(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) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:member12(T77, T78, .(T79, T80)) :- member12(T77, T78, T80).
member124(T128, X131, .(T129, T130)) :- member124(T128, X131, T130).
member34(T159, .(T160, T161)) :- member34(T159, T161).
reach1(T44, T45, .(T46, T47), T12) :- member12(T44, T45, T47).
reach1(T99, T100, T101, T102) :- member124(T99, X90, T101).
reach1(T99, T100, T101, T102) :- ','(member1c24(T99, T107, T101), member34(T107, T102)).
reach1(T99, T100, T101, T102) :- ','(member1c24(T99, T107, T101), ','(memberc34(T107, T102), reach1(T107, T100, T101, .(T107, T102)))).
Clauses:
memberc12(T66, T67, .(.(T66, .(T67, [])), T68)).
memberc12(T77, T78, .(T79, T80)) :- memberc12(T77, T78, T80).
member1c24(T120, X117, .(.(T120, .(X117, [])), T121)).
member1c24(T128, X131, .(T129, T130)) :- member1c24(T128, X131, T130).
reachc1(T25, T26, .(.(T25, .(T26, [])), T27), T12).
reachc1(T44, T45, .(T46, T47), T12) :- memberc12(T44, T45, T47).
reachc1(T99, T100, T101, T102) :- ','(member1c24(T99, T107, T101), ','(memberc34(T107, T102), reachc1(T107, T100, T101, .(T107, T102)))).
memberc34(T151, .(T151, T152)).
memberc34(T159, .(T160, T161)) :- memberc34(T159, T161).
Afs:
reach1(x1, x2, x3, x4) = reach1(x1, x2, x3, x4)
(3) TriplesToPiDPProof (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)
member1c24_in: (b,f,b)
member34_in: (b,b)
memberc34_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
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, member1c24_in_gag(T99, T107, T101))
U6_GGGG(T99, T100, T101, T102, member1c24_out_gag(T99, T107, T101)) → U7_GGGG(T99, T100, T101, T102, member34_in_gg(T107, T102))
U6_GGGG(T99, T100, T101, T102, member1c24_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)
U6_GGGG(T99, T100, T101, T102, member1c24_out_gag(T99, T107, T101)) → U8_GGGG(T99, T100, T101, T102, T107, memberc34_in_gg(T107, T102))
U8_GGGG(T99, T100, T101, T102, T107, memberc34_out_gg(T107, T102)) → U9_GGGG(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U8_GGGG(T99, T100, T101, T102, T107, memberc34_out_gg(T107, T102)) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102))
The TRS R consists of the following rules:
member1c24_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, X131, .(T129, T130)) → U12_gag(T128, X131, T129, T130, member1c24_in_gag(T128, X131, T130))
U12_gag(T128, X131, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
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)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member1c24_in_gag(x1, x2, x3) = member1c24_in_gag(x1, x3)
[] = []
member1c24_out_gag(x1, x2, x3) = member1c24_out_gag(x1, x2, x3)
U12_gag(x1, x2, x3, x4, x5) = U12_gag(x1, x3, x4, x5)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
memberc34_in_gg(x1, x2) = memberc34_in_gg(x1, x2)
memberc34_out_gg(x1, x2) = memberc34_out_gg(x1, x2)
U17_gg(x1, x2, x3, x4) = U17_gg(x1, x2, x3, x4)
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) = U7_GGGG(x1, x2, x3, x4, x5)
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, x6) = U8_GGGG(x1, x2, x3, x4, x5, x6)
U9_GGGG(x1, x2, x3, x4, x5) = U9_GGGG(x1, x2, x3, x4, x5)
We have to consider all (P,R,Pi)-chains
Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
(4) 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, member1c24_in_gag(T99, T107, T101))
U6_GGGG(T99, T100, T101, T102, member1c24_out_gag(T99, T107, T101)) → U7_GGGG(T99, T100, T101, T102, member34_in_gg(T107, T102))
U6_GGGG(T99, T100, T101, T102, member1c24_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)
U6_GGGG(T99, T100, T101, T102, member1c24_out_gag(T99, T107, T101)) → U8_GGGG(T99, T100, T101, T102, T107, memberc34_in_gg(T107, T102))
