(0) Obligation:
Clauses:color_map(.(Region, Regions), Colors) :- ','(color_region(Region, Colors), color_map(Regions, Colors)).
color_map([], Colors).
color_region(region(Name, Color, Neighbors), Colors) :- ','(select(Color, Colors, Colors1), members(Neighbors, Colors1)).
select(X, .(X, Xs), Xs).
select(X, .(Y, Ys), .(Y, Zs)) :- select(X, Ys, Zs).
members(.(X, Xs), Ys) :- ','(member(X, Ys), members(Xs, Ys)).
members([], Ys).
member(X, .(X, X1)).
member(X, .(X2, T)) :- member(X, T).
test_color(Name, Map) :- ','(map(Name, Map), ','(colors(Name, Colors), color_map(Map, Colors))).
map(test, .(region(a, A, .(B, .(C, .(D, [])))), .(region(b, B, .(A, .(C, .(E, [])))), .(region(c, C, .(A, .(B, .(D, .(E, .(F, [])))))), .(region(d, D, .(A, .(C, .(F, [])))), .(region(e, E, .(B, .(C, .(F, [])))), .(region(f, F, .(C, .(D, .(E, [])))), []))))))).
map(west_europe, .(region(portugal, P, .(E, [])), .(region(spain, E, .(F, .(P, []))), .(region(france, F, .(E, .(I, .(S, .(B, .(WG, .(L, []))))))), .(region(belgium, B, .(F, .(H, .(L, .(WG, []))))), .(region(holland, H, .(B, .(WG, []))), .(region(west_germany, WG, .(F, .(A, .(S, .(H, .(B, .(L, []))))))), .(region(luxembourg, L, .(F, .(B, .(WG, [])))), .(region(italy, I, .(F, .(A, .(S, [])))), .(region(switzerland, S, .(F, .(I, .(A, .(WG, []))))), .(region(austria, A, .(I, .(S, .(WG, [])))), []))))))))))).
colors(X, .(red, .(yellow, .(blue, .(white, []))))).
Queries:
color_map(a,g).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:select10(T64, .(T62, T63), .(T62, X73)) :- select10(T64, T63, X73).
members20(.(T91, T92), T90) :- member27(T91, T90).
members20(.(T91, T97), T90) :- ','(memberc27(T91, T90), members20(T97, T90)).
member27(T121, .(T119, T120)) :- member27(T121, T120).
color_map1(.(region(T27, T31, T32), T33), T30) :- select10(T31, T30, X34).
color_map1(.(region(T27, T31, T39), T40), T30) :- ','(selectc10(T31, T30, T38), members20(T39, T38)).
color_map1(.(region(T27, T31, T39), T72), T30) :- ','(selectc10(T31, T30, T38), ','(membersc20(T39, T38), color_map1(T72, T30))).
Clauses:
selectc10(T53, .(T53, T54), T54).
selectc10(T64, .(T62, T63), .(T62, X73)) :- selectc10(T64, T63, X73).
color_mapc1(.(region(T27, T31, T39), T72), T30) :- ','(selectc10(T31, T30, T38), ','(membersc20(T39, T38), color_mapc1(T72, T30))).
color_mapc1([], T137).
color_mapc1([], T139).
membersc20(.(T91, T97), T90) :- ','(memberc27(T91, T90), membersc20(T97, T90)).
membersc20([], T131).
memberc27(T110, .(T110, T111)).
memberc27(T121, .(T119, T120)) :- memberc27(T121, T120).
Afs:
color_map1(x1, x2) = color_map1(x2)
(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:color_map1_in: (f,b)
select10_in: (f,b,f)
selectc10_in: (f,b,f)
members20_in: (f,b)
member27_in: (f,b)
memberc27_in: (f,b)
membersc20_in: (f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(.(region(T27, T31, T32), T33), T30) → U6_AG(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
COLOR_MAP1_IN_AG(.(region(T27, T31, T32), T33), T30) → SELECT10_IN_AGA(T31, T30, X34)
SELECT10_IN_AGA(T64, .(T62, T63), .(T62, X73)) → U1_AGA(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
SELECT10_IN_AGA(T64, .(T62, T63), .(T62, X73)) → SELECT10_IN_AGA(T64, T63, X73)
COLOR_MAP1_IN_AG(.(region(T27, T31, T39), T40), T30) → U7_AG(T27, T31, T39, T40, T30, selectc10_in_aga(T31, T30, T38))
U7_AG(T27, T31, T39, T40, T30, selectc10_out_aga(T31, T30, T38)) → U8_AG(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
U7_AG(T27, T31, T39, T40, T30, selectc10_out_aga(T31, T30, T38)) → MEMBERS20_IN_AG(T39, T38)
MEMBERS20_IN_AG(.(T91, T92), T90) → U2_AG(T91, T92, T90, member27_in_ag(T91, T90))
MEMBERS20_IN_AG(.(T91, T92), T90) → MEMBER27_IN_AG(T91, T90)
MEMBER27_IN_AG(T121, .(T119, T120)) → U5_AG(T121, T119, T120, member27_in_ag(T121, T120))
MEMBER27_IN_AG(T121, .(T119, T120)) → MEMBER27_IN_AG(T121, T120)
MEMBERS20_IN_AG(.(T91, T97), T90) → U3_AG(T91, T97, T90, memberc27_in_ag(T91, T90))
U3_AG(T91, T97, T90, memberc27_out_ag(T91, T90)) → U4_AG(T91, T97, T90, members20_in_ag(T97, T90))
U3_AG(T91, T97, T90, memberc27_out_ag(T91, T90)) → MEMBERS20_IN_AG(T97, T90)
COLOR_MAP1_IN_AG(.(region(T27, T31, T39), T72), T30) → U9_AG(T27, T31, T39, T72, T30, selectc10_in_aga(T31, T30, T38))
U9_AG(T27, T31, T39, T72, T30, selectc10_out_aga(T31, T30, T38)) → U10_AG(T27, T31, T39, T72, T30, membersc20_in_ag(T39, T38))
U10_AG(T27, T31, T39, T72, T30, membersc20_out_ag(T39, T38)) → U11_AG(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
U10_AG(T27, T31, T39, T72, T30, membersc20_out_ag(T39, T38)) → COLOR_MAP1_IN_AG(T72, T30)
The TRS R consists of the following rules:
selectc10_in_aga(T53, .(T53, T54), T54) → selectc10_out_aga(T53, .(T53, T54), T54)
selectc10_in_aga(T64, .(T62, T63), .(T62, X73)) → U13_aga(T64, T62, T63, X73, selectc10_in_aga(T64, T63, X73))
U13_aga(T64, T62, T63, X73, selectc10_out_aga(T64, T63, X73)) → selectc10_out_aga(T64, .(T62, T63), .(T62, X73))
