T37, X71)) → delC_out_aga(X70, .(T36, T37), .(T36, X71))
U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
pK_in_agaaa(T46, T17, T47, X4, X5) → U10_agaaa(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
U10_agaaa(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_agaaa(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
pL_in_aag(T53, T54, T47) → U12_aag(T53, T54, T47, delD_in_aag(T53, T54, T47))
delD_in_aag(X121, .(X121, T60), T60) → delD_out_aag(X121, .(X121, T60), T60)
delD_in_aag(X141, .(T65, X143), .(T65, T66)) → U3_aag(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
U3_aag(X141, T65, X143, T66, delD_out_aag(X141, X143, T66)) → delD_out_aag(X141, .(T65, X143), .(T65, T66))
U12_aag(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_aag(T53, T54, T47, confF_in_g(T54))
confF_in_g(T74) → U5_g(T74, pG_in_gaaa(T74, X164, X165, X166))
pG_in_gaaa(T74, T76, X165, X166) → U14_gaaa(T74, T76, X165, X166, del2I_in_ga(T74, T76))
del2I_in_ga(T86, X199) → U7_ga(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
pJ_in_agaaa(T91, T86, T92, X198, X199) → U18_agaaa(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
delE_in_aga(T109, .(T109, T110), T110) → delE_out_aga(T109, .(T109, T110), T110)
delE_in_aga(X249, .(T115, T117), .(T115, X250)) → U4_aga(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
U4_aga(X249, T115, T117, X250, delE_out_aga(X249, T117, X250)) → delE_out_aga(X249, .(T115, T117), .(T115, X250))
U18_agaaa(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_agaaa(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U19_agaaa(T91, T86, T92, X198, X199, delE_out_aga(X198, T92, X199)) → pJ_out_agaaa(T91, T86, T92, X198, X199)
U7_ga(T86, X199, pJ_out_agaaa(X196, T86, X197, X198, X199)) → del2I_out_ga(T86, X199)
U14_gaaa(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_gaaa(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
pM_in_aag(T123, T124, T76) → U16_aag(T123, T124, T76, delH_in_aag(T123, T124, T76))
delH_in_aag(X288, .(X288, T130), T130) → delH_out_aag(X288, .(X288, T130), T130)
delH_in_aag(X308, .(T135, X310), .(T135, T137)) → U6_aag(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
U6_aag(X308, T135, X310, T137, delH_out_aag(X308, X310, T137)) → delH_out_aag(X308, .(T135, X310), .(T135, T137))
U16_aag(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_aag(T123, T124, T76, confF_in_g(T124))
U17_aag(T123, T124, T76, confF_out_g(T124)) → pM_out_aag(T123, T124, T76)
U15_gaaa(T74, T76, X165, X166, pM_out_aag(X165, X166, T76)) → pG_out_gaaa(T74, T76, X165, X166)
U5_g(T74, pG_out_gaaa(T74, X164, X165, X166)) → confF_out_g(T74)
U13_aag(T53, T54, T47, confF_out_g(T54)) → pL_out_aag(T53, T54, T47)
U11_agaaa(T46, T17, T47, X4, X5, pL_out_aag(X4, X5, T47)) → pK_out_agaaa(T46, T17, T47, X4, X5)
U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_out_agaaa(X25, T17, X26, X4, X5)) → pB_out_agaaaaa(T16, T11, T17, X25, X26, X4, X5)
U1_g(T11, pB_out_agaaaaa(X23, T11, X24, X25, X26, X4, X5)) → confA_out_g(T11)

The argument filtering Pi contains the following mapping:
confA_in_g(x1)  =  confA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
pB_in_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_in_agaaaaa(x2)
U8_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_agaaaaa(x2, x8)
delC_in_aga(x1, x2, x3)  =  delC_in_aga(x2)
.(x1, x2)  =  .(x2)
delC_out_aga(x1, x2, x3)  =  delC_out_aga(x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U9_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_agaaaaa(x2, x3, x8)
pK_in_agaaa(x1, x2, x3, x4, x5)  =  pK_in_agaaa(x2)
U10_agaaa(x1, x2, x3, x4, x5, x6)  =  U10_agaaa(x2, x6)
U11_agaaa(x1, x2, x3, x4, x5, x6)  =  U11_agaaa(x2, x3, x6)
pL_in_aag(x1, x2, x3)  =  pL_in_aag(x3)
U12_aag(x1, x2, x3, x4)  =  U12_aag(x3, x4)
delD_in_aag(x1, x2, x3)  =  delD_in_aag(x3)
delD_out_aag(x1, x2, x3)  =  delD_out_aag(x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x4, x5)
U13_aag(x1, x2, x3, x4)  =  U13_aag(x2, x3, x4)
confF_in_g(x1)  =  confF_in_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
pG_in_gaaa(x1, x2, x3, x4)  =  pG_in_gaaa(x1)
U14_gaaa(x1, x2, x3, x4, x5)  =  U14_gaaa(x1, x5)
del2I_in_ga(x1, x2)  =  del2I_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
pJ_in_agaaa(x1, x2, x3, x4, x5)  =  pJ_in_agaaa(x2)
U18_agaaa(x1, x2, x3, x4, x5, x6)  =  U18_agaaa(x2, x6)
delE_in_aga(x1, x2, x3)  =  delE_in_aga(x2)
delE_out_aga(x1, x2, x3)  =  delE_out_aga(x2, x3)
U4_aga(x1, x2, x3, x4, x5)  =  U4_aga(x3, x5)
U19_agaaa(x1, x2, x3, x4, x5, x6)  =  U19_agaaa(x2, x3, x6)
pJ_out_agaaa(x1, x2, x3, x4, x5)  =  pJ_out_agaaa(x2, x3, x5)
del2I_out_ga(x1, x2)  =  del2I_out_ga(x1, x2)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x2, x5)
pM_in_aag(x1, x2, x3)  =  pM_in_aag(x3)
U16_aag(x1, x2, x3, x4)  =  U16_aag(x3, x4)
delH_in_aag(x1, x2, x3)  =  delH_in_aag(x3)
delH_out_aag(x1, x2, x3)  =  delH_out_aag(x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x4, x5)
U17_aag(x1, x2, x3, x4)  =  U17_aag(x2, x3, x4)
confF_out_g(x1)  =  confF_out_g(x1)
pM_out_aag(x1, x2, x3)  =  pM_out_aag(x2, x3)
pG_out_gaaa(x1, x2, x3, x4)  =  pG_out_gaaa(x1, x2, x4)
pL_out_aag(x1, x2, x3)  =  pL_out_aag(x2, x3)
pK_out_agaaa(x1, x2, x3, x4, x5)  =  pK_out_agaaa(x2, x3, x5)
pB_out_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_out_agaaaaa(x2, x3, x5, x7)
confA_out_g(x1)  =  confA_out_g(x1)
DELH_IN_AAG(x1, x2, x3)  =  DELH_IN_AAG(x3)