U8_GGGG(T99, T100, T101, T102, T107, memberc34_out_gg(T107, T102)) → U9_GGGG(T99, T100, T101, T102, reach1_in_gggg(T107, T100, T101, .(T107, T102)))
U8_GGGG(T99, T100, T101, T102, T107, memberc34_out_gg(T107, T102)) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102))
The TRS R consists of the following rules:
member1c24_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, X131, .(T129, T130)) → U12_gag(T128, X131, T129, T130, member1c24_in_gag(T128, X131, T130))
U12_gag(T128, X131, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
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)
member12_in_ggg(x1, x2, x3) = member12_in_ggg(x1, x2, x3)
member124_in_gag(x1, x2, x3) = member124_in_gag(x1, x3)
member1c24_in_gag(x1, x2, x3) = member1c24_in_gag(x1, x3)
[] = []
member1c24_out_gag(x1, x2, x3) = member1c24_out_gag(x1, x2, x3)
U12_gag(x1, x2, x3, x4, x5) = U12_gag(x1, x3, x4, x5)
member34_in_gg(x1, x2) = member34_in_gg(x1, x2)
memberc34_in_gg(x1, x2) = memberc34_in_gg(x1, x2)
memberc34_out_gg(x1, x2) = memberc34_out_gg(x1, x2)
U17_gg(x1, x2, x3, x4) = U17_gg(x1, x2, x3, x4)
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) = U7_GGGG(x1, x2, x3, x4, x5)
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, x6) = U8_GGGG(x1, x2, x3, x4, x5, x6)
U9_GGGG(x1, x2, x3, x4, x5) = U9_GGGG(x1, x2, x3, x4, x5)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 10 less nodes.(6) Complex Obligation (AND)
(7) 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:
member1c24_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, X131, .(T129, T130)) → U12_gag(T128, X131, T129, T130, member1c24_in_gag(T128, X131, T130))
U12_gag(T128, X131, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
member1c24_in_gag(x1, x2, x3) = member1c24_in_gag(x1, x3)
[] = []
member1c24_out_gag(x1, x2, x3) = member1c24_out_gag(x1, x2, x3)
U12_gag(x1, x2, x3, x4, x5) = U12_gag(x1, x3, x4, x5)
memberc34_in_gg(x1, x2) = memberc34_in_gg(x1, x2)
memberc34_out_gg(x1, x2) = memberc34_out_gg(x1, x2)
U17_gg(x1, x2, x3, x4) = U17_gg(x1, x2, x3, x4)
MEMBER34_IN_GG(x1, x2) = MEMBER34_IN_GG(x1, x2)
We have to consider all (P,R,Pi)-chains
(8) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(9) Obligation:
Pi DP problem:The TRS P consists of the following rules:
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
(10) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(11) Obligation:
Q DP problem:The TRS P consists of the following rules:
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.
(12) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.From the DPs we obtained the following set of size-change graphs:
- MEMBER34_IN_GG(T159, .(T160, T161)) → MEMBER34_IN_GG(T159, T161)
The graph contains the following edges 1 >= 1, 2 > 2
(13) YES
(14) 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:
member1c24_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, X131, .(T129, T130)) → U12_gag(T128, X131, T129, T130, member1c24_in_gag(T128, X131, T130))
U12_gag(T128, X131, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
member1c24_in_gag(x1, x2, x3) = member1c24_in_gag(x1, x3)
[] = []
member1c24_out_gag(x1, x2, x3) = member1c24_out_gag(x1, x2, x3)
U12_gag(x1, x2, x3, x4, x5) = U12_gag(x1, x3, x4, x5)
memberc34_in_gg(x1, x2) = memberc34_in_gg(x1, x2)
memberc34_out_gg(x1, x2) = memberc34_out_gg(x1, x2)
U17_gg(x1, x2, x3, x4) = U17_gg(x1, x2, x3, x4)
MEMBER124_IN_GAG(x1, x2, x3) = MEMBER124_IN_GAG(x1, x3)
We have to consider all (P,R,Pi)-chains
(15) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(16) Obligation:
Pi DP problem:The TRS P consists of the following rules:
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
(17) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(18) Obligation:
Q DP problem:The TRS P consists of the following rules:
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.
