(0) Obligation:
Clauses:color_map(l(Region, Regions), Colors) :- ','(color_region(Region, Colors), color_map(Regions, Colors)).
color_map(nil, 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, l(region(a, A, .(B, .(C, .(D, [])))), l(region(b, B, .(A, .(C, .(E, [])))), l(region(c, C, .(A, .(B, .(D, .(E, .(F, [])))))), l(region(d, D, .(A, .(C, .(F, [])))), l(region(e, E, .(B, .(C, .(F, [])))), l(region(f, F, .(C, .(D, .(E, [])))), nil))))))).
colors(X, .(red, .(yellow, .(blue, .(white, []))))).
Queries:
color_map(a,g).
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph.
(2) Obligation:
Clauses:select10(T53, .(T53, T54), T54).
select10(T64, .(T62, T63), .(T62, X73)) :- select10(T64, T63, X73).
members20(.(T91, T92), T90) :- member27(T91, T90).
members20(.(T91, T97), T90) :- ','(member27(T91, T90), members20(T97, T90)).
members20([], T131).
member27(T110, .(T110, T111)).
member27(T121, .(T119, T120)) :- member27(T121, T120).
color_map1(l(region(T27, T31, T32), T33), T30) :- select10(T31, T30, X34).
color_map1(l(region(T27, T31, T39), T40), T30) :- ','(select10(T31, T30, T38), members20(T39, T38)).
color_map1(l(region(T27, T31, T39), T72), T30) :- ','(select10(T31, T30, T38), ','(members20(T39, T38), color_map1(T72, T30))).
color_map1(nil, T137).
color_map1(nil, T139).
Queries:
color_map1(a,g).
(3) PrologToPiTRSProof (SOUND transformation)
We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:color_map1_in: (f,b)
select10_in: (f,b,f)
members20_in: (f,b)
member27_in: (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
color_map1_in_ag(l(region(T27, T31, T32), T33), T30) → U6_ag(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
select10_in_aga(T53, .(T53, T54), T54) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(T64, .(T62, T63), .(T62, X73)) → U1_aga(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
U1_aga(T64, T62, T63, X73, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U6_ag(T27, T31, T32, T33, T30, select10_out_aga(T31, T30, X34)) → color_map1_out_ag(l(region(T27, T31, T32), T33), T30)
color_map1_in_ag(l(region(T27, T31, T39), T40), T30) → U7_ag(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_ag(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_ag(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
members20_in_ag(.(T91, T92), T90) → U2_ag(T91, T92, T90, member27_in_ag(T91, T90))
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
U2_ag(T91, T92, T90, member27_out_ag(T91, T90)) → members20_out_ag(.(T91, T92), T90)
members20_in_ag(.(T91, T97), T90) → U3_ag(T91, T97, T90, member27_in_ag(T91, T90))
U3_ag(T91, T97, T90, member27_out_ag(T91, T90)) → U4_ag(T91, T97, T90, members20_in_ag(T97, T90))
members20_in_ag([], T131) → members20_out_ag([], T131)
U4_ag(T91, T97, T90, members20_out_ag(T97, T90)) → members20_out_ag(.(T91, T97), T90)
U8_ag(T27, T31, T39, T40, T30, members20_out_ag(T39, T38)) → color_map1_out_ag(l(region(T27, T31, T39), T40), T30)
color_map1_in_ag(l(region(T27, T31, T39), T72), T30) → U9_ag(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_ag(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_ag(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_ag(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_ag(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
color_map1_in_ag(nil, T137) → color_map1_out_ag(nil, T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(l(region(T27, T31, T39), T72), T30)
The argument filtering Pi contains the following mapping:
color_map1_in_ag(x1, x2) = color_map1_in_ag(x2)
U6_ag(x1, x2, x3, x4, x5, x6) = U6_ag(x6)
select10_in_aga(x1, x2, x3) = select10_in_aga(x2)
.(x1, x2) = .(x1, x2)
select10_out_aga(x1, x2, x3) = select10_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x2, x5)
color_map1_out_ag(x1, x2) = color_map1_out_ag
region(x1, x2, x3) = region(x2, x3)
U7_ag(x1, x2, x3, x4, x5, x6) = U7_ag(x6)
U8_ag(x1, x2, x3, x4, x5, x6) = U8_ag(x6)
members20_in_ag(x1, x2) = members20_in_ag(x2)
U2_ag(x1, x2, x3, x4) = U2_ag(x4)
member27_in_ag(x1, x2) = member27_in_ag(x2)