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:

DELH_IN_AAG(X308, .(T135, X310), .(T135, T137)) → DELH_IN_AAG(X308, X310, T137)

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

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:

DELH_IN_AAG(.(T137)) → DELH_IN_AAG(T137)

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:

  • DELH_IN_AAG(.(T137)) → DELH_IN_AAG(T137)
    The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

DELE_IN_AGA(X249, .(T115, T117), .(T115, X250)) → DELE_IN_AGA(X249, T117, X250)

The TRS R consists of the following rules:

confA_in_g(T11) → U1_g(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
pB_in_agaaaaa(T16, T11, T17, X25, X26, X4, X5) → U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
delC_in_aga(T30, .(T30, T31), T31) → delC_out_aga(T30, .(T30, T31), T31)
delC_in_aga(X70, .(T36, T37), .(T36, X71)) → U2_aga(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
U2_aga(X70, T36, T37, X71, delC_out_aga(X70, T37, X71)) → delC_out_aga(X70, .(T36, T37), .(T36, X71))
U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
pK_in_agaaa(T46, T17, T47, X4, X5) → U10_agaaa(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
U10_agaaa(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_agaaa(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
pL_in_aag(T53, T54, T47) → U12_aag(T53, T54, T47, delD_in_aag(T53, T54, T47))
delD_in_aag(X121, .(X121, T60), T60) → delD_out_aag(X121, .(X121, T60), T60)
delD_in_aag(X141, .(T65, X143), .(T65, T66)) → U3_aag(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
U3_aag(X141, T65, X143, T66, delD_out_aag(X141, X143, T66)) → delD_out_aag(X141, .(T65, X143), .(T65, T66))
U12_aag(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_aag(T53, T54, T47, confF_in_g(T54))
confF_in_g(T74) → U5_g(T74, pG_in_gaaa(T74, X164, X165, X166))
pG_in_gaaa(T74, T76, X165, X166) → U14_gaaa(T74, T76, X165, X166, del2I_in_ga(T74, T76))
del2I_in_ga(T86, X199) → U7_ga(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
pJ_in_agaaa(T91, T86, T92, X198, X199) → U18_agaaa(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
delE_in_aga(T109, .(T109, T110), T110) → delE_out_aga(T109, .(T109, T110), T110)
delE_in_aga(X249, .(T115, T117), .(T115, X250)) → U4_aga(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
U4_aga(X249, T115, T117, X250, delE_out_aga(X249, T117, X250)) → delE_out_aga(X249, .(T115, T117), .(T115, X250))
U18_agaaa(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_agaaa(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U19_agaaa(T91, T86, T92, X198, X199, delE_out_aga(X198, T92, X199)) → pJ_out_agaaa(T91, T86, T92, X198, X199)
U7_ga(T86, X199, pJ_out_agaaa(X196, T86, X197, X198, X199)) → del2I_out_ga(T86, X199)
U14_gaaa(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_gaaa(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
pM_in_aag(T123, T124, T76) → U16_aag(T123, T124, T76, delH_in_aag(T123, T124, T76))
delH_in_aag(X288, .(X288, T130), T130) → delH_out_aag(X288, .(X288, T130), T130)
delH_in_aag(X308, .(T135, X310), .(T135, T137)) → U6_aag(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
U6_aag(X308, T135, X310, T137, delH_out_aag(X308, X310, T137)) → delH_out_aag(X308, .(T135, X310), .(T135, T137))
U16_aag(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_aag(T123, T124, T76, confF_in_g(T124))
U17_aag(T123, T124, T76, confF_out_g(T124)) → pM_out_aag(T123, T124, T76)
U15_gaaa(T74, T76, X165, X166, pM_out_aag(X165, X166, T76)) → pG_out_gaaa(T74, T76, X165, X166)
U5_g(T74, pG_out_gaaa(T74, X164, X165, X166)) → confF_out_g(T74)
U13_aag(T53, T54, T47, confF_out_g(T54)) → pL_out_aag(T53, T54, T47)
U11_agaaa(T46, T17, T47, X4, X5, pL_out_aag(X4, X5, T47)) → pK_out_agaaa(T46, T17, T47, X4, X5)
U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_out_agaaa(X25, T17, X26, X4, X5)) → pB_out_agaaaaa(T16, T11, T17, X25, X26, X4, X5)
U1_g(T11, pB_out_agaaaaa(X23, T11, X24, X25, X26, X4, X5)) → confA_out_g(T11)

