(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)).

Queries:

conf(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

del6(X53, .(X53, T20), T20).
del6(X72, .(T25, T26), .(T25, X73)) :- del6(X72, T26, X73).
del18(X127, .(X127, T43), T43).
del18(X147, .(X148, X149), .(X148, T46)) :- del18(X147, X149, T46).
del34(X242, .(X242, T77), T77).
del34(X261, .(T82, T84), .(T82, X262)) :- del34(X261, T84, X262).
conf19(T54) :- del229(T54, X174).
conf19(T54) :- ','(del229(T54, T56), del43(X175, X176, T56)).
conf19(T54) :- ','(del229(T54, T56), ','(del43(T88, T89, T56), conf19(T89))).
del43(X304, .(X304, T95), T95).
del43(X324, .(X325, X326), .(X325, T99)) :- del43(X324, X326, T99).
del229(T64, X209) :- del34(X206, T64, X207).
del229(T64, X209) :- ','(del34(T67, T64, T68), del34(X208, T68, X209)).
conf1(T9) :- del6(X23, T9, X24).
conf1(T9) :- ','(del6(T12, T9, T13), del6(X25, T13, X26)).
conf1(T9) :- ','(del6(T12, T9, T13), ','(del6(T31, T13, T32), del18(X4, X5, T32))).
conf1(T9) :- ','(del6(T12, T9, T13), ','(del6(T31, T13, T32), ','(del18(T36, T37, T32), conf19(T37)))).

Queries:

conf1(g).

(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:
conf1_in: (b)
del6_in: (f,b,f)
del18_in: (f,f,b)
conf19_in: (b) (f)
del229_in: (b,f) (f,f)
del34_in: (f,b,f) (f,f,f)
del43_in: (f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
conf1_out_g(x1)  =  conf1_out_g
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x5)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
del229_out_ga(x1, x2)  =  del229_out_ga
U11_ga(x1, x2, x3)  =  U11_ga(x3)
U12_ga(x1, x2, x3)  =  U12_ga(x3)
conf19_out_g(x1)  =  conf19_out_g
U5_g(x1, x2)  =  U5_g(x2)
U6_g(x1, x2)  =  U6_g(x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x2)
U8_g(x1, x2)  =  U8_g(x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
conf1_out_g(x1)  =  conf1_out_g
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x5)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
del229_out_ga(x1, x2)  =  del229_out_ga
U11_ga(x1, x2, x3)  =  U11_ga(x3)
U12_ga(x1, x2, x3)  =  U12_ga(x3)
conf19_out_g(x1)  =  conf19_out_g
U5_g(x1, x2)  =  U5_g(x2)
U6_g(x1, x2)  =  U6_g(x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x2)
U8_g(x1, x2)  =  U8_g(x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)

(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:

CONF1_IN_G(T9) → U13_G(T9, del6_in_aga(X23, T9, X24))
CONF1_IN_G(T9) → DEL6_IN_AGA(X23, T9, X24)
DEL6_IN_AGA(X72, .(T25, T26), .(T25, X73)) → U1_AGA(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
DEL6_IN_AGA(X72, .(T25, T26), .(T25, X73)) → DEL6_IN_AGA(X72, T26, X73)
CONF1_IN_G(T9) → U14_G(T9, del6_in_aga(T12, T9, T13))
U14_G(T9, del6_out_aga(T12, T9, T13)) → U15_G(T9, del6_in_aga(X25, T13, X26))
U14_G(T9, del6_out_aga(T12, T9, T13)) → DEL6_IN_AGA(X25, T13, X26)
U14_G(T9, del6_out_aga(T12, T9, T13)) → U16_G(T9, del6_in_aga(T31, T13, T32))
U16_G(T9, del6_out_aga(T31, T13, T32)) → U17_G(T9, del18_in_aag(X4, X5, T32))
U16_G(T9, del6_out_aga(T31, T13, T32)) → DEL18_IN_AAG(X4, X5, T32)
DEL18_IN_AAG(X147, .(X148, X149), .(X148, T46)) → U2_AAG(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
DEL18_IN_AAG(X147, .(X148, X149), .(X148, T46)) → DEL18_IN_AAG(X147, X149, T46)
U16_G(T9, del6_out_aga(T31, T13, T32)) → U18_G(T9, del18_in_aag(T36, T37, T32))
U18_G(T9, del18_out_aag(T36, T37, T32)) → U19_G(T9, conf19_in_g(T37))
U18_G(T9, del18_out_aag(T36, T37, T32)) → CONF19_IN_G(T37)
CONF19_IN_G(T54) → U4_G(T54, del229_in_ga(T54, X174))
CONF19_IN_G(T54) → DEL229_IN_GA(T54, X174)
DEL229_IN_GA(T64, X209) → U10_GA(T64, X209, del34_in_aga(X206, T64, X207))
DEL229_IN_GA(T64, X209) → DEL34_IN_AGA(X206, T64, X207)
DEL34_IN_AGA(X261, .(T82, T84), .(T82, X262)) → U3_AGA(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
DEL34_IN_AGA(X261, .(T82, T84), .(T82, X262)) → DEL34_IN_AGA(X261, T84, X262)
DEL229_IN_GA(T64, X209) → U11_GA(T64, X209, del34_in_aga(T67, T64, T68))
U11_GA(T64, X209, del34_out_aga(T67, T64, T68)) → U12_GA(T64, X209, del34_in_aga(X208, T68, X209))
U11_GA(T64, X209, del34_out_aga(T67, T64, T68)) → DEL34_IN_AGA(X208, T68, X209)
CONF19_IN_G(T54) → U5_G(T54, del229_in_ga(T54, T56))
U5_G(T54, del229_out_ga(T54, T56)) → U6_G(T54, del43_in_aaa(X175, X176, T56))
U5_G(T54, del229_out_ga(T54, T56)) → DEL43_IN_AAA(X175, X176, T56)
DEL43_IN_AAA(X324, .(X325, X326), .(X325, T99)) → U9_AAA(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
DEL43_IN_AAA(X324, .(X325, X326), .(X325, T99)) → DEL43_IN_AAA(X324, X326, T99)
U5_G(T54, del229_out_ga(T54, T56)) → U7_G(T54, del43_in_aaa(T88, T89, T56))
U7_G(T54, del43_out_aaa(T88, T89, T56)) → U8_G(T54, conf19_in_a(T89))
U7_G(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)
CONF19_IN_A(T54) → U4_A(T54, del229_in_aa(T54, X174))
CONF19_IN_A(T54) → DEL229_IN_AA(T54, X174)
DEL229_IN_AA(T64, X209) → U10_AA(T64, X209, del34_in_aaa(X206, T64, X207))
DEL229_IN_AA(T64, X209) → DEL34_IN_AAA(X206, T64, X207)
DEL34_IN_AAA(X261, .(T82, T84), .(T82, X262)) → U3_AAA(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
DEL34_IN_AAA(X261, .(T82, T84), .(T82, X262)) → DEL34_IN_AAA(X261, T84, X262)
DEL229_IN_AA(T64, X209) → U11_AA(T64, X209, del34_in_aaa(T67, T64, T68))
U11_AA(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_AA(T64, X209, del34_in_aaa(X208, T68, X209))
U11_AA(T64, X209, del34_out_aaa(T67, T64, T68)) → DEL34_IN_AAA(X208, T68, X209)
CONF19_IN_A(T54) → U5_A(T54, del229_in_aa(T54, T56))
U5_A(T54, del229_out_aa(T54, T56)) → U6_A(T54, del43_in_aaa(X175, X176, T56))
U5_A(T54, del229_out_aa(T54, T56)) → DEL43_IN_AAA(X175, X176, T56)
U5_A(T54, del229_out_aa(T54, T56)) → U7_A(T54, del43_in_aaa(T88, T89, T56))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → U8_A(T54, conf19_in_a(T89))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
conf1_out_g(x1)  =  conf1_out_g
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x5)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
del229_out_ga(x1, x2)  =  del229_out_ga
U11_ga(x1, x2, x3)  =  U11_ga(x3)
U12_ga(x1, x2, x3)  =  U12_ga(x3)
conf19_out_g(x1)  =  conf19_out_g
U5_g(x1, x2)  =  U5_g(x2)
U6_g(x1, x2)  =  U6_g(x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x2)
U8_g(x1, x2)  =  U8_g(x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)
CONF1_IN_G(x1)  =  CONF1_IN_G(x1)
U13_G(x1, x2)  =  U13_G(x2)
DEL6_IN_AGA(x1, x2, x3)  =  DEL6_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x5)
U14_G(x1, x2)  =  U14_G(x2)
U15_G(x1, x2)  =  U15_G(x2)
U16_G(x1, x2)  =  U16_G(x2)
U17_G(x1, x2)  =  U17_G(x2)
DEL18_IN_AAG(x1, x2, x3)  =  DEL18_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x5)
U18_G(x1, x2)  =  U18_G(x2)
U19_G(x1, x2)  =  U19_G(x2)
CONF19_IN_G(x1)  =  CONF19_IN_G(x1)
U4_G(x1, x2)  =  U4_G(x2)
DEL229_IN_GA(x1, x2)  =  DEL229_IN_GA(x1)
U10_GA(x1, x2, x3)  =  U10_GA(x3)
DEL34_IN_AGA(x1, x2, x3)  =  DEL34_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x5)
U11_GA(x1, x2, x3)  =  U11_GA(x3)
U12_GA(x1, x2, x3)  =  U12_GA(x3)
U5_G(x1, x2)  =  U5_G(x2)
U6_G(x1, x2)  =  U6_G(x2)
DEL43_IN_AAA(x1, x2, x3)  =  DEL43_IN_AAA
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
U7_G(x1, x2)  =  U7_G(x2)
U8_G(x1, x2)  =  U8_G(x2)
CONF19_IN_A(x1)  =  CONF19_IN_A
U4_A(x1, x2)  =  U4_A(x2)
DEL229_IN_AA(x1, x2)  =  DEL229_IN_AA
U10_AA(x1, x2, x3)  =  U10_AA(x3)
DEL34_IN_AAA(x1, x2, x3)  =  DEL34_IN_AAA
U3_AAA(x1, x2, x3, x4, x5)  =  U3_AAA(x5)
U11_AA(x1, x2, x3)  =  U11_AA(x3)
U12_AA(x1, x2, x3)  =  U12_AA(x3)
U5_A(x1, x2)  =  U5_A(x2)
U6_A(x1, x2)  =  U6_A(x2)
U7_A(x1, x2)  =  U7_A(x2)
U8_A(x1, x2)  =  U8_A(x2)