memberc27_in_ag(T110, .(T110, T111)) → memberc27_out_ag(T110, .(T110, T111))
memberc27_in_ag(T121, .(T119, T120)) → U19_ag(T121, T119, T120, memberc27_in_ag(T121, T120))
U19_ag(T121, T119, T120, memberc27_out_ag(T121, T120)) → memberc27_out_ag(T121, .(T119, T120))
membersc20_in_ag(.(T91, T97), T90) → U17_ag(T91, T97, T90, memberc27_in_ag(T91, T90))
U17_ag(T91, T97, T90, memberc27_out_ag(T91, T90)) → U18_ag(T91, T97, T90, membersc20_in_ag(T97, T90))
membersc20_in_ag([], T131) → membersc20_out_ag([], T131)
U18_ag(T91, T97, T90, membersc20_out_ag(T97, T90)) → membersc20_out_ag(.(T91, T97), T90)
The argument filtering Pi contains the following mapping:
color_map1_in_ag(x1, x2) = color_map1_in_ag(x2)
select10_in_aga(x1, x2, x3) = select10_in_aga(x2)
.(x1, x2) = .(x1, x2)
region(x1, x2, x3) = region(x2, x3)
selectc10_in_aga(x1, x2, x3) = selectc10_in_aga(x2)
selectc10_out_aga(x1, x2, x3) = selectc10_out_aga(x1, x2, x3)
U13_aga(x1, x2, x3, x4, x5) = U13_aga(x2, x3, x5)
members20_in_ag(x1, x2) = members20_in_ag(x2)
member27_in_ag(x1, x2) = member27_in_ag(x2)
memberc27_in_ag(x1, x2) = memberc27_in_ag(x2)
memberc27_out_ag(x1, x2) = memberc27_out_ag(x1, x2)
U19_ag(x1, x2, x3, x4) = U19_ag(x2, x3, x4)
membersc20_in_ag(x1, x2) = membersc20_in_ag(x2)
U17_ag(x1, x2, x3, x4) = U17_ag(x3, x4)
U18_ag(x1, x2, x3, x4) = U18_ag(x1, x3, x4)
membersc20_out_ag(x1, x2) = membersc20_out_ag(x1, x2)
COLOR_MAP1_IN_AG(x1, x2) = COLOR_MAP1_IN_AG(x2)
U6_AG(x1, x2, x3, x4, x5, x6) = U6_AG(x5, x6)
SELECT10_IN_AGA(x1, x2, x3) = SELECT10_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5) = U1_AGA(x2, x3, x5)
U7_AG(x1, x2, x3, x4, x5, x6) = U7_AG(x5, x6)
U8_AG(x1, x2, x3, x4, x5, x6) = U8_AG(x5, x6)
MEMBERS20_IN_AG(x1, x2) = MEMBERS20_IN_AG(x2)
U2_AG(x1, x2, x3, x4) = U2_AG(x3, x4)
MEMBER27_IN_AG(x1, x2) = MEMBER27_IN_AG(x2)
U5_AG(x1, x2, x3, x4) = U5_AG(x2, x3, x4)
U3_AG(x1, x2, x3, x4) = U3_AG(x3, x4)
U4_AG(x1, x2, x3, x4) = U4_AG(x3, x4)
U9_AG(x1, x2, x3, x4, x5, x6) = U9_AG(x5, x6)
U10_AG(x1, x2, x3, x4, x5, x6) = U10_AG(x5, x6)
U11_AG(x1, x2, x3, x4, x5, x6) = U11_AG(x5, x6)
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:
COLOR_MAP1_IN_AG(.(region(T27, T31, T32), T33), T30) → U6_AG(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
COLOR_MAP1_IN_AG(.(region(T27, T31, T32), T33), T30) → SELECT10_IN_AGA(T31, T30, X34)
SELECT10_IN_AGA(T64, .(T62, T63), .(T62, X73)) → U1_AGA(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
SELECT10_IN_AGA(T64, .(T62, T63), .(T62, X73)) → SELECT10_IN_AGA(T64, T63, X73)
COLOR_MAP1_IN_AG(.(region(T27, T31, T39), T40), T30) → U7_AG(T27, T31, T39, T40, T30, selectc10_in_aga(T31, T30, T38))
U7_AG(T27, T31, T39, T40, T30, selectc10_out_aga(T31, T30, T38)) → U8_AG(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
U7_AG(T27, T31, T39, T40, T30, selectc10_out_aga(T31, T30, T38)) → MEMBERS20_IN_AG(T39, T38)
MEMBERS20_IN_AG(.(T91, T92), T90) → U2_AG(T91, T92, T90, member27_in_ag(T91, T90))
MEMBERS20_IN_AG(.(T91, T92), T90) → MEMBER27_IN_AG(T91, T90)
MEMBER27_IN_AG(T121, .(T119, T120)) → U5_AG(T121, T119, T120, member27_in_ag(T121, T120))
MEMBER27_IN_AG(T121, .(T119, T120)) → MEMBER27_IN_AG(T121, T120)
MEMBERS20_IN_AG(.(T91, T97), T90) → U3_AG(T91, T97, T90, memberc27_in_ag(T91, T90))
U3_AG(T91, T97, T90, memberc27_out_ag(T91, T90)) → U4_AG(T91, T97, T90, members20_in_ag(T97, T90))
U3_AG(T91, T97, T90, memberc27_out_ag(T91, T90)) → MEMBERS20_IN_AG(T97, T90)
COLOR_MAP1_IN_AG(.(region(T27, T31, T39), T72), T30) → U9_AG(T27, T31, T39, T72, T30, selectc10_in_aga(T31, T30, T38))
U9_AG(T27, T31, T39, T72, T30, selectc10_out_aga(T31, T30, T38)) → U10_AG(T27, T31, T39, T72, T30, membersc20_in_ag(T39, T38))
U10_AG(T27, T31, T39, T72, T30, membersc20_out_ag(T39, T38)) → U11_AG(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
U10_AG(T27, T31, T39, T72, T30, membersc20_out_ag(T39, T38)) → COLOR_MAP1_IN_AG(T72, T30)
The TRS R consists of the following rules:
selectc10_in_aga(T53, .(T53, T54), T54) → selectc10_out_aga(T53, .(T53, T54), T54)
selectc10_in_aga(T64, .(T62, T63), .(T62, X73)) → U13_aga(T64, T62, T63, X73, selectc10_in_aga(T64, T63, X73))
U13_aga(T64, T62, T63, X73, selectc10_out_aga(T64, T63, X73)) → selectc10_out_aga(T64, .(T62, T63), .(T62, X73))
memberc27_in_ag(T110, .(T110, T111)) → memberc27_out_ag(T110, .(T110, T111))
memberc27_in_ag(T121, .(T119, T120)) → U19_ag(T121, T119, T120, memberc27_in_ag(T121, T120))
U19_ag(T121, T119, T120, memberc27_out_ag(T121, T120)) → memberc27_out_ag(T121, .(T119, T120))
membersc20_in_ag(.(T91, T97), T90) → U17_ag(T91, T97, T90, memberc27_in_ag(T91, T90))
U17_ag(T91, T97, T90, memberc27_out_ag(T91, T90)) → U18_ag(T91, T97, T90, membersc20_in_ag(T97, T90))
membersc20_in_ag([], T131) → membersc20_out_ag([], T131)
U18_ag(T91, T97, T90, membersc20_out_ag(T97, T90)) → membersc20_out_ag(.(T91, T97), T90)
The argument filtering Pi contains the following mapping:
color_map1_in_ag(x1, x2) = color_map1_in_ag(x2)