(19) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.From the DPs we obtained the following set of size-change graphs:
- MEMBER124_IN_GAG(T128, .(T129, T130)) → MEMBER124_IN_GAG(T128, T130)
The graph contains the following edges 1 >= 1, 2 > 2
(20) YES
(21) 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:
member1c24_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, X131, .(T129, T130)) → U12_gag(T128, X131, T129, T130, member1c24_in_gag(T128, X131, T130))
U12_gag(T128, X131, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
member1c24_in_gag(x1, x2, x3) = member1c24_in_gag(x1, x3)
[] = []
member1c24_out_gag(x1, x2, x3) = member1c24_out_gag(x1, x2, x3)
U12_gag(x1, x2, x3, x4, x5) = U12_gag(x1, x3, x4, x5)
memberc34_in_gg(x1, x2) = memberc34_in_gg(x1, x2)
memberc34_out_gg(x1, x2) = memberc34_out_gg(x1, x2)
U17_gg(x1, x2, x3, x4) = U17_gg(x1, x2, x3, x4)
MEMBER12_IN_GGG(x1, x2, x3) = MEMBER12_IN_GGG(x1, x2, x3)
We have to consider all (P,R,Pi)-chains
(22) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(23) Obligation:
Pi DP problem:The TRS P consists of the following rules:
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
(24) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(25) Obligation:
Q DP problem:The TRS P consists of the following rules:
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.
(26) 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
(27) YES
(28) 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, member1c24_in_gag(T99, T107, T101))
U6_GGGG(T99, T100, T101, T102, member1c24_out_gag(T99, T107, T101)) → U8_GGGG(T99, T100, T101, T102, T107, memberc34_in_gg(T107, T102))
U8_GGGG(T99, T100, T101, T102, T107, memberc34_out_gg(T107, T102)) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102))
The TRS R consists of the following rules:
member1c24_in_gag(T120, X117, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, X131, .(T129, T130)) → U12_gag(T128, X131, T129, T130, member1c24_in_gag(T128, X131, T130))
U12_gag(T128, X131, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
member1c24_in_gag(x1, x2, x3) = member1c24_in_gag(x1, x3)
[] = []
member1c24_out_gag(x1, x2, x3) = member1c24_out_gag(x1, x2, x3)
U12_gag(x1, x2, x3, x4, x5) = U12_gag(x1, x3, x4, x5)
memberc34_in_gg(x1, x2) = memberc34_in_gg(x1, x2)
memberc34_out_gg(x1, x2) = memberc34_out_gg(x1, x2)
U17_gg(x1, x2, x3, x4) = U17_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)
U8_GGGG(x1, x2, x3, x4, x5, x6) = U8_GGGG(x1, x2, x3, x4, x5, x6)
We have to consider all (P,R,Pi)-chains
(29) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(30) 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, member1c24_in_gag(T99, T101))
U6_GGGG(T99, T100, T101, T102, member1c24_out_gag(T99, T107, T101)) → U8_GGGG(T99, T100, T101, T102, T107, memberc34_in_gg(T107, T102))
U8_GGGG(T99, T100, T101, T102, T107, memberc34_out_gg(T107, T102)) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(31) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule REACH1_IN_GGGG(T99, T100, T101, T102) → U6_GGGG(T99, T100, T101, T102, member1c24_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, member1c24_out_gag(x0, x1, .(.(x0, .(x1, [])), x2)))
REACH1_IN_GGGG(x0, y1, .(x1, x2), y3) → U6_GGGG(x0, y1, .(x1, x2), y3, U12_gag(x0, x1, x2, member1c24_in_gag(x0, x2)))
(32) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(T99, T100, T101, T102, member1c24_out_gag(T99, T107, T101)) → U8_GGGG(T99, T100, T101, T102, T107, memberc34_in_gg(T107, T102))
U8_GGGG(T99, T100, T101, T102, T107, memberc34_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, member1c24_out_gag(x0, x1, .(.(x0, .(x1, [])), x2)))
REACH1_IN_GGGG(x0, y1, .(x1, x2), y3) → U6_GGGG(x0, y1, .(x1, x2), y3, U12_gag(x0, x1, x2, member1c24_in_gag(x0, x2)))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(33) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule U6_GGGG(T99, T100, T101, T102, member1c24_out_gag(T99, T107, T101)) → U8_GGGG(T99, T100, T101, T102, T107, memberc34_in_gg(T107, T102)) at position [5] we obtained the following new rules [LPAR04]:
U6_GGGG(y0, y1, y2, .(x0, x1), member1c24_out_gag(y0, x0, y2)) → U8_GGGG(y0, y1, y2, .(x0, x1), x0, memberc34_out_gg(x0, .(x0, x1)))
U6_GGGG(y0, y1, y2, .(x1, x2), member1c24_out_gag(y0, x0, y2)) → U8_GGGG(y0, y1, y2, .(x1, x2), x0, U17_gg(x0, x1, x2, memberc34_in_gg(x0, x2)))
(34) Obligation:
Q DP problem:The TRS P consists of the following rules:
U8_GGGG(T99, T100, T101, T102, T107, memberc34_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, member1c24_out_gag(x0, x1, .(.(x0, .(x1, [])), x2)))
REACH1_IN_GGGG(x0, y1, .(x1, x2), y3) → U6_GGGG(x0, y1, .(x1, x2), y3, U12_gag(x0, x1, x2, member1c24_in_gag(x0, x2)))
U6_GGGG(y0, y1, y2, .(x0, x1), member1c24_out_gag(y0, x0, y2)) → U8_GGGG(y0, y1, y2, .(x0, x1), x0, memberc34_out_gg(x0, .(x0, x1)))
U6_GGGG(y0, y1, y2, .(x1, x2), member1c24_out_gag(y0, x0, y2)) → U8_GGGG(y0, y1, y2, .(x1, x2), x0, U17_gg(x0, x1, x2, memberc34_in_gg(x0, x2)))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(35) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U8_GGGG(T99, T100, T101, T102, T107, memberc34_out_gg(T107, T102)) → REACH1_IN_GGGG(T107, T100, T101, .(T107, T102)) we obtained the following new rules [LPAR04]:
U8_GGGG(z0, z1, z2, .(z3, z4), z3, memberc34_out_gg(z3, .(z3, z4))) → REACH1_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))
U8_GGGG(z0, z1, z2, .(z3, z4), z5, memberc34_out_gg(z5, .(z3, z4))) → REACH1_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4)))
(36) 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, member1c24_out_gag(x0, x1, .(.(x0, .(x1, [])), x2)))
REACH1_IN_GGGG(x0, y1, .(x1, x2), y3) → U6_GGGG(x0, y1, .(x1, x2), y3, U12_gag(x0, x1, x2, member1c24_in_gag(x0, x2)))
U6_GGGG(y0, y1, y2, .(x0, x1), member1c24_out_gag(y0, x0, y2)) → U8_GGGG(y0, y1, y2, .(x0, x1), x0, memberc34_out_gg(x0, .(x0, x1)))
U6_GGGG(y0, y1, y2, .(x1, x2), member1c24_out_gag(y0, x0, y2)) → U8_GGGG(y0, y1, y2, .(x1, x2), x0, U17_gg(x0, x1, x2, memberc34_in_gg(x0, x2)))
U8_GGGG(z0, z1, z2, .(z3, z4), z3, memberc34_out_gg(z3, .(z3, z4))) → REACH1_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))