The argument filtering Pi contains the following mapping:
confA_in_g(x1)  =  confA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
pB_in_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_in_agaaaaa(x2)
U8_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_agaaaaa(x2, x8)
delC_in_aga(x1, x2, x3)  =  delC_in_aga(x2)
.(x1, x2)  =  .(x2)
delC_out_aga(x1, x2, x3)  =  delC_out_aga(x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U9_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_agaaaaa(x2, x3, x8)
pK_in_agaaa(x1, x2, x3, x4, x5)  =  pK_in_agaaa(x2)
U10_agaaa(x1, x2, x3, x4, x5, x6)  =  U10_agaaa(x2, x6)
U11_agaaa(x1, x2, x3, x4, x5, x6)  =  U11_agaaa(x2, x3, x6)
pL_in_aag(x1, x2, x3)  =  pL_in_aag(x3)
U12_aag(x1, x2, x3, x4)  =  U12_aag(x3, x4)
delD_in_aag(x1, x2, x3)  =  delD_in_aag(x3)
delD_out_aag(x1, x2, x3)  =  delD_out_aag(x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x4, x5)
U13_aag(x1, x2, x3, x4)  =  U13_aag(x2, x3, x4)
confF_in_g(x1)  =  confF_in_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
pG_in_gaaa(x1, x2, x3, x4)  =  pG_in_gaaa(x1)
U14_gaaa(x1, x2, x3, x4, x5)  =  U14_gaaa(x1, x5)
del2I_in_ga(x1, x2)  =  del2I_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
pJ_in_agaaa(x1, x2, x3, x4, x5)  =  pJ_in_agaaa(x2)
U18_agaaa(x1, x2, x3, x4, x5, x6)  =  U18_agaaa(x2, x6)
delE_in_aga(x1, x2, x3)  =  delE_in_aga(x2)
delE_out_aga(x1, x2, x3)  =  delE_out_aga(x2, x3)
U4_aga(x1, x2, x3, x4, x5)  =  U4_aga(x3, x5)
U19_agaaa(x1, x2, x3, x4, x5, x6)  =  U19_agaaa(x2, x3, x6)
pJ_out_agaaa(x1, x2, x3, x4, x5)  =  pJ_out_agaaa(x2, x3, x5)
del2I_out_ga(x1, x2)  =  del2I_out_ga(x1, x2)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x2, x5)
pM_in_aag(x1, x2, x3)  =  pM_in_aag(x3)
U16_aag(x1, x2, x3, x4)  =  U16_aag(x3, x4)
delH_in_aag(x1, x2, x3)  =  delH_in_aag(x3)
delH_out_aag(x1, x2, x3)  =  delH_out_aag(x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x4, x5)
U17_aag(x1, x2, x3, x4)  =  U17_aag(x2, x3, x4)
confF_out_g(x1)  =  confF_out_g(x1)
pM_out_aag(x1, x2, x3)  =  pM_out_aag(x2, x3)
pG_out_gaaa(x1, x2, x3, x4)  =  pG_out_gaaa(x1, x2, x4)
pL_out_aag(x1, x2, x3)  =  pL_out_aag(x2, x3)
pK_out_agaaa(x1, x2, x3, x4, x5)  =  pK_out_agaaa(x2, x3, x5)
pB_out_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_out_agaaaaa(x2, x3, x5, x7)
confA_out_g(x1)  =  confA_out_g(x1)
DELE_IN_AGA(x1, x2, x3)  =  DELE_IN_AGA(x2)

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:

DELE_IN_AGA(X249, .(T115, T117), .(T115, X250)) → DELE_IN_AGA(X249, T117, X250)

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

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:

DELE_IN_AGA(.(T117)) → DELE_IN_AGA(T117)

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

(19) 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:

  • DELE_IN_AGA(.(T117)) → DELE_IN_AGA(T117)
    The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

CONFF_IN_G(T74) → PG_IN_GAAA(T74, X164, X165, X166)
PG_IN_GAAA(T74, T76, X165, X166) → U14_GAAA(T74, T76, X165, X166, del2I_in_ga(T74, T76))
U14_GAAA(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → PM_IN_AAG(X165, X166, T76)
PM_IN_AAG(T123, T124, T76) → U16_AAG(T123, T124, T76, delH_in_aag(T123, T124, T76))
U16_AAG(T123, T124, T76, delH_out_aag(T123, T124, T76)) → CONFF_IN_G(T124)

The TRS R consists of the following rules:

confA_in_g(T11) → U1_g(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
pB_in_agaaaaa(T16, T11, T17, X25, X26, X4, X5) → U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
delC_in_aga(T30, .(T30, T31), T31) → delC_out_aga(T30, .(T30, T31), T31)
delC_in_aga(X70, .(T36, T37), .(T36, X71)) → U2_aga(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
U2_aga(X70, T36, T37, X71, delC_out_aga(X70, T37, X71)) → delC_out_aga(X70, .(T36, T37), .(T36, X71))
U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
pK_in_agaaa(T46, T17, T47, X4, X5) → U10_agaaa(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
U10_agaaa(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_agaaa(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
pL_in_aag(T53, T54, T47) → U12_aag(T53, T54, T47, delD_in_aag(T53, T54, T47))
delD_in_aag(X121, .(X121, T60), T60) → delD_out_aag(X121, .(X121, T60), T60)
delD_in_aag(X141, .(T65, X143), .(T65, T66)) → U3_aag(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
U3_aag(X141, T65, X143, T66, delD_out_aag(X141, X143, T66)) → delD_out_aag(X141, .(T65, X143), .(T65, T66))
U12_aag(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_aag(T53, T54, T47, confF_in_g(T54))
confF_in_g(T74) → U5_g(T74, pG_in_gaaa(T74, X164, X165, X166))
pG_in_gaaa(T74, T76, X165, X166) → U14_gaaa(T74, T76, X165, X166, del2I_in_ga(T74, T76))
del2I_in_ga(T86, X199) → U7_ga(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
pJ_in_agaaa(T91, T86, T92, X198, X199) → U18_agaaa(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
delE_in_aga(T109, .(T109, T110), T110) → delE_out_aga(T109, .(T109, T110), T110)
delE_in_aga(X249, .(T115, T117), .(T115, X250)) → U4_aga(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
U4_aga(X249, T115, T117, X250, delE_out_aga(X249, T117, X250)) → delE_out_aga(X249, .(T115, T117), .(T115, X250))
U18_agaaa(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_agaaa(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U19_agaaa(T91, T86, T92, X198, X199, delE_out_aga(X198, T92, X199)) → pJ_out_agaaa(T91, T86, T92, X198, X199)
U7_ga(T86, X199, pJ_out_agaaa(X196, T86, X197, X198, X199)) → del2I_out_ga(T86, X199)
U14_gaaa(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_gaaa(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
pM_in_aag(T123, T124, T76) → U16_aag(T123, T124, T76, delH_in_aag(T123, T124, T76))
delH_in_aag(X288, .(X288, T130), T130) → delH_out_aag(X288, .(X288, T130), T130)
delH_in_aag(X308, .(T135, X310), .(T135, T137)) → U6_aag(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
U6_aag(X308, T135, X310, T137, delH_out_aag(X308, X310, T137)) → delH_out_aag(X308, .(T135, X310), .(T135, T137))
U16_aag(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_aag(T123, T124, T76, confF_in_g(T124))
U17_aag(T123, T124, T76, confF_out_g(T124)) → pM_out_aag(T123, T124, T76)
U15_gaaa(T74, T76, X165, X166, pM_out_aag(X165, X166, T76)) → pG_out_gaaa(T74, T76, X165, X166)
U5_g(T74, pG_out_gaaa(T74, X164, X165, X166)) → confF_out_g(T74)
U13_aag(T53, T54, T47, confF_out_g(T54)) → pL_out_aag(T53, T54, T47)
U11_agaaa(T46, T17, T47, X4, X5, pL_out_aag(X4, X5, T47)) → pK_out_agaaa(T46, T17, T47, X4, X5)
U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_out_agaaa(X25, T17, X26, X4, X5)) → pB_out_agaaaaa(T16, T11, T17, X25, X26, X4, X5)
U1_g(T11, pB_out_agaaaaa(X23, T11, X24, X25, X26, X4, X5)) → confA_out_g(T11)