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

(6) Obligation:

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

CONF1_IN_G(T9) → U13_G(T9, del6_in_aga(X23, T9, X24))
CONF1_IN_G(T9) → DEL6_IN_AGA(X23, T9, X24)
DEL6_IN_AGA(X72, .(T25, T26), .(T25, X73)) → U1_AGA(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
DEL6_IN_AGA(X72, .(T25, T26), .(T25, X73)) → DEL6_IN_AGA(X72, T26, X73)
CONF1_IN_G(T9) → U14_G(T9, del6_in_aga(T12, T9, T13))
U14_G(T9, del6_out_aga(T12, T9, T13)) → U15_G(T9, del6_in_aga(X25, T13, X26))
U14_G(T9, del6_out_aga(T12, T9, T13)) → DEL6_IN_AGA(X25, T13, X26)
U14_G(T9, del6_out_aga(T12, T9, T13)) → U16_G(T9, del6_in_aga(T31, T13, T32))
U16_G(T9, del6_out_aga(T31, T13, T32)) → U17_G(T9, del18_in_aag(X4, X5, T32))
U16_G(T9, del6_out_aga(T31, T13, T32)) → DEL18_IN_AAG(X4, X5, T32)
DEL18_IN_AAG(X147, .(X148, X149), .(X148, T46)) → U2_AAG(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
DEL18_IN_AAG(X147, .(X148, X149), .(X148, T46)) → DEL18_IN_AAG(X147, X149, T46)
U16_G(T9, del6_out_aga(T31, T13, T32)) → U18_G(T9, del18_in_aag(T36, T37, T32))
U18_G(T9, del18_out_aag(T36, T37, T32)) → U19_G(T9, conf19_in_g(T37))
U18_G(T9, del18_out_aag(T36, T37, T32)) → CONF19_IN_G(T37)
CONF19_IN_G(T54) → U4_G(T54, del229_in_ga(T54, X174))
CONF19_IN_G(T54) → DEL229_IN_GA(T54, X174)
DEL229_IN_GA(T64, X209) → U10_GA(T64, X209, del34_in_aga(X206, T64, X207))
DEL229_IN_GA(T64, X209) → DEL34_IN_AGA(X206, T64, X207)
DEL34_IN_AGA(X261, .(T82, T84), .(T82, X262)) → U3_AGA(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
DEL34_IN_AGA(X261, .(T82, T84), .(T82, X262)) → DEL34_IN_AGA(X261, T84, X262)
DEL229_IN_GA(T64, X209) → U11_GA(T64, X209, del34_in_aga(T67, T64, T68))
U11_GA(T64, X209, del34_out_aga(T67, T64, T68)) → U12_GA(T64, X209, del34_in_aga(X208, T68, X209))
U11_GA(T64, X209, del34_out_aga(T67, T64, T68)) → DEL34_IN_AGA(X208, T68, X209)
CONF19_IN_G(T54) → U5_G(T54, del229_in_ga(T54, T56))
U5_G(T54, del229_out_ga(T54, T56)) → U6_G(T54, del43_in_aaa(X175, X176, T56))
U5_G(T54, del229_out_ga(T54, T56)) → DEL43_IN_AAA(X175, X176, T56)
DEL43_IN_AAA(X324, .(X325, X326), .(X325, T99)) → U9_AAA(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
DEL43_IN_AAA(X324, .(X325, X326), .(X325, T99)) → DEL43_IN_AAA(X324, X326, T99)
U5_G(T54, del229_out_ga(T54, T56)) → U7_G(T54, del43_in_aaa(T88, T89, T56))
U7_G(T54, del43_out_aaa(T88, T89, T56)) → U8_G(T54, conf19_in_a(T89))
U7_G(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)
CONF19_IN_A(T54) → U4_A(T54, del229_in_aa(T54, X174))
CONF19_IN_A(T54) → DEL229_IN_AA(T54, X174)
DEL229_IN_AA(T64, X209) → U10_AA(T64, X209, del34_in_aaa(X206, T64, X207))
DEL229_IN_AA(T64, X209) → DEL34_IN_AAA(X206, T64, X207)
DEL34_IN_AAA(X261, .(T82, T84), .(T82, X262)) → U3_AAA(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
DEL34_IN_AAA(X261, .(T82, T84), .(T82, X262)) → DEL34_IN_AAA(X261, T84, X262)
DEL229_IN_AA(T64, X209) → U11_AA(T64, X209, del34_in_aaa(T67, T64, T68))
U11_AA(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_AA(T64, X209, del34_in_aaa(X208, T68, X209))
U11_AA(T64, X209, del34_out_aaa(T67, T64, T68)) → DEL34_IN_AAA(X208, T68, X209)
CONF19_IN_A(T54) → U5_A(T54, del229_in_aa(T54, T56))
U5_A(T54, del229_out_aa(T54, T56)) → U6_A(T54, del43_in_aaa(X175, X176, T56))
U5_A(T54, del229_out_aa(T54, T56)) → DEL43_IN_AAA(X175, X176, T56)
U5_A(T54, del229_out_aa(T54, T56)) → U7_A(T54, del43_in_aaa(T88, T89, T56))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → U8_A(T54, conf19_in_a(T89))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
conf1_out_g(x1)  =  conf1_out_g
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x5)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
del229_out_ga(x1, x2)  =  del229_out_ga
U11_ga(x1, x2, x3)  =  U11_ga(x3)
U12_ga(x1, x2, x3)  =  U12_ga(x3)
conf19_out_g(x1)  =  conf19_out_g
U5_g(x1, x2)  =  U5_g(x2)
U6_g(x1, x2)  =  U6_g(x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x2)
U8_g(x1, x2)  =  U8_g(x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)
CONF1_IN_G(x1)  =  CONF1_IN_G(x1)
U13_G(x1, x2)  =  U13_G(x2)
DEL6_IN_AGA(x1, x2, x3)  =  DEL6_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x5)
U14_G(x1, x2)  =  U14_G(x2)
U15_G(x1, x2)  =  U15_G(x2)
U16_G(x1, x2)  =  U16_G(x2)
U17_G(x1, x2)  =  U17_G(x2)
DEL18_IN_AAG(x1, x2, x3)  =  DEL18_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x5)
U18_G(x1, x2)  =  U18_G(x2)
U19_G(x1, x2)  =  U19_G(x2)
CONF19_IN_G(x1)  =  CONF19_IN_G(x1)
U4_G(x1, x2)  =  U4_G(x2)
DEL229_IN_GA(x1, x2)  =  DEL229_IN_GA(x1)
U10_GA(x1, x2, x3)  =  U10_GA(x3)
DEL34_IN_AGA(x1, x2, x3)  =  DEL34_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x5)
U11_GA(x1, x2, x3)  =  U11_GA(x3)
U12_GA(x1, x2, x3)  =  U12_GA(x3)
U5_G(x1, x2)  =  U5_G(x2)
U6_G(x1, x2)  =  U6_G(x2)
DEL43_IN_AAA(x1, x2, x3)  =  DEL43_IN_AAA
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
U7_G(x1, x2)  =  U7_G(x2)
U8_G(x1, x2)  =  U8_G(x2)
CONF19_IN_A(x1)  =  CONF19_IN_A
U4_A(x1, x2)  =  U4_A(x2)
DEL229_IN_AA(x1, x2)  =  DEL229_IN_AA
U10_AA(x1, x2, x3)  =  U10_AA(x3)
DEL34_IN_AAA(x1, x2, x3)  =  DEL34_IN_AAA
U3_AAA(x1, x2, x3, x4, x5)  =  U3_AAA(x5)
U11_AA(x1, x2, x3)  =  U11_AA(x3)
U12_AA(x1, x2, x3)  =  U12_AA(x3)
U5_A(x1, x2)  =  U5_A(x2)
U6_A(x1, x2)  =  U6_A(x2)
U7_A(x1, x2)  =  U7_A(x2)
U8_A(x1, x2)  =  U8_A(x2)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 39 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