U8_GGGG(z0, z1, z2, .(z3, z4), z5, memberc34_out_gg(z5, .(z3, z4))) → REACH1_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4)))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(37) 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, member1c24_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)), member1c24_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)), member1c24_out_gag(z5, x2, .(.(z5, .(x2, [])), x3)))
(38) 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, U12_gag(x0, x1, x2, member1c24_in_gag(x0, x2)))
U6_GGGG(y0, y1, y2, .(x0, x1), member1c24_out_gag(y0, x0, y2)) → U8_GGGG(y0, y1, y2, .(x0, x1), x0, memberc34_out_gg(x0, .(x0, x1)))
U6_GGGG(y0, y1, y2, .(x1, x2), member1c24_out_gag(y0, x0, y2)) → U8_GGGG(y0, y1, y2, .(x1, x2), x0, U17_gg(x0, x1, x2, memberc34_in_gg(x0, x2)))
U8_GGGG(z0, z1, z2, .(z3, z4), z3, memberc34_out_gg(z3, .(z3, z4))) → REACH1_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))
U8_GGGG(z0, z1, z2, .(z3, z4), z5, memberc34_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)), member1c24_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)), member1c24_out_gag(z5, x2, .(.(z5, .(x2, [])), x3)))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(39) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule REACH1_IN_GGGG(x0, y1, .(x1, x2), y3) → U6_GGGG(x0, y1, .(x1, x2), y3, U12_gag(x0, x1, x2, member1c24_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)), U12_gag(z3, x2, x3, member1c24_in_gag(z3, x3)))
REACH1_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1c24_in_gag(z5, x3)))
(40) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(y0, y1, y2, .(x0, x1), member1c24_out_gag(y0, x0, y2)) → U8_GGGG(y0, y1, y2, .(x0, x1), x0, memberc34_out_gg(x0, .(x0, x1)))
U6_GGGG(y0, y1, y2, .(x1, x2), member1c24_out_gag(y0, x0, y2)) → U8_GGGG(y0, y1, y2, .(x1, x2), x0, U17_gg(x0, x1, x2, memberc34_in_gg(x0, x2)))
U8_GGGG(z0, z1, z2, .(z3, z4), z3, memberc34_out_gg(z3, .(z3, z4))) → REACH1_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))
U8_GGGG(z0, z1, z2, .(z3, z4), z5, memberc34_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)), member1c24_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)), member1c24_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)), U12_gag(z3, x2, x3, member1c24_in_gag(z3, x3)))
REACH1_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1c24_in_gag(z5, x3)))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(41) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(y0, y1, y2, .(x0, x1), member1c24_out_gag(y0, x0, y2)) → U8_GGGG(y0, y1, y2, .(x0, x1), x0, memberc34_out_gg(x0, .(x0, x1))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
(42) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(y0, y1, y2, .(x1, x2), member1c24_out_gag(y0, x0, y2)) → U8_GGGG(y0, y1, y2, .(x1, x2), x0, U17_gg(x0, x1, x2, memberc34_in_gg(x0, x2)))
U8_GGGG(z0, z1, z2, .(z3, z4), z3, memberc34_out_gg(z3, .(z3, z4))) → REACH1_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))
U8_GGGG(z0, z1, z2, .(z3, z4), z5, memberc34_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)), member1c24_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)), member1c24_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)), U12_gag(z3, x2, x3, member1c24_in_gag(z3, x3)))
REACH1_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1c24_in_gag(z5, x3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(43) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(y0, y1, y2, .(x1, x2), member1c24_out_gag(y0, x0, y2)) → U8_GGGG(y0, y1, y2, .(x1, x2), x0, U17_gg(x0, x1, x2, memberc34_in_gg(x0, x2))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), memberc34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), memberc34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), memberc34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), memberc34_in_gg(x5, .(z4, z5))))
(44) Obligation:
Q DP problem:The TRS P consists of the following rules:
U8_GGGG(z0, z1, z2, .(z3, z4), z3, memberc34_out_gg(z3, .(z3, z4))) → REACH1_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))
U8_GGGG(z0, z1, z2, .(z3, z4), z5, memberc34_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)), member1c24_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)), member1c24_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)), U12_gag(z3, x2, x3, member1c24_in_gag(z3, x3)))
REACH1_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1c24_in_gag(z5, x3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), memberc34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), memberc34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), memberc34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), memberc34_in_gg(x5, .(z4, z5))))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(45) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U8_GGGG(z0, z1, z2, .(z3, z4), z3, memberc34_out_gg(z3, .(z3, z4))) → REACH1_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4))) we obtained the following new rules [LPAR04]:
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, memberc34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, memberc34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
(46) Obligation:
Q DP problem:The TRS P consists of the following rules:
U8_GGGG(z0, z1, z2, .(z3, z4), z5, memberc34_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)), member1c24_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)), member1c24_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)), U12_gag(z3, x2, x3, member1c24_in_gag(z3, x3)))
REACH1_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1c24_in_gag(z5, x3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), memberc34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), memberc34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), memberc34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), memberc34_in_gg(x5, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, memberc34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, memberc34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_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:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(47) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U8_GGGG(z0, z1, z2, .(z3, z4), z5, memberc34_out_gg(z5, .(z3, z4))) → REACH1_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) we obtained the following new rules [LPAR04]:
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, memberc34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, memberc34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, memberc34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, memberc34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, memberc34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, memberc34_out_gg(z6, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))))
(48) 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)), member1c24_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)), member1c24_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)), U12_gag(z3, x2, x3, member1c24_in_gag(z3, x3)))
REACH1_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1c24_in_gag(z5, x3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), memberc34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), memberc34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), memberc34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), memberc34_in_gg(x5, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, memberc34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, memberc34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, memberc34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, memberc34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, memberc34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, memberc34_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:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(49) 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)), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
(50) 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)), member1c24_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)), U12_gag(z3, x2, x3, member1c24_in_gag(z3, x3)))
REACH1_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1c24_in_gag(z5, x3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), memberc34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), memberc34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), memberc34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), memberc34_in_gg(x5, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, memberc34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, memberc34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, memberc34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, memberc34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, memberc34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, memberc34_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_out_gag(z0, x2, .(.(z0, .(x2, [])), z3)))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(51) 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)), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_out_gag(z6, x2, .(.(z6, .(x2, [])), z3)))
(52) 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)), U12_gag(z3, x2, x3, member1c24_in_gag(z3, x3)))
REACH1_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U6_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U12_gag(z5, x2, x3, member1c24_in_gag(z5, x3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), memberc34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), memberc34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), memberc34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), memberc34_in_gg(x5, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, memberc34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, memberc34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, memberc34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, memberc34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, memberc34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, memberc34_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_out_gag(z6, x2, .(.(z6, .(x2, [])), z3)))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(53) 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)), U12_gag(z3, x2, x3, member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z0, z2, z3, member1c24_in_gag(z0, z3)))
(54) 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)), U12_gag(z5, x2, x3, member1c24_in_gag(z5, x3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), memberc34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), memberc34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), memberc34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), memberc34_in_gg(x5, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, memberc34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, memberc34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, memberc34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, memberc34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, memberc34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, memberc34_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z0, z2, z3, member1c24_in_gag(z0, z3)))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(55) 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)), U12_gag(z5, x2, x3, member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z5, z2, z3, member1c24_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))), U12_gag(z6, z2, z3, member1c24_in_gag(z6, z3)))
(56) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), memberc34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), memberc34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), memberc34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), memberc34_in_gg(x5, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, memberc34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, memberc34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, memberc34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, memberc34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, memberc34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, memberc34_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z5, z2, z3, member1c24_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))), U12_gag(z6, z2, z3, member1c24_in_gag(z6, z3)))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(57) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