member27_out_ag(x1, x2) = member27_out_ag(x1)
U5_ag(x1, x2, x3, x4) = U5_ag(x4)
members20_out_ag(x1, x2) = members20_out_ag
U3_ag(x1, x2, x3, x4) = U3_ag(x3, x4)
U4_ag(x1, x2, x3, x4) = U4_ag(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(x6)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(4) Obligation:
Pi-finite rewrite system:The TRS R consists of the following rules:
color_map1_in_ag(l(region(T27, T31, T32), T33), T30) → U6_ag(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
select10_in_aga(T53, .(T53, T54), T54) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(T64, .(T62, T63), .(T62, X73)) → U1_aga(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
U1_aga(T64, T62, T63, X73, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U6_ag(T27, T31, T32, T33, T30, select10_out_aga(T31, T30, X34)) → color_map1_out_ag(l(region(T27, T31, T32), T33), T30)
color_map1_in_ag(l(region(T27, T31, T39), T40), T30) → U7_ag(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_ag(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_ag(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
members20_in_ag(.(T91, T92), T90) → U2_ag(T91, T92, T90, member27_in_ag(T91, T90))
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
U2_ag(T91, T92, T90, member27_out_ag(T91, T90)) → members20_out_ag(.(T91, T92), T90)
members20_in_ag(.(T91, T97), T90) → U3_ag(T91, T97, T90, member27_in_ag(T91, T90))
U3_ag(T91, T97, T90, member27_out_ag(T91, T90)) → U4_ag(T91, T97, T90, members20_in_ag(T97, T90))
members20_in_ag([], T131) → members20_out_ag([], T131)
U4_ag(T91, T97, T90, members20_out_ag(T97, T90)) → members20_out_ag(.(T91, T97), T90)
U8_ag(T27, T31, T39, T40, T30, members20_out_ag(T39, T38)) → color_map1_out_ag(l(region(T27, T31, T39), T40), T30)
color_map1_in_ag(l(region(T27, T31, T39), T72), T30) → U9_ag(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_ag(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_ag(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_ag(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_ag(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
color_map1_in_ag(nil, T137) → color_map1_out_ag(nil, T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(l(region(T27, T31, T39), T72), T30)
The argument filtering Pi contains the following mapping:
color_map1_in_ag(x1, x2) = color_map1_in_ag(x2)
U6_ag(x1, x2, x3, x4, x5, x6) = U6_ag(x6)
select10_in_aga(x1, x2, x3) = select10_in_aga(x2)
.(x1, x2) = .(x1, x2)
select10_out_aga(x1, x2, x3) = select10_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x2, x5)
color_map1_out_ag(x1, x2) = color_map1_out_ag
region(x1, x2, x3) = region(x2, x3)
U7_ag(x1, x2, x3, x4, x5, x6) = U7_ag(x6)
U8_ag(x1, x2, x3, x4, x5, x6) = U8_ag(x6)
members20_in_ag(x1, x2) = members20_in_ag(x2)
U2_ag(x1, x2, x3, x4) = U2_ag(x4)
member27_in_ag(x1, x2) = member27_in_ag(x2)
member27_out_ag(x1, x2) = member27_out_ag(x1)
U5_ag(x1, x2, x3, x4) = U5_ag(x4)
members20_out_ag(x1, x2) = members20_out_ag
U3_ag(x1, x2, x3, x4) = U3_ag(x3, x4)
U4_ag(x1, x2, x3, x4) = U4_ag(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(x6)
(5) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:Pi DP problem:
The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(l(region(T27, T31, T32), T33), T30) → U6_AG(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
COLOR_MAP1_IN_AG(l(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(l(region(T27, T31, T39), T40), T30) → U7_AG(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_AG(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_AG(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
U7_AG(T27, T31, T39, T40, T30, select10_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, member27_in_ag(T91, T90))
U3_AG(T91, T97, T90, member27_out_ag(T91, T90)) → U4_AG(T91, T97, T90, members20_in_ag(T97, T90))
U3_AG(T91, T97, T90, member27_out_ag(T91, T90)) → MEMBERS20_IN_AG(T97, T90)