select10_in_aga(x1, x2, x3) = select10_in_aga(x2)
.(x1, x2) = .(x1, x2)
region(x1, x2, x3) = region(x2, x3)
selectc10_in_aga(x1, x2, x3) = selectc10_in_aga(x2)
selectc10_out_aga(x1, x2, x3) = selectc10_out_aga(x1, x2, x3)
U13_aga(x1, x2, x3, x4, x5) = U13_aga(x2, x3, x5)
members20_in_ag(x1, x2) = members20_in_ag(x2)
member27_in_ag(x1, x2) = member27_in_ag(x2)
memberc27_in_ag(x1, x2) = memberc27_in_ag(x2)
memberc27_out_ag(x1, x2) = memberc27_out_ag(x1, x2)
U19_ag(x1, x2, x3, x4) = U19_ag(x2, x3, x4)
membersc20_in_ag(x1, x2) = membersc20_in_ag(x2)
U17_ag(x1, x2, x3, x4) = U17_ag(x3, x4)
U18_ag(x1, x2, x3, x4) = U18_ag(x1, x3, x4)
membersc20_out_ag(x1, x2) = membersc20_out_ag(x1, x2)
COLOR_MAP1_IN_AG(x1, x2) = COLOR_MAP1_IN_AG(x2)
U6_AG(x1, x2, x3, x4, x5, x6) = U6_AG(x5, x6)
SELECT10_IN_AGA(x1, x2, x3) = SELECT10_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5) = U1_AGA(x2, x3, x5)
U7_AG(x1, x2, x3, x4, x5, x6) = U7_AG(x5, x6)
U8_AG(x1, x2, x3, x4, x5, x6) = U8_AG(x5, x6)
MEMBERS20_IN_AG(x1, x2) = MEMBERS20_IN_AG(x2)
U2_AG(x1, x2, x3, x4) = U2_AG(x3, x4)
MEMBER27_IN_AG(x1, x2) = MEMBER27_IN_AG(x2)
U5_AG(x1, x2, x3, x4) = U5_AG(x2, x3, x4)
U3_AG(x1, x2, x3, x4) = U3_AG(x3, x4)
U4_AG(x1, x2, x3, x4) = U4_AG(x3, x4)
U9_AG(x1, x2, x3, x4, x5, x6) = U9_AG(x5, x6)
U10_AG(x1, x2, x3, x4, x5, x6) = U10_AG(x5, x6)
U11_AG(x1, x2, x3, x4, x5, x6) = U11_AG(x5, x6)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 11 less nodes.(6) Complex Obligation (AND)
(7) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MEMBER27_IN_AG(T121, .(T119, T120)) → MEMBER27_IN_AG(T121, T120)
The TRS R consists of the following rules:
selectc10_in_aga(T53, .(T53, T54), T54) → selectc10_out_aga(T53, .(T53, T54), T54)
selectc10_in_aga(T64, .(T62, T63), .(T62, X73)) → U13_aga(T64, T62, T63, X73, selectc10_in_aga(T64, T63, X73))
U13_aga(T64, T62, T63, X73, selectc10_out_aga(T64, T63, X73)) → selectc10_out_aga(T64, .(T62, T63), .(T62, X73))
memberc27_in_ag(T110, .(T110, T111)) → memberc27_out_ag(T110, .(T110, T111))
memberc27_in_ag(T121, .(T119, T120)) → U19_ag(T121, T119, T120, memberc27_in_ag(T121, T120))
U19_ag(T121, T119, T120, memberc27_out_ag(T121, T120)) → memberc27_out_ag(T121, .(T119, T120))
membersc20_in_ag(.(T91, T97), T90) → U17_ag(T91, T97, T90, memberc27_in_ag(T91, T90))
U17_ag(T91, T97, T90, memberc27_out_ag(T91, T90)) → U18_ag(T91, T97, T90, membersc20_in_ag(T97, T90))
membersc20_in_ag([], T131) → membersc20_out_ag([], T131)
U18_ag(T91, T97, T90, membersc20_out_ag(T97, T90)) → membersc20_out_ag(.(T91, T97), T90)
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
selectc10_in_aga(x1, x2, x3) = selectc10_in_aga(x2)
selectc10_out_aga(x1, x2, x3) = selectc10_out_aga(x1, x2, x3)
U13_aga(x1, x2, x3, x4, x5) = U13_aga(x2, x3, x5)
memberc27_in_ag(x1, x2) = memberc27_in_ag(x2)
memberc27_out_ag(x1, x2) = memberc27_out_ag(x1, x2)
U19_ag(x1, x2, x3, x4) = U19_ag(x2, x3, x4)
membersc20_in_ag(x1, x2) = membersc20_in_ag(x2)
U17_ag(x1, x2, x3, x4) = U17_ag(x3, x4)
U18_ag(x1, x2, x3, x4) = U18_ag(x1, x3, x4)
membersc20_out_ag(x1, x2) = membersc20_out_ag(x1, x2)
MEMBER27_IN_AG(x1, x2) = MEMBER27_IN_AG(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:
MEMBER27_IN_AG(T121, .(T119, T120)) → MEMBER27_IN_AG(T121, T120)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
MEMBER27_IN_AG(x1, x2) = MEMBER27_IN_AG(x2)
We have to consider all (P,R,Pi)-chains
(10) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(11) Obligation:
Q DP problem:The TRS P consists of the following rules:
MEMBER27_IN_AG(.(T119, T120)) → MEMBER27_IN_AG(T120)
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:
- MEMBER27_IN_AG(.(T119, T120)) → MEMBER27_IN_AG(T120)
The graph contains the following edges 1 > 1
(13) YES
(14) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MEMBERS20_IN_AG(.(T91, T97), T90) → U3_AG(T91, T97, T90, memberc27_in_ag(T91, T90))
U3_AG(T91, T97, T90, memberc27_out_ag(T91, T90)) → MEMBERS20_IN_AG(T97, T90)
The TRS R consists of the following rules:
selectc10_in_aga(T53, .(T53, T54), T54) → selectc10_out_aga(T53, .(T53, T54), T54)
selectc10_in_aga(T64, .(T62, T63), .(T62, X73)) → U13_aga(T64, T62, T63, X73, selectc10_in_aga(T64, T63, X73))
U13_aga(T64, T62, T63, X73, selectc10_out_aga(T64, T63, X73)) → selectc10_out_aga(T64, .(T62, T63), .(T62, X73))
memberc27_in_ag(T110, .(T110, T111)) → memberc27_out_ag(T110, .(T110, T111))
memberc27_in_ag(T121, .(T119, T120)) → U19_ag(T121, T119, T120, memberc27_in_ag(T121, T120))
U19_ag(T121, T119, T120, memberc27_out_ag(T121, T120)) → memberc27_out_ag(T121, .(T119, T120))
membersc20_in_ag(.(T91, T97), T90) → U17_ag(T91, T97, T90, memberc27_in_ag(T91, T90))
U17_ag(T91, T97, T90, memberc27_out_ag(T91, T90)) → U18_ag(T91, T97, T90, membersc20_in_ag(T97, T90))
membersc20_in_ag([], T131) → membersc20_out_ag([], T131)
U18_ag(T91, T97, T90, membersc20_out_ag(T97, T90)) → membersc20_out_ag(.(T91, T97), T90)