(58) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), memberc34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), memberc34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), memberc34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), memberc34_in_gg(x5, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, memberc34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, memberc34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, memberc34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, memberc34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, memberc34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, memberc34_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z5, z2, z3, member1c24_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))), U12_gag(z6, z2, z3, member1c24_in_gag(z6, z3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(59) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
(60) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), memberc34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), memberc34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), memberc34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), memberc34_in_gg(x5, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, memberc34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, memberc34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, memberc34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, memberc34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, memberc34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, memberc34_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z5, z2, z3, member1c24_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))), U12_gag(z6, z2, z3, member1c24_in_gag(z6, z3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(61) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z4, z5)))))
(62) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), memberc34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), memberc34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), memberc34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), memberc34_in_gg(x5, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, memberc34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, memberc34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, memberc34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, memberc34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, memberc34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, memberc34_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z5, z2, z3, member1c24_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))), U12_gag(z6, z2, z3, member1c24_in_gag(z6, z3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z4, z5)))))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(63) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1c24_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, memberc34_out_gg(z0, .(z0, .(z2, .(z2, z4)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1c24_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z2, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
(64) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), memberc34_in_gg(z2, .(z0, z4))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), memberc34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), memberc34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), memberc34_in_gg(x5, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, memberc34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, memberc34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, memberc34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, memberc34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, memberc34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, memberc34_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z5, z2, z3, member1c24_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))), U12_gag(z6, z2, z3, member1c24_in_gag(z6, z3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1c24_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, memberc34_out_gg(z0, .(z0, .(z2, .(z2, z4)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1c24_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z2, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(65) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, U17_gg(z2, z0, .(z0, z4), memberc34_in_gg(z2, .(z0, z4)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), memberc34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), memberc34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), memberc34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), memberc34_in_gg(z2, .(z0, .(z4, z5)))))
(66) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), memberc34_in_gg(z2, .(z4, z5))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), memberc34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), memberc34_in_gg(x5, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, memberc34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, memberc34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, memberc34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, memberc34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, memberc34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, memberc34_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z5, z2, z3, member1c24_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))), U12_gag(z6, z2, z3, member1c24_in_gag(z6, z3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1c24_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, memberc34_out_gg(z0, .(z0, .(z2, .(z2, z4)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1c24_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z2, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), memberc34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), memberc34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), memberc34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), memberc34_in_gg(z2, .(z0, .(z4, z5)))))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(67) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, U17_gg(z2, z0, .(z4, z5), memberc34_in_gg(z2, .(z4, z5)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), memberc34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), memberc34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), memberc34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), memberc34_in_gg(z2, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U17_gg(z2, z0, .(z4, .(z4, z5)), memberc34_in_gg(z2, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U17_gg(z2, z0, .(z4, .(z5, z6)), memberc34_in_gg(z2, .(z4, .(z5, z6)))))