COLOR_MAP1_IN_AG(l(region(T27, T31, T39), T72), T30) → U9_AG(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_AG(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_AG(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U9_AG(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → MEMBERS20_IN_AG(T39, T38)
U10_AG(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_AG(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
U10_AG(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → COLOR_MAP1_IN_AG(T72, T30)
The TRS R consists of the following rules:
color_map1_in_ag(l(region(T27, T31, T32), T33), T30) → U6_ag(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
select10_in_aga(T53, .(T53, T54), T54) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(T64, .(T62, T63), .(T62, X73)) → U1_aga(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
U1_aga(T64, T62, T63, X73, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U6_ag(T27, T31, T32, T33, T30, select10_out_aga(T31, T30, X34)) → color_map1_out_ag(l(region(T27, T31, T32), T33), T30)
color_map1_in_ag(l(region(T27, T31, T39), T40), T30) → U7_ag(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_ag(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_ag(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
members20_in_ag(.(T91, T92), T90) → U2_ag(T91, T92, T90, member27_in_ag(T91, T90))
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
U2_ag(T91, T92, T90, member27_out_ag(T91, T90)) → members20_out_ag(.(T91, T92), T90)
members20_in_ag(.(T91, T97), T90) → U3_ag(T91, T97, T90, member27_in_ag(T91, T90))
U3_ag(T91, T97, T90, member27_out_ag(T91, T90)) → U4_ag(T91, T97, T90, members20_in_ag(T97, T90))
members20_in_ag([], T131) → members20_out_ag([], T131)
U4_ag(T91, T97, T90, members20_out_ag(T97, T90)) → members20_out_ag(.(T91, T97), T90)
U8_ag(T27, T31, T39, T40, T30, members20_out_ag(T39, T38)) → color_map1_out_ag(l(region(T27, T31, T39), T40), T30)
color_map1_in_ag(l(region(T27, T31, T39), T72), T30) → U9_ag(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_ag(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_ag(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_ag(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_ag(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
color_map1_in_ag(nil, T137) → color_map1_out_ag(nil, T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(l(region(T27, T31, T39), T72), T30)
The argument filtering Pi contains the following mapping:
color_map1_in_ag(x1, x2) = color_map1_in_ag(x2)
U6_ag(x1, x2, x3, x4, x5, x6) = U6_ag(x6)
select10_in_aga(x1, x2, x3) = select10_in_aga(x2)
.(x1, x2) = .(x1, x2)
select10_out_aga(x1, x2, x3) = select10_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x2, x5)
color_map1_out_ag(x1, x2) = color_map1_out_ag
region(x1, x2, x3) = region(x2, x3)
U7_ag(x1, x2, x3, x4, x5, x6) = U7_ag(x6)
U8_ag(x1, x2, x3, x4, x5, x6) = U8_ag(x6)
members20_in_ag(x1, x2) = members20_in_ag(x2)
U2_ag(x1, x2, x3, x4) = U2_ag(x4)
member27_in_ag(x1, x2) = member27_in_ag(x2)
member27_out_ag(x1, x2) = member27_out_ag(x1)
U5_ag(x1, x2, x3, x4) = U5_ag(x4)
members20_out_ag(x1, x2) = members20_out_ag
U3_ag(x1, x2, x3, x4) = U3_ag(x3, x4)
U4_ag(x1, x2, x3, x4) = U4_ag(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(x6)
COLOR_MAP1_IN_AG(x1, x2) = COLOR_MAP1_IN_AG(x2)
U6_AG(x1, x2, x3, x4, x5, x6) = U6_AG(x6)
SELECT10_IN_AGA(x1, x2, x3) = SELECT10_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5) = U1_AGA(x2, x5)
U7_AG(x1, x2, x3, x4, x5, x6) = U7_AG(x6)
U8_AG(x1, x2, x3, x4, x5, x6) = U8_AG(x6)
MEMBERS20_IN_AG(x1, x2) = MEMBERS20_IN_AG(x2)
U2_AG(x1, x2, x3, x4) = U2_AG(x4)
MEMBER27_IN_AG(x1, x2) = MEMBER27_IN_AG(x2)
U5_AG(x1, x2, x3, x4) = U5_AG(x4)
U3_AG(x1, x2, x3, x4) = U3_AG(x3, x4)
U4_AG(x1, x2, x3, x4) = U4_AG(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(x6)
We have to consider all (P,R,Pi)-chains
(6) Obligation:
Pi DP problem:The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(l(region(T27, T31, T32), T33), T30) → U6_AG(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