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
selectc10_in_aga(x1, x2, x3) = selectc10_in_aga(x2)
selectc10_out_aga(x1, x2, x3) = selectc10_out_aga(x1, x2, x3)
U13_aga(x1, x2, x3, x4, x5) = U13_aga(x2, x3, x5)
memberc27_in_ag(x1, x2) = memberc27_in_ag(x2)
memberc27_out_ag(x1, x2) = memberc27_out_ag(x1, x2)
U19_ag(x1, x2, x3, x4) = U19_ag(x2, x3, x4)
membersc20_in_ag(x1, x2) = membersc20_in_ag(x2)
U17_ag(x1, x2, x3, x4) = U17_ag(x3, x4)
U18_ag(x1, x2, x3, x4) = U18_ag(x1, x3, x4)
membersc20_out_ag(x1, x2) = membersc20_out_ag(x1, x2)
MEMBERS20_IN_AG(x1, x2) = MEMBERS20_IN_AG(x2)
U3_AG(x1, x2, x3, x4) = U3_AG(x3, x4)
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:
MEMBERS20_IN_AG(.(T91, T97), T90) → U3_AG(T91, T97, T90, memberc27_in_ag(T91, T90))
U3_AG(T91, T97, T90, memberc27_out_ag(T91, T90)) → MEMBERS20_IN_AG(T97, T90)
The TRS R consists of the following rules:
memberc27_in_ag(T110, .(T110, T111)) → memberc27_out_ag(T110, .(T110, T111))
memberc27_in_ag(T121, .(T119, T120)) → U19_ag(T121, T119, T120, memberc27_in_ag(T121, T120))
U19_ag(T121, T119, T120, memberc27_out_ag(T121, T120)) → memberc27_out_ag(T121, .(T119, T120))
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
memberc27_in_ag(x1, x2) = memberc27_in_ag(x2)
memberc27_out_ag(x1, x2) = memberc27_out_ag(x1, x2)
U19_ag(x1, x2, x3, x4) = U19_ag(x2, x3, x4)
MEMBERS20_IN_AG(x1, x2) = MEMBERS20_IN_AG(x2)
U3_AG(x1, x2, x3, x4) = U3_AG(x3, x4)
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:
MEMBERS20_IN_AG(T90) → U3_AG(T90, memberc27_in_ag(T90))
U3_AG(T90, memberc27_out_ag(T91, T90)) → MEMBERS20_IN_AG(T90)
The TRS R consists of the following rules:
memberc27_in_ag(.(T110, T111)) → memberc27_out_ag(T110, .(T110, T111))
memberc27_in_ag(.(T119, T120)) → U19_ag(T119, T120, memberc27_in_ag(T120))
U19_ag(T119, T120, memberc27_out_ag(T121, T120)) → memberc27_out_ag(T121, .(T119, T120))
The set Q consists of the following terms:
memberc27_in_ag(x0)
U19_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(19) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule MEMBERS20_IN_AG(T90) → U3_AG(T90, memberc27_in_ag(T90)) at position [1] we obtained the following new rules [LPAR04]:
MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), memberc27_out_ag(x0, .(x0, x1)))
MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), U19_ag(x0, x1, memberc27_in_ag(x1)))
(20) Obligation:
Q DP problem:The TRS P consists of the following rules:
U3_AG(T90, memberc27_out_ag(T91, T90)) → MEMBERS20_IN_AG(T90)
MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), memberc27_out_ag(x0, .(x0, x1)))
MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), U19_ag(x0, x1, memberc27_in_ag(x1)))
The TRS R consists of the following rules:
memberc27_in_ag(.(T110, T111)) → memberc27_out_ag(T110, .(T110, T111))
memberc27_in_ag(.(T119, T120)) → U19_ag(T119, T120, memberc27_in_ag(T120))
U19_ag(T119, T120, memberc27_out_ag(T121, T120)) → memberc27_out_ag(T121, .(T119, T120))
The set Q consists of the following terms:
memberc27_in_ag(x0)
U19_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(21) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U3_AG(T90, memberc27_out_ag(T91, T90)) → MEMBERS20_IN_AG(T90) we obtained the following new rules [LPAR04]:
U3_AG(.(z0, z1), memberc27_out_ag(z0, .(z0, z1))) → MEMBERS20_IN_AG(.(z0, z1))
U3_AG(.(z0, z1), memberc27_out_ag(x1, .(z0, z1))) → MEMBERS20_IN_AG(.(z0, z1))
(22) Obligation:
Q DP problem:The TRS P consists of the following rules:
MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), memberc27_out_ag(x0, .(x0, x1)))
MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), U19_ag(x0, x1, memberc27_in_ag(x1)))
U3_AG(.(z0, z1), memberc27_out_ag(z0, .(z0, z1))) → MEMBERS20_IN_AG(.(z0, z1))
U3_AG(.(z0, z1), memberc27_out_ag(x1, .(z0, z1))) → MEMBERS20_IN_AG(.(z0, z1))
The TRS R consists of the following rules:
memberc27_in_ag(.(T110, T111)) → memberc27_out_ag(T110, .(T110, T111))
memberc27_in_ag(.(T119, T120)) → U19_ag(T119, T120, memberc27_in_ag(T120))
U19_ag(T119, T120, memberc27_out_ag(T121, T120)) → memberc27_out_ag(T121, .(T119, T120))
The set Q consists of the following terms:
memberc27_in_ag(x0)
U19_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(23) 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 = U3_AG(.(z0, z1), memberc27_out_ag(z0, .(z0, z1))) evaluates to t =U3_AG(.(z0, z1), memberc27_out_ag(z0, .(z0, z1)))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [ ]
- Semiunifier: [ ]
Rewriting sequence
U3_AG(.(z0, z1), memberc27_out_ag(z0, .(z0, z1))) → MEMBERS20_IN_AG(.(z0, z1))
with rule U3_AG(.(z0', z1'), memberc27_out_ag(z0', .(z0', z1'))) → MEMBERS20_IN_AG(.(z0', z1')) at position [] and matcher [z0' / z0, z1' / z1]
MEMBERS20_IN_AG(.(z0, z1)) → U3_AG(.(z0, z1), memberc27_out_ag(z0, .(z0, z1)))
with rule MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), memberc27_out_ag(x0, .(x0, x1)))
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.
