(0) Obligation:
Clauses:reach(X, Y, Edges, Visited) :- member(.(X, .(Y, [])), Edges).
reach(X, Z, Edges, Visited) :- ','(member1(.(X, .(Y, [])), Edges), ','(member(Y, Visited), reach(Y, Z, Edges, .(Y, Visited)))).
member(H, .(H, L)).
member(X, .(H, L)) :- member(X, L).
member1(H, .(H, L)).
member1(X, .(H, L)) :- member1(X, L).
Queries:
reach(g,g,g,g).
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph.
(2) Obligation:
Clauses:member12(T66, T67, .(.(T66, .(T67, [])), T68)).
member12(T77, T78, .(T79, T80)) :- member12(T77, T78, T80).
member124(T120, X117, .(.(T120, .(X117, [])), T121)).
member124(T128, X131, .(T129, T130)) :- member124(T128, X131, T130).
member34(T151, .(T151, T152)).
member34(T159, .(T160, T161)) :- member34(T159, T161).
reach1(T25, T26, .(.(T25, .(T26, [])), T27), T12).
reach1(T44, T45, .(T46, T47), T12) :- member12(T44, T45, T47).
reach1(T99, T100, T101, T102) :- member124(T99, X90, T101).
reach1(T99, T100, T101, T102) :- ','(member124(T99, T107, T101), member34(T107, T102)).
reach1(T99, T100, T101, T102) :- ','(member124(T99, T107, T101), ','(member34(T107, T102), reach1(T107, T100, T101, .(T107, T102)))).
Queries:
reach1(g,g,g,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:reach1_in: (b,b,b,b)
member12_in: (b,b,b)
member124_in: (b,f,b)
member34_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
reach1_in_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12) → reach1_out_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12)
reach1_in_gggg(T44, T45, .(T46, T47), T12) → U4_gggg(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
member12_in_ggg(T66, T67, .(.(T66, .(T67, [])), T68)) → member12_out_ggg(T66, T67, .(.(T66, .(T67, [])), T68))
member12_in_ggg(T77, T78, .(T79, T80)) → U1_ggg(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
U1_ggg(T77, T78, T79, T80, member12_out_ggg(T77, T78, T80)) → member12_out_ggg(T77, T78, .(T79, T80))
U4_gggg(T44, T45, T46, T47, T12, member12_out_ggg(T44, T45, T47)) → reach1_out_gggg(T44, T45, .(T46, T47), T12)
reach1_in_gggg(T99, T100, T101, T102) → U5_gggg(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
member124_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, X131, .(T129, T130)) → U2_gag(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
U2_gag(T128, X131, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U5_gggg(T99, T100, T101, T102, member124_out_gag(T99, X90, T101)) → reach1_out_gggg(T99, T100, T101, T102)
reach1_in_gggg(T99, T100, T101, T102) → U6_gggg(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_gggg(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_gggg(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → reach1_out_gggg(T99, T100, T101, T102)
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_gggg(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U8_gggg(T99, T100, T101, T102, reach1_out_gggg(T107, T100, T101, .(T107, T102))) → reach1_out_gggg(T99, T100, T101, T102)
The argument filtering Pi contains the following mapping:
reach1_in_gggg(x1, x2, x3, x4) = reach1_in_gggg(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
[] = []
reach1_out_gggg(x1, x2, x3, x4) = reach1_out_gggg
U4_gggg(x1, x2, x3, x4, x5, x6) = U4_gggg(x6)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member12_out_ggg(x1, x2, x3) = member12_out_ggg
U1_ggg(x1, x2, x3, x4, x5) = U1_ggg(x5)
U5_gggg(x1, x2, x3, x4, x5) = U5_gggg(x5)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member124_out_gag(x1, x2, x3) = member124_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5)
U6_gggg(x1, x2, x3, x4, x5) = U6_gggg(x2, x3, x4, x5)
U7_gggg(x1, x2, x3, x4, x5, x6) = U7_gggg(x2, x3, x4, x5, x6)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
member34_out_gg(x1, x2) = member34_out_gg
U3_gg(x1, x2, x3, x4) = U3_gg(x4)
U8_gggg(x1, x2, x3, x4, x5) = U8_gggg(x5)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(4) Obligation:
Pi-finite rewrite system:The TRS R consists of the following rules:
reach1_in_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12) → reach1_out_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12)
reach1_in_gggg(T44, T45, .(T46, T47), T12) → U4_gggg(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
member12_in_ggg(T66, T67, .(.(T66, .(T67, [])), T68)) → member12_out_ggg(T66, T67, .(.(T66, .(T67, [])), T68))
member12_in_ggg(T77, T78, .(T79, T80)) → U1_ggg(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
U1_ggg(T77, T78, T79, T80, member12_out_ggg(T77, T78, T80)) → member12_out_ggg(T77, T78, .(T79, T80))
U4_gggg(T44, T45, T46, T47, T12, member12_out_ggg(T44, T45, T47)) → reach1_out_gggg(T44, T45, .(T46, T47), T12)
reach1_in_gggg(T99, T100, T101, T102) → U5_gggg(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
member124_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, X131, .(T129, T130)) → U2_gag(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
U2_gag(T128, X131, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U5_gggg(T99, T100, T101, T102, member124_out_gag(T99, X90, T101)) → reach1_out_gggg(T99, T100, T101, T102)
reach1_in_gggg(T99, T100, T101, T102) → U6_gggg(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_gggg(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_gggg(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → reach1_out_gggg(T99, T100, T101, T102)
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_gggg(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U8_gggg(T99, T100, T101, T102, reach1_out_gggg(T107, T100, T101, .(T107, T102))) → reach1_out_gggg(T99, T100, T101, T102)
The argument filtering Pi contains the following mapping:
reach1_in_gggg(x1, x2, x3, x4) = reach1_in_gggg(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
[] = []
reach1_out_gggg(x1, x2, x3, x4) = reach1_out_gggg
U4_gggg(x1, x2, x3, x4, x5, x6) = U4_gggg(x6)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member12_out_ggg(x1, x2, x3) = member12_out_ggg
U1_ggg(x1, x2, x3, x4, x5) = U1_ggg(x5)
U5_gggg(x1, x2, x3, x4, x5) = U5_gggg(x5)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member124_out_gag(x1, x2, x3) = member124_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5)
U6_gggg(x1, x2, x3, x4, x5) = U6_gggg(x2, x3, x4, x5)
U7_gggg(x1, x2, x3, x4, x5, x6) = U7_gggg(x2, x3, x4, x5, x6)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
member34_out_gg(x1, x2) = member34_out_gg
U3_gg(x1, x2, x3, x4) = U3_gg(x4)
U8_gggg(x1, x2, x3, x4, x5) = U8_gggg(x5)
(5) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:Pi DP problem:
The TRS P consists of the following rules:
REACH1_IN_GGGG(T44, T45, .(T46, T47), T12) → U4_GGGG(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
REACH1_IN_GGGG(T44, T45, .(T46, T47), T12) → MEMBER12_IN_GGG(T44, T45, T47)
MEMBER12_IN_GGG(T77, T78, .(T79, T80)) → U1_GGG(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
MEMBER12_IN_GGG(T77, T78, .(T79, T80)) → MEMBER12_IN_GGG(T77, T78, T80)
REACH1_IN_GGGG(T99, T100, T101, T102) → U5_GGGG(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
REACH1_IN_GGGG(T99, T100, T101, T102) → MEMBER124_IN_GAG(T99, X90, T101)
MEMBER124_IN_GAG(T128, X131, .(T129, T130)) → U2_GAG(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
MEMBER124_IN_GAG(T128, X131, .(T129, T130)) → MEMBER124_IN_GAG(T128, X131, T130)
REACH1_IN_GGGG(T99, T100, T101, T102) → U6_GGGG(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_GGGG(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_GGGG(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
U6_GGGG(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → MEMBER34_IN_GG(T107, T102)
MEMBER34_IN_GG(T159, .(T160, T161)) → U3_GG(T159, T160, T161, member34_in_gg(T159, T161))
MEMBER34_IN_GG(T159, .(T160, T161)) → MEMBER34_IN_GG(T159, T161)
U7_GGGG(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_GGGG(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U7_GGGG(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102))
The TRS R consists of the following rules:
reach1_in_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12) → reach1_out_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12)
reach1_in_gggg(T44, T45, .(T46, T47), T12) → U4_gggg(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
member12_in_ggg(T66, T67, .(.(T66, .(T67, [])), T68)) → member12_out_ggg(T66, T67, .(.(T66, .(T67, [])), T68))
member12_in_ggg(T77, T78, .(T79, T80)) → U1_ggg(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
U1_ggg(T77, T78, T79, T80, member12_out_ggg(T77, T78, T80)) → member12_out_ggg(T77, T78, .(T79, T80))
U4_gggg(T44, T45, T46, T47, T12, member12_out_ggg(T44, T45, T47)) → reach1_out_gggg(T44, T45, .(T46, T47), T12)
reach1_in_gggg(T99, T100, T101, T102) → U5_gggg(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
member124_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, X131, .(T129, T130)) → U2_gag(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
U2_gag(T128, X131, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U5_gggg(T99, T100, T101, T102, member124_out_gag(T99, X90, T101)) → reach1_out_gggg(T99, T100, T101, T102)
reach1_in_gggg(T99, T100, T101, T102) → U6_gggg(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_gggg(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_gggg(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → reach1_out_gggg(T99, T100, T101, T102)
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_gggg(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U8_gggg(T99, T100, T101, T102, reach1_out_gggg(T107, T100, T101, .(T107, T102))) → reach1_out_gggg(T99, T100, T101, T102)
The argument filtering Pi contains the following mapping:
reach1_in_gggg(x1, x2, x3, x4) = reach1_in_gggg(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
[] = []
reach1_out_gggg(x1, x2, x3, x4) = reach1_out_gggg
U4_gggg(x1, x2, x3, x4, x5, x6) = U4_gggg(x6)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member12_out_ggg(x1, x2, x3) = member12_out_ggg
U1_ggg(x1, x2, x3, x4, x5) = U1_ggg(x5)
U5_gggg(x1, x2, x3, x4, x5) = U5_gggg(x5)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member124_out_gag(x1, x2, x3) = member124_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5)
U6_gggg(x1, x2, x3, x4, x5) = U6_gggg(x2, x3, x4, x5)
U7_gggg(x1, x2, x3, x4, x5, x6) = U7_gggg(x2, x3, x4, x5, x6)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
member34_out_gg(x1, x2) = member34_out_gg
U3_gg(x1, x2, x3, x4) = U3_gg(x4)
U8_gggg(x1, x2, x3, x4, x5) = U8_gggg(x5)
REACH1_IN_GGGG(x1, x2, x3, x4) = REACH1_IN_GGGG(x1, x2, x3, x4)
U4_GGGG(x1, x2, x3, x4, x5, x6) = U4_GGGG(x6)
MEMBER12_IN_GGG(x1, x2, x3) = MEMBER12_IN_GGG(x1, x2, x3)
U1_GGG(x1, x2, x3, x4, x5) = U1_GGG(x5)
U5_GGGG(x1, x2, x3, x4, x5) = U5_GGGG(x5)
MEMBER124_IN_GAG(x1, x2, x3) = MEMBER124_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5) = U2_GAG(x5)
U6_GGGG(x1, x2, x3, x4, x5) = U6_GGGG(x2, x3, x4, x5)
U7_GGGG(x1, x2, x3, x4, x5, x6) = U7_GGGG(x2, x3, x4, x5, x6)
MEMBER34_IN_GG(x1, x2) = MEMBER34_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4) = U3_GG(x4)
U8_GGGG(x1, x2, x3, x4, x5) = U8_GGGG(x5)
We have to consider all (P,R,Pi)-chains
(6) Obligation:
Pi DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(T44, T45, .(T46, T47), T12) → U4_GGGG(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
REACH1_IN_GGGG(T44, T45, .(T46, T47), T12) → MEMBER12_IN_GGG(T44, T45, T47)
MEMBER12_IN_GGG(T77, T78, .(T79, T80)) → U1_GGG(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
MEMBER12_IN_GGG(T77, T78, .(T79, T80)) → MEMBER12_IN_GGG(T77, T78, T80)
REACH1_IN_GGGG(T99, T100, T101, T102) → U5_GGGG(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
REACH1_IN_GGGG(T99, T100, T101, T102) → MEMBER124_IN_GAG(T99, X90, T101)
MEMBER124_IN_GAG(T128, X131, .(T129, T130)) → U2_GAG(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
MEMBER124_IN_GAG(T128, X131, .(T129, T130)) → MEMBER124_IN_GAG(T128, X131, T130)
REACH1_IN_GGGG(T99, T100, T101, T102) → U6_GGGG(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_GGGG(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_GGGG(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
U6_GGGG(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → MEMBER34_IN_GG(T107, T102)
MEMBER34_IN_GG(T159, .(T160, T161)) → U3_GG(T159, T160, T161, member34_in_gg(T159, T161))
MEMBER34_IN_GG(T159, .(T160, T161)) → MEMBER34_IN_GG(T159, T161)
U7_GGGG(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_GGGG(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U7_GGGG(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102))
The TRS R consists of the following rules:
reach1_in_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12) → reach1_out_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12)
reach1_in_gggg(T44, T45, .(T46, T47), T12) → U4_gggg(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
member12_in_ggg(T66, T67, .(.(T66, .(T67, [])), T68)) → member12_out_ggg(T66, T67, .(.(T66, .(T67, [])), T68))
member12_in_ggg(T77, T78, .(T79, T80)) → U1_ggg(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
U1_ggg(T77, T78, T79, T80, member12_out_ggg(T77, T78, T80)) → member12_out_ggg(T77, T78, .(T79, T80))
U4_gggg(T44, T45, T46, T47, T12, member12_out_ggg(T44, T45, T47)) → reach1_out_gggg(T44, T45, .(T46, T47), T12)
reach1_in_gggg(T99, T100, T101, T102) → U5_gggg(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
member124_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, X131, .(T129, T130)) → U2_gag(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
U2_gag(T128, X131, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U5_gggg(T99, T100, T101, T102, member124_out_gag(T99, X90, T101)) → reach1_out_gggg(T99, T100, T101, T102)
reach1_in_gggg(T99, T100, T101, T102) → U6_gggg(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_gggg(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_gggg(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → reach1_out_gggg(T99, T100, T101, T102)
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_gggg(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U8_gggg(T99, T100, T101, T102, reach1_out_gggg(T107, T100, T101, .(T107, T102))) → reach1_out_gggg(T99, T100, T101, T102)
The argument filtering Pi contains the following mapping:
reach1_in_gggg(x1, x2, x3, x4) = reach1_in_gggg(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
[] = []
reach1_out_gggg(x1, x2, x3, x4) = reach1_out_gggg
U4_gggg(x1, x2, x3, x4, x5, x6) = U4_gggg(x6)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member12_out_ggg(x1, x2, x3) = member12_out_ggg
U1_ggg(x1, x2, x3, x4, x5) = U1_ggg(x5)
U5_gggg(x1, x2, x3, x4, x5) = U5_gggg(x5)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member124_out_gag(x1, x2, x3) = member124_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5)
U6_gggg(x1, x2, x3, x4, x5) = U6_gggg(x2, x3, x4, x5)
U7_gggg(x1, x2, x3, x4, x5, x6) = U7_gggg(x2, x3, x4, x5, x6)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
member34_out_gg(x1, x2) = member34_out_gg
U3_gg(x1, x2, x3, x4) = U3_gg(x4)
U8_gggg(x1, x2, x3, x4, x5) = U8_gggg(x5)
REACH1_IN_GGGG(x1, x2, x3, x4) = REACH1_IN_GGGG(x1, x2, x3, x4)
U4_GGGG(x1, x2, x3, x4, x5, x6) = U4_GGGG(x6)
MEMBER12_IN_GGG(x1, x2, x3) = MEMBER12_IN_GGG(x1, x2, x3)
U1_GGG(x1, x2, x3, x4, x5) = U1_GGG(x5)
U5_GGGG(x1, x2, x3, x4, x5) = U5_GGGG(x5)
MEMBER124_IN_GAG(x1, x2, x3) = MEMBER124_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5) = U2_GAG(x5)
U6_GGGG(x1, x2, x3, x4, x5) = U6_GGGG(x2, x3, x4, x5)
U7_GGGG(x1, x2, x3, x4, x5, x6) = U7_GGGG(x2, x3, x4, x5, x6)
MEMBER34_IN_GG(x1, x2) = MEMBER34_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4) = U3_GG(x4)
U8_GGGG(x1, x2, x3, x4, x5) = U8_GGGG(x5)
We have to consider all (P,R,Pi)-chains
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 9 less nodes.(8) Complex Obligation (AND)
(9) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MEMBER34_IN_GG(T159, .(T160, T161)) → MEMBER34_IN_GG(T159, T161)
The TRS R consists of the following rules:
reach1_in_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12) → reach1_out_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12)
reach1_in_gggg(T44, T45, .(T46, T47), T12) → U4_gggg(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
member12_in_ggg(T66, T67, .(.(T66, .(T67, [])), T68)) → member12_out_ggg(T66, T67, .(.(T66, .(T67, [])), T68))
member12_in_ggg(T77, T78, .(T79, T80)) → U1_ggg(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
U1_ggg(T77, T78, T79, T80, member12_out_ggg(T77, T78, T80)) → member12_out_ggg(T77, T78, .(T79, T80))
U4_gggg(T44, T45, T46, T47, T12, member12_out_ggg(T44, T45, T47)) → reach1_out_gggg(T44, T45, .(T46, T47), T12)
reach1_in_gggg(T99, T100, T101, T102) → U5_gggg(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
member124_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, X131, .(T129, T130)) → U2_gag(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
U2_gag(T128, X131, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U5_gggg(T99, T100, T101, T102, member124_out_gag(T99, X90, T101)) → reach1_out_gggg(T99, T100, T101, T102)
reach1_in_gggg(T99, T100, T101, T102) → U6_gggg(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_gggg(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_gggg(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → reach1_out_gggg(T99, T100, T101, T102)