COLOR_MAP1_IN_AG(l(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(l(region(T27, T31, T39), T40), T30) → U7_AG(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_AG(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_AG(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
U7_AG(T27, T31, T39, T40, T30, select10_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, member27_in_ag(T91, T90))
U3_AG(T91, T97, T90, member27_out_ag(T91, T90)) → U4_AG(T91, T97, T90, members20_in_ag(T97, T90))
U3_AG(T91, T97, T90, member27_out_ag(T91, T90)) → MEMBERS20_IN_AG(T97, T90)
COLOR_MAP1_IN_AG(l(region(T27, T31, T39), T72), T30) → U9_AG(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_AG(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_AG(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U9_AG(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → MEMBERS20_IN_AG(T39, T38)
U10_AG(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_AG(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
U10_AG(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → COLOR_MAP1_IN_AG(T72, T30)
The TRS R consists of the following rules:
color_map1_in_ag(l(region(T27, T31, T32), T33), T30) → U6_ag(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
select10_in_aga(T53, .(T53, T54), T54) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(T64, .(T62, T63), .(T62, X73)) → U1_aga(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
U1_aga(T64, T62, T63, X73, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U6_ag(T27, T31, T32, T33, T30, select10_out_aga(T31, T30, X34)) → color_map1_out_ag(l(region(T27, T31, T32), T33), T30)
color_map1_in_ag(l(region(T27, T31, T39), T40), T30) → U7_ag(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_ag(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_ag(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
members20_in_ag(.(T91, T92), T90) → U2_ag(T91, T92, T90, member27_in_ag(T91, T90))
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
U2_ag(T91, T92, T90, member27_out_ag(T91, T90)) → members20_out_ag(.(T91, T92), T90)
members20_in_ag(.(T91, T97), T90) → U3_ag(T91, T97, T90, member27_in_ag(T91, T90))
U3_ag(T91, T97, T90, member27_out_ag(T91, T90)) → U4_ag(T91, T97, T90, members20_in_ag(T97, T90))
members20_in_ag([], T131) → members20_out_ag([], T131)
U4_ag(T91, T97, T90, members20_out_ag(T97, T90)) → members20_out_ag(.(T91, T97), T90)
U8_ag(T27, T31, T39, T40, T30, members20_out_ag(T39, T38)) → color_map1_out_ag(l(region(T27, T31, T39), T40), T30)
color_map1_in_ag(l(region(T27, T31, T39), T72), T30) → U9_ag(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_ag(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_ag(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_ag(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_ag(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
color_map1_in_ag(nil, T137) → color_map1_out_ag(nil, T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(l(region(T27, T31, T39), T72), T30)
The argument filtering Pi contains the following mapping:
color_map1_in_ag(x1, x2) = color_map1_in_ag(x2)
U6_ag(x1, x2, x3, x4, x5, x6) = U6_ag(x6)
select10_in_aga(x1, x2, x3) = select10_in_aga(x2)
.(x1, x2) = .(x1, x2)
select10_out_aga(x1, x2, x3) = select10_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x2, x5)
color_map1_out_ag(x1, x2) = color_map1_out_ag
region(x1, x2, x3) = region(x2, x3)
U7_ag(x1, x2, x3, x4, x5, x6) = U7_ag(x6)
U8_ag(x1, x2, x3, x4, x5, x6) = U8_ag(x6)
members20_in_ag(x1, x2) = members20_in_ag(x2)
U2_ag(x1, x2, x3, x4) = U2_ag(x4)
member27_in_ag(x1, x2) = member27_in_ag(x2)
member27_out_ag(x1, x2) = member27_out_ag(x1)
U5_ag(x1, x2, x3, x4) = U5_ag(x4)
members20_out_ag(x1, x2) = members20_out_ag
U3_ag(x1, x2, x3, x4) = U3_ag(x3, x4)
U4_ag(x1, x2, x3, x4) = U4_ag(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(x6)
COLOR_MAP1_IN_AG(x1, x2) = COLOR_MAP1_IN_AG(x2)
U6_AG(x1, x2, x3, x4, x5, x6) = U6_AG(x6)
SELECT10_IN_AGA(x1, x2, x3) = SELECT10_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5) = U1_AGA(x2, x5)
U7_AG(x1, x2, x3, x4, x5, x6) = U7_AG(x6)
U8_AG(x1, x2, x3, x4, x5, x6) = U8_AG(x6)