DEL34_IN_AAA(X261, .(T82, T84), .(T82, X262)) → DEL34_IN_AAA(X261, T84, X262)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
conf1_out_g(x1)  =  conf1_out_g
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x5)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
del229_out_ga(x1, x2)  =  del229_out_ga
U11_ga(x1, x2, x3)  =  U11_ga(x3)
U12_ga(x1, x2, x3)  =  U12_ga(x3)
conf19_out_g(x1)  =  conf19_out_g
U5_g(x1, x2)  =  U5_g(x2)
U6_g(x1, x2)  =  U6_g(x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x2)
U8_g(x1, x2)  =  U8_g(x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)
DEL34_IN_AAA(x1, x2, x3)  =  DEL34_IN_AAA

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:

DEL34_IN_AAA(X261, .(T82, T84), .(T82, X262)) → DEL34_IN_AAA(X261, T84, X262)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
DEL34_IN_AAA(x1, x2, x3)  =  DEL34_IN_AAA

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:

DEL34_IN_AAADEL34_IN_AAA

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

(14) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = DEL34_IN_AAA evaluates to t =DEL34_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from DEL34_IN_AAA to DEL34_IN_AAA.



(15) NO

(16) Obligation:

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

DEL43_IN_AAA(X324, .(X325, X326), .(X325, T99)) → DEL43_IN_AAA(X324, X326, T99)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
conf1_out_g(x1)  =  conf1_out_g
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x5)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
del229_out_ga(x1, x2)  =  del229_out_ga
U11_ga(x1, x2, x3)  =  U11_ga(x3)
U12_ga(x1, x2, x3)  =  U12_ga(x3)
conf19_out_g(x1)  =  conf19_out_g
U5_g(x1, x2)  =  U5_g(x2)
U6_g(x1, x2)  =  U6_g(x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x2)
U8_g(x1, x2)  =  U8_g(x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)
DEL43_IN_AAA(x1, x2, x3)  =  DEL43_IN_AAA

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:

DEL43_IN_AAA(X324, .(X325, X326), .(X325, T99)) → DEL43_IN_AAA(X324, X326, T99)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
DEL43_IN_AAA(x1, x2, x3)  =  DEL43_IN_AAA

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:

DEL43_IN_AAADEL43_IN_AAA

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

(21) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = DEL43_IN_AAA evaluates to t =DEL43_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from DEL43_IN_AAA to DEL43_IN_AAA.



(22) NO

(23) Obligation:

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

U5_A(T54, del229_out_aa(T54, T56)) → U7_A(T54, del43_in_aaa(T88, T89, T56))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)
CONF19_IN_A(T54) → U5_A(T54, del229_in_aa(T54, T56))

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
conf1_out_g(x1)  =  conf1_out_g
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x5)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
del229_out_ga(x1, x2)  =  del229_out_ga
U11_ga(x1, x2, x3)  =  U11_ga(x3)
U12_ga(x1, x2, x3)  =  U12_ga(x3)
conf19_out_g(x1)  =  conf19_out_g
U5_g(x1, x2)  =  U5_g(x2)
U6_g(x1, x2)  =  U6_g(x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x2)
U8_g(x1, x2)  =  U8_g(x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)
CONF19_IN_A(x1)  =  CONF19_IN_A
U5_A(x1, x2)  =  U5_A(x2)
U7_A(x1, x2)  =  U7_A(x2)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

U5_A(T54, del229_out_aa(T54, T56)) → U7_A(T54, del43_in_aaa(T88, T89, T56))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)
CONF19_IN_A(T54) → U5_A(T54, del229_in_aa(T54, T56))

The TRS R consists of the following rules:

del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
CONF19_IN_A(x1)  =  CONF19_IN_A
U5_A(x1, x2)  =  U5_A(x2)
U7_A(x1, x2)  =  U7_A(x2)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

U5_A(del229_out_aa) → U7_A(del43_in_aaa)
U7_A(del43_out_aaa) → CONF19_IN_A
CONF19_IN_AU5_A(del229_in_aa)

The TRS R consists of the following rules:

del43_in_aaadel43_out_aaa
del43_in_aaaU9_aaa(del43_in_aaa)
del229_in_aaU10_aa(del34_in_aaa)
del229_in_aaU11_aa(del34_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
del34_in_aaadel34_out_aaa
del34_in_aaaU3_aaa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa

The set Q consists of the following terms:

del43_in_aaa
del229_in_aa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(28) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U5_A(del229_out_aa) → U7_A(del43_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))

(29) Obligation:

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

U7_A(del43_out_aaa) → CONF19_IN_A
CONF19_IN_AU5_A(del229_in_aa)
U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))

The TRS R consists of the following rules:

del43_in_aaadel43_out_aaa
del43_in_aaaU9_aaa(del43_in_aaa)
del229_in_aaU10_aa(del34_in_aaa)
del229_in_aaU11_aa(del34_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
del34_in_aaadel34_out_aaa
del34_in_aaaU3_aaa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa

The set Q consists of the following terms:

del43_in_aaa
del229_in_aa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(30) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule CONF19_IN_AU5_A(del229_in_aa) at position [0] we obtained the following new rules [LPAR04]:

CONF19_IN_AU5_A(U10_aa(del34_in_aaa))
CONF19_IN_AU5_A(U11_aa(del34_in_aaa))

(31) Obligation:

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

U7_A(del43_out_aaa) → CONF19_IN_A
U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))
CONF19_IN_AU5_A(U10_aa(del34_in_aaa))
CONF19_IN_AU5_A(U11_aa(del34_in_aaa))

The TRS R consists of the following rules:

del43_in_aaadel43_out_aaa
del43_in_aaaU9_aaa(del43_in_aaa)
del229_in_aaU10_aa(del34_in_aaa)
del229_in_aaU11_aa(del34_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
del34_in_aaadel34_out_aaa
del34_in_aaaU3_aaa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa

The set Q consists of the following terms:

del43_in_aaa
del229_in_aa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(32) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(33) Obligation:

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

U7_A(del43_out_aaa) → CONF19_IN_A
U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))
CONF19_IN_AU5_A(U10_aa(del34_in_aaa))
CONF19_IN_AU5_A(U11_aa(del34_in_aaa))

The TRS R consists of the following rules:

del34_in_aaadel34_out_aaa
del34_in_aaaU3_aaa(del34_in_aaa)
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
del43_in_aaadel43_out_aaa
del43_in_aaaU9_aaa(del43_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa

The set Q consists of the following terms:

del43_in_aaa
del229_in_aa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(34) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

del229_in_aa

(35) Obligation:

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

U7_A(del43_out_aaa) → CONF19_IN_A
U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))
CONF19_IN_AU5_A(U10_aa(del34_in_aaa))
CONF19_IN_AU5_A(U11_aa(del34_in_aaa))

The TRS R consists of the following rules:

del34_in_aaadel34_out_aaa
del34_in_aaaU3_aaa(del34_in_aaa)
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
del43_in_aaadel43_out_aaa
del43_in_aaaU9_aaa(del43_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa

The set Q consists of the following terms:

del43_in_aaa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(36) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U5_A(U10_aa(del34_in_aaa)) evaluates to t =U5_A(U10_aa(del34_in_aaa))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]




Rewriting sequence

U5_A(U10_aa(del34_in_aaa))U5_A(U10_aa(del34_out_aaa))
with rule del34_in_aaadel34_out_aaa at position [0,0] and matcher [ ]

U5_A(U10_aa(del34_out_aaa))U5_A(del229_out_aa)
with rule U10_aa(del34_out_aaa) → del229_out_aa at position [0] and matcher [ ]

U5_A(del229_out_aa)U7_A(del43_out_aaa)
with rule U5_A(del229_out_aa) → U7_A(del43_out_aaa) at position [] and matcher [ ]

U7_A(del43_out_aaa)CONF19_IN_A
with rule U7_A(del43_out_aaa) → CONF19_IN_A at position [] and matcher [ ]

CONF19_IN_AU5_A(U10_aa(del34_in_aaa))
with rule CONF19_IN_AU5_A(U10_aa(del34_in_aaa))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.