The argument filtering Pi contains the following mapping:
confA_in_g(x1)  =  confA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
pB_in_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_in_agaaaaa(x2)
U8_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_agaaaaa(x2, x8)
delC_in_aga(x1, x2, x3)  =  delC_in_aga(x2)
.(x1, x2)  =  .(x2)
delC_out_aga(x1, x2, x3)  =  delC_out_aga(x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U9_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_agaaaaa(x2, x3, x8)
pK_in_agaaa(x1, x2, x3, x4, x5)  =  pK_in_agaaa(x2)
U10_agaaa(x1, x2, x3, x4, x5, x6)  =  U10_agaaa(x2, x6)
U11_agaaa(x1, x2, x3, x4, x5, x6)  =  U11_agaaa(x2, x3, x6)
pL_in_aag(x1, x2, x3)  =  pL_in_aag(x3)
U12_aag(x1, x2, x3, x4)  =  U12_aag(x3, x4)
delD_in_aag(x1, x2, x3)  =  delD_in_aag(x3)
delD_out_aag(x1, x2, x3)  =  delD_out_aag(x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x4, x5)
U13_aag(x1, x2, x3, x4)  =  U13_aag(x2, x3, x4)
confF_in_g(x1)  =  confF_in_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
pG_in_gaaa(x1, x2, x3, x4)  =  pG_in_gaaa(x1)
U14_gaaa(x1, x2, x3, x4, x5)  =  U14_gaaa(x1, x5)
del2I_in_ga(x1, x2)  =  del2I_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
pJ_in_agaaa(x1, x2, x3, x4, x5)  =  pJ_in_agaaa(x2)
U18_agaaa(x1, x2, x3, x4, x5, x6)  =  U18_agaaa(x2, x6)
delE_in_aga(x1, x2, x3)  =  delE_in_aga(x2)
delE_out_aga(x1, x2, x3)  =  delE_out_aga(x2, x3)
U4_aga(x1, x2, x3, x4, x5)  =  U4_aga(x3, x5)
U19_agaaa(x1, x2, x3, x4, x5, x6)  =  U19_agaaa(x2, x3, x6)
pJ_out_agaaa(x1, x2, x3, x4, x5)  =  pJ_out_agaaa(x2, x3, x5)
del2I_out_ga(x1, x2)  =  del2I_out_ga(x1, x2)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x2, x5)
pM_in_aag(x1, x2, x3)  =  pM_in_aag(x3)
U16_aag(x1, x2, x3, x4)  =  U16_aag(x3, x4)
delH_in_aag(x1, x2, x3)  =  delH_in_aag(x3)
delH_out_aag(x1, x2, x3)  =  delH_out_aag(x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x4, x5)
U17_aag(x1, x2, x3, x4)  =  U17_aag(x2, x3, x4)
confF_out_g(x1)  =  confF_out_g(x1)
pM_out_aag(x1, x2, x3)  =  pM_out_aag(x2, x3)
pG_out_gaaa(x1, x2, x3, x4)  =  pG_out_gaaa(x1, x2, x4)
pL_out_aag(x1, x2, x3)  =  pL_out_aag(x2, x3)
pK_out_agaaa(x1, x2, x3, x4, x5)  =  pK_out_agaaa(x2, x3, x5)
pB_out_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_out_agaaaaa(x2, x3, x5, x7)
confA_out_g(x1)  =  confA_out_g(x1)
CONFF_IN_G(x1)  =  CONFF_IN_G(x1)
PG_IN_GAAA(x1, x2, x3, x4)  =  PG_IN_GAAA(x1)
U14_GAAA(x1, x2, x3, x4, x5)  =  U14_GAAA(x1, x5)
PM_IN_AAG(x1, x2, x3)  =  PM_IN_AAG(x3)
U16_AAG(x1, x2, x3, x4)  =  U16_AAG(x3, x4)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

CONFF_IN_G(T74) → PG_IN_GAAA(T74, X164, X165, X166)
PG_IN_GAAA(T74, T76, X165, X166) → U14_GAAA(T74, T76, X165, X166, del2I_in_ga(T74, T76))
U14_GAAA(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → PM_IN_AAG(X165, X166, T76)
PM_IN_AAG(T123, T124, T76) → U16_AAG(T123, T124, T76, delH_in_aag(T123, T124, T76))
U16_AAG(T123, T124, T76, delH_out_aag(T123, T124, T76)) → CONFF_IN_G(T124)

The TRS R consists of the following rules:

del2I_in_ga(T86, X199) → U7_ga(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
delH_in_aag(X288, .(X288, T130), T130) → delH_out_aag(X288, .(X288, T130), T130)
delH_in_aag(X308, .(T135, X310), .(T135, T137)) → U6_aag(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
U7_ga(T86, X199, pJ_out_agaaa(X196, T86, X197, X198, X199)) → del2I_out_ga(T86, X199)
U6_aag(X308, T135, X310, T137, delH_out_aag(X308, X310, T137)) → delH_out_aag(X308, .(T135, X310), .(T135, T137))
pJ_in_agaaa(T91, T86, T92, X198, X199) → U18_agaaa(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
U18_agaaa(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_agaaa(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
delE_in_aga(T109, .(T109, T110), T110) → delE_out_aga(T109, .(T109, T110), T110)
delE_in_aga(X249, .(T115, T117), .(T115, X250)) → U4_aga(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
U19_agaaa(T91, T86, T92, X198, X199, delE_out_aga(X198, T92, X199)) → pJ_out_agaaa(T91, T86, T92, X198, X199)
U4_aga(X249, T115, T117, X250, delE_out_aga(X249, T117, X250)) → delE_out_aga(X249, .(T115, T117), .(T115, X250))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
del2I_in_ga(x1, x2)  =  del2I_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
pJ_in_agaaa(x1, x2, x3, x4, x5)  =  pJ_in_agaaa(x2)
U18_agaaa(x1, x2, x3, x4, x5, x6)  =  U18_agaaa(x2, x6)
delE_in_aga(x1, x2, x3)  =  delE_in_aga(x2)
delE_out_aga(x1, x2, x3)  =  delE_out_aga(x2, x3)
U4_aga(x1, x2, x3, x4, x5)  =  U4_aga(x3, x5)
U19_agaaa(x1, x2, x3, x4, x5, x6)  =  U19_agaaa(x2, x3, x6)
pJ_out_agaaa(x1, x2, x3, x4, x5)  =  pJ_out_agaaa(x2, x3, x5)
del2I_out_ga(x1, x2)  =  del2I_out_ga(x1, x2)
delH_in_aag(x1, x2, x3)  =  delH_in_aag(x3)
delH_out_aag(x1, x2, x3)  =  delH_out_aag(x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x4, x5)
CONFF_IN_G(x1)  =  CONFF_IN_G(x1)
PG_IN_GAAA(x1, x2, x3, x4)  =  PG_IN_GAAA(x1)
U14_GAAA(x1, x2, x3, x4, x5)  =  U14_GAAA(x1, x5)
PM_IN_AAG(x1, x2, x3)  =  PM_IN_AAG(x3)
U16_AAG(x1, x2, x3, x4)  =  U16_AAG(x3, x4)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

CONFF_IN_G(T74) → PG_IN_GAAA(T74)
PG_IN_GAAA(T74) → U14_GAAA(T74, del2I_in_ga(T74))
U14_GAAA(T74, del2I_out_ga(T74, T76)) → PM_IN_AAG(T76)
PM_IN_AAG(T76) → U16_AAG(T76, delH_in_aag(T76))
U16_AAG(T76, delH_out_aag(T124, T76)) → CONFF_IN_G(T124)

The TRS R consists of the following rules:

del2I_in_ga(T86) → U7_ga(T86, pJ_in_agaaa(T86))
delH_in_aag(T130) → delH_out_aag(.(T130), T130)
delH_in_aag(.(T137)) → U6_aag(T137, delH_in_aag(T137))
U7_ga(T86, pJ_out_agaaa(T86, X197, X199)) → del2I_out_ga(T86, X199)
U6_aag(T137, delH_out_aag(X310, T137)) → delH_out_aag(.(X310), .(T137))
pJ_in_agaaa(T86) → U18_agaaa(T86, delE_in_aga(T86))
U18_agaaa(T86, delE_out_aga(T86, T92)) → U19_agaaa(T86, T92, delE_in_aga(T92))
delE_in_aga(.(T110)) → delE_out_aga(.(T110), T110)
delE_in_aga(.(T117)) → U4_aga(T117, delE_in_aga(T117))
U19_agaaa(T86, T92, delE_out_aga(T92, X199)) → pJ_out_agaaa(T86, T92, X199)
U4_aga(T117, delE_out_aga(T117, X250)) → delE_out_aga(.(T117), .(X250))

The set Q consists of the following terms:

del2I_in_ga(x0)
delH_in_aag(x0)
U7_ga(x0, x1)
U6_aag(x0, x1)
pJ_in_agaaa(x0)
U18_agaaa(x0, x1)
delE_in_aga(x0)
U19_agaaa(x0, x1, x2)
U4_aga(x0, x1)

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

(26) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


U14_GAAA(T74, del2I_out_ga(T74, T76)) → PM_IN_AAG(T76)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U14_GAAA(x1, x2) ) = max{0, 2x2 - 1}


POL( del2I_in_ga(x1) ) = x1


POL( U7_ga(x1, x2) ) = x2


POL( pJ_in_agaaa(x1) ) = x1


POL( U16_AAG(x1, x2) ) = x2


POL( delH_in_aag(x1) ) = 2x1 + 2


POL( delH_out_aag(x1, x2) ) = 2x1


POL( .(x1) ) = x1 + 1


POL( U6_aag(x1, x2) ) = x2 + 2


POL( U18_agaaa(x1, x2) ) = max{0, x2 - 1}


POL( delE_in_aga(x1) ) = x1 + 1


POL( pJ_out_agaaa(x1, ..., x3) ) = x3 + 2


POL( del2I_out_ga(x1, x2) ) = x2 + 2


POL( U19_agaaa(x1, ..., x3) ) = x3


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


POL( U4_aga(x1, x2) ) = x2 + 1


POL( CONFF_IN_G(x1) ) = 2x1


POL( PG_IN_GAAA(x1) ) = 2x1


POL( PM_IN_AAG(x1) ) = 2x1 + 2



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

del2I_in_ga(T86) → U7_ga(T86, pJ_in_agaaa(T86))
delH_in_aag(T130) → delH_out_aag(.(T130), T130)
delH_in_aag(.(T137)) → U6_aag(T137, delH_in_aag(T137))
pJ_in_agaaa(T86) → U18_agaaa(T86, delE_in_aga(T86))
U7_ga(T86, pJ_out_agaaa(T86, X197, X199)) → del2I_out_ga(T86, X199)
U6_aag(T137, delH_out_aag(X310, T137)) → delH_out_aag(.(X310), .(T137))
delE_in_aga(.(T110)) → delE_out_aga(.(T110), T110)
delE_in_aga(.(T117)) → U4_aga(T117, delE_in_aga(T117))
U18_agaaa(T86, delE_out_aga(T86, T92)) → U19_agaaa(T86, T92, delE_in_aga(T92))
U19_agaaa(T86, T92, delE_out_aga(T92, X199)) → pJ_out_agaaa(T86, T92, X199)
U4_aga(T117, delE_out_aga(T117, X250)) → delE_out_aga(.(T117), .(X250))

(27) Obligation:

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

CONFF_IN_G(T74) → PG_IN_GAAA(T74)
PG_IN_GAAA(T74) → U14_GAAA(T74, del2I_in_ga(T74))
PM_IN_AAG(T76) → U16_AAG(T76, delH_in_aag(T76))
U16_AAG(T76, delH_out_aag(T124, T76)) → CONFF_IN_G(T124)