MEMBERS20_IN_AG(x1, x2) = MEMBERS20_IN_AG(x2)
U2_AG(x1, x2, x3, x4) = U2_AG(x4)
MEMBER27_IN_AG(x1, x2) = MEMBER27_IN_AG(x2)
U5_AG(x1, x2, x3, x4) = U5_AG(x4)
U3_AG(x1, x2, x3, x4) = U3_AG(x3, x4)
U4_AG(x1, x2, x3, x4) = U4_AG(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(x6)
We have to consider all (P,R,Pi)-chains
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 12 less nodes.(8) Complex Obligation (AND)
(9) 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:
color_map1_in_ag(l(region(T27, T31, T32), T33), T30) → U6_ag(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
select10_in_aga(T53, .(T53, T54), T54) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(T64, .(T62, T63), .(T62, X73)) → U1_aga(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
U1_aga(T64, T62, T63, X73, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U6_ag(T27, T31, T32, T33, T30, select10_out_aga(T31, T30, X34)) → color_map1_out_ag(l(region(T27, T31, T32), T33), T30)
color_map1_in_ag(l(region(T27, T31, T39), T40), T30) → U7_ag(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_ag(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_ag(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
members20_in_ag(.(T91, T92), T90) → U2_ag(T91, T92, T90, member27_in_ag(T91, T90))
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
U2_ag(T91, T92, T90, member27_out_ag(T91, T90)) → members20_out_ag(.(T91, T92), T90)
members20_in_ag(.(T91, T97), T90) → U3_ag(T91, T97, T90, member27_in_ag(T91, T90))
U3_ag(T91, T97, T90, member27_out_ag(T91, T90)) → U4_ag(T91, T97, T90, members20_in_ag(T97, T90))
members20_in_ag([], T131) → members20_out_ag([], T131)
U4_ag(T91, T97, T90, members20_out_ag(T97, T90)) → members20_out_ag(.(T91, T97), T90)
U8_ag(T27, T31, T39, T40, T30, members20_out_ag(T39, T38)) → color_map1_out_ag(l(region(T27, T31, T39), T40), T30)
color_map1_in_ag(l(region(T27, T31, T39), T72), T30) → U9_ag(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_ag(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_ag(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_ag(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_ag(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
color_map1_in_ag(nil, T137) → color_map1_out_ag(nil, T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(l(region(T27, T31, T39), T72), T30)
The argument filtering Pi contains the following mapping:
color_map1_in_ag(x1, x2) = color_map1_in_ag(x2)
U6_ag(x1, x2, x3, x4, x5, x6) = U6_ag(x6)
select10_in_aga(x1, x2, x3) = select10_in_aga(x2)
.(x1, x2) = .(x1, x2)
select10_out_aga(x1, x2, x3) = select10_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x2, x5)
color_map1_out_ag(x1, x2) = color_map1_out_ag
region(x1, x2, x3) = region(x2, x3)
U7_ag(x1, x2, x3, x4, x5, x6) = U7_ag(x6)
U8_ag(x1, x2, x3, x4, x5, x6) = U8_ag(x6)
members20_in_ag(x1, x2) = members20_in_ag(x2)
U2_ag(x1, x2, x3, x4) = U2_ag(x4)
member27_in_ag(x1, x2) = member27_in_ag(x2)
member27_out_ag(x1, x2) = member27_out_ag(x1)
U5_ag(x1, x2, x3, x4) = U5_ag(x4)
members20_out_ag(x1, x2) = members20_out_ag
U3_ag(x1, x2, x3, x4) = U3_ag(x3, x4)
U4_ag(x1, x2, x3, x4) = U4_ag(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(x6)
MEMBER27_IN_AG(x1, x2) = MEMBER27_IN_AG(x2)
We have to consider all (P,R,Pi)-chains
(10) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(11) Obligation:
Pi DP problem:The TRS P consists of the following rules:
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
(12) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(13) Obligation:
Q DP problem:The TRS P consists of the following rules:
MEMBER27_IN_AG(.(T119, T120)) → MEMBER27_IN_AG(T120)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(14) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.From the DPs we obtained the following set of size-change graphs:
- MEMBER27_IN_AG(.(T119, T120)) → MEMBER27_IN_AG(T120)
The graph contains the following edges 1 > 1
(15) YES
(16) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MEMBERS20_IN_AG(.(T91, T97), T90) → U3_AG(T91, T97, T90, member27_in_ag(T91, T90))