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_gggg(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U8_gggg(T99, T100, T101, T102, reach1_out_gggg(T107, T100, T101, .(T107, T102))) → reach1_out_gggg(T99, T100, T101, T102)
The argument filtering Pi contains the following mapping:
reach1_in_gggg(x1, x2, x3, x4) = reach1_in_gggg(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
[] = []
reach1_out_gggg(x1, x2, x3, x4) = reach1_out_gggg
U4_gggg(x1, x2, x3, x4, x5, x6) = U4_gggg(x6)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member12_out_ggg(x1, x2, x3) = member12_out_ggg
U1_ggg(x1, x2, x3, x4, x5) = U1_ggg(x5)
U5_gggg(x1, x2, x3, x4, x5) = U5_gggg(x5)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member124_out_gag(x1, x2, x3) = member124_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5)
U6_gggg(x1, x2, x3, x4, x5) = U6_gggg(x2, x3, x4, x5)
U7_gggg(x1, x2, x3, x4, x5, x6) = U7_gggg(x2, x3, x4, x5, x6)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
member34_out_gg(x1, x2) = member34_out_gg
U3_gg(x1, x2, x3, x4) = U3_gg(x4)
U8_gggg(x1, x2, x3, x4, x5) = U8_gggg(x5)
MEMBER34_IN_GG(x1, x2) = MEMBER34_IN_GG(x1, x2)
We have to consider all (P,R,Pi)-chains
(10) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(11) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MEMBER34_IN_GG(T159, .(T160, T161)) → MEMBER34_IN_GG(T159, T161)
R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
(12) PiDPToQDPProof (EQUIVALENT 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:
MEMBER34_IN_GG(T159, .(T160, T161)) → MEMBER34_IN_GG(T159, T161)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(14) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.From the DPs we obtained the following set of size-change graphs:
- MEMBER34_IN_GG(T159, .(T160, T161)) → MEMBER34_IN_GG(T159, T161)
The graph contains the following edges 1 >= 1, 2 > 2
(15) YES
(16) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MEMBER124_IN_GAG(T128, X131, .(T129, T130)) → MEMBER124_IN_GAG(T128, X131, T130)
The TRS R consists of the following rules:
reach1_in_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12) → reach1_out_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12)
reach1_in_gggg(T44, T45, .(T46, T47), T12) → U4_gggg(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
member12_in_ggg(T66, T67, .(.(T66, .(T67, [])), T68)) → member12_out_ggg(T66, T67, .(.(T66, .(T67, [])), T68))
member12_in_ggg(T77, T78, .(T79, T80)) → U1_ggg(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
U1_ggg(T77, T78, T79, T80, member12_out_ggg(T77, T78, T80)) → member12_out_ggg(T77, T78, .(T79, T80))
U4_gggg(T44, T45, T46, T47, T12, member12_out_ggg(T44, T45, T47)) → reach1_out_gggg(T44, T45, .(T46, T47), T12)
reach1_in_gggg(T99, T100, T101, T102) → U5_gggg(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
member124_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, X131, .(T129, T130)) → U2_gag(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
U2_gag(T128, X131, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U5_gggg(T99, T100, T101, T102, member124_out_gag(T99, X90, T101)) → reach1_out_gggg(T99, T100, T101, T102)
reach1_in_gggg(T99, T100, T101, T102) → U6_gggg(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_gggg(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_gggg(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → reach1_out_gggg(T99, T100, T101, T102)
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_gggg(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U8_gggg(T99, T100, T101, T102, reach1_out_gggg(T107, T100, T101, .(T107, T102))) → reach1_out_gggg(T99, T100, T101, T102)
The argument filtering Pi contains the following mapping:
reach1_in_gggg(x1, x2, x3, x4) = reach1_in_gggg(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
[] = []
reach1_out_gggg(x1, x2, x3, x4) = reach1_out_gggg
U4_gggg(x1, x2, x3, x4, x5, x6) = U4_gggg(x6)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member12_out_ggg(x1, x2, x3) = member12_out_ggg
U1_ggg(x1, x2, x3, x4, x5) = U1_ggg(x5)
U5_gggg(x1, x2, x3, x4, x5) = U5_gggg(x5)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member124_out_gag(x1, x2, x3) = member124_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5)
U6_gggg(x1, x2, x3, x4, x5) = U6_gggg(x2, x3, x4, x5)
U7_gggg(x1, x2, x3, x4, x5, x6) = U7_gggg(x2, x3, x4, x5, x6)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
member34_out_gg(x1, x2) = member34_out_gg
U3_gg(x1, x2, x3, x4) = U3_gg(x4)
U8_gggg(x1, x2, x3, x4, x5) = U8_gggg(x5)
MEMBER124_IN_GAG(x1, x2, x3) = MEMBER124_IN_GAG(x1, x3)
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:
MEMBER124_IN_GAG(T128, X131, .(T129, T130)) → MEMBER124_IN_GAG(T128, X131, T130)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
MEMBER124_IN_GAG(x1, x2, x3) = MEMBER124_IN_GAG(x1, x3)
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:
MEMBER124_IN_GAG(T128, .(T129, T130)) → MEMBER124_IN_GAG(T128, T130)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(21) 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:
- MEMBER124_IN_GAG(T128, .(T129, T130)) → MEMBER124_IN_GAG(T128, T130)
The graph contains the following edges 1 >= 1, 2 > 2
(22) YES
(23) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MEMBER12_IN_GGG(T77, T78, .(T79, T80)) → MEMBER12_IN_GGG(T77, T78, T80)
The TRS R consists of the following rules:
reach1_in_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12) → reach1_out_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12)
reach1_in_gggg(T44, T45, .(T46, T47), T12) → U4_gggg(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
member12_in_ggg(T66, T67, .(.(T66, .(T67, [])), T68)) → member12_out_ggg(T66, T67, .(.(T66, .(T67, [])), T68))
member12_in_ggg(T77, T78, .(T79, T80)) → U1_ggg(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
U1_ggg(T77, T78, T79, T80, member12_out_ggg(T77, T78, T80)) → member12_out_ggg(T77, T78, .(T79, T80))
U4_gggg(T44, T45, T46, T47, T12, member12_out_ggg(T44, T45, T47)) → reach1_out_gggg(T44, T45, .(T46, T47), T12)
reach1_in_gggg(T99, T100, T101, T102) → U5_gggg(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
member124_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, X131, .(T129, T130)) → U2_gag(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
U2_gag(T128, X131, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U5_gggg(T99, T100, T101, T102, member124_out_gag(T99, X90, T101)) → reach1_out_gggg(T99, T100, T101, T102)
reach1_in_gggg(T99, T100, T101, T102) → U6_gggg(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_gggg(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_gggg(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → reach1_out_gggg(T99, T100, T101, T102)
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_gggg(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U8_gggg(T99, T100, T101, T102, reach1_out_gggg(T107, T100, T101, .(T107, T102))) → reach1_out_gggg(T99, T100, T101, T102)
The argument filtering Pi contains the following mapping:
reach1_in_gggg(x1, x2, x3, x4) = reach1_in_gggg(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
[] = []
reach1_out_gggg(x1, x2, x3, x4) = reach1_out_gggg
U4_gggg(x1, x2, x3, x4, x5, x6) = U4_gggg(x6)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member12_out_ggg(x1, x2, x3) = member12_out_ggg
U1_ggg(x1, x2, x3, x4, x5) = U1_ggg(x5)
U5_gggg(x1, x2, x3, x4, x5) = U5_gggg(x5)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member124_out_gag(x1, x2, x3) = member124_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5)
U6_gggg(x1, x2, x3, x4, x5) = U6_gggg(x2, x3, x4, x5)
U7_gggg(x1, x2, x3, x4, x5, x6) = U7_gggg(x2, x3, x4, x5, x6)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
member34_out_gg(x1, x2) = member34_out_gg
U3_gg(x1, x2, x3, x4) = U3_gg(x4)
U8_gggg(x1, x2, x3, x4, x5) = U8_gggg(x5)
MEMBER12_IN_GGG(x1, x2, x3) = MEMBER12_IN_GGG(x1, x2, x3)
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:
MEMBER12_IN_GGG(T77, T78, .(T79, T80)) → MEMBER12_IN_GGG(T77, T78, T80)
R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
(26) PiDPToQDPProof (EQUIVALENT 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:
MEMBER12_IN_GGG(T77, T78, .(T79, T80)) → MEMBER12_IN_GGG(T77, T78, T80)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(28) 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:
- MEMBER12_IN_GGG(T77, T78, .(T79, T80)) → MEMBER12_IN_GGG(T77, T78, T80)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
(29) YES
(30) Obligation:
Pi DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(T99, T100, T101, T102) → U6_GGGG(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_GGGG(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_GGGG(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
U7_GGGG(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102))
The TRS R consists of the following rules:
reach1_in_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12) → reach1_out_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12)
reach1_in_gggg(T44, T45, .(T46, T47), T12) → U4_gggg(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
member12_in_ggg(T66, T67, .(.(T66, .(T67, [])), T68)) → member12_out_ggg(T66, T67, .(.(T66, .(T67, [])), T68))
member12_in_ggg(T77, T78, .(T79, T80)) → U1_ggg(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
U1_ggg(T77, T78, T79, T80, member12_out_ggg(T77, T78, T80)) → member12_out_ggg(T77, T78, .(T79, T80))
U4_gggg(T44, T45, T46, T47, T12, member12_out_ggg(T44, T45, T47)) → reach1_out_gggg(T44, T45, .(T46, T47), T12)
reach1_in_gggg(T99, T100, T101, T102) → U5_gggg(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
member124_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, X131, .(T129, T130)) → U2_gag(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
U2_gag(T128, X131, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U5_gggg(T99, T100, T101, T102, member124_out_gag(T99, X90, T101)) → reach1_out_gggg(T99, T100, T101, T102)
reach1_in_gggg(T99, T100, T101, T102) → U6_gggg(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_gggg(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_gggg(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → reach1_out_gggg(T99, T100, T101, T102)
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_gggg(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U8_gggg(T99, T100, T101, T102, reach1_out_gggg(T107, T100, T101, .(T107, T102))) → reach1_out_gggg(T99, T100, T101, T102)
The argument filtering Pi contains the following mapping:
reach1_in_gggg(x1, x2, x3, x4) = reach1_in_gggg(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
[] = []
reach1_out_gggg(x1, x2, x3, x4) = reach1_out_gggg
U4_gggg(x1, x2, x3, x4, x5, x6) = U4_gggg(x6)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member12_out_ggg(x1, x2, x3) = member12_out_ggg
U1_ggg(x1, x2, x3, x4, x5) = U1_ggg(x5)
U5_gggg(x1, x2, x3, x4, x5) = U5_gggg(x5)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member124_out_gag(x1, x2, x3) = member124_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5)
U6_gggg(x1, x2, x3, x4, x5) = U6_gggg(x2, x3, x4, x5)
U7_gggg(x1, x2, x3, x4, x5, x6) = U7_gggg(x2, x3, x4, x5, x6)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
member34_out_gg(x1, x2) = member34_out_gg
U3_gg(x1, x2, x3, x4) = U3_gg(x4)
U8_gggg(x1, x2, x3, x4, x5) = U8_gggg(x5)
REACH1_IN_GGGG(x1, x2, x3, x4) = REACH1_IN_GGGG(x1, x2, x3, x4)
U6_GGGG(x1, x2, x3, x4, x5) = U6_GGGG(x2, x3, x4, x5)
U7_GGGG(x1, x2, x3, x4, x5, x6) = U7_GGGG(x2, x3, x4, x5, x6)
We have to consider all (P,R,Pi)-chains
(31) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(32) Obligation:
Pi DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(T99, T100, T101, T102) → U6_GGGG(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_GGGG(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_GGGG(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
U7_GGGG(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102))
The TRS R consists of the following rules:
member124_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, X131, .(T129, T130)) → U2_gag(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, X131, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member124_out_gag(x1, x2, x3) = member124_out_gag(x2)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x5)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
member34_out_gg(x1, x2) = member34_out_gg
U3_gg(x1, x2, x3, x4) = U3_gg(x4)
REACH1_IN_GGGG(x1, x2, x3, x4) = REACH1_IN_GGGG(x1, x2, x3, x4)
U6_GGGG(x1, x2, x3, x4, x5) = U6_GGGG(x2, x3, x4, x5)
U7_GGGG(x1, x2, x3, x4, x5, x6) = U7_GGGG(x2, x3, x4, x5, x6)
We have to consider all (P,R,Pi)-chains
(33) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(34) Obligation:
Q DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(T99, T100, T101, T102) → U6_GGGG(T100, T101, T102, member124_in_gag(T99, T101))
U6_GGGG(T100, T101, T102, member124_out_gag(T107)) → U7_GGGG(T100, T101, T102, T107, member34_in_gg(T107, T102))
U7_GGGG(T100, T101, T102, T107, member34_out_gg) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(35) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule REACH1_IN_GGGG(T99, T100, T101, T102) → U6_GGGG(T100, T101, T102, member124_in_gag(T99, T101)) at position [3] we obtained the following new rules [LPAR04]:
REACH1_IN_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3) → U6_GGGG(y1, .(.(x0, .(x1, [])), x2), y3, member124_out_gag(x1))
REACH1_IN_GGGG(x0, y1, .(x1, x2), y3) → U6_GGGG(y1, .(x1, x2), y3, U2_gag(member124_in_gag(x0, x2)))
(36) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(T100, T101, T102, member124_out_gag(T107)) → U7_GGGG(T100, T101, T102, T107, member34_in_gg(T107, T102))
U7_GGGG(T100, T101, T102, T107, member34_out_gg) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102))
REACH1_IN_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3) → U6_GGGG(y1, .(.(x0, .(x1, [])), x2), y3, member124_out_gag(x1))
REACH1_IN_GGGG(x0, y1, .(x1, x2), y3) → U6_GGGG(y1, .(x1, x2), y3, U2_gag(member124_in_gag(x0, x2)))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(37) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule U6_GGGG(T100, T101, T102, member124_out_gag(T107)) → U7_GGGG(T100, T101, T102, T107, member34_in_gg(T107, T102)) at position [4] we obtained the following new rules [LPAR04]:
U6_GGGG(y0, y1, .(x0, x1), member124_out_gag(x0)) → U7_GGGG(y0, y1, .(x0, x1), x0, member34_out_gg)
U6_GGGG(y0, y1, .(x1, x2), member124_out_gag(x0)) → U7_GGGG(y0, y1, .(x1, x2), x0, U3_gg(member34_in_gg(x0, x2)))
(38) Obligation:
Q DP problem:The TRS P consists of the following rules:
U7_GGGG(T100, T101, T102, T107, member34_out_gg) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102))
REACH1_IN_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3) → U6_GGGG(y1, .(.(x0, .(x1, [])), x2), y3, member124_out_gag(x1))
REACH1_IN_GGGG(x0, y1, .(x1, x2), y3) → U6_GGGG(y1, .(x1, x2), y3, U2_gag(member124_in_gag(x0, x2)))
U6_GGGG(y0, y1, .(x0, x1), member124_out_gag(x0)) → U7_GGGG(y0, y1, .(x0, x1), x0, member34_out_gg)
U6_GGGG(y0, y1, .(x1, x2), member124_out_gag(x0)) → U7_GGGG(y0, y1, .(x1, x2), x0, U3_gg(member34_in_gg(x0, x2)))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(39) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U7_GGGG(T100, T101, T102, T107, member34_out_gg) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102)) we obtained the following new rules [LPAR04]:
U7_GGGG(z0, z1, .(z2, z3), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3)))
U7_GGGG(z0, z1, .(z2, z3), z4, member34_out_gg) → REACH1_IN_GGGG(z4, z0, z1, .(z4, .(z2, z3)))
(40) Obligation:
Q DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3) → U6_GGGG(y1, .(.(x0, .(x1, [])), x2), y3, member124_out_gag(x1))
REACH1_IN_GGGG(x0, y1, .(x1, x2), y3) → U6_GGGG(y1, .(x1, x2), y3, U2_gag(member124_in_gag(x0, x2)))
U6_GGGG(y0, y1, .(x0, x1), member124_out_gag(x0)) → U7_GGGG(y0, y1, .(x0, x1), x0, member34_out_gg)
U6_GGGG(y0, y1, .(x1, x2), member124_out_gag(x0)) → U7_GGGG(y0, y1, .(x1, x2), x0, U3_gg(member34_in_gg(x0, x2)))
U7_GGGG(z0, z1, .(z2, z3), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3)))
U7_GGGG(z0, z1, .(z2, z3), z4, member34_out_gg) → REACH1_IN_GGGG(z4, z0, z1, .(z4, .(z2, z3)))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(41) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule REACH1_IN_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3) → U6_GGGG(y1, .(.(x0, .(x1, [])), x2), y3, member124_out_gag(x1)) we obtained the following new rules [LPAR04]:
REACH1_IN_GGGG(z2, z0, .(.(z2, .(x2, [])), x3), .(z2, .(z2, z3))) → U6_GGGG(z0, .(.(z2, .(x2, [])), x3), .(z2, .(z2, z3)), member124_out_gag(x2))
REACH1_IN_GGGG(z4, z0, .(.(z4, .(x2, [])), x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(.(z4, .(x2, [])), x3), .(z4, .(z2, z3)), member124_out_gag(x2))
(42) Obligation:
Q DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(x0, y1, .(x1, x2), y3) → U6_GGGG(y1, .(x1, x2), y3, U2_gag(member124_in_gag(x0, x2)))
U6_GGGG(y0, y1, .(x0, x1), member124_out_gag(x0)) → U7_GGGG(y0, y1, .(x0, x1), x0, member34_out_gg)