(68) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), memberc34_in_gg(x5, .(z0, z4))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), memberc34_in_gg(x5, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, memberc34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, memberc34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, memberc34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, memberc34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, memberc34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, memberc34_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z5, z2, z3, member1c24_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))), U12_gag(z6, z2, z3, member1c24_in_gag(z6, z3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1c24_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, memberc34_out_gg(z0, .(z0, .(z2, .(z2, z4)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1c24_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z2, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), memberc34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), memberc34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), memberc34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), memberc34_in_gg(z2, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U17_gg(z2, z0, .(z4, .(z4, z5)), memberc34_in_gg(z2, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U17_gg(z2, z0, .(z4, .(z5, z6)), memberc34_in_gg(z2, .(z4, .(z5, z6)))))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(69) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U17_gg(x5, z0, .(z0, z4), memberc34_in_gg(x5, .(z0, z4)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), memberc34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), memberc34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), memberc34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), memberc34_in_gg(z2, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U17_gg(x5, z0, .(z0, .(z0, z3)), memberc34_in_gg(x5, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U17_gg(x5, z0, .(z0, .(z3, z4)), memberc34_in_gg(x5, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U17_gg(x5, z0, .(z0, .(z0, z4)), memberc34_in_gg(x5, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U17_gg(x5, z0, .(z0, .(z4, z5)), memberc34_in_gg(x5, .(z0, .(z4, z5)))))
(70) Obligation:
Q DP problem:The TRS P consists of the following rules:
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), memberc34_in_gg(x5, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, memberc34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, memberc34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, memberc34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, memberc34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, memberc34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, memberc34_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z5, z2, z3, member1c24_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))), U12_gag(z6, z2, z3, member1c24_in_gag(z6, z3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1c24_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, memberc34_out_gg(z0, .(z0, .(z2, .(z2, z4)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1c24_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z2, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), memberc34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), memberc34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), memberc34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), memberc34_in_gg(z2, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U17_gg(z2, z0, .(z4, .(z4, z5)), memberc34_in_gg(z2, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U17_gg(z2, z0, .(z4, .(z5, z6)), memberc34_in_gg(z2, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U17_gg(x5, z0, .(z0, .(z0, z3)), memberc34_in_gg(x5, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U17_gg(x5, z0, .(z0, .(z3, z4)), memberc34_in_gg(x5, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U17_gg(x5, z0, .(z0, .(z0, z4)), memberc34_in_gg(x5, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U17_gg(x5, z0, .(z0, .(z4, z5)), memberc34_in_gg(x5, .(z0, .(z4, z5)))))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(71) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U17_gg(x5, z0, .(z4, z5), memberc34_in_gg(x5, .(z4, z5)))) we obtained the following new rules [LPAR04]:
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), memberc34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), memberc34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), memberc34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), memberc34_in_gg(z2, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U17_gg(z2, z0, .(z4, .(z4, z5)), memberc34_in_gg(z2, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U17_gg(z2, z0, .(z4, .(z5, z6)), memberc34_in_gg(z2, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, x6, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x6, U17_gg(x6, z0, .(z0, .(z0, z3)), memberc34_in_gg(x6, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, x6, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x6, U17_gg(x6, z0, .(z0, .(z3, z4)), memberc34_in_gg(x6, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, x6, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U17_gg(x6, z0, .(z0, .(z0, z4)), memberc34_in_gg(x6, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, x6, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U17_gg(x6, z0, .(z0, .(z4, z5)), memberc34_in_gg(x6, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1c24_out_gag(z0, x6, .(.(z2, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), x6, U17_gg(x6, z0, .(z2, .(z2, z4)), memberc34_in_gg(x6, .(z2, .(z2, z4)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1c24_out_gag(z0, x6, .(.(z2, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), x6, U17_gg(x6, z0, .(z2, .(z4, z5)), memberc34_in_gg(x6, .(z2, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, x6, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x6, U17_gg(x6, z0, .(z4, .(z4, z5)), memberc34_in_gg(x6, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, x6, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x6, U17_gg(x6, z0, .(z4, .(z5, z6)), memberc34_in_gg(x6, .(z4, .(z5, z6)))))