(37) NO

(38) Obligation:

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

DEL34_IN_AGA(X261, .(T82, T84), .(T82, X262)) → DEL34_IN_AGA(X261, T84, X262)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
conf1_out_g(x1)  =  conf1_out_g
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x5)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
del229_out_ga(x1, x2)  =  del229_out_ga
U11_ga(x1, x2, x3)  =  U11_ga(x3)
U12_ga(x1, x2, x3)  =  U12_ga(x3)
conf19_out_g(x1)  =  conf19_out_g
U5_g(x1, x2)  =  U5_g(x2)
U6_g(x1, x2)  =  U6_g(x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x2)
U8_g(x1, x2)  =  U8_g(x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)
DEL34_IN_AGA(x1, x2, x3)  =  DEL34_IN_AGA(x2)

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

(39) UsableRulesProof (EQUIVALENT transformation)

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

(40) Obligation:

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

DEL34_IN_AGA(X261, .(T82, T84), .(T82, X262)) → DEL34_IN_AGA(X261, T84, X262)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
DEL34_IN_AGA(x1, x2, x3)  =  DEL34_IN_AGA(x2)

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

(41) PiDPToQDPProof (SOUND transformation)

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

(42) Obligation:

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

DEL34_IN_AGA(.(T84)) → DEL34_IN_AGA(T84)

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

(43) 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:

  • DEL34_IN_AGA(.(T84)) → DEL34_IN_AGA(T84)
    The graph contains the following edges 1 > 1

(44) YES

(45) Obligation:

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

DEL18_IN_AAG(X147, .(X148, X149), .(X148, T46)) → DEL18_IN_AAG(X147, X149, T46)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
conf1_out_g(x1)  =  conf1_out_g
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x5)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
del229_out_ga(x1, x2)  =  del229_out_ga
U11_ga(x1, x2, x3)  =  U11_ga(x3)
U12_ga(x1, x2, x3)  =  U12_ga(x3)
conf19_out_g(x1)  =  conf19_out_g
U5_g(x1, x2)  =  U5_g(x2)
U6_g(x1, x2)  =  U6_g(x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x2)
U8_g(x1, x2)  =  U8_g(x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)
DEL18_IN_AAG(x1, x2, x3)  =  DEL18_IN_AAG(x3)

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

(46) UsableRulesProof (EQUIVALENT transformation)

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

(47) Obligation:

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

DEL18_IN_AAG(X147, .(X148, X149), .(X148, T46)) → DEL18_IN_AAG(X147, X149, T46)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
DEL18_IN_AAG(x1, x2, x3)  =  DEL18_IN_AAG(x3)

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

(48) PiDPToQDPProof (SOUND transformation)

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

(49) Obligation:

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

DEL18_IN_AAG(.(T46)) → DEL18_IN_AAG(T46)

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

(50) 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:

  • DEL18_IN_AAG(.(T46)) → DEL18_IN_AAG(T46)
    The graph contains the following edges 1 > 1

(51) YES

(52) Obligation:

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

DEL6_IN_AGA(X72, .(T25, T26), .(T25, X73)) → DEL6_IN_AGA(X72, T26, X73)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
conf1_out_g(x1)  =  conf1_out_g
U14_g(x1, x2)  =  U14_g(x2)
U15_g(x1, x2)  =  U15_g(x2)
U16_g(x1, x2)  =  U16_g(x2)
U17_g(x1, x2)  =  U17_g(x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x5)
U18_g(x1, x2)  =  U18_g(x2)
U19_g(x1, x2)  =  U19_g(x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
del229_out_ga(x1, x2)  =  del229_out_ga
U11_ga(x1, x2, x3)  =  U11_ga(x3)
U12_ga(x1, x2, x3)  =  U12_ga(x3)
conf19_out_g(x1)  =  conf19_out_g
U5_g(x1, x2)  =  U5_g(x2)
U6_g(x1, x2)  =  U6_g(x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x2)
U8_g(x1, x2)  =  U8_g(x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)
DEL6_IN_AGA(x1, x2, x3)  =  DEL6_IN_AGA(x2)

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

(53) UsableRulesProof (EQUIVALENT transformation)

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

(54) Obligation:

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

DEL6_IN_AGA(X72, .(T25, T26), .(T25, X73)) → DEL6_IN_AGA(X72, T26, X73)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
DEL6_IN_AGA(x1, x2, x3)  =  DEL6_IN_AGA(x2)

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

(55) PiDPToQDPProof (SOUND transformation)

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

(56) Obligation:

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

DEL6_IN_AGA(.(T26)) → DEL6_IN_AGA(T26)

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

(57) 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:

  • DEL6_IN_AGA(.(T26)) → DEL6_IN_AGA(T26)
    The graph contains the following edges 1 > 1

(58) YES

(59) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
conf1_in: (b)
del6_in: (f,b,f)
del18_in: (f,f,b)
conf19_in: (b) (f)
del229_in: (b,f) (f,f)
del34_in: (f,b,f) (f,f,f)
del43_in: (f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x1, x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
conf1_out_g(x1)  =  conf1_out_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2, x3)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x4, x5)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x3, x5)
del229_out_ga(x1, x2)  =  del229_out_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
conf19_out_g(x1)  =  conf19_out_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
U6_g(x1, x2)  =  U6_g(x1, x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x1, x2)
U8_g(x1, x2)  =  U8_g(x1, x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(60) Obligation:

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

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x1, x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
conf1_out_g(x1)  =  conf1_out_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2, x3)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x4, x5)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x3, x5)
del229_out_ga(x1, x2)  =  del229_out_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
conf19_out_g(x1)  =  conf19_out_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
U6_g(x1, x2)  =  U6_g(x1, x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x1, x2)
U8_g(x1, x2)  =  U8_g(x1, x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)

(61) 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:

CONF1_IN_G(T9) → U13_G(T9, del6_in_aga(X23, T9, X24))
CONF1_IN_G(T9) → DEL6_IN_AGA(X23, T9, X24)
DEL6_IN_AGA(X72, .(T25, T26), .(T25, X73)) → U1_AGA(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
DEL6_IN_AGA(X72, .(T25, T26), .(T25, X73)) → DEL6_IN_AGA(X72, T26, X73)
CONF1_IN_G(T9) → U14_G(T9, del6_in_aga(T12, T9, T13))
U14_G(T9, del6_out_aga(T12, T9, T13)) → U15_G(T9, del6_in_aga(X25, T13, X26))
U14_G(T9, del6_out_aga(T12, T9, T13)) → DEL6_IN_AGA(X25, T13, X26)
U14_G(T9, del6_out_aga(T12, T9, T13)) → U16_G(T9, del6_in_aga(T31, T13, T32))
U16_G(T9, del6_out_aga(T31, T13, T32)) → U17_G(T9, del18_in_aag(X4, X5, T32))
U16_G(T9, del6_out_aga(T31, T13, T32)) → DEL18_IN_AAG(X4, X5, T32)
DEL18_IN_AAG(X147, .(X148, X149), .(X148, T46)) → U2_AAG(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
DEL18_IN_AAG(X147, .(X148, X149), .(X148, T46)) → DEL18_IN_AAG(X147, X149, T46)
U16_G(T9, del6_out_aga(T31, T13, T32)) → U18_G(T9, del18_in_aag(T36, T37, T32))
U18_G(T9, del18_out_aag(T36, T37, T32)) → U19_G(T9, conf19_in_g(T37))
U18_G(T9, del18_out_aag(T36, T37, T32)) → CONF19_IN_G(T37)
CONF19_IN_G(T54) → U4_G(T54, del229_in_ga(T54, X174))
CONF19_IN_G(T54) → DEL229_IN_GA(T54, X174)
DEL229_IN_GA(T64, X209) → U10_GA(T64, X209, del34_in_aga(X206, T64, X207))
DEL229_IN_GA(T64, X209) → DEL34_IN_AGA(X206, T64, X207)
DEL34_IN_AGA(X261, .(T82, T84), .(T82, X262)) → U3_AGA(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
DEL34_IN_AGA(X261, .(T82, T84), .(T82, X262)) → DEL34_IN_AGA(X261, T84, X262)
DEL229_IN_GA(T64, X209) → U11_GA(T64, X209, del34_in_aga(T67, T64, T68))
U11_GA(T64, X209, del34_out_aga(T67, T64, T68)) → U12_GA(T64, X209, del34_in_aga(X208, T68, X209))
U11_GA(T64, X209, del34_out_aga(T67, T64, T68)) → DEL34_IN_AGA(X208, T68, X209)
CONF19_IN_G(T54) → U5_G(T54, del229_in_ga(T54, T56))
U5_G(T54, del229_out_ga(T54, T56)) → U6_G(T54, del43_in_aaa(X175, X176, T56))
U5_G(T54, del229_out_ga(T54, T56)) → DEL43_IN_AAA(X175, X176, T56)
DEL43_IN_AAA(X324, .(X325, X326), .(X325, T99)) → U9_AAA(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
DEL43_IN_AAA(X324, .(X325, X326), .(X325, T99)) → DEL43_IN_AAA(X324, X326, T99)
U5_G(T54, del229_out_ga(T54, T56)) → U7_G(T54, del43_in_aaa(T88, T89, T56))
U7_G(T54, del43_out_aaa(T88, T89, T56)) → U8_G(T54, conf19_in_a(T89))
U7_G(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)
CONF19_IN_A(T54) → U4_A(T54, del229_in_aa(T54, X174))
CONF19_IN_A(T54) → DEL229_IN_AA(T54, X174)
DEL229_IN_AA(T64, X209) → U10_AA(T64, X209, del34_in_aaa(X206, T64, X207))
DEL229_IN_AA(T64, X209) → DEL34_IN_AAA(X206, T64, X207)
DEL34_IN_AAA(X261, .(T82, T84), .(T82, X262)) → U3_AAA(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
DEL34_IN_AAA(X261, .(T82, T84), .(T82, X262)) → DEL34_IN_AAA(X261, T84, X262)
DEL229_IN_AA(T64, X209) → U11_AA(T64, X209, del34_in_aaa(T67, T64, T68))
U11_AA(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_AA(T64, X209, del34_in_aaa(X208, T68, X209))
U11_AA(T64, X209, del34_out_aaa(T67, T64, T68)) → DEL34_IN_AAA(X208, T68, X209)
CONF19_IN_A(T54) → U5_A(T54, del229_in_aa(T54, T56))
U5_A(T54, del229_out_aa(T54, T56)) → U6_A(T54, del43_in_aaa(X175, X176, T56))
U5_A(T54, del229_out_aa(T54, T56)) → DEL43_IN_AAA(X175, X176, T56)
U5_A(T54, del229_out_aa(T54, T56)) → U7_A(T54, del43_in_aaa(T88, T89, T56))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → U8_A(T54, conf19_in_a(T89))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x1, x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
conf1_out_g(x1)  =  conf1_out_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2, x3)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x4, x5)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x3, x5)
del229_out_ga(x1, x2)  =  del229_out_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
conf19_out_g(x1)  =  conf19_out_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
U6_g(x1, x2)  =  U6_g(x1, x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x1, x2)
U8_g(x1, x2)  =  U8_g(x1, x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)
CONF1_IN_G(x1)  =  CONF1_IN_G(x1)
U13_G(x1, x2)  =  U13_G(x1, x2)
DEL6_IN_AGA(x1, x2, x3)  =  DEL6_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, x5)
U14_G(x1, x2)  =  U14_G(x1, x2)
U15_G(x1, x2)  =  U15_G(x1, x2)
U16_G(x1, x2)  =  U16_G(x1, x2)
U17_G(x1, x2)  =  U17_G(x1, x2)
DEL18_IN_AAG(x1, x2, x3)  =  DEL18_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x4, x5)
U18_G(x1, x2)  =  U18_G(x1, x2)
U19_G(x1, x2)  =  U19_G(x1, x2)
CONF19_IN_G(x1)  =  CONF19_IN_G(x1)
U4_G(x1, x2)  =  U4_G(x1, x2)
DEL229_IN_GA(x1, x2)  =  DEL229_IN_GA(x1)
U10_GA(x1, x2, x3)  =  U10_GA(x1, x3)
DEL34_IN_AGA(x1, x2, x3)  =  DEL34_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x3, x5)
U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)
U12_GA(x1, x2, x3)  =  U12_GA(x1, x3)
U5_G(x1, x2)  =  U5_G(x1, x2)
U6_G(x1, x2)  =  U6_G(x1, x2)
DEL43_IN_AAA(x1, x2, x3)  =  DEL43_IN_AAA
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
U7_G(x1, x2)  =  U7_G(x1, x2)
U8_G(x1, x2)  =  U8_G(x1, x2)
CONF19_IN_A(x1)  =  CONF19_IN_A
U4_A(x1, x2)  =  U4_A(x2)
DEL229_IN_AA(x1, x2)  =  DEL229_IN_AA
U10_AA(x1, x2, x3)  =  U10_AA(x3)
DEL34_IN_AAA(x1, x2, x3)  =  DEL34_IN_AAA
U3_AAA(x1, x2, x3, x4, x5)  =  U3_AAA(x5)
U11_AA(x1, x2, x3)  =  U11_AA(x3)
U12_AA(x1, x2, x3)  =  U12_AA(x3)
U5_A(x1, x2)  =  U5_A(x2)
U6_A(x1, x2)  =  U6_A(x2)
U7_A(x1, x2)  =  U7_A(x2)
U8_A(x1, x2)  =  U8_A(x2)

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

(62) Obligation:

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

CONF1_IN_G(T9) → U13_G(T9, del6_in_aga(X23, T9, X24))
CONF1_IN_G(T9) → DEL6_IN_AGA(X23, T9, X24)
DEL6_IN_AGA(X72, .(T25, T26), .(T25, X73)) → U1_AGA(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
DEL6_IN_AGA(X72, .(T25, T26), .(T25, X73)) → DEL6_IN_AGA(X72, T26, X73)
CONF1_IN_G(T9) → U14_G(T9, del6_in_aga(T12, T9, T13))
U14_G(T9, del6_out_aga(T12, T9, T13)) → U15_G(T9, del6_in_aga(X25, T13, X26))
U14_G(T9, del6_out_aga(T12, T9, T13)) → DEL6_IN_AGA(X25, T13, X26)
U14_G(T9, del6_out_aga(T12, T9, T13)) → U16_G(T9, del6_in_aga(T31, T13, T32))
U16_G(T9, del6_out_aga(T31, T13, T32)) → U17_G(T9, del18_in_aag(X4, X5, T32))
U16_G(T9, del6_out_aga(T31, T13, T32)) → DEL18_IN_AAG(X4, X5, T32)
DEL18_IN_AAG(X147, .(X148, X149), .(X148, T46)) → U2_AAG(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
DEL18_IN_AAG(X147, .(X148, X149), .(X148, T46)) → DEL18_IN_AAG(X147, X149, T46)
U16_G(T9, del6_out_aga(T31, T13, T32)) → U18_G(T9, del18_in_aag(T36, T37, T32))
U18_G(T9, del18_out_aag(T36, T37, T32)) → U19_G(T9, conf19_in_g(T37))
U18_G(T9, del18_out_aag(T36, T37, T32)) → CONF19_IN_G(T37)
CONF19_IN_G(T54) → U4_G(T54, del229_in_ga(T54, X174))
CONF19_IN_G(T54) → DEL229_IN_GA(T54, X174)
DEL229_IN_GA(T64, X209) → U10_GA(T64, X209, del34_in_aga(X206, T64, X207))
DEL229_IN_GA(T64, X209) → DEL34_IN_AGA(X206, T64, X207)
DEL34_IN_AGA(X261, .(T82, T84), .(T82, X262)) → U3_AGA(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
DEL34_IN_AGA(X261, .(T82, T84), .(T82, X262)) → DEL34_IN_AGA(X261, T84, X262)
DEL229_IN_GA(T64, X209) → U11_GA(T64, X209, del34_in_aga(T67, T64, T68))
U11_GA(T64, X209, del34_out_aga(T67, T64, T68)) → U12_GA(T64, X209, del34_in_aga(X208, T68, X209))
U11_GA(T64, X209, del34_out_aga(T67, T64, T68)) → DEL34_IN_AGA(X208, T68, X209)
CONF19_IN_G(T54) → U5_G(T54, del229_in_ga(T54, T56))
U5_G(T54, del229_out_ga(T54, T56)) → U6_G(T54, del43_in_aaa(X175, X176, T56))
U5_G(T54, del229_out_ga(T54, T56)) → DEL43_IN_AAA(X175, X176, T56)
DEL43_IN_AAA(X324, .(X325, X326), .(X325, T99)) → U9_AAA(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
DEL43_IN_AAA(X324, .(X325, X326), .(X325, T99)) → DEL43_IN_AAA(X324, X326, T99)
U5_G(T54, del229_out_ga(T54, T56)) → U7_G(T54, del43_in_aaa(T88, T89, T56))
U7_G(T54, del43_out_aaa(T88, T89, T56)) → U8_G(T54, conf19_in_a(T89))
U7_G(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)
CONF19_IN_A(T54) → U4_A(T54, del229_in_aa(T54, X174))
CONF19_IN_A(T54) → DEL229_IN_AA(T54, X174)
DEL229_IN_AA(T64, X209) → U10_AA(T64, X209, del34_in_aaa(X206, T64, X207))
DEL229_IN_AA(T64, X209) → DEL34_IN_AAA(X206, T64, X207)
DEL34_IN_AAA(X261, .(T82, T84), .(T82, X262)) → U3_AAA(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
DEL34_IN_AAA(X261, .(T82, T84), .(T82, X262)) → DEL34_IN_AAA(X261, T84, X262)
DEL229_IN_AA(T64, X209) → U11_AA(T64, X209, del34_in_aaa(T67, T64, T68))
U11_AA(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_AA(T64, X209, del34_in_aaa(X208, T68, X209))
U11_AA(T64, X209, del34_out_aaa(T67, T64, T68)) → DEL34_IN_AAA(X208, T68, X209)
CONF19_IN_A(T54) → U5_A(T54, del229_in_aa(T54, T56))
U5_A(T54, del229_out_aa(T54, T56)) → U6_A(T54, del43_in_aaa(X175, X176, T56))
U5_A(T54, del229_out_aa(T54, T56)) → DEL43_IN_AAA(X175, X176, T56)
U5_A(T54, del229_out_aa(T54, T56)) → U7_A(T54, del43_in_aaa(T88, T89, T56))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → U8_A(T54, conf19_in_a(T89))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x1, x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
conf1_out_g(x1)  =  conf1_out_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2, x3)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x4, x5)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x3, x5)
del229_out_ga(x1, x2)  =  del229_out_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
conf19_out_g(x1)  =  conf19_out_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
U6_g(x1, x2)  =  U6_g(x1, x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x1, x2)
U8_g(x1, x2)  =  U8_g(x1, x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)
CONF1_IN_G(x1)  =  CONF1_IN_G(x1)
U13_G(x1, x2)  =  U13_G(x1, x2)
DEL6_IN_AGA(x1, x2, x3)  =  DEL6_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, x5)
U14_G(x1, x2)  =  U14_G(x1, x2)
U15_G(x1, x2)  =  U15_G(x1, x2)
U16_G(x1, x2)  =  U16_G(x1, x2)
U17_G(x1, x2)  =  U17_G(x1, x2)
DEL18_IN_AAG(x1, x2, x3)  =  DEL18_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x4, x5)
U18_G(x1, x2)  =  U18_G(x1, x2)
U19_G(x1, x2)  =  U19_G(x1, x2)
CONF19_IN_G(x1)  =  CONF19_IN_G(x1)
U4_G(x1, x2)  =  U4_G(x1, x2)
DEL229_IN_GA(x1, x2)  =  DEL229_IN_GA(x1)
U10_GA(x1, x2, x3)  =  U10_GA(x1, x3)
DEL34_IN_AGA(x1, x2, x3)  =  DEL34_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x3, x5)
U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)
U12_GA(x1, x2, x3)  =  U12_GA(x1, x3)
U5_G(x1, x2)  =  U5_G(x1, x2)
U6_G(x1, x2)  =  U6_G(x1, x2)
DEL43_IN_AAA(x1, x2, x3)  =  DEL43_IN_AAA
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
U7_G(x1, x2)  =  U7_G(x1, x2)
U8_G(x1, x2)  =  U8_G(x1, x2)
CONF19_IN_A(x1)  =  CONF19_IN_A
U4_A(x1, x2)  =  U4_A(x2)
DEL229_IN_AA(x1, x2)  =  DEL229_IN_AA
U10_AA(x1, x2, x3)  =  U10_AA(x3)
DEL34_IN_AAA(x1, x2, x3)  =  DEL34_IN_AAA
U3_AAA(x1, x2, x3, x4, x5)  =  U3_AAA(x5)
U11_AA(x1, x2, x3)  =  U11_AA(x3)
U12_AA(x1, x2, x3)  =  U12_AA(x3)
U5_A(x1, x2)  =  U5_A(x2)
U6_A(x1, x2)  =  U6_A(x2)
U7_A(x1, x2)  =  U7_A(x2)
U8_A(x1, x2)  =  U8_A(x2)

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

(63) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 39 less nodes.

(64) Complex Obligation (AND)

(65) Obligation:

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

DEL34_IN_AAA(X261, .(T82, T84), .(T82, X262)) → DEL34_IN_AAA(X261, T84, X262)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x1, x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
conf1_out_g(x1)  =  conf1_out_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2, x3)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x4, x5)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x3, x5)
del229_out_ga(x1, x2)  =  del229_out_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
conf19_out_g(x1)  =  conf19_out_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
U6_g(x1, x2)  =  U6_g(x1, x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x1, x2)
U8_g(x1, x2)  =  U8_g(x1, x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)
DEL34_IN_AAA(x1, x2, x3)  =  DEL34_IN_AAA

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

(66) UsableRulesProof (EQUIVALENT transformation)

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

(67) Obligation:

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

DEL34_IN_AAA(X261, .(T82, T84), .(T82, X262)) → DEL34_IN_AAA(X261, T84, X262)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
DEL34_IN_AAA(x1, x2, x3)  =  DEL34_IN_AAA

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

(68) PiDPToQDPProof (SOUND transformation)

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

(69) Obligation:

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

DEL34_IN_AAADEL34_IN_AAA

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

(70) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = DEL34_IN_AAA evaluates to t =DEL34_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from DEL34_IN_AAA to DEL34_IN_AAA.



(71) NO

(72) Obligation:

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

DEL43_IN_AAA(X324, .(X325, X326), .(X325, T99)) → DEL43_IN_AAA(X324, X326, T99)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x1, x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
conf1_out_g(x1)  =  conf1_out_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2, x3)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x4, x5)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x3, x5)
del229_out_ga(x1, x2)  =  del229_out_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
conf19_out_g(x1)  =  conf19_out_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
U6_g(x1, x2)  =  U6_g(x1, x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x1, x2)
U8_g(x1, x2)  =  U8_g(x1, x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)
DEL43_IN_AAA(x1, x2, x3)  =  DEL43_IN_AAA

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

(73) UsableRulesProof (EQUIVALENT transformation)

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

(74) Obligation:

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

DEL43_IN_AAA(X324, .(X325, X326), .(X325, T99)) → DEL43_IN_AAA(X324, X326, T99)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
DEL43_IN_AAA(x1, x2, x3)  =  DEL43_IN_AAA

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

(75) PiDPToQDPProof (SOUND transformation)

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

(76) Obligation:

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

DEL43_IN_AAADEL43_IN_AAA

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

(77) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = DEL43_IN_AAA evaluates to t =DEL43_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from DEL43_IN_AAA to DEL43_IN_AAA.



(78) NO

(79) Obligation:

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

U5_A(T54, del229_out_aa(T54, T56)) → U7_A(T54, del43_in_aaa(T88, T89, T56))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)
CONF19_IN_A(T54) → U5_A(T54, del229_in_aa(T54, T56))

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x1, x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
conf1_out_g(x1)  =  conf1_out_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2, x3)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x4, x5)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x3, x5)
del229_out_ga(x1, x2)  =  del229_out_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
conf19_out_g(x1)  =  conf19_out_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
U6_g(x1, x2)  =  U6_g(x1, x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x1, x2)
U8_g(x1, x2)  =  U8_g(x1, x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)
CONF19_IN_A(x1)  =  CONF19_IN_A
U5_A(x1, x2)  =  U5_A(x2)
U7_A(x1, x2)  =  U7_A(x2)

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

(80) UsableRulesProof (EQUIVALENT transformation)

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

(81) Obligation:

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

U5_A(T54, del229_out_aa(T54, T56)) → U7_A(T54, del43_in_aaa(T88, T89, T56))
U7_A(T54, del43_out_aaa(T88, T89, T56)) → CONF19_IN_A(T89)
CONF19_IN_A(T54) → U5_A(T54, del229_in_aa(T54, T56))

The TRS R consists of the following rules:

del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
CONF19_IN_A(x1)  =  CONF19_IN_A
U5_A(x1, x2)  =  U5_A(x2)
U7_A(x1, x2)  =  U7_A(x2)

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

(82) PiDPToQDPProof (SOUND transformation)

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

(83) Obligation:

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

U5_A(del229_out_aa) → U7_A(del43_in_aaa)
U7_A(del43_out_aaa) → CONF19_IN_A
CONF19_IN_AU5_A(del229_in_aa)