U6_GGGG(y0, y1, .(x1, x2), member124_out_gag(x0)) → U7_GGGG(y0, y1, .(x1, x2), x0, U3_gg(member34_in_gg(x0, x2)))
U7_GGGG(z0, z1, .(z2, z3), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3)))
U7_GGGG(z0, z1, .(z2, z3), z4, member34_out_gg) → REACH1_IN_GGGG(z4, z0, z1, .(z4, .(z2, z3)))
REACH1_IN_GGGG(z2, z0, .(.(z2, .(x2, [])), x3), .(z2, .(z2, z3))) → U6_GGGG(z0, .(.(z2, .(x2, [])), x3), .(z2, .(z2, z3)), member124_out_gag(x2))
REACH1_IN_GGGG(z4, z0, .(.(z4, .(x2, [])), x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(.(z4, .(x2, [])), x3), .(z4, .(z2, z3)), member124_out_gag(x2))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(43) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule REACH1_IN_GGGG(x0, y1, .(x1, x2), y3) → U6_GGGG(y1, .(x1, x2), y3, U2_gag(member124_in_gag(x0, x2))) we obtained the following new rules [LPAR04]:
REACH1_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U2_gag(member124_in_gag(z2, x3)))
REACH1_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U2_gag(member124_in_gag(z4, x3)))
(44) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(y0, y1, .(x0, x1), member124_out_gag(x0)) → U7_GGGG(y0, y1, .(x0, x1), x0, member34_out_gg)
U6_GGGG(y0, y1, .(x1, x2), member124_out_gag(x0)) → U7_GGGG(y0, y1, .(x1, x2), x0, U3_gg(member34_in_gg(x0, x2)))
U7_GGGG(z0, z1, .(z2, z3), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3)))
U7_GGGG(z0, z1, .(z2, z3), z4, member34_out_gg) → REACH1_IN_GGGG(z4, z0, z1, .(z4, .(z2, z3)))
REACH1_IN_GGGG(z2, z0, .(.(z2, .(x2, [])), x3), .(z2, .(z2, z3))) → U6_GGGG(z0, .(.(z2, .(x2, [])), x3), .(z2, .(z2, z3)), member124_out_gag(x2))
REACH1_IN_GGGG(z4, z0, .(.(z4, .(x2, [])), x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(.(z4, .(x2, [])), x3), .(z4, .(z2, z3)), member124_out_gag(x2))
REACH1_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U2_gag(member124_in_gag(z2, x3)))
REACH1_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U2_gag(member124_in_gag(z4, x3)))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(45) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(y0, y1, .(x0, x1), member124_out_gag(x0)) → U7_GGGG(y0, y1, .(x0, x1), x0, member34_out_gg) we obtained the following new rules [LPAR04]:
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg)
(46) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(y0, y1, .(x1, x2), member124_out_gag(x0)) → U7_GGGG(y0, y1, .(x1, x2), x0, U3_gg(member34_in_gg(x0, x2)))
U7_GGGG(z0, z1, .(z2, z3), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3)))
U7_GGGG(z0, z1, .(z2, z3), z4, member34_out_gg) → REACH1_IN_GGGG(z4, z0, z1, .(z4, .(z2, z3)))
REACH1_IN_GGGG(z2, z0, .(.(z2, .(x2, [])), x3), .(z2, .(z2, z3))) → U6_GGGG(z0, .(.(z2, .(x2, [])), x3), .(z2, .(z2, z3)), member124_out_gag(x2))
REACH1_IN_GGGG(z4, z0, .(.(z4, .(x2, [])), x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(.(z4, .(x2, [])), x3), .(z4, .(z2, z3)), member124_out_gag(x2))
REACH1_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U2_gag(member124_in_gag(z2, x3)))
REACH1_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U2_gag(member124_in_gag(z4, x3)))
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg)
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(47) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(y0, y1, .(x1, x2), member124_out_gag(x0)) → U7_GGGG(y0, y1, .(x1, x2), x0, U3_gg(member34_in_gg(x0, x2))) we obtained the following new rules [LPAR04]:
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x4, U3_gg(member34_in_gg(x4, .(z0, z4))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x4, U3_gg(member34_in_gg(x4, .(z4, z5))))
(48) Obligation:
Q DP problem:The TRS P consists of the following rules:
U7_GGGG(z0, z1, .(z2, z3), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3)))
U7_GGGG(z0, z1, .(z2, z3), z4, member34_out_gg) → REACH1_IN_GGGG(z4, z0, z1, .(z4, .(z2, z3)))
REACH1_IN_GGGG(z2, z0, .(.(z2, .(x2, [])), x3), .(z2, .(z2, z3))) → U6_GGGG(z0, .(.(z2, .(x2, [])), x3), .(z2, .(z2, z3)), member124_out_gag(x2))
REACH1_IN_GGGG(z4, z0, .(.(z4, .(x2, [])), x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(.(z4, .(x2, [])), x3), .(z4, .(z2, z3)), member124_out_gag(x2))
REACH1_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U2_gag(member124_in_gag(z2, x3)))
REACH1_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U2_gag(member124_in_gag(z4, x3)))
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x4, U3_gg(member34_in_gg(x4, .(z0, z4))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x4, U3_gg(member34_in_gg(x4, .(z4, z5))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(49) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U7_GGGG(z0, z1, .(z2, z3), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3))) we obtained the following new rules [LPAR04]:
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, z3)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z3, z4)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
(50) Obligation:
Q DP problem:The TRS P consists of the following rules:
U7_GGGG(z0, z1, .(z2, z3), z4, member34_out_gg) → REACH1_IN_GGGG(z4, z0, z1, .(z4, .(z2, z3)))
REACH1_IN_GGGG(z2, z0, .(.(z2, .(x2, [])), x3), .(z2, .(z2, z3))) → U6_GGGG(z0, .(.(z2, .(x2, [])), x3), .(z2, .(z2, z3)), member124_out_gag(x2))
REACH1_IN_GGGG(z4, z0, .(.(z4, .(x2, [])), x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(.(z4, .(x2, [])), x3), .(z4, .(z2, z3)), member124_out_gag(x2))
REACH1_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U2_gag(member124_in_gag(z2, x3)))
REACH1_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U2_gag(member124_in_gag(z4, x3)))
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x4, U3_gg(member34_in_gg(x4, .(z0, z4))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x4, U3_gg(member34_in_gg(x4, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, z3)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z3, z4)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(51) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U7_GGGG(z0, z1, .(z2, z3), z4, member34_out_gg) → REACH1_IN_GGGG(z4, z0, z1, .(z4, .(z2, z3))) we obtained the following new rules [LPAR04]:
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, z3)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z3, z4)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z1, z4)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z4, z5)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z5, member34_out_gg) → REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z6, member34_out_gg) → REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5))))
(52) Obligation:
Q DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(z2, z0, .(.(z2, .(x2, [])), x3), .(z2, .(z2, z3))) → U6_GGGG(z0, .(.(z2, .(x2, [])), x3), .(z2, .(z2, z3)), member124_out_gag(x2))
REACH1_IN_GGGG(z4, z0, .(.(z4, .(x2, [])), x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(.(z4, .(x2, [])), x3), .(z4, .(z2, z3)), member124_out_gag(x2))
REACH1_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U2_gag(member124_in_gag(z2, x3)))
REACH1_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U2_gag(member124_in_gag(z4, x3)))
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x4, U3_gg(member34_in_gg(x4, .(z0, z4))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x4, U3_gg(member34_in_gg(x4, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, z3)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z3, z4)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z1, z4)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z4, z5)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z5, member34_out_gg) → REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z6, member34_out_gg) → REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(53) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule REACH1_IN_GGGG(z2, z0, .(.(z2, .(x2, [])), x3), .(z2, .(z2, z3))) → U6_GGGG(z0, .(.(z2, .(x2, [])), x3), .(z2, .(z2, z3)), member124_out_gag(x2)) we obtained the following new rules [LPAR04]:
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), member124_out_gag(z1))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), member124_out_gag(z1))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5))), member124_out_gag(x2))
(54) Obligation:
Q DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(z4, z0, .(.(z4, .(x2, [])), x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(.(z4, .(x2, [])), x3), .(z4, .(z2, z3)), member124_out_gag(x2))
REACH1_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U2_gag(member124_in_gag(z2, x3)))
REACH1_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U2_gag(member124_in_gag(z4, x3)))
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x4, U3_gg(member34_in_gg(x4, .(z0, z4))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x4, U3_gg(member34_in_gg(x4, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, z3)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z3, z4)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z1, z4)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z4, z5)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z5, member34_out_gg) → REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z6, member34_out_gg) → REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5))))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), member124_out_gag(z1))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), member124_out_gag(z1))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5))), member124_out_gag(x2))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(55) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule REACH1_IN_GGGG(z4, z0, .(.(z4, .(x2, [])), x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(.(z4, .(x2, [])), x3), .(z4, .(z2, z3)), member124_out_gag(x2)) we obtained the following new rules [LPAR04]:
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), member124_out_gag(z1))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), member124_out_gag(z1))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z5, z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z6, z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5))), member124_out_gag(x2))
(56) Obligation:
Q DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U2_gag(member124_in_gag(z2, x3)))
REACH1_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U2_gag(member124_in_gag(z4, x3)))
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x4, U3_gg(member34_in_gg(x4, .(z0, z4))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x4, U3_gg(member34_in_gg(x4, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, z3)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z3, z4)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z1, z4)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z4, z5)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z5, member34_out_gg) → REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z6, member34_out_gg) → REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5))))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), member124_out_gag(z1))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), member124_out_gag(z1))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z5, z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z6, z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5))), member124_out_gag(x2))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(57) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule REACH1_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U2_gag(member124_in_gag(z2, x3))) we obtained the following new rules [LPAR04]:
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z3, z2)))
(58) Obligation:
Q DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U2_gag(member124_in_gag(z4, x3)))
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x4, U3_gg(member34_in_gg(x4, .(z0, z4))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x4, U3_gg(member34_in_gg(x4, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, z3)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z3, z4)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z1, z4)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z4, z5)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z5, member34_out_gg) → REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z6, member34_out_gg) → REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5))))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), member124_out_gag(z1))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), member124_out_gag(z1))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z5, z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z6, z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z3, z2)))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(59) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule REACH1_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U6_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U2_gag(member124_in_gag(z4, x3))) we obtained the following new rules [LPAR04]:
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z5, z2)))
REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z6, z2)))
(60) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x4, U3_gg(member34_in_gg(x4, .(z0, z4))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x4, U3_gg(member34_in_gg(x4, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, z3)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z3, z4)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z1, z4)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z4, z5)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z5, member34_out_gg) → REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z6, member34_out_gg) → REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5))))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), member124_out_gag(z1))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), member124_out_gag(z1))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z5, z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z6, z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z5, z2)))
REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z6, z2)))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(61) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg) we obtained the following new rules [LPAR04]:
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg)
(62) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x4, U3_gg(member34_in_gg(x4, .(z0, z4))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x4, U3_gg(member34_in_gg(x4, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, z3)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z3, z4)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z1, z4)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z4, z5)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z5, member34_out_gg) → REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z6, member34_out_gg) → REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5))))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), member124_out_gag(z1))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), member124_out_gag(z1))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z5, z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z6, z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z5, z2)))
REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z6, z2)))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg)
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(63) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg) we obtained the following new rules [LPAR04]:
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg)
(64) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x4, U3_gg(member34_in_gg(x4, .(z0, z4))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x4, U3_gg(member34_in_gg(x4, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, z3)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z3, z4)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z1, z4)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z4, z5)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z5, member34_out_gg) → REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z6, member34_out_gg) → REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5))))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), member124_out_gag(z1))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), member124_out_gag(z1))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z5, z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z6, z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z5, z2)))
REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z6, z2)))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg)
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(65) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg) we obtained the following new rules [LPAR04]:
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member34_out_gg)