The TRS R consists of the following rules:

del2I_in_ga(T86) → U7_ga(T86, pJ_in_agaaa(T86))
delH_in_aag(T130) → delH_out_aag(.(T130), T130)
delH_in_aag(.(T137)) → U6_aag(T137, delH_in_aag(T137))
U7_ga(T86, pJ_out_agaaa(T86, X197, X199)) → del2I_out_ga(T86, X199)
U6_aag(T137, delH_out_aag(X310, T137)) → delH_out_aag(.(X310), .(T137))
pJ_in_agaaa(T86) → U18_agaaa(T86, delE_in_aga(T86))
U18_agaaa(T86, delE_out_aga(T86, T92)) → U19_agaaa(T86, T92, delE_in_aga(T92))
delE_in_aga(.(T110)) → delE_out_aga(.(T110), T110)
delE_in_aga(.(T117)) → U4_aga(T117, delE_in_aga(T117))
U19_agaaa(T86, T92, delE_out_aga(T92, X199)) → pJ_out_agaaa(T86, T92, X199)
U4_aga(T117, delE_out_aga(T117, X250)) → delE_out_aga(.(T117), .(X250))

The set Q consists of the following terms:

del2I_in_ga(x0)
delH_in_aag(x0)
U7_ga(x0, x1)
U6_aag(x0, x1)
pJ_in_agaaa(x0)
U18_agaaa(x0, x1)
delE_in_aga(x0)
U19_agaaa(x0, x1, x2)
U4_aga(x0, x1)

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

(28) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.

(29) TRUE

(30) Obligation:

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

DELD_IN_AAG(X141, .(T65, X143), .(T65, T66)) → DELD_IN_AAG(X141, X143, T66)

The TRS R consists of the following rules:

confA_in_g(T11) → U1_g(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
pB_in_agaaaaa(T16, T11, T17, X25, X26, X4, X5) → U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
delC_in_aga(T30, .(T30, T31), T31) → delC_out_aga(T30, .(T30, T31), T31)
delC_in_aga(X70, .(T36, T37), .(T36, X71)) → U2_aga(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
U2_aga(X70, T36, T37, X71, delC_out_aga(X70, T37, X71)) → delC_out_aga(X70, .(T36, T37), .(T36, X71))
U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
pK_in_agaaa(T46, T17, T47, X4, X5) → U10_agaaa(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
U10_agaaa(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_agaaa(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
pL_in_aag(T53, T54, T47) → U12_aag(T53, T54, T47, delD_in_aag(T53, T54, T47))
delD_in_aag(X121, .(X121, T60), T60) → delD_out_aag(X121, .(X121, T60), T60)
delD_in_aag(X141, .(T65, X143), .(T65, T66)) → U3_aag(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
U3_aag(X141, T65, X143, T66, delD_out_aag(X141, X143, T66)) → delD_out_aag(X141, .(T65, X143), .(T65, T66))
U12_aag(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_aag(T53, T54, T47, confF_in_g(T54))
confF_in_g(T74) → U5_g(T74, pG_in_gaaa(T74, X164, X165, X166))
pG_in_gaaa(T74, T76, X165, X166) → U14_gaaa(T74, T76, X165, X166, del2I_in_ga(T74, T76))
del2I_in_ga(T86, X199) → U7_ga(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
pJ_in_agaaa(T91, T86, T92, X198, X199) → U18_agaaa(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
delE_in_aga(T109, .(T109, T110), T110) → delE_out_aga(T109, .(T109, T110), T110)
delE_in_aga(X249, .(T115, T117), .(T115, X250)) → U4_aga(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
U4_aga(X249, T115, T117, X250, delE_out_aga(X249, T117, X250)) → delE_out_aga(X249, .(T115, T117), .(T115, X250))
U18_agaaa(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_agaaa(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U19_agaaa(T91, T86, T92, X198, X199, delE_out_aga(X198, T92, X199)) → pJ_out_agaaa(T91, T86, T92, X198, X199)
U7_ga(T86, X199, pJ_out_agaaa(X196, T86, X197, X198, X199)) → del2I_out_ga(T86, X199)
U14_gaaa(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_gaaa(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
pM_in_aag(T123, T124, T76) → U16_aag(T123, T124, T76, delH_in_aag(T123, T124, T76))
delH_in_aag(X288, .(X288, T130), T130) → delH_out_aag(X288, .(X288, T130), T130)
delH_in_aag(X308, .(T135, X310), .(T135, T137)) → U6_aag(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
U6_aag(X308, T135, X310, T137, delH_out_aag(X308, X310, T137)) → delH_out_aag(X308, .(T135, X310), .(T135, T137))
U16_aag(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_aag(T123, T124, T76, confF_in_g(T124))
U17_aag(T123, T124, T76, confF_out_g(T124)) → pM_out_aag(T123, T124, T76)
U15_gaaa(T74, T76, X165, X166, pM_out_aag(X165, X166, T76)) → pG_out_gaaa(T74, T76, X165, X166)
U5_g(T74, pG_out_gaaa(T74, X164, X165, X166)) → confF_out_g(T74)
U13_aag(T53, T54, T47, confF_out_g(T54)) → pL_out_aag(T53, T54, T47)
U11_agaaa(T46, T17, T47, X4, X5, pL_out_aag(X4, X5, T47)) → pK_out_agaaa(T46, T17, T47, X4, X5)
U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_out_agaaa(X25, T17, X26, X4, X5)) → pB_out_agaaaaa(T16, T11, T17, X25, X26, X4, X5)
U1_g(T11, pB_out_agaaaaa(X23, T11, X24, X25, X26, X4, X5)) → confA_out_g(T11)