(72) Obligation:
Q DP problem:The TRS P consists of the following rules:
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, memberc34_out_gg(z0, .(z0, .(z0, z3)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, memberc34_out_gg(z0, .(z0, .(z3, z4)))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, memberc34_out_gg(z0, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, memberc34_out_gg(z0, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, memberc34_out_gg(z2, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, memberc34_out_gg(z2, .(z0, .(z4, z5)))) → REACH1_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, memberc34_out_gg(z5, .(z0, .(z0, z4)))) → REACH1_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, memberc34_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, .(z0, .(z0, [])), z2, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z0, z2, z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z2, .(z0, .(z2, [])), z3, member1c24_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))), U12_gag(z5, z2, z3, member1c24_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))), U12_gag(z6, z2, z3, member1c24_in_gag(z6, z3)))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1c24_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, memberc34_out_gg(z0, .(z0, .(z2, .(z2, z4)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1c24_out_gag(z0, z0, .(.(z2, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z2, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z0, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, memberc34_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U17_gg(z0, z0, .(z0, .(z0, z3)), memberc34_in_gg(z0, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U17_gg(z0, z0, .(z0, .(z3, z4)), memberc34_in_gg(z0, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z0, z4))), z2, U17_gg(z2, z0, .(z0, .(z0, z4)), memberc34_in_gg(z2, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, .(z4, z5))), z2, U17_gg(z2, z0, .(z0, .(z4, z5)), memberc34_in_gg(z2, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z4, z5))), z2, U17_gg(z2, z0, .(z4, .(z4, z5)), memberc34_in_gg(z2, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, z2, .(.(z0, .(z2, [])), z3))) → U8_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, .(z5, z6))), z2, U17_gg(z2, z0, .(z4, .(z5, z6)), memberc34_in_gg(z2, .(z4, .(z5, z6)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1c24_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x5, U17_gg(x5, z0, .(z0, .(z0, z3)), memberc34_in_gg(x5, .(z0, .(z0, z3)))))
U6_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1c24_out_gag(z0, x5, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x5, U17_gg(x5, z0, .(z0, .(z3, z4)), memberc34_in_gg(x5, .(z0, .(z3, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x5, U17_gg(x5, z0, .(z0, .(z0, z4)), memberc34_in_gg(x5, .(z0, .(z0, z4)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1c24_out_gag(z0, x5, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x5, U17_gg(x5, z0, .(z0, .(z4, z5)), memberc34_in_gg(x5, .(z0, .(z4, z5)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1c24_out_gag(z0, x6, .(.(z2, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), x6, U17_gg(x6, z0, .(z2, .(z2, z4)), memberc34_in_gg(x6, .(z2, .(z2, z4)))))
U6_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1c24_out_gag(z0, x6, .(.(z2, .(z0, [])), z3))) → U8_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), x6, U17_gg(x6, z0, .(z2, .(z4, z5)), memberc34_in_gg(x6, .(z2, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1c24_out_gag(z0, x6, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x6, U17_gg(x6, z0, .(z4, .(z4, z5)), memberc34_in_gg(x6, .(z4, .(z4, z5)))))
U6_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1c24_out_gag(z0, x6, .(z2, z3))) → U8_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x6, U17_gg(x6, z0, .(z4, .(z5, z6)), memberc34_in_gg(x6, .(z4, .(z5, z6)))))
The TRS R consists of the following rules:
member1c24_in_gag(T120, .(.(T120, .(X117, [])), T121)) → member1c24_out_gag(T120, X117, .(.(T120, .(X117, [])), T121))
member1c24_in_gag(T128, .(T129, T130)) → U12_gag(T128, T129, T130, member1c24_in_gag(T128, T130))
U12_gag(T128, T129, T130, member1c24_out_gag(T128, X131, T130)) → member1c24_out_gag(T128, X131, .(T129, T130))
memberc34_in_gg(T151, .(T151, T152)) → memberc34_out_gg(T151, .(T151, T152))
memberc34_in_gg(T159, .(T160, T161)) → U17_gg(T159, T160, T161, memberc34_in_gg(T159, T161))
U17_gg(T159, T160, T161, memberc34_out_gg(T159, T161)) → memberc34_out_gg(T159, .(T160, T161))
The set Q consists of the following terms:
member1c24_in_gag(x0, x1)
U12_gag(x0, x1, x2, x3)
memberc34_in_gg(x0, x1)
U17_gg(x0, x1, x2, x3)
We have to consider all (P,Q,R)-chains.
(73) 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))), member1c24_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'))), member1c24_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))), member1c24_out_gag(z0, z0, .(.(z0, .(z0, [])), z2))) → U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
with rule U6_GGGG(z0', z1', .(.(z0', .(z0', [])), z2'), .(z0', .(z0', .(z0', z3'))), member1c24_out_gag(z0', z0', .(.(z0', .(z0', [])), z2'))) → U8_GGGG(z0', z1', .(.(z0', .(z0', [])), z2'), .(z0', .(z0', .(z0', z3'))), z0', memberc34_out_gg(z0', .(z0', .(z0', .(z0', z3'))))) at position [] and matcher [z0' / z0, z1' / z1, z2' / z2, z3' / z3]
U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, memberc34_out_gg(z0, .(z0, .(z0, .(z0, z3))))) → REACH1_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, .(z0, z3)))))
with rule U8_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, memberc34_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.