(66) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x4, U3_gg(member34_in_gg(x4, .(z0, z4))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x4, U3_gg(member34_in_gg(x4, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, z3)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z3, z4)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z1, z4)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z4, z5)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z5, member34_out_gg) → REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z6, member34_out_gg) → REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5))))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), member124_out_gag(z1))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), member124_out_gag(z1))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z5, z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z6, z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z5, z2)))
REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z6, z2)))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member34_out_gg)
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(67) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg) we obtained the following new rules [LPAR04]:
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg)
(68) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x4, U3_gg(member34_in_gg(x4, .(z0, z4))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x4, U3_gg(member34_in_gg(x4, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, z3)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z3, z4)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z1, z4)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z4, z5)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z5, member34_out_gg) → REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z6, member34_out_gg) → REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5))))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), member124_out_gag(z1))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), member124_out_gag(z1))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z5, z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z6, z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z5, z2)))
REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z6, z2)))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg)
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(69) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(member34_in_gg(z2, .(z0, z4)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U3_gg(member34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U3_gg(member34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U3_gg(member34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U3_gg(member34_in_gg(z2, .(z0, .(z4, z5)))))
(70) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x4, U3_gg(member34_in_gg(x4, .(z0, z4))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x4, U3_gg(member34_in_gg(x4, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, z3)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z3, z4)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z1, z4)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z4, z5)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z5, member34_out_gg) → REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z6, member34_out_gg) → REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5))))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), member124_out_gag(z1))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), member124_out_gag(z1))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z5, z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z6, z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z5, z2)))
REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z6, z2)))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U3_gg(member34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U3_gg(member34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U3_gg(member34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U3_gg(member34_in_gg(z2, .(z0, .(z4, z5)))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(71) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(member34_in_gg(z2, .(z4, z5)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U3_gg(member34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U3_gg(member34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U3_gg(member34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U3_gg(member34_in_gg(z2, .(z0, .(z4, z5)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U3_gg(member34_in_gg(z2, .(z4, .(z4, z5)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U3_gg(member34_in_gg(z2, .(z4, .(z5, z6)))))
(72) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x4, U3_gg(member34_in_gg(x4, .(z0, z4))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x4, U3_gg(member34_in_gg(x4, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, z3)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z3, z4)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z1, z4)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z4, z5)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z5, member34_out_gg) → REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z6, member34_out_gg) → REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5))))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), member124_out_gag(z1))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), member124_out_gag(z1))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z5, z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z6, z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z5, z2)))
REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z6, z2)))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U3_gg(member34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U3_gg(member34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U3_gg(member34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U3_gg(member34_in_gg(z2, .(z0, .(z4, z5)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U3_gg(member34_in_gg(z2, .(z4, .(z4, z5)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U3_gg(member34_in_gg(z2, .(z4, .(z5, z6)))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(73) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x4, U3_gg(member34_in_gg(x4, .(z0, z4)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U3_gg(member34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U3_gg(member34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U3_gg(member34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U3_gg(member34_in_gg(z2, .(z0, .(z4, z5)))))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(x5)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U3_gg(member34_in_gg(x5, .(z0, .(z0, z3)))))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(x5)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U3_gg(member34_in_gg(x5, .(z0, .(z3, z4)))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(x5)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U3_gg(member34_in_gg(x5, .(z0, .(z0, z4)))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(x5)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U3_gg(member34_in_gg(x5, .(z0, .(z4, z5)))))
(74) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x4, U3_gg(member34_in_gg(x4, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, z3)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z3, z4)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z1, z4)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z4, z5)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z5, member34_out_gg) → REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z6, member34_out_gg) → REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5))))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), member124_out_gag(z1))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), member124_out_gag(z1))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z5, z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z6, z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z5, z2)))
REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z6, z2)))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U3_gg(member34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U3_gg(member34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U3_gg(member34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U3_gg(member34_in_gg(z2, .(z0, .(z4, z5)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U3_gg(member34_in_gg(z2, .(z4, .(z4, z5)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U3_gg(member34_in_gg(z2, .(z4, .(z5, z6)))))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(x5)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U3_gg(member34_in_gg(x5, .(z0, .(z0, z3)))))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(x5)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U3_gg(member34_in_gg(x5, .(z0, .(z3, z4)))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(x5)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U3_gg(member34_in_gg(x5, .(z0, .(z0, z4)))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(x5)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U3_gg(member34_in_gg(x5, .(z0, .(z4, z5)))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
We have to consider all (P,Q,R)-chains.
(75) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(x4)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x4, U3_gg(member34_in_gg(x4, .(z4, z5)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U3_gg(member34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U3_gg(member34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U3_gg(member34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U3_gg(member34_in_gg(z2, .(z0, .(z4, z5)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U3_gg(member34_in_gg(z2, .(z4, .(z4, z5)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U3_gg(member34_in_gg(z2, .(z4, .(z5, z6)))))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(x6)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x6, U3_gg(member34_in_gg(x6, .(z0, .(z0, z3)))))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(x6)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x6, U3_gg(member34_in_gg(x6, .(z0, .(z3, z4)))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(x6)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U3_gg(member34_in_gg(x6, .(z0, .(z0, z4)))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(x6)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U3_gg(member34_in_gg(x6, .(z0, .(z4, z5)))))
U6_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member124_out_gag(x6)) → U7_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), x6, U3_gg(member34_in_gg(x6, .(z2, .(z2, z4)))))
U6_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member124_out_gag(x6)) → U7_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), x6, U3_gg(member34_in_gg(x6, .(z2, .(z4, z5)))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(x6)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x6, U3_gg(member34_in_gg(x6, .(z4, .(z4, z5)))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(x6)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x6, U3_gg(member34_in_gg(x6, .(z4, .(z5, z6)))))
(76) Obligation:
Q DP problem:The TRS P consists of the following rules:
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, z3)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))))
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z3, z4)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member34_out_gg) → REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z1, z4)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))))
U7_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z1, .(z4, z5)), z2, member34_out_gg) → REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z5, member34_out_gg) → REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4))))
U7_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z6, member34_out_gg) → REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5))))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), member124_out_gag(z1))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), member124_out_gag(z1))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z3, z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z3, .(x2, [])), z2), .(z3, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z5, z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(.(z5, .(x2, [])), z2), .(z5, .(z3, .(z3, z4))), member124_out_gag(x2))
REACH1_IN_GGGG(z6, z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(.(z6, .(x2, [])), z2), .(z6, .(z3, .(z4, z5))), member124_out_gag(x2))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z3, z4))), U2_gag(member124_in_gag(z1, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z3, z2)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z1, z4))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5)))) → U6_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z1, .(z4, z5))), U2_gag(member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U6_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U2_gag(member124_in_gag(z5, z2)))
REACH1_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U6_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U2_gag(member124_in_gag(z6, z2)))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg)
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg)
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U3_gg(member34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U3_gg(member34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U3_gg(member34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U3_gg(member34_in_gg(z2, .(z0, .(z4, z5)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U3_gg(member34_in_gg(z2, .(z4, .(z4, z5)))))
U6_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z2)) → U7_GGGG(z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U3_gg(member34_in_gg(z2, .(z4, .(z5, z6)))))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(x5)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U3_gg(member34_in_gg(x5, .(z0, .(z0, z3)))))
U6_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(x5)) → U7_GGGG(z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U3_gg(member34_in_gg(x5, .(z0, .(z3, z4)))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(x5)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U3_gg(member34_in_gg(x5, .(z0, .(z0, z4)))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(x5)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U3_gg(member34_in_gg(x5, .(z0, .(z4, z5)))))
U6_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member124_out_gag(x6)) → U7_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), x6, U3_gg(member34_in_gg(x6, .(z2, .(z2, z4)))))
U6_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member124_out_gag(x6)) → U7_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), x6, U3_gg(member34_in_gg(x6, .(z2, .(z4, z5)))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(x6)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x6, U3_gg(member34_in_gg(x6, .(z4, .(z4, z5)))))
U6_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(x6)) → U7_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x6, U3_gg(member34_in_gg(x6, .(z4, .(z5, z6)))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(X117)
member124_in_gag(T128, .(T129, T130)) → U2_gag(member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg
member34_in_gg(T159, .(T160, T161)) → U3_gg(member34_in_gg(T159, T161))
U2_gag(member124_out_gag(X131)) → member124_out_gag(X131)
U3_gg(member34_out_gg) → member34_out_gg
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0)
U3_gg(x0)
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 narrowing to the left:
s = REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) evaluates to t =REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, .(z1, z3)))))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [z3 / .(z1, z3)]
- Semiunifier: [ ]
Rewriting sequence
REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3)))) → U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), member124_out_gag(z1))
with rule REACH1_IN_GGGG(z1', z0', .(.(z1', .(z1', [])), z2'), .(z1', .(z1', .(z1', z3')))) → U6_GGGG(z0', .(.(z1', .(z1', [])), z2'), .(z1', .(z1', .(z1', z3'))), member124_out_gag(z1')) at position [] and matcher [z1' / z1, z0' / z0, z2' / z2, z3' / z3]
U6_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), member124_out_gag(z1)) → U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), z1, member34_out_gg)
with rule U6_GGGG(z1', .(.(z0', .(z0', [])), z2'), .(z0', .(z0', .(z0', z3'))), member124_out_gag(z0')) → U7_GGGG(z1', .(.(z0', .(z0', [])), z2'), .(z0', .(z0', .(z0', z3'))), z0', member34_out_gg) at position [] and matcher [z1' / z0, z0' / z1, z2' / z2, z3' / z3]
U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, .(z1, z3)))))
with rule U7_GGGG(z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, z3)), z1, member34_out_gg) → REACH1_IN_GGGG(z1, z0, .(.(z1, .(z1, [])), z2), .(z1, .(z1, .(z1, z3))))
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.