The TRS R consists of the following rules:

del43_in_aaadel43_out_aaa
del43_in_aaaU9_aaa(del43_in_aaa)
del229_in_aaU10_aa(del34_in_aaa)
del229_in_aaU11_aa(del34_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
del34_in_aaadel34_out_aaa
del34_in_aaaU3_aaa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa

The set Q consists of the following terms:

del43_in_aaa
del229_in_aa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(84) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U5_A(del229_out_aa) → U7_A(del43_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))

(85) Obligation:

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

U7_A(del43_out_aaa) → CONF19_IN_A
CONF19_IN_AU5_A(del229_in_aa)
U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))

The TRS R consists of the following rules:

del43_in_aaadel43_out_aaa
del43_in_aaaU9_aaa(del43_in_aaa)
del229_in_aaU10_aa(del34_in_aaa)
del229_in_aaU11_aa(del34_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
del34_in_aaadel34_out_aaa
del34_in_aaaU3_aaa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa

The set Q consists of the following terms:

del43_in_aaa
del229_in_aa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(86) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule CONF19_IN_AU5_A(del229_in_aa) at position [0] we obtained the following new rules [LPAR04]:

CONF19_IN_AU5_A(U10_aa(del34_in_aaa))
CONF19_IN_AU5_A(U11_aa(del34_in_aaa))

(87) Obligation:

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

U7_A(del43_out_aaa) → CONF19_IN_A
U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))
CONF19_IN_AU5_A(U10_aa(del34_in_aaa))
CONF19_IN_AU5_A(U11_aa(del34_in_aaa))

The TRS R consists of the following rules:

del43_in_aaadel43_out_aaa
del43_in_aaaU9_aaa(del43_in_aaa)
del229_in_aaU10_aa(del34_in_aaa)
del229_in_aaU11_aa(del34_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
del34_in_aaadel34_out_aaa
del34_in_aaaU3_aaa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa

The set Q consists of the following terms:

del43_in_aaa
del229_in_aa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(88) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(89) Obligation:

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

U7_A(del43_out_aaa) → CONF19_IN_A
U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))
CONF19_IN_AU5_A(U10_aa(del34_in_aaa))
CONF19_IN_AU5_A(U11_aa(del34_in_aaa))

The TRS R consists of the following rules:

del34_in_aaadel34_out_aaa
del34_in_aaaU3_aaa(del34_in_aaa)
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
del43_in_aaadel43_out_aaa
del43_in_aaaU9_aaa(del43_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa

The set Q consists of the following terms:

del43_in_aaa
del229_in_aa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(90) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

del229_in_aa

(91) Obligation:

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

U7_A(del43_out_aaa) → CONF19_IN_A
U5_A(del229_out_aa) → U7_A(del43_out_aaa)
U5_A(del229_out_aa) → U7_A(U9_aaa(del43_in_aaa))
CONF19_IN_AU5_A(U10_aa(del34_in_aaa))
CONF19_IN_AU5_A(U11_aa(del34_in_aaa))

The TRS R consists of the following rules:

del34_in_aaadel34_out_aaa
del34_in_aaaU3_aaa(del34_in_aaa)
U11_aa(del34_out_aaa) → U12_aa(del34_in_aaa)
U12_aa(del34_out_aaa) → del229_out_aa
U3_aaa(del34_out_aaa) → del34_out_aaa
U10_aa(del34_out_aaa) → del229_out_aa
del43_in_aaadel43_out_aaa
del43_in_aaaU9_aaa(del43_in_aaa)
U9_aaa(del43_out_aaa) → del43_out_aaa

The set Q consists of the following terms:

del43_in_aaa
U9_aaa(x0)
U10_aa(x0)
U11_aa(x0)
del34_in_aaa
U12_aa(x0)
U3_aaa(x0)

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

(92) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U5_A(U10_aa(del34_in_aaa)) evaluates to t =U5_A(U10_aa(del34_in_aaa))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]




Rewriting sequence

U5_A(U10_aa(del34_in_aaa))U5_A(U10_aa(del34_out_aaa))
with rule del34_in_aaadel34_out_aaa at position [0,0] and matcher [ ]

U5_A(U10_aa(del34_out_aaa))U5_A(del229_out_aa)
with rule U10_aa(del34_out_aaa) → del229_out_aa at position [0] and matcher [ ]

U5_A(del229_out_aa)U7_A(del43_out_aaa)
with rule U5_A(del229_out_aa) → U7_A(del43_out_aaa) at position [] and matcher [ ]

U7_A(del43_out_aaa)CONF19_IN_A
with rule U7_A(del43_out_aaa) → CONF19_IN_A at position [] and matcher [ ]

CONF19_IN_AU5_A(U10_aa(del34_in_aaa))
with rule CONF19_IN_AU5_A(U10_aa(del34_in_aaa))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.



(93) NO

(94) Obligation:

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

DEL34_IN_AGA(X261, .(T82, T84), .(T82, X262)) → DEL34_IN_AGA(X261, T84, X262)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x1, x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
conf1_out_g(x1)  =  conf1_out_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2, x3)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x4, x5)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x3, x5)
del229_out_ga(x1, x2)  =  del229_out_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
conf19_out_g(x1)  =  conf19_out_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
U6_g(x1, x2)  =  U6_g(x1, x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x1, x2)
U8_g(x1, x2)  =  U8_g(x1, x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)
DEL34_IN_AGA(x1, x2, x3)  =  DEL34_IN_AGA(x2)

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

(95) UsableRulesProof (EQUIVALENT transformation)

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

(96) Obligation:

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

DEL34_IN_AGA(X261, .(T82, T84), .(T82, X262)) → DEL34_IN_AGA(X261, T84, X262)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
DEL34_IN_AGA(x1, x2, x3)  =  DEL34_IN_AGA(x2)

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

(97) PiDPToQDPProof (SOUND transformation)

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

(98) Obligation:

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

DEL34_IN_AGA(.(T84)) → DEL34_IN_AGA(T84)

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

(99) 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:

  • DEL34_IN_AGA(.(T84)) → DEL34_IN_AGA(T84)
    The graph contains the following edges 1 > 1

(100) YES

(101) Obligation:

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

DEL18_IN_AAG(X147, .(X148, X149), .(X148, T46)) → DEL18_IN_AAG(X147, X149, T46)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x1, x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
conf1_out_g(x1)  =  conf1_out_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2, x3)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x4, x5)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x3, x5)
del229_out_ga(x1, x2)  =  del229_out_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
conf19_out_g(x1)  =  conf19_out_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
U6_g(x1, x2)  =  U6_g(x1, x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x1, x2)
U8_g(x1, x2)  =  U8_g(x1, x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)
DEL18_IN_AAG(x1, x2, x3)  =  DEL18_IN_AAG(x3)

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

(102) UsableRulesProof (EQUIVALENT transformation)

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

(103) Obligation:

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

DEL18_IN_AAG(X147, .(X148, X149), .(X148, T46)) → DEL18_IN_AAG(X147, X149, T46)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
DEL18_IN_AAG(x1, x2, x3)  =  DEL18_IN_AAG(x3)

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

(104) PiDPToQDPProof (SOUND transformation)

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

(105) Obligation:

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

DEL18_IN_AAG(.(T46)) → DEL18_IN_AAG(T46)

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

(106) 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:

  • DEL18_IN_AAG(.(T46)) → DEL18_IN_AAG(T46)
    The graph contains the following edges 1 > 1

(107) YES

(108) Obligation:

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

DEL6_IN_AGA(X72, .(T25, T26), .(T25, X73)) → DEL6_IN_AGA(X72, T26, X73)