(24) NO
(25) Obligation:
Pi DP problem:The TRS P consists of the following rules:
SELECT10_IN_AGA(T64, .(T62, T63), .(T62, X73)) → SELECT10_IN_AGA(T64, T63, X73)
The TRS R consists of the following rules:
selectc10_in_aga(T53, .(T53, T54), T54) → selectc10_out_aga(T53, .(T53, T54), T54)
selectc10_in_aga(T64, .(T62, T63), .(T62, X73)) → U13_aga(T64, T62, T63, X73, selectc10_in_aga(T64, T63, X73))
U13_aga(T64, T62, T63, X73, selectc10_out_aga(T64, T63, X73)) → selectc10_out_aga(T64, .(T62, T63), .(T62, X73))
memberc27_in_ag(T110, .(T110, T111)) → memberc27_out_ag(T110, .(T110, T111))
memberc27_in_ag(T121, .(T119, T120)) → U19_ag(T121, T119, T120, memberc27_in_ag(T121, T120))
U19_ag(T121, T119, T120, memberc27_out_ag(T121, T120)) → memberc27_out_ag(T121, .(T119, T120))
membersc20_in_ag(.(T91, T97), T90) → U17_ag(T91, T97, T90, memberc27_in_ag(T91, T90))
U17_ag(T91, T97, T90, memberc27_out_ag(T91, T90)) → U18_ag(T91, T97, T90, membersc20_in_ag(T97, T90))
membersc20_in_ag([], T131) → membersc20_out_ag([], T131)
U18_ag(T91, T97, T90, membersc20_out_ag(T97, T90)) → membersc20_out_ag(.(T91, T97), T90)
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
selectc10_in_aga(x1, x2, x3) = selectc10_in_aga(x2)
selectc10_out_aga(x1, x2, x3) = selectc10_out_aga(x1, x2, x3)
U13_aga(x1, x2, x3, x4, x5) = U13_aga(x2, x3, x5)
memberc27_in_ag(x1, x2) = memberc27_in_ag(x2)
memberc27_out_ag(x1, x2) = memberc27_out_ag(x1, x2)
U19_ag(x1, x2, x3, x4) = U19_ag(x2, x3, x4)
membersc20_in_ag(x1, x2) = membersc20_in_ag(x2)
U17_ag(x1, x2, x3, x4) = U17_ag(x3, x4)
U18_ag(x1, x2, x3, x4) = U18_ag(x1, x3, x4)
membersc20_out_ag(x1, x2) = membersc20_out_ag(x1, x2)
SELECT10_IN_AGA(x1, x2, x3) = SELECT10_IN_AGA(x2)
We have to consider all (P,R,Pi)-chains
(26) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(27) Obligation:
Pi DP problem:The TRS P consists of the following rules:
SELECT10_IN_AGA(T64, .(T62, T63), .(T62, X73)) → SELECT10_IN_AGA(T64, T63, X73)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
SELECT10_IN_AGA(x1, x2, x3) = SELECT10_IN_AGA(x2)
We have to consider all (P,R,Pi)-chains
(28) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(29) Obligation:
Q DP problem:The TRS P consists of the following rules:
SELECT10_IN_AGA(.(T62, T63)) → SELECT10_IN_AGA(T63)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(30) 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:
- SELECT10_IN_AGA(.(T62, T63)) → SELECT10_IN_AGA(T63)
The graph contains the following edges 1 > 1
(31) YES
(32) Obligation:
Pi DP problem:The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(.(region(T27, T31, T39), T72), T30) → U9_AG(T27, T31, T39, T72, T30, selectc10_in_aga(T31, T30, T38))
U9_AG(T27, T31, T39, T72, T30, selectc10_out_aga(T31, T30, T38)) → U10_AG(T27, T31, T39, T72, T30, membersc20_in_ag(T39, T38))
U10_AG(T27, T31, T39, T72, T30, membersc20_out_ag(T39, T38)) → COLOR_MAP1_IN_AG(T72, T30)
The TRS R consists of the following rules:
selectc10_in_aga(T53, .(T53, T54), T54) → selectc10_out_aga(T53, .(T53, T54), T54)
selectc10_in_aga(T64, .(T62, T63), .(T62, X73)) → U13_aga(T64, T62, T63, X73, selectc10_in_aga(T64, T63, X73))
U13_aga(T64, T62, T63, X73, selectc10_out_aga(T64, T63, X73)) → selectc10_out_aga(T64, .(T62, T63), .(T62, X73))
memberc27_in_ag(T110, .(T110, T111)) → memberc27_out_ag(T110, .(T110, T111))
memberc27_in_ag(T121, .(T119, T120)) → U19_ag(T121, T119, T120, memberc27_in_ag(T121, T120))
U19_ag(T121, T119, T120, memberc27_out_ag(T121, T120)) → memberc27_out_ag(T121, .(T119, T120))
membersc20_in_ag(.(T91, T97), T90) → U17_ag(T91, T97, T90, memberc27_in_ag(T91, T90))
U17_ag(T91, T97, T90, memberc27_out_ag(T91, T90)) → U18_ag(T91, T97, T90, membersc20_in_ag(T97, T90))
membersc20_in_ag([], T131) → membersc20_out_ag([], T131)
U18_ag(T91, T97, T90, membersc20_out_ag(T97, T90)) → membersc20_out_ag(.(T91, T97), T90)
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
region(x1, x2, x3) = region(x2, x3)
selectc10_in_aga(x1, x2, x3) = selectc10_in_aga(x2)
selectc10_out_aga(x1, x2, x3) = selectc10_out_aga(x1, x2, x3)
U13_aga(x1, x2, x3, x4, x5) = U13_aga(x2, x3, x5)
memberc27_in_ag(x1, x2) = memberc27_in_ag(x2)
memberc27_out_ag(x1, x2) = memberc27_out_ag(x1, x2)
U19_ag(x1, x2, x3, x4) = U19_ag(x2, x3, x4)
membersc20_in_ag(x1, x2) = membersc20_in_ag(x2)
U17_ag(x1, x2, x3, x4) = U17_ag(x3, x4)
U18_ag(x1, x2, x3, x4) = U18_ag(x1, x3, x4)
membersc20_out_ag(x1, x2) = membersc20_out_ag(x1, x2)
COLOR_MAP1_IN_AG(x1, x2) = COLOR_MAP1_IN_AG(x2)
U9_AG(x1, x2, x3, x4, x5, x6) = U9_AG(x5, x6)
U10_AG(x1, x2, x3, x4, x5, x6) = U10_AG(x5, x6)
We have to consider all (P,R,Pi)-chains
(33) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(34) Obligation:
Q DP problem:The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(T30) → U9_AG(T30, selectc10_in_aga(T30))