(78) NO
(79) PrologToPiTRSProof (SOUND transformation)
We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:reach1_in: (b,b,b,b)
member12_in: (b,b,b)
member124_in: (b,f,b)
member34_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
reach1_in_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12) → reach1_out_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12)
reach1_in_gggg(T44, T45, .(T46, T47), T12) → U4_gggg(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
member12_in_ggg(T66, T67, .(.(T66, .(T67, [])), T68)) → member12_out_ggg(T66, T67, .(.(T66, .(T67, [])), T68))
member12_in_ggg(T77, T78, .(T79, T80)) → U1_ggg(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
U1_ggg(T77, T78, T79, T80, member12_out_ggg(T77, T78, T80)) → member12_out_ggg(T77, T78, .(T79, T80))
U4_gggg(T44, T45, T46, T47, T12, member12_out_ggg(T44, T45, T47)) → reach1_out_gggg(T44, T45, .(T46, T47), T12)
reach1_in_gggg(T99, T100, T101, T102) → U5_gggg(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
member124_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, X131, .(T129, T130)) → U2_gag(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
U2_gag(T128, X131, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U5_gggg(T99, T100, T101, T102, member124_out_gag(T99, X90, T101)) → reach1_out_gggg(T99, T100, T101, T102)
reach1_in_gggg(T99, T100, T101, T102) → U6_gggg(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_gggg(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_gggg(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → reach1_out_gggg(T99, T100, T101, T102)
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_gggg(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U8_gggg(T99, T100, T101, T102, reach1_out_gggg(T107, T100, T101, .(T107, T102))) → reach1_out_gggg(T99, T100, T101, T102)
The argument filtering Pi contains the following mapping:
reach1_in_gggg(x1, x2, x3, x4) = reach1_in_gggg(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
[] = []
reach1_out_gggg(x1, x2, x3, x4) = reach1_out_gggg(x1, x2, x3, x4)
U4_gggg(x1, x2, x3, x4, x5, x6) = U4_gggg(x1, x2, x3, x4, x5, x6)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member12_out_ggg(x1, x2, x3) = member12_out_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4, x5) = U1_ggg(x1, x2, x3, x4, x5)
U5_gggg(x1, x2, x3, x4, x5) = U5_gggg(x1, x2, x3, x4, x5)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member124_out_gag(x1, x2, x3) = member124_out_gag(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x1, x3, x4, x5)
U6_gggg(x1, x2, x3, x4, x5) = U6_gggg(x1, x2, x3, x4, x5)
U7_gggg(x1, x2, x3, x4, x5, x6) = U7_gggg(x1, x2, x3, x4, x5, x6)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
member34_out_gg(x1, x2) = member34_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4) = U3_gg(x1, x2, x3, x4)
U8_gggg(x1, x2, x3, x4, x5) = U8_gggg(x1, x2, x3, x4, x5)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(80) Obligation:
Pi-finite rewrite system:The TRS R consists of the following rules:
reach1_in_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12) → reach1_out_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12)
reach1_in_gggg(T44, T45, .(T46, T47), T12) → U4_gggg(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
member12_in_ggg(T66, T67, .(.(T66, .(T67, [])), T68)) → member12_out_ggg(T66, T67, .(.(T66, .(T67, [])), T68))
member12_in_ggg(T77, T78, .(T79, T80)) → U1_ggg(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
U1_ggg(T77, T78, T79, T80, member12_out_ggg(T77, T78, T80)) → member12_out_ggg(T77, T78, .(T79, T80))
U4_gggg(T44, T45, T46, T47, T12, member12_out_ggg(T44, T45, T47)) → reach1_out_gggg(T44, T45, .(T46, T47), T12)
reach1_in_gggg(T99, T100, T101, T102) → U5_gggg(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
member124_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, X131, .(T129, T130)) → U2_gag(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
U2_gag(T128, X131, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U5_gggg(T99, T100, T101, T102, member124_out_gag(T99, X90, T101)) → reach1_out_gggg(T99, T100, T101, T102)
reach1_in_gggg(T99, T100, T101, T102) → U6_gggg(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_gggg(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_gggg(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → reach1_out_gggg(T99, T100, T101, T102)
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_gggg(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U8_gggg(T99, T100, T101, T102, reach1_out_gggg(T107, T100, T101, .(T107, T102))) → reach1_out_gggg(T99, T100, T101, T102)
The argument filtering Pi contains the following mapping:
reach1_in_gggg(x1, x2, x3, x4) = reach1_in_gggg(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
[] = []
reach1_out_gggg(x1, x2, x3, x4) = reach1_out_gggg(x1, x2, x3, x4)
U4_gggg(x1, x2, x3, x4, x5, x6) = U4_gggg(x1, x2, x3, x4, x5, x6)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member12_out_ggg(x1, x2, x3) = member12_out_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4, x5) = U1_ggg(x1, x2, x3, x4, x5)
U5_gggg(x1, x2, x3, x4, x5) = U5_gggg(x1, x2, x3, x4, x5)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member124_out_gag(x1, x2, x3) = member124_out_gag(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x1, x3, x4, x5)
U6_gggg(x1, x2, x3, x4, x5) = U6_gggg(x1, x2, x3, x4, x5)
U7_gggg(x1, x2, x3, x4, x5, x6) = U7_gggg(x1, x2, x3, x4, x5, x6)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
member34_out_gg(x1, x2) = member34_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4) = U3_gg(x1, x2, x3, x4)
U8_gggg(x1, x2, x3, x4, x5) = U8_gggg(x1, x2, x3, x4, x5)
(81) 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:
REACH1_IN_GGGG(T44, T45, .(T46, T47), T12) → U4_GGGG(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
REACH1_IN_GGGG(T44, T45, .(T46, T47), T12) → MEMBER12_IN_GGG(T44, T45, T47)
MEMBER12_IN_GGG(T77, T78, .(T79, T80)) → U1_GGG(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
MEMBER12_IN_GGG(T77, T78, .(T79, T80)) → MEMBER12_IN_GGG(T77, T78, T80)
REACH1_IN_GGGG(T99, T100, T101, T102) → U5_GGGG(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
REACH1_IN_GGGG(T99, T100, T101, T102) → MEMBER124_IN_GAG(T99, X90, T101)
MEMBER124_IN_GAG(T128, X131, .(T129, T130)) → U2_GAG(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
MEMBER124_IN_GAG(T128, X131, .(T129, T130)) → MEMBER124_IN_GAG(T128, X131, T130)
REACH1_IN_GGGG(T99, T100, T101, T102) → U6_GGGG(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_GGGG(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_GGGG(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
U6_GGGG(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → MEMBER34_IN_GG(T107, T102)
MEMBER34_IN_GG(T159, .(T160, T161)) → U3_GG(T159, T160, T161, member34_in_gg(T159, T161))
MEMBER34_IN_GG(T159, .(T160, T161)) → MEMBER34_IN_GG(T159, T161)
U7_GGGG(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_GGGG(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U7_GGGG(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102))
The TRS R consists of the following rules:
reach1_in_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12) → reach1_out_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12)
reach1_in_gggg(T44, T45, .(T46, T47), T12) → U4_gggg(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
member12_in_ggg(T66, T67, .(.(T66, .(T67, [])), T68)) → member12_out_ggg(T66, T67, .(.(T66, .(T67, [])), T68))
member12_in_ggg(T77, T78, .(T79, T80)) → U1_ggg(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
U1_ggg(T77, T78, T79, T80, member12_out_ggg(T77, T78, T80)) → member12_out_ggg(T77, T78, .(T79, T80))
U4_gggg(T44, T45, T46, T47, T12, member12_out_ggg(T44, T45, T47)) → reach1_out_gggg(T44, T45, .(T46, T47), T12)
reach1_in_gggg(T99, T100, T101, T102) → U5_gggg(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
member124_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, X131, .(T129, T130)) → U2_gag(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
U2_gag(T128, X131, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U5_gggg(T99, T100, T101, T102, member124_out_gag(T99, X90, T101)) → reach1_out_gggg(T99, T100, T101, T102)
reach1_in_gggg(T99, T100, T101, T102) → U6_gggg(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_gggg(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_gggg(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → reach1_out_gggg(T99, T100, T101, T102)
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_gggg(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U8_gggg(T99, T100, T101, T102, reach1_out_gggg(T107, T100, T101, .(T107, T102))) → reach1_out_gggg(T99, T100, T101, T102)
The argument filtering Pi contains the following mapping:
reach1_in_gggg(x1, x2, x3, x4) = reach1_in_gggg(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
[] = []
reach1_out_gggg(x1, x2, x3, x4) = reach1_out_gggg(x1, x2, x3, x4)
U4_gggg(x1, x2, x3, x4, x5, x6) = U4_gggg(x1, x2, x3, x4, x5, x6)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member12_out_ggg(x1, x2, x3) = member12_out_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4, x5) = U1_ggg(x1, x2, x3, x4, x5)
U5_gggg(x1, x2, x3, x4, x5) = U5_gggg(x1, x2, x3, x4, x5)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member124_out_gag(x1, x2, x3) = member124_out_gag(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x1, x3, x4, x5)
U6_gggg(x1, x2, x3, x4, x5) = U6_gggg(x1, x2, x3, x4, x5)
U7_gggg(x1, x2, x3, x4, x5, x6) = U7_gggg(x1, x2, x3, x4, x5, x6)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
member34_out_gg(x1, x2) = member34_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4) = U3_gg(x1, x2, x3, x4)
U8_gggg(x1, x2, x3, x4, x5) = U8_gggg(x1, x2, x3, x4, x5)
REACH1_IN_GGGG(x1, x2, x3, x4) = REACH1_IN_GGGG(x1, x2, x3, x4)
U4_GGGG(x1, x2, x3, x4, x5, x6) = U4_GGGG(x1, x2, x3, x4, x5, x6)
MEMBER12_IN_GGG(x1, x2, x3) = MEMBER12_IN_GGG(x1, x2, x3)
U1_GGG(x1, x2, x3, x4, x5) = U1_GGG(x1, x2, x3, x4, x5)
U5_GGGG(x1, x2, x3, x4, x5) = U5_GGGG(x1, x2, x3, x4, x5)
MEMBER124_IN_GAG(x1, x2, x3) = MEMBER124_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5) = U2_GAG(x1, x3, x4, x5)
U6_GGGG(x1, x2, x3, x4, x5) = U6_GGGG(x1, x2, x3, x4, x5)
U7_GGGG(x1, x2, x3, x4, x5, x6) = U7_GGGG(x1, x2, x3, x4, x5, x6)
MEMBER34_IN_GG(x1, x2) = MEMBER34_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4) = U3_GG(x1, x2, x3, x4)
U8_GGGG(x1, x2, x3, x4, x5) = U8_GGGG(x1, x2, x3, x4, x5)
We have to consider all (P,R,Pi)-chains
(82) Obligation:
Pi DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(T44, T45, .(T46, T47), T12) → U4_GGGG(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
REACH1_IN_GGGG(T44, T45, .(T46, T47), T12) → MEMBER12_IN_GGG(T44, T45, T47)
MEMBER12_IN_GGG(T77, T78, .(T79, T80)) → U1_GGG(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
MEMBER12_IN_GGG(T77, T78, .(T79, T80)) → MEMBER12_IN_GGG(T77, T78, T80)
REACH1_IN_GGGG(T99, T100, T101, T102) → U5_GGGG(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
REACH1_IN_GGGG(T99, T100, T101, T102) → MEMBER124_IN_GAG(T99, X90, T101)
MEMBER124_IN_GAG(T128, X131, .(T129, T130)) → U2_GAG(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
MEMBER124_IN_GAG(T128, X131, .(T129, T130)) → MEMBER124_IN_GAG(T128, X131, T130)
REACH1_IN_GGGG(T99, T100, T101, T102) → U6_GGGG(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_GGGG(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_GGGG(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
U6_GGGG(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → MEMBER34_IN_GG(T107, T102)
MEMBER34_IN_GG(T159, .(T160, T161)) → U3_GG(T159, T160, T161, member34_in_gg(T159, T161))
MEMBER34_IN_GG(T159, .(T160, T161)) → MEMBER34_IN_GG(T159, T161)
U7_GGGG(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_GGGG(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U7_GGGG(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102))
The TRS R consists of the following rules:
reach1_in_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12) → reach1_out_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12)
reach1_in_gggg(T44, T45, .(T46, T47), T12) → U4_gggg(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
member12_in_ggg(T66, T67, .(.(T66, .(T67, [])), T68)) → member12_out_ggg(T66, T67, .(.(T66, .(T67, [])), T68))
member12_in_ggg(T77, T78, .(T79, T80)) → U1_ggg(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
U1_ggg(T77, T78, T79, T80, member12_out_ggg(T77, T78, T80)) → member12_out_ggg(T77, T78, .(T79, T80))
U4_gggg(T44, T45, T46, T47, T12, member12_out_ggg(T44, T45, T47)) → reach1_out_gggg(T44, T45, .(T46, T47), T12)
reach1_in_gggg(T99, T100, T101, T102) → U5_gggg(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
member124_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, X131, .(T129, T130)) → U2_gag(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
U2_gag(T128, X131, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U5_gggg(T99, T100, T101, T102, member124_out_gag(T99, X90, T101)) → reach1_out_gggg(T99, T100, T101, T102)
reach1_in_gggg(T99, T100, T101, T102) → U6_gggg(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_gggg(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_gggg(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → reach1_out_gggg(T99, T100, T101, T102)
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_gggg(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U8_gggg(T99, T100, T101, T102, reach1_out_gggg(T107, T100, T101, .(T107, T102))) → reach1_out_gggg(T99, T100, T101, T102)
The argument filtering Pi contains the following mapping:
reach1_in_gggg(x1, x2, x3, x4) = reach1_in_gggg(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
[] = []
reach1_out_gggg(x1, x2, x3, x4) = reach1_out_gggg(x1, x2, x3, x4)
U4_gggg(x1, x2, x3, x4, x5, x6) = U4_gggg(x1, x2, x3, x4, x5, x6)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member12_out_ggg(x1, x2, x3) = member12_out_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4, x5) = U1_ggg(x1, x2, x3, x4, x5)
U5_gggg(x1, x2, x3, x4, x5) = U5_gggg(x1, x2, x3, x4, x5)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member124_out_gag(x1, x2, x3) = member124_out_gag(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x1, x3, x4, x5)
U6_gggg(x1, x2, x3, x4, x5) = U6_gggg(x1, x2, x3, x4, x5)
U7_gggg(x1, x2, x3, x4, x5, x6) = U7_gggg(x1, x2, x3, x4, x5, x6)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
member34_out_gg(x1, x2) = member34_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4) = U3_gg(x1, x2, x3, x4)
U8_gggg(x1, x2, x3, x4, x5) = U8_gggg(x1, x2, x3, x4, x5)
REACH1_IN_GGGG(x1, x2, x3, x4) = REACH1_IN_GGGG(x1, x2, x3, x4)
U4_GGGG(x1, x2, x3, x4, x5, x6) = U4_GGGG(x1, x2, x3, x4, x5, x6)
MEMBER12_IN_GGG(x1, x2, x3) = MEMBER12_IN_GGG(x1, x2, x3)
U1_GGG(x1, x2, x3, x4, x5) = U1_GGG(x1, x2, x3, x4, x5)
U5_GGGG(x1, x2, x3, x4, x5) = U5_GGGG(x1, x2, x3, x4, x5)
MEMBER124_IN_GAG(x1, x2, x3) = MEMBER124_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4, x5) = U2_GAG(x1, x3, x4, x5)
U6_GGGG(x1, x2, x3, x4, x5) = U6_GGGG(x1, x2, x3, x4, x5)
U7_GGGG(x1, x2, x3, x4, x5, x6) = U7_GGGG(x1, x2, x3, x4, x5, x6)
MEMBER34_IN_GG(x1, x2) = MEMBER34_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4) = U3_GG(x1, x2, x3, x4)
U8_GGGG(x1, x2, x3, x4, x5) = U8_GGGG(x1, x2, x3, x4, x5)
We have to consider all (P,R,Pi)-chains
(83) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 9 less nodes.(84) Complex Obligation (AND)
(85) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MEMBER34_IN_GG(T159, .(T160, T161)) → MEMBER34_IN_GG(T159, T161)
The TRS R consists of the following rules:
reach1_in_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12) → reach1_out_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12)
reach1_in_gggg(T44, T45, .(T46, T47), T12) → U4_gggg(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
member12_in_ggg(T66, T67, .(.(T66, .(T67, [])), T68)) → member12_out_ggg(T66, T67, .(.(T66, .(T67, [])), T68))
member12_in_ggg(T77, T78, .(T79, T80)) → U1_ggg(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
U1_ggg(T77, T78, T79, T80, member12_out_ggg(T77, T78, T80)) → member12_out_ggg(T77, T78, .(T79, T80))
U4_gggg(T44, T45, T46, T47, T12, member12_out_ggg(T44, T45, T47)) → reach1_out_gggg(T44, T45, .(T46, T47), T12)
reach1_in_gggg(T99, T100, T101, T102) → U5_gggg(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
member124_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, X131, .(T129, T130)) → U2_gag(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
U2_gag(T128, X131, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U5_gggg(T99, T100, T101, T102, member124_out_gag(T99, X90, T101)) → reach1_out_gggg(T99, T100, T101, T102)