U3_AG(T91, T97, T90, member27_out_ag(T91, T90)) → MEMBERS20_IN_AG(T97, T90)
The TRS R consists of the following rules:
color_map1_in_ag(l(region(T27, T31, T32), T33), T30) → U6_ag(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
select10_in_aga(T53, .(T53, T54), T54) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(T64, .(T62, T63), .(T62, X73)) → U1_aga(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
U1_aga(T64, T62, T63, X73, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U6_ag(T27, T31, T32, T33, T30, select10_out_aga(T31, T30, X34)) → color_map1_out_ag(l(region(T27, T31, T32), T33), T30)
color_map1_in_ag(l(region(T27, T31, T39), T40), T30) → U7_ag(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_ag(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_ag(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
members20_in_ag(.(T91, T92), T90) → U2_ag(T91, T92, T90, member27_in_ag(T91, T90))
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
U2_ag(T91, T92, T90, member27_out_ag(T91, T90)) → members20_out_ag(.(T91, T92), T90)
members20_in_ag(.(T91, T97), T90) → U3_ag(T91, T97, T90, member27_in_ag(T91, T90))
U3_ag(T91, T97, T90, member27_out_ag(T91, T90)) → U4_ag(T91, T97, T90, members20_in_ag(T97, T90))
members20_in_ag([], T131) → members20_out_ag([], T131)
U4_ag(T91, T97, T90, members20_out_ag(T97, T90)) → members20_out_ag(.(T91, T97), T90)
U8_ag(T27, T31, T39, T40, T30, members20_out_ag(T39, T38)) → color_map1_out_ag(l(region(T27, T31, T39), T40), T30)
color_map1_in_ag(l(region(T27, T31, T39), T72), T30) → U9_ag(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_ag(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_ag(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_ag(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_ag(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
color_map1_in_ag(nil, T137) → color_map1_out_ag(nil, T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(l(region(T27, T31, T39), T72), T30)
The argument filtering Pi contains the following mapping:
color_map1_in_ag(x1, x2) = color_map1_in_ag(x2)
U6_ag(x1, x2, x3, x4, x5, x6) = U6_ag(x6)
select10_in_aga(x1, x2, x3) = select10_in_aga(x2)
.(x1, x2) = .(x1, x2)
select10_out_aga(x1, x2, x3) = select10_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x2, x5)
color_map1_out_ag(x1, x2) = color_map1_out_ag
region(x1, x2, x3) = region(x2, x3)
U7_ag(x1, x2, x3, x4, x5, x6) = U7_ag(x6)
U8_ag(x1, x2, x3, x4, x5, x6) = U8_ag(x6)
members20_in_ag(x1, x2) = members20_in_ag(x2)
U2_ag(x1, x2, x3, x4) = U2_ag(x4)
member27_in_ag(x1, x2) = member27_in_ag(x2)
member27_out_ag(x1, x2) = member27_out_ag(x1)
U5_ag(x1, x2, x3, x4) = U5_ag(x4)
members20_out_ag(x1, x2) = members20_out_ag
U3_ag(x1, x2, x3, x4) = U3_ag(x3, x4)
U4_ag(x1, x2, x3, x4) = U4_ag(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(x6)
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) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(18) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MEMBERS20_IN_AG(.(T91, T97), T90) → U3_AG(T91, T97, T90, member27_in_ag(T91, T90))
U3_AG(T91, T97, T90, member27_out_ag(T91, T90)) → MEMBERS20_IN_AG(T97, T90)
The TRS R consists of the following rules:
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
member27_in_ag(x1, x2) = member27_in_ag(x2)
member27_out_ag(x1, x2) = member27_out_ag(x1)
U5_ag(x1, x2, x3, x4) = U5_ag(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
(19) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(20) Obligation:
Q DP problem:The TRS P consists of the following rules:
MEMBERS20_IN_AG(T90) → U3_AG(T90, member27_in_ag(T90))
U3_AG(T90, member27_out_ag(T91)) → MEMBERS20_IN_AG(T90)
The TRS R consists of the following rules:
member27_in_ag(.(T110, T111)) → member27_out_ag(T110)
member27_in_ag(.(T119, T120)) → U5_ag(member27_in_ag(T120))
U5_ag(member27_out_ag(T121)) → member27_out_ag(T121)
The set Q consists of the following terms:
member27_in_ag(x0)
U5_ag(x0)
We have to consider all (P,Q,R)-chains.