The argument filtering Pi contains the following mapping:
confA_in_g(x1)  =  confA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
pB_in_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_in_agaaaaa(x2)
U8_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_agaaaaa(x2, x8)
delC_in_aga(x1, x2, x3)  =  delC_in_aga(x2)
.(x1, x2)  =  .(x2)
delC_out_aga(x1, x2, x3)  =  delC_out_aga(x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U9_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_agaaaaa(x2, x3, x8)
pK_in_agaaa(x1, x2, x3, x4, x5)  =  pK_in_agaaa(x2)
U10_agaaa(x1, x2, x3, x4, x5, x6)  =  U10_agaaa(x2, x6)
U11_agaaa(x1, x2, x3, x4, x5, x6)  =  U11_agaaa(x2, x3, x6)
pL_in_aag(x1, x2, x3)  =  pL_in_aag(x3)
U12_aag(x1, x2, x3, x4)  =  U12_aag(x3, x4)
delD_in_aag(x1, x2, x3)  =  delD_in_aag(x3)
delD_out_aag(x1, x2, x3)  =  delD_out_aag(x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x4, x5)
U13_aag(x1, x2, x3, x4)  =  U13_aag(x2, x3, x4)
confF_in_g(x1)  =  confF_in_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
pG_in_gaaa(x1, x2, x3, x4)  =  pG_in_gaaa(x1)
U14_gaaa(x1, x2, x3, x4, x5)  =  U14_gaaa(x1, x5)
del2I_in_ga(x1, x2)  =  del2I_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
pJ_in_agaaa(x1, x2, x3, x4, x5)  =  pJ_in_agaaa(x2)
U18_agaaa(x1, x2, x3, x4, x5, x6)  =  U18_agaaa(x2, x6)
delE_in_aga(x1, x2, x3)  =  delE_in_aga(x2)
delE_out_aga(x1, x2, x3)  =  delE_out_aga(x2, x3)
U4_aga(x1, x2, x3, x4, x5)  =  U4_aga(x3, x5)
U19_agaaa(x1, x2, x3, x4, x5, x6)  =  U19_agaaa(x2, x3, x6)
pJ_out_agaaa(x1, x2, x3, x4, x5)  =  pJ_out_agaaa(x2, x3, x5)
del2I_out_ga(x1, x2)  =  del2I_out_ga(x1, x2)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x2, x5)
pM_in_aag(x1, x2, x3)  =  pM_in_aag(x3)
U16_aag(x1, x2, x3, x4)  =  U16_aag(x3, x4)
delH_in_aag(x1, x2, x3)  =  delH_in_aag(x3)
delH_out_aag(x1, x2, x3)  =  delH_out_aag(x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x4, x5)
U17_aag(x1, x2, x3, x4)  =  U17_aag(x2, x3, x4)
confF_out_g(x1)  =  confF_out_g(x1)
pM_out_aag(x1, x2, x3)  =  pM_out_aag(x2, x3)
pG_out_gaaa(x1, x2, x3, x4)  =  pG_out_gaaa(x1, x2, x4)
pL_out_aag(x1, x2, x3)  =  pL_out_aag(x2, x3)
pK_out_agaaa(x1, x2, x3, x4, x5)  =  pK_out_agaaa(x2, x3, x5)
pB_out_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_out_agaaaaa(x2, x3, x5, x7)
confA_out_g(x1)  =  confA_out_g(x1)
DELD_IN_AAG(x1, x2, x3)  =  DELD_IN_AAG(x3)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

DELD_IN_AAG(X141, .(T65, X143), .(T65, T66)) → DELD_IN_AAG(X141, X143, T66)

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

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

DELD_IN_AAG(.(T66)) → DELD_IN_AAG(T66)

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

(35) 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:

  • DELD_IN_AAG(.(T66)) → DELD_IN_AAG(T66)
    The graph contains the following edges 1 > 1

(36) YES

(37) Obligation:

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

DELC_IN_AGA(X70, .(T36, T37), .(T36, X71)) → DELC_IN_AGA(X70, T37, X71)

The TRS R consists of the following rules:

confA_in_g(T11) → U1_g(T11, pB_in_agaaaaa(X23, T11, X24, X25, X26, X4, X5))
pB_in_agaaaaa(T16, T11, T17, X25, X26, X4, X5) → U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_in_aga(T16, T11, T17))
delC_in_aga(T30, .(T30, T31), T31) → delC_out_aga(T30, .(T30, T31), T31)
delC_in_aga(X70, .(T36, T37), .(T36, X71)) → U2_aga(X70, T36, T37, X71, delC_in_aga(X70, T37, X71))
U2_aga(X70, T36, T37, X71, delC_out_aga(X70, T37, X71)) → delC_out_aga(X70, .(T36, T37), .(T36, X71))
U8_agaaaaa(T16, T11, T17, X25, X26, X4, X5, delC_out_aga(T16, T11, T17)) → U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_in_agaaa(X25, T17, X26, X4, X5))
pK_in_agaaa(T46, T17, T47, X4, X5) → U10_agaaa(T46, T17, T47, X4, X5, delC_in_aga(T46, T17, T47))
U10_agaaa(T46, T17, T47, X4, X5, delC_out_aga(T46, T17, T47)) → U11_agaaa(T46, T17, T47, X4, X5, pL_in_aag(X4, X5, T47))
pL_in_aag(T53, T54, T47) → U12_aag(T53, T54, T47, delD_in_aag(T53, T54, T47))
delD_in_aag(X121, .(X121, T60), T60) → delD_out_aag(X121, .(X121, T60), T60)
delD_in_aag(X141, .(T65, X143), .(T65, T66)) → U3_aag(X141, T65, X143, T66, delD_in_aag(X141, X143, T66))
U3_aag(X141, T65, X143, T66, delD_out_aag(X141, X143, T66)) → delD_out_aag(X141, .(T65, X143), .(T65, T66))
U12_aag(T53, T54, T47, delD_out_aag(T53, T54, T47)) → U13_aag(T53, T54, T47, confF_in_g(T54))
confF_in_g(T74) → U5_g(T74, pG_in_gaaa(T74, X164, X165, X166))
pG_in_gaaa(T74, T76, X165, X166) → U14_gaaa(T74, T76, X165, X166, del2I_in_ga(T74, T76))
del2I_in_ga(T86, X199) → U7_ga(T86, X199, pJ_in_agaaa(X196, T86, X197, X198, X199))
pJ_in_agaaa(T91, T86, T92, X198, X199) → U18_agaaa(T91, T86, T92, X198, X199, delE_in_aga(T91, T86, T92))
delE_in_aga(T109, .(T109, T110), T110) → delE_out_aga(T109, .(T109, T110), T110)
delE_in_aga(X249, .(T115, T117), .(T115, X250)) → U4_aga(X249, T115, T117, X250, delE_in_aga(X249, T117, X250))
U4_aga(X249, T115, T117, X250, delE_out_aga(X249, T117, X250)) → delE_out_aga(X249, .(T115, T117), .(T115, X250))
U18_agaaa(T91, T86, T92, X198, X199, delE_out_aga(T91, T86, T92)) → U19_agaaa(T91, T86, T92, X198, X199, delE_in_aga(X198, T92, X199))
U19_agaaa(T91, T86, T92, X198, X199, delE_out_aga(X198, T92, X199)) → pJ_out_agaaa(T91, T86, T92, X198, X199)
U7_ga(T86, X199, pJ_out_agaaa(X196, T86, X197, X198, X199)) → del2I_out_ga(T86, X199)
U14_gaaa(T74, T76, X165, X166, del2I_out_ga(T74, T76)) → U15_gaaa(T74, T76, X165, X166, pM_in_aag(X165, X166, T76))
pM_in_aag(T123, T124, T76) → U16_aag(T123, T124, T76, delH_in_aag(T123, T124, T76))
delH_in_aag(X288, .(X288, T130), T130) → delH_out_aag(X288, .(X288, T130), T130)
delH_in_aag(X308, .(T135, X310), .(T135, T137)) → U6_aag(X308, T135, X310, T137, delH_in_aag(X308, X310, T137))
U6_aag(X308, T135, X310, T137, delH_out_aag(X308, X310, T137)) → delH_out_aag(X308, .(T135, X310), .(T135, T137))
U16_aag(T123, T124, T76, delH_out_aag(T123, T124, T76)) → U17_aag(T123, T124, T76, confF_in_g(T124))
U17_aag(T123, T124, T76, confF_out_g(T124)) → pM_out_aag(T123, T124, T76)
U15_gaaa(T74, T76, X165, X166, pM_out_aag(X165, X166, T76)) → pG_out_gaaa(T74, T76, X165, X166)
U5_g(T74, pG_out_gaaa(T74, X164, X165, X166)) → confF_out_g(T74)
U13_aag(T53, T54, T47, confF_out_g(T54)) → pL_out_aag(T53, T54, T47)
U11_agaaa(T46, T17, T47, X4, X5, pL_out_aag(X4, X5, T47)) → pK_out_agaaa(T46, T17, T47, X4, X5)
U9_agaaaaa(T16, T11, T17, X25, X26, X4, X5, pK_out_agaaa(X25, T17, X26, X4, X5)) → pB_out_agaaaaa(T16, T11, T17, X25, X26, X4, X5)
U1_g(T11, pB_out_agaaaaa(X23, T11, X24, X25, X26, X4, X5)) → confA_out_g(T11)