reach1_in_gggg(T99, T100, T101, T102) → U6_gggg(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_gggg(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_gggg(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → reach1_out_gggg(T99, T100, T101, T102)
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_gggg(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U8_gggg(T99, T100, T101, T102, reach1_out_gggg(T107, T100, T101, .(T107, T102))) → reach1_out_gggg(T99, T100, T101, T102)
The argument filtering Pi contains the following mapping:
reach1_in_gggg(x1, x2, x3, x4) = reach1_in_gggg(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
[] = []
reach1_out_gggg(x1, x2, x3, x4) = reach1_out_gggg(x1, x2, x3, x4)
U4_gggg(x1, x2, x3, x4, x5, x6) = U4_gggg(x1, x2, x3, x4, x5, x6)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member12_out_ggg(x1, x2, x3) = member12_out_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4, x5) = U1_ggg(x1, x2, x3, x4, x5)
U5_gggg(x1, x2, x3, x4, x5) = U5_gggg(x1, x2, x3, x4, x5)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member124_out_gag(x1, x2, x3) = member124_out_gag(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x1, x3, x4, x5)
U6_gggg(x1, x2, x3, x4, x5) = U6_gggg(x1, x2, x3, x4, x5)
U7_gggg(x1, x2, x3, x4, x5, x6) = U7_gggg(x1, x2, x3, x4, x5, x6)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
member34_out_gg(x1, x2) = member34_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4) = U3_gg(x1, x2, x3, x4)
U8_gggg(x1, x2, x3, x4, x5) = U8_gggg(x1, x2, x3, x4, x5)
MEMBER34_IN_GG(x1, x2) = MEMBER34_IN_GG(x1, x2)
We have to consider all (P,R,Pi)-chains
(86) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(87) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MEMBER34_IN_GG(T159, .(T160, T161)) → MEMBER34_IN_GG(T159, T161)
R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
(88) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(89) Obligation:
Q DP problem:The TRS P consists of the following rules:
MEMBER34_IN_GG(T159, .(T160, T161)) → MEMBER34_IN_GG(T159, T161)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(90) 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:
- MEMBER34_IN_GG(T159, .(T160, T161)) → MEMBER34_IN_GG(T159, T161)
The graph contains the following edges 1 >= 1, 2 > 2
(91) YES
(92) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MEMBER124_IN_GAG(T128, X131, .(T129, T130)) → MEMBER124_IN_GAG(T128, X131, T130)
The TRS R consists of the following rules:
reach1_in_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12) → reach1_out_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12)
reach1_in_gggg(T44, T45, .(T46, T47), T12) → U4_gggg(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
member12_in_ggg(T66, T67, .(.(T66, .(T67, [])), T68)) → member12_out_ggg(T66, T67, .(.(T66, .(T67, [])), T68))
member12_in_ggg(T77, T78, .(T79, T80)) → U1_ggg(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
U1_ggg(T77, T78, T79, T80, member12_out_ggg(T77, T78, T80)) → member12_out_ggg(T77, T78, .(T79, T80))
U4_gggg(T44, T45, T46, T47, T12, member12_out_ggg(T44, T45, T47)) → reach1_out_gggg(T44, T45, .(T46, T47), T12)
reach1_in_gggg(T99, T100, T101, T102) → U5_gggg(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
member124_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, X131, .(T129, T130)) → U2_gag(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
U2_gag(T128, X131, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U5_gggg(T99, T100, T101, T102, member124_out_gag(T99, X90, T101)) → reach1_out_gggg(T99, T100, T101, T102)
reach1_in_gggg(T99, T100, T101, T102) → U6_gggg(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_gggg(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_gggg(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → reach1_out_gggg(T99, T100, T101, T102)
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_gggg(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U8_gggg(T99, T100, T101, T102, reach1_out_gggg(T107, T100, T101, .(T107, T102))) → reach1_out_gggg(T99, T100, T101, T102)
The argument filtering Pi contains the following mapping:
reach1_in_gggg(x1, x2, x3, x4) = reach1_in_gggg(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
[] = []
reach1_out_gggg(x1, x2, x3, x4) = reach1_out_gggg(x1, x2, x3, x4)
U4_gggg(x1, x2, x3, x4, x5, x6) = U4_gggg(x1, x2, x3, x4, x5, x6)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member12_out_ggg(x1, x2, x3) = member12_out_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4, x5) = U1_ggg(x1, x2, x3, x4, x5)
U5_gggg(x1, x2, x3, x4, x5) = U5_gggg(x1, x2, x3, x4, x5)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member124_out_gag(x1, x2, x3) = member124_out_gag(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x1, x3, x4, x5)
U6_gggg(x1, x2, x3, x4, x5) = U6_gggg(x1, x2, x3, x4, x5)
U7_gggg(x1, x2, x3, x4, x5, x6) = U7_gggg(x1, x2, x3, x4, x5, x6)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
member34_out_gg(x1, x2) = member34_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4) = U3_gg(x1, x2, x3, x4)
U8_gggg(x1, x2, x3, x4, x5) = U8_gggg(x1, x2, x3, x4, x5)
MEMBER124_IN_GAG(x1, x2, x3) = MEMBER124_IN_GAG(x1, x3)
We have to consider all (P,R,Pi)-chains
(93) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(94) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MEMBER124_IN_GAG(T128, X131, .(T129, T130)) → MEMBER124_IN_GAG(T128, X131, T130)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
MEMBER124_IN_GAG(x1, x2, x3) = MEMBER124_IN_GAG(x1, x3)
We have to consider all (P,R,Pi)-chains
(95) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(96) Obligation:
Q DP problem:The TRS P consists of the following rules:
MEMBER124_IN_GAG(T128, .(T129, T130)) → MEMBER124_IN_GAG(T128, T130)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(97) 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:
- MEMBER124_IN_GAG(T128, .(T129, T130)) → MEMBER124_IN_GAG(T128, T130)
The graph contains the following edges 1 >= 1, 2 > 2
(98) YES
(99) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MEMBER12_IN_GGG(T77, T78, .(T79, T80)) → MEMBER12_IN_GGG(T77, T78, T80)
The TRS R consists of the following rules:
reach1_in_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12) → reach1_out_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12)
reach1_in_gggg(T44, T45, .(T46, T47), T12) → U4_gggg(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
member12_in_ggg(T66, T67, .(.(T66, .(T67, [])), T68)) → member12_out_ggg(T66, T67, .(.(T66, .(T67, [])), T68))
member12_in_ggg(T77, T78, .(T79, T80)) → U1_ggg(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
U1_ggg(T77, T78, T79, T80, member12_out_ggg(T77, T78, T80)) → member12_out_ggg(T77, T78, .(T79, T80))
U4_gggg(T44, T45, T46, T47, T12, member12_out_ggg(T44, T45, T47)) → reach1_out_gggg(T44, T45, .(T46, T47), T12)
reach1_in_gggg(T99, T100, T101, T102) → U5_gggg(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
member124_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, X131, .(T129, T130)) → U2_gag(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
U2_gag(T128, X131, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U5_gggg(T99, T100, T101, T102, member124_out_gag(T99, X90, T101)) → reach1_out_gggg(T99, T100, T101, T102)
reach1_in_gggg(T99, T100, T101, T102) → U6_gggg(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_gggg(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_gggg(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → reach1_out_gggg(T99, T100, T101, T102)
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_gggg(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U8_gggg(T99, T100, T101, T102, reach1_out_gggg(T107, T100, T101, .(T107, T102))) → reach1_out_gggg(T99, T100, T101, T102)
The argument filtering Pi contains the following mapping:
reach1_in_gggg(x1, x2, x3, x4) = reach1_in_gggg(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
[] = []
reach1_out_gggg(x1, x2, x3, x4) = reach1_out_gggg(x1, x2, x3, x4)
U4_gggg(x1, x2, x3, x4, x5, x6) = U4_gggg(x1, x2, x3, x4, x5, x6)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member12_out_ggg(x1, x2, x3) = member12_out_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4, x5) = U1_ggg(x1, x2, x3, x4, x5)
U5_gggg(x1, x2, x3, x4, x5) = U5_gggg(x1, x2, x3, x4, x5)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member124_out_gag(x1, x2, x3) = member124_out_gag(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x1, x3, x4, x5)
U6_gggg(x1, x2, x3, x4, x5) = U6_gggg(x1, x2, x3, x4, x5)
U7_gggg(x1, x2, x3, x4, x5, x6) = U7_gggg(x1, x2, x3, x4, x5, x6)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
member34_out_gg(x1, x2) = member34_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4) = U3_gg(x1, x2, x3, x4)
U8_gggg(x1, x2, x3, x4, x5) = U8_gggg(x1, x2, x3, x4, x5)
MEMBER12_IN_GGG(x1, x2, x3) = MEMBER12_IN_GGG(x1, x2, x3)
We have to consider all (P,R,Pi)-chains
(100) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(101) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MEMBER12_IN_GGG(T77, T78, .(T79, T80)) → MEMBER12_IN_GGG(T77, T78, T80)
R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
(102) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(103) Obligation:
Q DP problem:The TRS P consists of the following rules:
MEMBER12_IN_GGG(T77, T78, .(T79, T80)) → MEMBER12_IN_GGG(T77, T78, T80)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(104) 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:
- MEMBER12_IN_GGG(T77, T78, .(T79, T80)) → MEMBER12_IN_GGG(T77, T78, T80)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
(105) YES
(106) Obligation:
Pi DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(T99, T100, T101, T102) → U6_GGGG(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_GGGG(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_GGGG(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
U7_GGGG(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102))
The TRS R consists of the following rules:
reach1_in_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12) → reach1_out_gggg(T25, T26, .(.(T25, .(T26, [])), T27), T12)
reach1_in_gggg(T44, T45, .(T46, T47), T12) → U4_gggg(T44, T45, T46, T47, T12, member12_in_ggg(T44, T45, T47))
member12_in_ggg(T66, T67, .(.(T66, .(T67, [])), T68)) → member12_out_ggg(T66, T67, .(.(T66, .(T67, [])), T68))
member12_in_ggg(T77, T78, .(T79, T80)) → U1_ggg(T77, T78, T79, T80, member12_in_ggg(T77, T78, T80))
U1_ggg(T77, T78, T79, T80, member12_out_ggg(T77, T78, T80)) → member12_out_ggg(T77, T78, .(T79, T80))
U4_gggg(T44, T45, T46, T47, T12, member12_out_ggg(T44, T45, T47)) → reach1_out_gggg(T44, T45, .(T46, T47), T12)
reach1_in_gggg(T99, T100, T101, T102) → U5_gggg(T99, T100, T101, T102, member124_in_gag(T99, X90, T101))
member124_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, X131, .(T129, T130)) → U2_gag(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
U2_gag(T128, X131, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U5_gggg(T99, T100, T101, T102, member124_out_gag(T99, X90, T101)) → reach1_out_gggg(T99, T100, T101, T102)
reach1_in_gggg(T99, T100, T101, T102) → U6_gggg(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_gggg(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_gggg(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → reach1_out_gggg(T99, T100, T101, T102)
U7_gggg(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → U8_gggg(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U8_gggg(T99, T100, T101, T102, reach1_out_gggg(T107, T100, T101, .(T107, T102))) → reach1_out_gggg(T99, T100, T101, T102)
The argument filtering Pi contains the following mapping:
reach1_in_gggg(x1, x2, x3, x4) = reach1_in_gggg(x1, x2, x3, x4)
.(x1, x2) = .(x1, x2)
[] = []
reach1_out_gggg(x1, x2, x3, x4) = reach1_out_gggg(x1, x2, x3, x4)
U4_gggg(x1, x2, x3, x4, x5, x6) = U4_gggg(x1, x2, x3, x4, x5, x6)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member12_out_ggg(x1, x2, x3) = member12_out_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4, x5) = U1_ggg(x1, x2, x3, x4, x5)
U5_gggg(x1, x2, x3, x4, x5) = U5_gggg(x1, x2, x3, x4, x5)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member124_out_gag(x1, x2, x3) = member124_out_gag(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x1, x3, x4, x5)
U6_gggg(x1, x2, x3, x4, x5) = U6_gggg(x1, x2, x3, x4, x5)
U7_gggg(x1, x2, x3, x4, x5, x6) = U7_gggg(x1, x2, x3, x4, x5, x6)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
member34_out_gg(x1, x2) = member34_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4) = U3_gg(x1, x2, x3, x4)
U8_gggg(x1, x2, x3, x4, x5) = U8_gggg(x1, x2, x3, x4, x5)
REACH1_IN_GGGG(x1, x2, x3, x4) = REACH1_IN_GGGG(x1, x2, x3, x4)
U6_GGGG(x1, x2, x3, x4, x5) = U6_GGGG(x1, x2, x3, x4, x5)
U7_GGGG(x1, x2, x3, x4, x5, x6) = U7_GGGG(x1, x2, x3, x4, x5, x6)
We have to consider all (P,R,Pi)-chains
(107) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(108) Obligation:
Pi DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(T99, T100, T101, T102) → U6_GGGG(T99, T100, T101, T102, member124_in_gag(T99, T107, T101))
U6_GGGG(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_GGGG(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
U7_GGGG(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102))
The TRS R consists of the following rules:
member124_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, X131, .(T129, T130)) → U2_gag(T128, X131, T129, T130, member124_in_gag(T128, X131, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, X131, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member124_out_gag(x1, x2, x3) = member124_out_gag(x1, x2, x3)
U2_gag(x1, x2, x3, x4, x5) = U2_gag(x1, x3, x4, x5)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
member34_out_gg(x1, x2) = member34_out_gg(x1, x2)
U3_gg(x1, x2, x3, x4) = U3_gg(x1, x2, x3, x4)
REACH1_IN_GGGG(x1, x2, x3, x4) = REACH1_IN_GGGG(x1, x2, x3, x4)
U6_GGGG(x1, x2, x3, x4, x5) = U6_GGGG(x1, x2, x3, x4, x5)
U7_GGGG(x1, x2, x3, x4, x5, x6) = U7_GGGG(x1, x2, x3, x4, x5, x6)
We have to consider all (P,R,Pi)-chains
(109) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(110) Obligation:
Q DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(T99, T100, T101, T102) → U6_GGGG(T99, T100, T101, T102, member124_in_gag(T99, T101))
U6_GGGG(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_GGGG(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
U7_GGGG(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(111) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule REACH1_IN_GGGG(T99, T100, T101, T102) → U6_GGGG(T99, T100, T101, T102, member124_in_gag(T99, T101)) at position [4] we obtained the following new rules [LPAR04]:
REACH1_IN_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3) → U6_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3, member124_out_gag(x0, x1, .(.(x0, .(x1, [])), x2)))
REACH1_IN_GGGG(x0, y1, .(x1, x2), y3) → U6_GGGG(x0, y1, .(x1, x2), y3, U2_gag(x0, x1, x2, member124_in_gag(x0, x2)))
(112) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_GGGG(T99, T100, T101, T102, T107, member34_in_gg(T107, T102))
U7_GGGG(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102))
REACH1_IN_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3) → U6_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3, member124_out_gag(x0, x1, .(.(x0, .(x1, [])), x2)))
REACH1_IN_GGGG(x0, y1, .(x1, x2), y3) → U6_GGGG(x0, y1, .(x1, x2), y3, U2_gag(x0, x1, x2, member124_in_gag(x0, x2)))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(113) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule U6_GGGG(T99, T100, T101, T102, member124_out_gag(T99, T107, T101)) → U7_GGGG(T99, T100, T101, T102, T107, member34_in_gg(T107, T102)) at position [5] we obtained the following new rules [LPAR04]:
U6_GGGG(y0, y1, y2, .(x0, x1), member124_out_gag(y0, x0, y2)) → U7_GGGG(y0, y1, y2, .(x0, x1), x0, member34_out_gg(x0, .(x0, x1)))
U6_GGGG(y0, y1, y2, .(x1, x2), member124_out_gag(y0, x0, y2)) → U7_GGGG(y0, y1, y2, .(x1, x2), x0, U3_gg(x0, x1, x2, member34_in_gg(x0, x2)))
(114) Obligation:
Q DP problem:The TRS P consists of the following rules:
U7_GGGG(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102))
REACH1_IN_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3) → U6_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3, member124_out_gag(x0, x1, .(.(x0, .(x1, [])), x2)))
REACH1_IN_GGGG(x0, y1, .(x1, x2), y3) → U6_GGGG(x0, y1, .(x1, x2), y3, U2_gag(x0, x1, x2, member124_in_gag(x0, x2)))
U6_GGGG(y0, y1, y2, .(x0, x1), member124_out_gag(y0, x0, y2)) → U7_GGGG(y0, y1, y2, .(x0, x1), x0, member34_out_gg(x0, .(x0, x1)))
U6_GGGG(y0, y1, y2, .(x1, x2), member124_out_gag(y0, x0, y2)) → U7_GGGG(y0, y1, y2, .(x1, x2), x0, U3_gg(x0, x1, x2, member34_in_gg(x0, x2)))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(115) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U7_GGGG(T99, T100, T101, T102, T107, member34_out_gg(T107, T102)) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102)) we obtained the following new rules [LPAR04]:
U7_GGGG(z0, z1, z2, .(z3, z4), z3, member34_out_gg(z3, .(z3, z4))) → REACH1_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))
U7_GGGG(z0, z1, z2, .(z3, z4), z5, member34_out_gg(z5, .(z3, z4))) → REACH1_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4)))
(116) Obligation:
Q DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3) → U6_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3, member124_out_gag(x0, x1, .(.(x0, .(x1, [])), x2)))
REACH1_IN_GGGG(x0, y1, .(x1, x2), y3) → U6_GGGG(x0, y1, .(x1, x2), y3, U2_gag(x0, x1, x2, member124_in_gag(x0, x2)))
U6_GGGG(y0, y1, y2, .(x0, x1), member124_out_gag(y0, x0, y2)) → U7_GGGG(y0, y1, y2, .(x0, x1), x0, member34_out_gg(x0, .(x0, x1)))
U6_GGGG(y0, y1, y2, .(x1, x2), member124_out_gag(y0, x0, y2)) → U7_GGGG(y0, y1, y2, .(x1, x2), x0, U3_gg(x0, x1, x2, member34_in_gg(x0, x2)))
U7_GGGG(z0, z1, z2, .(z3, z4), z3, member34_out_gg(z3, .(z3, z4))) → REACH1_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))
U7_GGGG(z0, z1, z2, .(z3, z4), z5, member34_out_gg(z5, .(z3, z4))) → REACH1_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4)))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(117) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule REACH1_IN_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3) → U6_GGGG(x0, y1, .(.(x0, .(x1, [])), x2), y3, member124_out_gag(x0, x1, .(.(x0, .(x1, [])), x2))) we obtained the following new rules [LPAR04]:
REACH1_IN_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4))) → U6_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4)), member124_out_gag(z3, x2, .(.(z3, .(x2, [])), x3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), x3)))
(118) Obligation:
Q DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(x0, y1, .(x1, x2), y3) → U6_GGGG(x0, y1, .(x1, x2), y3, U2_gag(x0, x1, x2, member124_in_gag(x0, x2)))
U6_GGGG(y0, y1, y2, .(x0, x1), member124_out_gag(y0, x0, y2)) → U7_GGGG(y0, y1, y2, .(x0, x1), x0, member34_out_gg(x0, .(x0, x1)))
U6_GGGG(y0, y1, y2, .(x1, x2), member124_out_gag(y0, x0, y2)) → U7_GGGG(y0, y1, y2, .(x1, x2), x0, U3_gg(x0, x1, x2, member34_in_gg(x0, x2)))
U7_GGGG(z0, z1, z2, .(z3, z4), z3, member34_out_gg(z3, .(z3, z4))) → REACH1_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))
U7_GGGG(z0, z1, z2, .(z3, z4), z5, member34_out_gg(z5, .(z3, z4))) → REACH1_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4)))
REACH1_IN_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4))) → U6_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4)), member124_out_gag(z3, x2, .(.(z3, .(x2, [])), x3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), x3)))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(119) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule REACH1_IN_GGGG(x0, y1, .(x1, x2), y3) → U6_GGGG(x0, y1, .(x1, x2), y3, U2_gag(x0, x1, x2, member124_in_gag(x0, x2))) we obtained the following new rules [LPAR04]:
REACH1_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U2_gag(z3, x2, x3, member124_in_gag(z3, x3)))
REACH1_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U2_gag(z5, x2, x3, member124_in_gag(z5, x3)))
(120) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(y0, y1, y2, .(x0, x1), member124_out_gag(y0, x0, y2)) → U7_GGGG(y0, y1, y2, .(x0, x1), x0, member34_out_gg(x0, .(x0, x1)))
U6_GGGG(y0, y1, y2, .(x1, x2), member124_out_gag(y0, x0, y2)) → U7_GGGG(y0, y1, y2, .(x1, x2), x0, U3_gg(x0, x1, x2, member34_in_gg(x0, x2)))
U7_GGGG(z0, z1, z2, .(z3, z4), z3, member34_out_gg(z3, .(z3, z4))) → REACH1_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))
U7_GGGG(z0, z1, z2, .(z3, z4), z5, member34_out_gg(z5, .(z3, z4))) → REACH1_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4)))
REACH1_IN_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4))) → U6_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4)), member124_out_gag(z3, x2, .(.(z3, .(x2, [])), x3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), x3)))
REACH1_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U2_gag(z3, x2, x3, member124_in_gag(z3, x3)))
REACH1_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U2_gag(z5, x2, x3, member124_in_gag(z5, x3)))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(121) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(y0, y1, y2, .(x0, x1), member124_out_gag(y0, x0, y2)) → U7_GGGG(y0, y1, y2, .(x0, x1), x0, member34_out_gg(x0, .(x0, x1))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
(122) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(y0, y1, y2, .(x1, x2), member124_out_gag(y0, x0, y2)) → U7_GGGG(y0, y1, y2, .(x1, x2), x0, U3_gg(x0, x1, x2, member34_in_gg(x0, x2)))
U7_GGGG(z0, z1, z2, .(z3, z4), z3, member34_out_gg(z3, .(z3, z4))) → REACH1_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))
U7_GGGG(z0, z1, z2, .(z3, z4), z5, member34_out_gg(z5, .(z3, z4))) → REACH1_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4)))
REACH1_IN_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4))) → U6_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4)), member124_out_gag(z3, x2, .(.(z3, .(x2, [])), x3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), x3)))
REACH1_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U2_gag(z3, x2, x3, member124_in_gag(z3, x3)))
REACH1_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U2_gag(z5, x2, x3, member124_in_gag(z5, x3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(123) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(y0, y1, y2, .(x1, x2), member124_out_gag(y0, x0, y2)) → U7_GGGG(y0, y1, y2, .(x1, x2), x0, U3_gg(x0, x1, x2, member34_in_gg(x0, x2))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(z2, z0, .(z0, z4), member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(z2, z0, .(z4, z5), member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U3_gg(x5, z0, .(z0, z4), member34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U3_gg(x5, z0, .(z4, z5), member34_in_gg(x5, .(z4, z5))))
(124) Obligation:
Q DP problem:The TRS P consists of the following rules:
U7_GGGG(z0, z1, z2, .(z3, z4), z3, member34_out_gg(z3, .(z3, z4))) → REACH1_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))
U7_GGGG(z0, z1, z2, .(z3, z4), z5, member34_out_gg(z5, .(z3, z4))) → REACH1_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4)))
REACH1_IN_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4))) → U6_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4)), member124_out_gag(z3, x2, .(.(z3, .(x2, [])), x3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), x3)))
REACH1_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U2_gag(z3, x2, x3, member124_in_gag(z3, x3)))
REACH1_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U2_gag(z5, x2, x3, member124_in_gag(z5, x3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(z2, z0, .(z0, z4), member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(z2, z0, .(z4, z5), member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U3_gg(x5, z0, .(z0, z4), member34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U3_gg(x5, z0, .(z4, z5), member34_in_gg(x5, .(z4, z5))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(125) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U7_GGGG(z0, z1, z2, .(z3, z4), z3, member34_out_gg(z3, .(z3, z4))) → REACH1_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4))) we obtained the following new rules [LPAR04]:
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
(126) Obligation:
Q DP problem:The TRS P consists of the following rules:
U7_GGGG(z0, z1, z2, .(z3, z4), z5, member34_out_gg(z5, .(z3, z4))) → REACH1_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4)))
REACH1_IN_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4))) → U6_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4)), member124_out_gag(z3, x2, .(.(z3, .(x2, [])), x3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), x3)))
REACH1_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U2_gag(z3, x2, x3, member124_in_gag(z3, x3)))
REACH1_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U2_gag(z5, x2, x3, member124_in_gag(z5, x3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(z2, z0, .(z0, z4), member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(z2, z0, .(z4, z5), member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U3_gg(x5, z0, .(z0, z4), member34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U3_gg(x5, z0, .(z4, z5), member34_in_gg(x5, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(127) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U7_GGGG(z0, z1, z2, .(z3, z4), z5, member34_out_gg(z5, .(z3, z4))) → REACH1_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) we obtained the following new rules [LPAR04]:
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member34_out_gg(z6, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))))
(128) Obligation:
Q DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4))) → U6_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4)), member124_out_gag(z3, x2, .(.(z3, .(x2, [])), x3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), x3)))
REACH1_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U2_gag(z3, x2, x3, member124_in_gag(z3, x3)))
REACH1_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U2_gag(z5, x2, x3, member124_in_gag(z5, x3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(z2, z0, .(z0, z4), member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(z2, z0, .(z4, z5), member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U3_gg(x5, z0, .(z0, z4), member34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U3_gg(x5, z0, .(z4, z5), member34_in_gg(x5, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member34_out_gg(z6, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(129) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule REACH1_IN_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4))) → U6_GGGG(z3, z1, .(.(z3, .(x2, [])), x3), .(z3, .(z3, z4)), member124_out_gag(z3, x2, .(.(z3, .(x2, [])), x3))) we obtained the following new rules [LPAR04]:
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
(130) Obligation:
Q DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), x3)))
REACH1_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U2_gag(z3, x2, x3, member124_in_gag(z3, x3)))
REACH1_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U2_gag(z5, x2, x3, member124_in_gag(z5, x3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(z2, z0, .(z0, z4), member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(z2, z0, .(z4, z5), member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U3_gg(x5, z0, .(z0, z4), member34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U3_gg(x5, z0, .(z4, z5), member34_in_gg(x5, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member34_out_gg(z6, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(131) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), x3), .(z5, .(z3, z4)), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), x3))) we obtained the following new rules [LPAR04]:
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), z3)))
REACH1_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member124_out_gag(z6, x2, .(.(z6, .(x2, [])), z3)))
(132) Obligation:
Q DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U2_gag(z3, x2, x3, member124_in_gag(z3, x3)))
REACH1_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U2_gag(z5, x2, x3, member124_in_gag(z5, x3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(z2, z0, .(z0, z4), member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(z2, z0, .(z4, z5), member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U3_gg(x5, z0, .(z0, z4), member34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U3_gg(x5, z0, .(z4, z5), member34_in_gg(x5, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member34_out_gg(z6, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), z3)))
REACH1_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member124_out_gag(z6, x2, .(.(z6, .(x2, [])), z3)))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(133) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule REACH1_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U6_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U2_gag(z3, x2, x3, member124_in_gag(z3, x3))) we obtained the following new rules [LPAR04]:
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
(134) Obligation:
Q DP problem:The TRS P consists of the following rules:
REACH1_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U2_gag(z5, x2, x3, member124_in_gag(z5, x3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(z2, z0, .(z0, z4), member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(z2, z0, .(z4, z5), member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U3_gg(x5, z0, .(z0, z4), member34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U3_gg(x5, z0, .(z4, z5), member34_in_gg(x5, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member34_out_gg(z6, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), z3)))
REACH1_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member124_out_gag(z6, x2, .(.(z6, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(135) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule REACH1_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U2_gag(z5, x2, x3, member124_in_gag(z5, x3))) we obtained the following new rules [LPAR04]:
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U2_gag(z5, z2, z3, member124_in_gag(z5, z3)))
REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U2_gag(z6, z2, z3, member124_in_gag(z6, z3)))
(136) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(z2, z0, .(z0, z4), member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(z2, z0, .(z4, z5), member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U3_gg(x5, z0, .(z0, z4), member34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U3_gg(x5, z0, .(z4, z5), member34_in_gg(x5, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member34_out_gg(z6, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), z3)))
REACH1_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member124_out_gag(z6, x2, .(.(z6, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U2_gag(z5, z2, z3, member124_in_gag(z5, z3)))
REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U2_gag(z6, z2, z3, member124_in_gag(z6, z3)))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(137) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
(138) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(z2, z0, .(z0, z4), member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(z2, z0, .(z4, z5), member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U3_gg(x5, z0, .(z0, z4), member34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U3_gg(x5, z0, .(z4, z5), member34_in_gg(x5, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member34_out_gg(z6, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), z3)))
REACH1_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member124_out_gag(z6, x2, .(.(z6, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U2_gag(z5, z2, z3, member124_in_gag(z5, z3)))
REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U2_gag(z6, z2, z3, member124_in_gag(z6, z3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(139) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
(140) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(z2, z0, .(z0, z4), member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(z2, z0, .(z4, z5), member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U3_gg(x5, z0, .(z0, z4), member34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U3_gg(x5, z0, .(z4, z5), member34_in_gg(x5, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member34_out_gg(z6, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), z3)))
REACH1_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member124_out_gag(z6, x2, .(.(z6, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U2_gag(z5, z2, z3, member124_in_gag(z5, z3)))
REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U2_gag(z6, z2, z3, member124_in_gag(z6, z3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(141) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z0, .(z4, z5)))))
(142) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(z2, z0, .(z0, z4), member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(z2, z0, .(z4, z5), member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U3_gg(x5, z0, .(z0, z4), member34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U3_gg(x5, z0, .(z4, z5), member34_in_gg(x5, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member34_out_gg(z6, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), z3)))