(21) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule MEMBERS20_IN_AG(T90) → U3_AG(T90, member27_in_ag(T90)) at position [1] we obtained the following new rules [LPAR04]:
MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), member27_out_ag(x0))
MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), U5_ag(member27_in_ag(x1)))
(22) Obligation:
Q DP problem:The TRS P consists of the following rules:
U3_AG(T90, member27_out_ag(T91)) → MEMBERS20_IN_AG(T90)
MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), member27_out_ag(x0))
MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), U5_ag(member27_in_ag(x1)))
The TRS R consists of the following rules:
member27_in_ag(.(T110, T111)) → member27_out_ag(T110)
member27_in_ag(.(T119, T120)) → U5_ag(member27_in_ag(T120))
U5_ag(member27_out_ag(T121)) → member27_out_ag(T121)
The set Q consists of the following terms:
member27_in_ag(x0)
U5_ag(x0)
We have to consider all (P,Q,R)-chains.
(23) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U3_AG(T90, member27_out_ag(T91)) → MEMBERS20_IN_AG(T90) we obtained the following new rules [LPAR04]:
U3_AG(.(z0, z1), member27_out_ag(z0)) → MEMBERS20_IN_AG(.(z0, z1))
U3_AG(.(z0, z1), member27_out_ag(x1)) → MEMBERS20_IN_AG(.(z0, z1))
(24) Obligation:
Q DP problem:The TRS P consists of the following rules:
MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), member27_out_ag(x0))
MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), U5_ag(member27_in_ag(x1)))
U3_AG(.(z0, z1), member27_out_ag(z0)) → MEMBERS20_IN_AG(.(z0, z1))
U3_AG(.(z0, z1), member27_out_ag(x1)) → MEMBERS20_IN_AG(.(z0, z1))
The TRS R consists of the following rules:
member27_in_ag(.(T110, T111)) → member27_out_ag(T110)
member27_in_ag(.(T119, T120)) → U5_ag(member27_in_ag(T120))
U5_ag(member27_out_ag(T121)) → member27_out_ag(T121)
The set Q consists of the following terms:
member27_in_ag(x0)
U5_ag(x0)
We have to consider all (P,Q,R)-chains.
(25) 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), member27_out_ag(z0)) evaluates to t =U3_AG(.(z0, z1), member27_out_ag(z0))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequence
U3_AG(.(z0, z1), member27_out_ag(z0)) → MEMBERS20_IN_AG(.(z0, z1))
with rule U3_AG(.(z0', z1'), member27_out_ag(z0')) → MEMBERS20_IN_AG(.(z0', z1')) at position [] and matcher [z0' / z0, z1' / z1]
MEMBERS20_IN_AG(.(z0, z1)) → U3_AG(.(z0, z1), member27_out_ag(z0))
with rule MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), member27_out_ag(x0))
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.