U9_AG(T30, selectc10_out_aga(T31, T30, T38)) → U10_AG(T30, membersc20_in_ag(T38))
U10_AG(T30, membersc20_out_ag(T39, T38)) → COLOR_MAP1_IN_AG(T30)
The TRS R consists of the following rules:
selectc10_in_aga(.(T53, T54)) → selectc10_out_aga(T53, .(T53, T54), T54)
selectc10_in_aga(.(T62, T63)) → U13_aga(T62, T63, selectc10_in_aga(T63))
U13_aga(T62, T63, selectc10_out_aga(T64, T63, X73)) → selectc10_out_aga(T64, .(T62, T63), .(T62, X73))
memberc27_in_ag(.(T110, T111)) → memberc27_out_ag(T110, .(T110, T111))
memberc27_in_ag(.(T119, T120)) → U19_ag(T119, T120, memberc27_in_ag(T120))
U19_ag(T119, T120, memberc27_out_ag(T121, T120)) → memberc27_out_ag(T121, .(T119, T120))
membersc20_in_ag(T90) → U17_ag(T90, memberc27_in_ag(T90))
U17_ag(T90, memberc27_out_ag(T91, T90)) → U18_ag(T91, T90, membersc20_in_ag(T90))
membersc20_in_ag(T131) → membersc20_out_ag([], T131)
U18_ag(T91, T90, membersc20_out_ag(T97, T90)) → membersc20_out_ag(.(T91, T97), T90)
The set Q consists of the following terms:
selectc10_in_aga(x0)
U13_aga(x0, x1, x2)
memberc27_in_ag(x0)
U19_ag(x0, x1, x2)
membersc20_in_ag(x0)
U17_ag(x0, x1)
U18_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(35) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule COLOR_MAP1_IN_AG(T30) → U9_AG(T30, selectc10_in_aga(T30)) at position [1] we obtained the following new rules [LPAR04]:
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), selectc10_out_aga(x0, .(x0, x1), x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U13_aga(x0, x1, selectc10_in_aga(x1)))
(36) Obligation:
Q DP problem:The TRS P consists of the following rules:
U9_AG(T30, selectc10_out_aga(T31, T30, T38)) → U10_AG(T30, membersc20_in_ag(T38))
U10_AG(T30, membersc20_out_ag(T39, T38)) → COLOR_MAP1_IN_AG(T30)
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), selectc10_out_aga(x0, .(x0, x1), x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U13_aga(x0, x1, selectc10_in_aga(x1)))
The TRS R consists of the following rules:
selectc10_in_aga(.(T53, T54)) → selectc10_out_aga(T53, .(T53, T54), T54)
selectc10_in_aga(.(T62, T63)) → U13_aga(T62, T63, selectc10_in_aga(T63))
U13_aga(T62, T63, selectc10_out_aga(T64, T63, X73)) → selectc10_out_aga(T64, .(T62, T63), .(T62, X73))
memberc27_in_ag(.(T110, T111)) → memberc27_out_ag(T110, .(T110, T111))
memberc27_in_ag(.(T119, T120)) → U19_ag(T119, T120, memberc27_in_ag(T120))
U19_ag(T119, T120, memberc27_out_ag(T121, T120)) → memberc27_out_ag(T121, .(T119, T120))
membersc20_in_ag(T90) → U17_ag(T90, memberc27_in_ag(T90))
U17_ag(T90, memberc27_out_ag(T91, T90)) → U18_ag(T91, T90, membersc20_in_ag(T90))
membersc20_in_ag(T131) → membersc20_out_ag([], T131)
U18_ag(T91, T90, membersc20_out_ag(T97, T90)) → membersc20_out_ag(.(T91, T97), T90)
The set Q consists of the following terms:
selectc10_in_aga(x0)
U13_aga(x0, x1, x2)
memberc27_in_ag(x0)
U19_ag(x0, x1, x2)
membersc20_in_ag(x0)
U17_ag(x0, x1)
U18_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(37) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule U9_AG(T30, selectc10_out_aga(T31, T30, T38)) → U10_AG(T30, membersc20_in_ag(T38)) at position [1] we obtained the following new rules [LPAR04]:
U9_AG(y0, selectc10_out_aga(y1, y0, x0)) → U10_AG(y0, U17_ag(x0, memberc27_in_ag(x0)))
U9_AG(y0, selectc10_out_aga(y1, y0, x0)) → U10_AG(y0, membersc20_out_ag([], x0))
(38) Obligation:
Q DP problem:The TRS P consists of the following rules:
U10_AG(T30, membersc20_out_ag(T39, T38)) → COLOR_MAP1_IN_AG(T30)
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), selectc10_out_aga(x0, .(x0, x1), x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U13_aga(x0, x1, selectc10_in_aga(x1)))
U9_AG(y0, selectc10_out_aga(y1, y0, x0)) → U10_AG(y0, U17_ag(x0, memberc27_in_ag(x0)))
U9_AG(y0, selectc10_out_aga(y1, y0, x0)) → U10_AG(y0, membersc20_out_ag([], x0))
The TRS R consists of the following rules:
selectc10_in_aga(.(T53, T54)) → selectc10_out_aga(T53, .(T53, T54), T54)
selectc10_in_aga(.(T62, T63)) → U13_aga(T62, T63, selectc10_in_aga(T63))
U13_aga(T62, T63, selectc10_out_aga(T64, T63, X73)) → selectc10_out_aga(T64, .(T62, T63), .(T62, X73))
memberc27_in_ag(.(T110, T111)) → memberc27_out_ag(T110, .(T110, T111))
memberc27_in_ag(.(T119, T120)) → U19_ag(T119, T120, memberc27_in_ag(T120))
U19_ag(T119, T120, memberc27_out_ag(T121, T120)) → memberc27_out_ag(T121, .(T119, T120))
membersc20_in_ag(T90) → U17_ag(T90, memberc27_in_ag(T90))
U17_ag(T90, memberc27_out_ag(T91, T90)) → U18_ag(T91, T90, membersc20_in_ag(T90))
membersc20_in_ag(T131) → membersc20_out_ag([], T131)
U18_ag(T91, T90, membersc20_out_ag(T97, T90)) → membersc20_out_ag(.(T91, T97), T90)
The set Q consists of the following terms:
selectc10_in_aga(x0)
U13_aga(x0, x1, x2)
memberc27_in_ag(x0)
U19_ag(x0, x1, x2)
membersc20_in_ag(x0)
U17_ag(x0, x1)