REACH1_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member124_out_gag(z6, x2, .(.(z6, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U2_gag(z5, z2, z3, member124_in_gag(z5, z3)))
REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U2_gag(z6, z2, z3, member124_in_gag(z6, z3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z0, .(z4, z5)))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(143) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member124_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member34_out_gg(z0, .(z0, .(z2, .(z2, z4)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member124_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z2, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
(144) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(z2, z0, .(z0, z4), member34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(z2, z0, .(z4, z5), member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U3_gg(x5, z0, .(z0, z4), member34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U3_gg(x5, z0, .(z4, z5), member34_in_gg(x5, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member34_out_gg(z6, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), z3)))
REACH1_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member124_out_gag(z6, x2, .(.(z6, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U2_gag(z5, z2, z3, member124_in_gag(z5, z3)))
REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U2_gag(z6, z2, z3, member124_in_gag(z6, z3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member124_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member34_out_gg(z0, .(z0, .(z2, .(z2, z4)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member124_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z2, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(145) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U3_gg(z2, z0, .(z0, z4), member34_in_gg(z2, .(z0, z4)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U3_gg(z0, z0, .(z0, .(z0, z3)), member34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U3_gg(z0, z0, .(z0, .(z3, z4)), member34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U3_gg(z2, z0, .(z0, .(z0, z4)), member34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U3_gg(z2, z0, .(z0, .(z4, z5)), member34_in_gg(z2, .(z0, .(z4, z5)))))
(146) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(z2, z0, .(z4, z5), member34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U3_gg(x5, z0, .(z0, z4), member34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U3_gg(x5, z0, .(z4, z5), member34_in_gg(x5, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member34_out_gg(z6, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), z3)))
REACH1_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member124_out_gag(z6, x2, .(.(z6, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U2_gag(z5, z2, z3, member124_in_gag(z5, z3)))
REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U2_gag(z6, z2, z3, member124_in_gag(z6, z3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member124_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member34_out_gg(z0, .(z0, .(z2, .(z2, z4)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member124_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z2, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U3_gg(z0, z0, .(z0, .(z0, z3)), member34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U3_gg(z0, z0, .(z0, .(z3, z4)), member34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U3_gg(z2, z0, .(z0, .(z0, z4)), member34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U3_gg(z2, z0, .(z0, .(z4, z5)), member34_in_gg(z2, .(z0, .(z4, z5)))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(147) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U3_gg(z2, z0, .(z4, z5), member34_in_gg(z2, .(z4, z5)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U3_gg(z0, z0, .(z0, .(z0, z3)), member34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U3_gg(z0, z0, .(z0, .(z3, z4)), member34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U3_gg(z2, z0, .(z0, .(z0, z4)), member34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U3_gg(z2, z0, .(z0, .(z4, z5)), member34_in_gg(z2, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U3_gg(z2, z0, .(z4, .(z4, z5)), member34_in_gg(z2, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U3_gg(z2, z0, .(z4, .(z5, z6)), member34_in_gg(z2, .(z4, .(z5, z6)))))
(148) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U3_gg(x5, z0, .(z0, z4), member34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U3_gg(x5, z0, .(z4, z5), member34_in_gg(x5, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member34_out_gg(z6, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), z3)))
REACH1_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member124_out_gag(z6, x2, .(.(z6, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U2_gag(z5, z2, z3, member124_in_gag(z5, z3)))
REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U2_gag(z6, z2, z3, member124_in_gag(z6, z3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member124_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member34_out_gg(z0, .(z0, .(z2, .(z2, z4)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member124_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z2, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U3_gg(z0, z0, .(z0, .(z0, z3)), member34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U3_gg(z0, z0, .(z0, .(z3, z4)), member34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U3_gg(z2, z0, .(z0, .(z0, z4)), member34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U3_gg(z2, z0, .(z0, .(z4, z5)), member34_in_gg(z2, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U3_gg(z2, z0, .(z4, .(z4, z5)), member34_in_gg(z2, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U3_gg(z2, z0, .(z4, .(z5, z6)), member34_in_gg(z2, .(z4, .(z5, z6)))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(149) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U3_gg(x5, z0, .(z0, z4), member34_in_gg(x5, .(z0, z4)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U3_gg(z0, z0, .(z0, .(z0, z3)), member34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U3_gg(z0, z0, .(z0, .(z3, z4)), member34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U3_gg(z2, z0, .(z0, .(z0, z4)), member34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U3_gg(z2, z0, .(z0, .(z4, z5)), member34_in_gg(z2, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U3_gg(x5, z0, .(z0, .(z0, z3)), member34_in_gg(x5, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U3_gg(x5, z0, .(z0, .(z3, z4)), member34_in_gg(x5, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U3_gg(x5, z0, .(z0, .(z0, z4)), member34_in_gg(x5, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U3_gg(x5, z0, .(z0, .(z4, z5)), member34_in_gg(x5, .(z0, .(z4, z5)))))
(150) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U3_gg(x5, z0, .(z4, z5), member34_in_gg(x5, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member34_out_gg(z6, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), z3)))
REACH1_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member124_out_gag(z6, x2, .(.(z6, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U2_gag(z5, z2, z3, member124_in_gag(z5, z3)))
REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U2_gag(z6, z2, z3, member124_in_gag(z6, z3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member124_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member34_out_gg(z0, .(z0, .(z2, .(z2, z4)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member124_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z2, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U3_gg(z0, z0, .(z0, .(z0, z3)), member34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U3_gg(z0, z0, .(z0, .(z3, z4)), member34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U3_gg(z2, z0, .(z0, .(z0, z4)), member34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U3_gg(z2, z0, .(z0, .(z4, z5)), member34_in_gg(z2, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U3_gg(z2, z0, .(z4, .(z4, z5)), member34_in_gg(z2, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U3_gg(z2, z0, .(z4, .(z5, z6)), member34_in_gg(z2, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U3_gg(x5, z0, .(z0, .(z0, z3)), member34_in_gg(x5, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U3_gg(x5, z0, .(z0, .(z3, z4)), member34_in_gg(x5, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U3_gg(x5, z0, .(z0, .(z0, z4)), member34_in_gg(x5, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U3_gg(x5, z0, .(z0, .(z4, z5)), member34_in_gg(x5, .(z0, .(z4, z5)))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(151) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U3_gg(x5, z0, .(z4, z5), member34_in_gg(x5, .(z4, z5)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U3_gg(z0, z0, .(z0, .(z0, z3)), member34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U3_gg(z0, z0, .(z0, .(z3, z4)), member34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U3_gg(z2, z0, .(z0, .(z0, z4)), member34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U3_gg(z2, z0, .(z0, .(z4, z5)), member34_in_gg(z2, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U3_gg(z2, z0, .(z4, .(z4, z5)), member34_in_gg(z2, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U3_gg(z2, z0, .(z4, .(z5, z6)), member34_in_gg(z2, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, x6, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x6, U3_gg(x6, z0, .(z0, .(z0, z3)), member34_in_gg(x6, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, x6, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x6, U3_gg(x6, z0, .(z0, .(z3, z4)), member34_in_gg(x6, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, x6, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U3_gg(x6, z0, .(z0, .(z0, z4)), member34_in_gg(x6, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, x6, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U3_gg(x6, z0, .(z0, .(z4, z5)), member34_in_gg(x6, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member124_out_gag(z0, x6, .(.(z2, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), x6, U3_gg(x6, z0, .(z2, .(z2, z4)), member34_in_gg(x6, .(z2, .(z2, z4)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member124_out_gag(z0, x6, .(.(z2, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), x6, U3_gg(x6, z0, .(z2, .(z4, z5)), member34_in_gg(x6, .(z2, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, x6, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x6, U3_gg(x6, z0, .(z4, .(z4, z5)), member34_in_gg(x6, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, x6, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x6, U3_gg(x6, z0, .(z4, .(z5, z6)), member34_in_gg(x6, .(z4, .(z5, z6)))))
(152) Obligation:
Q DP problem:The TRS P consists of the following rules:
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member34_out_gg(z6, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
REACH1_IN_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(.(z5, .(x2, [])), z3), .(z5, .(z0, .(z0, z4))), member124_out_gag(z5, x2, .(.(z5, .(x2, [])), z3)))
REACH1_IN_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(.(z6, .(x2, [])), z3), .(z6, .(z0, .(z4, z5))), member124_out_gag(z6, x2, .(.(z6, .(x2, [])), z3)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U2_gag(z0, .(z0, .(z0, [])), z2, member124_in_gag(z0, z2)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) → U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U2_gag(z0, z2, z3, member124_in_gag(z0, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) → U6_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U2_gag(z2, .(z0, .(z2, [])), z3, member124_in_gag(z2, z3)))
REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) → U6_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U2_gag(z5, z2, z3, member124_in_gag(z5, z3)))
REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) → U6_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U2_gag(z6, z2, z3, member124_in_gag(z6, z3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member124_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member34_out_gg(z0, .(z0, .(z2, .(z2, z4)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member124_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z2, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z0, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U3_gg(z0, z0, .(z0, .(z0, z3)), member34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U3_gg(z0, z0, .(z0, .(z3, z4)), member34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U3_gg(z2, z0, .(z0, .(z0, z4)), member34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U3_gg(z2, z0, .(z0, .(z4, z5)), member34_in_gg(z2, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U3_gg(z2, z0, .(z4, .(z4, z5)), member34_in_gg(z2, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U7_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U3_gg(z2, z0, .(z4, .(z5, z6)), member34_in_gg(z2, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U3_gg(x5, z0, .(z0, .(z0, z3)), member34_in_gg(x5, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member124_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U3_gg(x5, z0, .(z0, .(z3, z4)), member34_in_gg(x5, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U3_gg(x5, z0, .(z0, .(z0, z4)), member34_in_gg(x5, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member124_out_gag(z0, x5, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U3_gg(x5, z0, .(z0, .(z4, z5)), member34_in_gg(x5, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member124_out_gag(z0, x6, .(.(z2, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), x6, U3_gg(x6, z0, .(z2, .(z2, z4)), member34_in_gg(x6, .(z2, .(z2, z4)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member124_out_gag(z0, x6, .(.(z2, .(z0, [])), z3))) → U7_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), x6, U3_gg(x6, z0, .(z2, .(z4, z5)), member34_in_gg(x6, .(z2, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member124_out_gag(z0, x6, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x6, U3_gg(x6, z0, .(z4, .(z4, z5)), member34_in_gg(x6, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member124_out_gag(z0, x6, .(z2, z3))) → U7_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x6, U3_gg(x6, z0, .(z4, .(z5, z6)), member34_in_gg(x6, .(z4, .(z5, z6)))))
The TRS R consists of the following rules:
member124_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member124_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member124_in_gag(T128, .(T129, T130)) → U2_gag(T128, T129, T130, member124_in_gag(T128, T130))
member34_in_gg(T151, .(T151, T152)) → member34_out_gg(T151, .(T151, T152))
member34_in_gg(T159, .(T160, T161)) → U3_gg(T159, T160, T161, member34_in_gg(T159, T161))
U2_gag(T128, T129, T130, member124_out_gag(T128, X131, T130)) → member124_out_gag(T128, X131, .(T129, T130))
U3_gg(T159, T160, T161, member34_out_gg(T159, T161)) → member34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member124_in_gag(x0, x1)
member34_in_gg(x0, x1)
U2_gag(x0, x1, x2, x3)
U3_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(153) 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 = REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) evaluates to t =REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, .(z0, z3)))))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [z3 / .(z0, z3)]
- Semiunifier: [ ]
Rewriting sequence
REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2)))
with rule REACH1_IN_GGGG(z0', z1', .(.(z0', .(z0', [])), z2'), .(z0', .(z0', .(z0', z3')))) → U6_GGGG(z0', z1', .(.(z0', .(z0', [])), z2'), .(z0', .(z0', .(z0', z3'))), member124_out_gag(z0', z0', .(.(z0', .(z0', [])), z2'))) at position [] and matcher [z0' / z0, z1' / z1, z2' / z2, z3' / z3]
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member124_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
with rule U6_GGGG(z0', z1', .(.(z0', .(z0', [])), z2'), .(z0', .(z0', .(z0', z3'))), member124_out_gag(z0', z0', .(.(z0', .(z0', [])), z2'))) → U7_GGGG(z0', z1', .(.(z0', .(z0', [])), z2'), .(z0', .(z0', .(z0', z3'))), z0', member34_out_gg(z0', .(z0', .(z0', .(z0', z3'))))) at position [] and matcher [z0' / z0, z1' / z1, z2' / z2, z3' / z3]
U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member34_out_gg(z0, .(z0, .(z0, .(z0, z3))))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, .(z0, z3)))))
with rule U7_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
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.