The TRS R consists of the following rules:

conf1_in_g(T9) → U13_g(T9, del6_in_aga(X23, T9, X24))
del6_in_aga(X53, .(X53, T20), T20) → del6_out_aga(X53, .(X53, T20), T20)
del6_in_aga(X72, .(T25, T26), .(T25, X73)) → U1_aga(X72, T25, T26, X73, del6_in_aga(X72, T26, X73))
U1_aga(X72, T25, T26, X73, del6_out_aga(X72, T26, X73)) → del6_out_aga(X72, .(T25, T26), .(T25, X73))
U13_g(T9, del6_out_aga(X23, T9, X24)) → conf1_out_g(T9)
conf1_in_g(T9) → U14_g(T9, del6_in_aga(T12, T9, T13))
U14_g(T9, del6_out_aga(T12, T9, T13)) → U15_g(T9, del6_in_aga(X25, T13, X26))
U15_g(T9, del6_out_aga(X25, T13, X26)) → conf1_out_g(T9)
U14_g(T9, del6_out_aga(T12, T9, T13)) → U16_g(T9, del6_in_aga(T31, T13, T32))
U16_g(T9, del6_out_aga(T31, T13, T32)) → U17_g(T9, del18_in_aag(X4, X5, T32))
del18_in_aag(X127, .(X127, T43), T43) → del18_out_aag(X127, .(X127, T43), T43)
del18_in_aag(X147, .(X148, X149), .(X148, T46)) → U2_aag(X147, X148, X149, T46, del18_in_aag(X147, X149, T46))
U2_aag(X147, X148, X149, T46, del18_out_aag(X147, X149, T46)) → del18_out_aag(X147, .(X148, X149), .(X148, T46))
U17_g(T9, del18_out_aag(X4, X5, T32)) → conf1_out_g(T9)
U16_g(T9, del6_out_aga(T31, T13, T32)) → U18_g(T9, del18_in_aag(T36, T37, T32))
U18_g(T9, del18_out_aag(T36, T37, T32)) → U19_g(T9, conf19_in_g(T37))
conf19_in_g(T54) → U4_g(T54, del229_in_ga(T54, X174))
del229_in_ga(T64, X209) → U10_ga(T64, X209, del34_in_aga(X206, T64, X207))
del34_in_aga(X242, .(X242, T77), T77) → del34_out_aga(X242, .(X242, T77), T77)
del34_in_aga(X261, .(T82, T84), .(T82, X262)) → U3_aga(X261, T82, T84, X262, del34_in_aga(X261, T84, X262))
U3_aga(X261, T82, T84, X262, del34_out_aga(X261, T84, X262)) → del34_out_aga(X261, .(T82, T84), .(T82, X262))
U10_ga(T64, X209, del34_out_aga(X206, T64, X207)) → del229_out_ga(T64, X209)
del229_in_ga(T64, X209) → U11_ga(T64, X209, del34_in_aga(T67, T64, T68))
U11_ga(T64, X209, del34_out_aga(T67, T64, T68)) → U12_ga(T64, X209, del34_in_aga(X208, T68, X209))
U12_ga(T64, X209, del34_out_aga(X208, T68, X209)) → del229_out_ga(T64, X209)
U4_g(T54, del229_out_ga(T54, X174)) → conf19_out_g(T54)
conf19_in_g(T54) → U5_g(T54, del229_in_ga(T54, T56))
U5_g(T54, del229_out_ga(T54, T56)) → U6_g(T54, del43_in_aaa(X175, X176, T56))
del43_in_aaa(X304, .(X304, T95), T95) → del43_out_aaa(X304, .(X304, T95), T95)
del43_in_aaa(X324, .(X325, X326), .(X325, T99)) → U9_aaa(X324, X325, X326, T99, del43_in_aaa(X324, X326, T99))
U9_aaa(X324, X325, X326, T99, del43_out_aaa(X324, X326, T99)) → del43_out_aaa(X324, .(X325, X326), .(X325, T99))
U6_g(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_g(T54)
U5_g(T54, del229_out_ga(T54, T56)) → U7_g(T54, del43_in_aaa(T88, T89, T56))
U7_g(T54, del43_out_aaa(T88, T89, T56)) → U8_g(T54, conf19_in_a(T89))
conf19_in_a(T54) → U4_a(T54, del229_in_aa(T54, X174))
del229_in_aa(T64, X209) → U10_aa(T64, X209, del34_in_aaa(X206, T64, X207))
del34_in_aaa(X242, .(X242, T77), T77) → del34_out_aaa(X242, .(X242, T77), T77)
del34_in_aaa(X261, .(T82, T84), .(T82, X262)) → U3_aaa(X261, T82, T84, X262, del34_in_aaa(X261, T84, X262))
U3_aaa(X261, T82, T84, X262, del34_out_aaa(X261, T84, X262)) → del34_out_aaa(X261, .(T82, T84), .(T82, X262))
U10_aa(T64, X209, del34_out_aaa(X206, T64, X207)) → del229_out_aa(T64, X209)
del229_in_aa(T64, X209) → U11_aa(T64, X209, del34_in_aaa(T67, T64, T68))
U11_aa(T64, X209, del34_out_aaa(T67, T64, T68)) → U12_aa(T64, X209, del34_in_aaa(X208, T68, X209))
U12_aa(T64, X209, del34_out_aaa(X208, T68, X209)) → del229_out_aa(T64, X209)
U4_a(T54, del229_out_aa(T54, X174)) → conf19_out_a(T54)
conf19_in_a(T54) → U5_a(T54, del229_in_aa(T54, T56))
U5_a(T54, del229_out_aa(T54, T56)) → U6_a(T54, del43_in_aaa(X175, X176, T56))
U6_a(T54, del43_out_aaa(X175, X176, T56)) → conf19_out_a(T54)
U5_a(T54, del229_out_aa(T54, T56)) → U7_a(T54, del43_in_aaa(T88, T89, T56))
U7_a(T54, del43_out_aaa(T88, T89, T56)) → U8_a(T54, conf19_in_a(T89))
U8_a(T54, conf19_out_a(T89)) → conf19_out_a(T54)
U8_g(T54, conf19_out_a(T89)) → conf19_out_g(T54)
U19_g(T9, conf19_out_g(T37)) → conf1_out_g(T9)

The argument filtering Pi contains the following mapping:
conf1_in_g(x1)  =  conf1_in_g(x1)
U13_g(x1, x2)  =  U13_g(x1, x2)
del6_in_aga(x1, x2, x3)  =  del6_in_aga(x2)
.(x1, x2)  =  .(x2)
del6_out_aga(x1, x2, x3)  =  del6_out_aga(x2, x3)
U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
conf1_out_g(x1)  =  conf1_out_g(x1)
U14_g(x1, x2)  =  U14_g(x1, x2)
U15_g(x1, x2)  =  U15_g(x1, x2)
U16_g(x1, x2)  =  U16_g(x1, x2)
U17_g(x1, x2)  =  U17_g(x1, x2)
del18_in_aag(x1, x2, x3)  =  del18_in_aag(x3)
del18_out_aag(x1, x2, x3)  =  del18_out_aag(x2, x3)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x4, x5)
U18_g(x1, x2)  =  U18_g(x1, x2)
U19_g(x1, x2)  =  U19_g(x1, x2)
conf19_in_g(x1)  =  conf19_in_g(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
del229_in_ga(x1, x2)  =  del229_in_ga(x1)
U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)
del34_in_aga(x1, x2, x3)  =  del34_in_aga(x2)
del34_out_aga(x1, x2, x3)  =  del34_out_aga(x2, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x3, x5)
del229_out_ga(x1, x2)  =  del229_out_ga(x1)
U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)
U12_ga(x1, x2, x3)  =  U12_ga(x1, x3)
conf19_out_g(x1)  =  conf19_out_g(x1)
U5_g(x1, x2)  =  U5_g(x1, x2)
U6_g(x1, x2)  =  U6_g(x1, x2)
del43_in_aaa(x1, x2, x3)  =  del43_in_aaa
del43_out_aaa(x1, x2, x3)  =  del43_out_aaa
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U7_g(x1, x2)  =  U7_g(x1, x2)
U8_g(x1, x2)  =  U8_g(x1, x2)
conf19_in_a(x1)  =  conf19_in_a
U4_a(x1, x2)  =  U4_a(x2)
del229_in_aa(x1, x2)  =  del229_in_aa
U10_aa(x1, x2, x3)  =  U10_aa(x3)
del34_in_aaa(x1, x2, x3)  =  del34_in_aaa
del34_out_aaa(x1, x2, x3)  =  del34_out_aaa
U3_aaa(x1, x2, x3, x4, x5)  =  U3_aaa(x5)
del229_out_aa(x1, x2)  =  del229_out_aa
U11_aa(x1, x2, x3)  =  U11_aa(x3)
U12_aa(x1, x2, x3)  =  U12_aa(x3)
conf19_out_a(x1)  =  conf19_out_a
U5_a(x1, x2)  =  U5_a(x2)
U6_a(x1, x2)  =  U6_a(x2)
U7_a(x1, x2)  =  U7_a(x2)
U8_a(x1, x2)  =  U8_a(x2)
DEL6_IN_AGA(x1, x2, x3)  =  DEL6_IN_AGA(x2)

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

(109) UsableRulesProof (EQUIVALENT transformation)

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

(110) Obligation:

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

DEL6_IN_AGA(X72, .(T25, T26), .(T25, X73)) → DEL6_IN_AGA(X72, T26, X73)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
DEL6_IN_AGA(x1, x2, x3)  =  DEL6_IN_AGA(x2)

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

(111) PiDPToQDPProof (SOUND transformation)

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

(112) Obligation:

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

DEL6_IN_AGA(.(T26)) → DEL6_IN_AGA(T26)

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

(113) 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:

  • DEL6_IN_AGA(.(T26)) → DEL6_IN_AGA(T26)
    The graph contains the following edges 1 > 1

(114) YES