U18_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(39) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U9_AG(y0, selectc10_out_aga(y1, y0, x0)) → U10_AG(y0, U17_ag(x0, memberc27_in_ag(x0))) we obtained the following new rules [LPAR04]:
U9_AG(.(z0, z1), selectc10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), U17_ag(z1, memberc27_in_ag(z1)))
U9_AG(.(z0, z1), selectc10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), U17_ag(x2, memberc27_in_ag(x2)))
(40) Obligation:
Q DP problem:The TRS P consists of the following rules:
U10_AG(T30, membersc20_out_ag(T39, T38)) → COLOR_MAP1_IN_AG(T30)
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), selectc10_out_aga(x0, .(x0, x1), x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U13_aga(x0, x1, selectc10_in_aga(x1)))
U9_AG(y0, selectc10_out_aga(y1, y0, x0)) → U10_AG(y0, membersc20_out_ag([], x0))
U9_AG(.(z0, z1), selectc10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), U17_ag(z1, memberc27_in_ag(z1)))
U9_AG(.(z0, z1), selectc10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), U17_ag(x2, memberc27_in_ag(x2)))
The TRS R consists of the following rules:
selectc10_in_aga(.(T53, T54)) → selectc10_out_aga(T53, .(T53, T54), T54)
selectc10_in_aga(.(T62, T63)) → U13_aga(T62, T63, selectc10_in_aga(T63))
U13_aga(T62, T63, selectc10_out_aga(T64, T63, X73)) → selectc10_out_aga(T64, .(T62, T63), .(T62, X73))
memberc27_in_ag(.(T110, T111)) → memberc27_out_ag(T110, .(T110, T111))
memberc27_in_ag(.(T119, T120)) → U19_ag(T119, T120, memberc27_in_ag(T120))
U19_ag(T119, T120, memberc27_out_ag(T121, T120)) → memberc27_out_ag(T121, .(T119, T120))
membersc20_in_ag(T90) → U17_ag(T90, memberc27_in_ag(T90))
U17_ag(T90, memberc27_out_ag(T91, T90)) → U18_ag(T91, T90, membersc20_in_ag(T90))
membersc20_in_ag(T131) → membersc20_out_ag([], T131)
U18_ag(T91, T90, membersc20_out_ag(T97, T90)) → membersc20_out_ag(.(T91, T97), T90)
The set Q consists of the following terms:
selectc10_in_aga(x0)
U13_aga(x0, x1, x2)
memberc27_in_ag(x0)
U19_ag(x0, x1, x2)
membersc20_in_ag(x0)
U17_ag(x0, x1)
U18_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(41) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U10_AG(T30, membersc20_out_ag(T39, T38)) → COLOR_MAP1_IN_AG(T30) we obtained the following new rules [LPAR04]:
U10_AG(z0, membersc20_out_ag([], z2)) → COLOR_MAP1_IN_AG(z0)
U10_AG(.(z0, z1), membersc20_out_ag(x1, x2)) → COLOR_MAP1_IN_AG(.(z0, z1))
(42) Obligation:
Q DP problem:The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), selectc10_out_aga(x0, .(x0, x1), x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U13_aga(x0, x1, selectc10_in_aga(x1)))
U9_AG(y0, selectc10_out_aga(y1, y0, x0)) → U10_AG(y0, membersc20_out_ag([], x0))
U9_AG(.(z0, z1), selectc10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), U17_ag(z1, memberc27_in_ag(z1)))
U9_AG(.(z0, z1), selectc10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), U17_ag(x2, memberc27_in_ag(x2)))
U10_AG(z0, membersc20_out_ag([], z2)) → COLOR_MAP1_IN_AG(z0)
U10_AG(.(z0, z1), membersc20_out_ag(x1, x2)) → COLOR_MAP1_IN_AG(.(z0, z1))
The TRS R consists of the following rules:
selectc10_in_aga(.(T53, T54)) → selectc10_out_aga(T53, .(T53, T54), T54)
selectc10_in_aga(.(T62, T63)) → U13_aga(T62, T63, selectc10_in_aga(T63))
U13_aga(T62, T63, selectc10_out_aga(T64, T63, X73)) → selectc10_out_aga(T64, .(T62, T63), .(T62, X73))
memberc27_in_ag(.(T110, T111)) → memberc27_out_ag(T110, .(T110, T111))
memberc27_in_ag(.(T119, T120)) → U19_ag(T119, T120, memberc27_in_ag(T120))
U19_ag(T119, T120, memberc27_out_ag(T121, T120)) → memberc27_out_ag(T121, .(T119, T120))
membersc20_in_ag(T90) → U17_ag(T90, memberc27_in_ag(T90))
U17_ag(T90, memberc27_out_ag(T91, T90)) → U18_ag(T91, T90, membersc20_in_ag(T90))
membersc20_in_ag(T131) → membersc20_out_ag([], T131)
U18_ag(T91, T90, membersc20_out_ag(T97, T90)) → membersc20_out_ag(.(T91, T97), T90)
The set Q consists of the following terms:
selectc10_in_aga(x0)
U13_aga(x0, x1, x2)
memberc27_in_ag(x0)
U19_ag(x0, x1, x2)
membersc20_in_ag(x0)
U17_ag(x0, x1)
U18_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(43) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U9_AG(y0, selectc10_out_aga(y1, y0, x0)) → U10_AG(y0, membersc20_out_ag([], x0)) we obtained the following new rules [LPAR04]:
U9_AG(.(z0, z1), selectc10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), membersc20_out_ag([], z1))
U9_AG(.(z0, z1), selectc10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), membersc20_out_ag([], x2))
(44) Obligation:
Q DP problem:The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), selectc10_out_aga(x0, .(x0, x1), x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U13_aga(x0, x1, selectc10_in_aga(x1)))
U9_AG(.(z0, z1), selectc10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), U17_ag(z1, memberc27_in_ag(z1)))