The argument filtering Pi contains the following mapping:
confA_in_g(x1)  =  confA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
pB_in_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_in_agaaaaa(x2)
U8_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_agaaaaa(x2, x8)
delC_in_aga(x1, x2, x3)  =  delC_in_aga(x2)
.(x1, x2)  =  .(x2)
delC_out_aga(x1, x2, x3)  =  delC_out_aga(x2, x3)
U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
U9_agaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_agaaaaa(x2, x3, x8)
pK_in_agaaa(x1, x2, x3, x4, x5)  =  pK_in_agaaa(x2)
U10_agaaa(x1, x2, x3, x4, x5, x6)  =  U10_agaaa(x2, x6)
U11_agaaa(x1, x2, x3, x4, x5, x6)  =  U11_agaaa(x2, x3, x6)
pL_in_aag(x1, x2, x3)  =  pL_in_aag(x3)
U12_aag(x1, x2, x3, x4)  =  U12_aag(x3, x4)
delD_in_aag(x1, x2, x3)  =  delD_in_aag(x3)
delD_out_aag(x1, x2, x3)  =  delD_out_aag(x2, x3)
U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x4, x5)
U13_aag(x1, x2, x3, x4)  =  U13_aag(x2, x3, x4)
confF_in_g(x1)  =  confF_in_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
pG_in_gaaa(x1, x2, x3, x4)  =  pG_in_gaaa(x1)
U14_gaaa(x1, x2, x3, x4, x5)  =  U14_gaaa(x1, x5)
del2I_in_ga(x1, x2)  =  del2I_in_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
pJ_in_agaaa(x1, x2, x3, x4, x5)  =  pJ_in_agaaa(x2)
U18_agaaa(x1, x2, x3, x4, x5, x6)  =  U18_agaaa(x2, x6)
delE_in_aga(x1, x2, x3)  =  delE_in_aga(x2)
delE_out_aga(x1, x2, x3)  =  delE_out_aga(x2, x3)
U4_aga(x1, x2, x3, x4, x5)  =  U4_aga(x3, x5)
U19_agaaa(x1, x2, x3, x4, x5, x6)  =  U19_agaaa(x2, x3, x6)
pJ_out_agaaa(x1, x2, x3, x4, x5)  =  pJ_out_agaaa(x2, x3, x5)
del2I_out_ga(x1, x2)  =  del2I_out_ga(x1, x2)
U15_gaaa(x1, x2, x3, x4, x5)  =  U15_gaaa(x1, x2, x5)
pM_in_aag(x1, x2, x3)  =  pM_in_aag(x3)
U16_aag(x1, x2, x3, x4)  =  U16_aag(x3, x4)
delH_in_aag(x1, x2, x3)  =  delH_in_aag(x3)
delH_out_aag(x1, x2, x3)  =  delH_out_aag(x2, x3)
U6_aag(x1, x2, x3, x4, x5)  =  U6_aag(x4, x5)
U17_aag(x1, x2, x3, x4)  =  U17_aag(x2, x3, x4)
confF_out_g(x1)  =  confF_out_g(x1)
pM_out_aag(x1, x2, x3)  =  pM_out_aag(x2, x3)
pG_out_gaaa(x1, x2, x3, x4)  =  pG_out_gaaa(x1, x2, x4)
pL_out_aag(x1, x2, x3)  =  pL_out_aag(x2, x3)
pK_out_agaaa(x1, x2, x3, x4, x5)  =  pK_out_agaaa(x2, x3, x5)
pB_out_agaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pB_out_agaaaaa(x2, x3, x5, x7)
confA_out_g(x1)  =  confA_out_g(x1)
DELC_IN_AGA(x1, x2, x3)  =  DELC_IN_AGA(x2)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

DELC_IN_AGA(X70, .(T36, T37), .(T36, X71)) → DELC_IN_AGA(X70, T37, X71)

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

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

DELC_IN_AGA(.(T37)) → DELC_IN_AGA(T37)

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

(42) 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:

  • DELC_IN_AGA(.(T37)) → DELC_IN_AGA(T37)
    The graph contains the following edges 1 > 1

(43) YES