U9_AG(.(z0, z1), selectc10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), U17_ag(x2, memberc27_in_ag(x2)))
U10_AG(z0, membersc20_out_ag([], z2)) → COLOR_MAP1_IN_AG(z0)
U10_AG(.(z0, z1), membersc20_out_ag(x1, x2)) → COLOR_MAP1_IN_AG(.(z0, z1))
U9_AG(.(z0, z1), selectc10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), membersc20_out_ag([], z1))
U9_AG(.(z0, z1), selectc10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), membersc20_out_ag([], x2))
The TRS R consists of the following rules:
selectc10_in_aga(.(T53, T54)) → selectc10_out_aga(T53, .(T53, T54), T54)
selectc10_in_aga(.(T62, T63)) → U13_aga(T62, T63, selectc10_in_aga(T63))
U13_aga(T62, T63, selectc10_out_aga(T64, T63, X73)) → selectc10_out_aga(T64, .(T62, T63), .(T62, X73))
memberc27_in_ag(.(T110, T111)) → memberc27_out_ag(T110, .(T110, T111))
memberc27_in_ag(.(T119, T120)) → U19_ag(T119, T120, memberc27_in_ag(T120))
U19_ag(T119, T120, memberc27_out_ag(T121, T120)) → memberc27_out_ag(T121, .(T119, T120))
membersc20_in_ag(T90) → U17_ag(T90, memberc27_in_ag(T90))
U17_ag(T90, memberc27_out_ag(T91, T90)) → U18_ag(T91, T90, membersc20_in_ag(T90))
membersc20_in_ag(T131) → membersc20_out_ag([], T131)
U18_ag(T91, T90, membersc20_out_ag(T97, T90)) → membersc20_out_ag(.(T91, T97), T90)
The set Q consists of the following terms:
selectc10_in_aga(x0)
U13_aga(x0, x1, x2)
memberc27_in_ag(x0)
U19_ag(x0, x1, x2)
membersc20_in_ag(x0)
U17_ag(x0, x1)
U18_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(45) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U10_AG(z0, membersc20_out_ag([], z2)) → COLOR_MAP1_IN_AG(z0) we obtained the following new rules [LPAR04]:
U10_AG(.(z0, z1), membersc20_out_ag([], x1)) → COLOR_MAP1_IN_AG(.(z0, z1))
U10_AG(.(z0, z1), membersc20_out_ag([], z1)) → COLOR_MAP1_IN_AG(.(z0, z1))
(46) Obligation:
Q DP problem:The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), selectc10_out_aga(x0, .(x0, x1), x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U13_aga(x0, x1, selectc10_in_aga(x1)))
U9_AG(.(z0, z1), selectc10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), U17_ag(z1, memberc27_in_ag(z1)))
U9_AG(.(z0, z1), selectc10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), U17_ag(x2, memberc27_in_ag(x2)))
U10_AG(.(z0, z1), membersc20_out_ag(x1, x2)) → COLOR_MAP1_IN_AG(.(z0, z1))
U9_AG(.(z0, z1), selectc10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), membersc20_out_ag([], z1))
U9_AG(.(z0, z1), selectc10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), membersc20_out_ag([], x2))
U10_AG(.(z0, z1), membersc20_out_ag([], x1)) → COLOR_MAP1_IN_AG(.(z0, z1))
U10_AG(.(z0, z1), membersc20_out_ag([], z1)) → COLOR_MAP1_IN_AG(.(z0, z1))
The TRS R consists of the following rules:
selectc10_in_aga(.(T53, T54)) → selectc10_out_aga(T53, .(T53, T54), T54)
selectc10_in_aga(.(T62, T63)) → U13_aga(T62, T63, selectc10_in_aga(T63))
U13_aga(T62, T63, selectc10_out_aga(T64, T63, X73)) → selectc10_out_aga(T64, .(T62, T63), .(T62, X73))
memberc27_in_ag(.(T110, T111)) → memberc27_out_ag(T110, .(T110, T111))
memberc27_in_ag(.(T119, T120)) → U19_ag(T119, T120, memberc27_in_ag(T120))
U19_ag(T119, T120, memberc27_out_ag(T121, T120)) → memberc27_out_ag(T121, .(T119, T120))
membersc20_in_ag(T90) → U17_ag(T90, memberc27_in_ag(T90))
U17_ag(T90, memberc27_out_ag(T91, T90)) → U18_ag(T91, T90, membersc20_in_ag(T90))
membersc20_in_ag(T131) → membersc20_out_ag([], T131)
U18_ag(T91, T90, membersc20_out_ag(T97, T90)) → membersc20_out_ag(.(T91, T97), T90)
The set Q consists of the following terms:
selectc10_in_aga(x0)
U13_aga(x0, x1, x2)
memberc27_in_ag(x0)
U19_ag(x0, x1, x2)
membersc20_in_ag(x0)
U17_ag(x0, x1)
U18_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(47) 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 = U9_AG(.(z0', z1'), selectc10_out_aga(z0', .(z0', z1'), z1')) evaluates to t =U9_AG(.(z0', z1'), selectc10_out_aga(z0', .(z0', z1'), z1'))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequence
U9_AG(.(z0', z1'), selectc10_out_aga(z0', .(z0', z1'), z1')) → U10_AG(.(z0', z1'), membersc20_out_ag([], z1'))
with rule U9_AG(.(z0'', z1''), selectc10_out_aga(z0'', .(z0'', z1''), z1'')) → U10_AG(.(z0'', z1''), membersc20_out_ag([], z1'')) at position [] and matcher [z0'' / z0', z1'' / z1']
U10_AG(.(z0', z1'), membersc20_out_ag([], z1')) → COLOR_MAP1_IN_AG(.(z0', z1'))
with rule U10_AG(.(z0, z1), membersc20_out_ag(x1', x2)) → COLOR_MAP1_IN_AG(.(z0, z1)) at position [] and matcher [z0 / z0', z1 / z1', x1' / [], x2 / z1']
COLOR_MAP1_IN_AG(.(z0', z1')) → U9_AG(.(z0', z1'), selectc10_out_aga(z0', .(z0', z1'), z1'))
with rule COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), selectc10_out_aga(x0, .(x0, x1), x1))
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.