(26) NO
(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)
The TRS R consists of the following rules:
color_map1_in_ag(l(region(T27, T31, T32), T33), T30) → U6_ag(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
select10_in_aga(T53, .(T53, T54), T54) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(T64, .(T62, T63), .(T62, X73)) → U1_aga(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
U1_aga(T64, T62, T63, X73, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U6_ag(T27, T31, T32, T33, T30, select10_out_aga(T31, T30, X34)) → color_map1_out_ag(l(region(T27, T31, T32), T33), T30)
color_map1_in_ag(l(region(T27, T31, T39), T40), T30) → U7_ag(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_ag(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_ag(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
members20_in_ag(.(T91, T92), T90) → U2_ag(T91, T92, T90, member27_in_ag(T91, T90))
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
U2_ag(T91, T92, T90, member27_out_ag(T91, T90)) → members20_out_ag(.(T91, T92), T90)
members20_in_ag(.(T91, T97), T90) → U3_ag(T91, T97, T90, member27_in_ag(T91, T90))
U3_ag(T91, T97, T90, member27_out_ag(T91, T90)) → U4_ag(T91, T97, T90, members20_in_ag(T97, T90))
members20_in_ag([], T131) → members20_out_ag([], T131)
U4_ag(T91, T97, T90, members20_out_ag(T97, T90)) → members20_out_ag(.(T91, T97), T90)
U8_ag(T27, T31, T39, T40, T30, members20_out_ag(T39, T38)) → color_map1_out_ag(l(region(T27, T31, T39), T40), T30)
color_map1_in_ag(l(region(T27, T31, T39), T72), T30) → U9_ag(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_ag(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_ag(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_ag(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_ag(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
color_map1_in_ag(nil, T137) → color_map1_out_ag(nil, T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(l(region(T27, T31, T39), T72), T30)
The argument filtering Pi contains the following mapping:
color_map1_in_ag(x1, x2) = color_map1_in_ag(x2)
U6_ag(x1, x2, x3, x4, x5, x6) = U6_ag(x6)
select10_in_aga(x1, x2, x3) = select10_in_aga(x2)
.(x1, x2) = .(x1, x2)
select10_out_aga(x1, x2, x3) = select10_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x2, x5)
color_map1_out_ag(x1, x2) = color_map1_out_ag
region(x1, x2, x3) = region(x2, x3)
U7_ag(x1, x2, x3, x4, x5, x6) = U7_ag(x6)
U8_ag(x1, x2, x3, x4, x5, x6) = U8_ag(x6)
members20_in_ag(x1, x2) = members20_in_ag(x2)
U2_ag(x1, x2, x3, x4) = U2_ag(x4)
member27_in_ag(x1, x2) = member27_in_ag(x2)
member27_out_ag(x1, x2) = member27_out_ag(x1)
U5_ag(x1, x2, x3, x4) = U5_ag(x4)
members20_out_ag(x1, x2) = members20_out_ag
U3_ag(x1, x2, x3, x4) = U3_ag(x3, x4)
U4_ag(x1, x2, x3, x4) = U4_ag(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(x6)
SELECT10_IN_AGA(x1, x2, x3) = SELECT10_IN_AGA(x2)
We have to consider all (P,R,Pi)-chains
(28) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(29) 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
(30) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(31) 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.
(32) 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
(33) YES
(34) Obligation:
Pi DP problem:The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(l(region(T27, T31, T39), T72), T30) → U9_AG(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_AG(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_AG(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_AG(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → COLOR_MAP1_IN_AG(T72, T30)
The TRS R consists of the following rules:
color_map1_in_ag(l(region(T27, T31, T32), T33), T30) → U6_ag(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
select10_in_aga(T53, .(T53, T54), T54) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(T64, .(T62, T63), .(T62, X73)) → U1_aga(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
U1_aga(T64, T62, T63, X73, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U6_ag(T27, T31, T32, T33, T30, select10_out_aga(T31, T30, X34)) → color_map1_out_ag(l(region(T27, T31, T32), T33), T30)
color_map1_in_ag(l(region(T27, T31, T39), T40), T30) → U7_ag(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_ag(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_ag(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
members20_in_ag(.(T91, T92), T90) → U2_ag(T91, T92, T90, member27_in_ag(T91, T90))
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
U2_ag(T91, T92, T90, member27_out_ag(T91, T90)) → members20_out_ag(.(T91, T92), T90)
members20_in_ag(.(T91, T97), T90) → U3_ag(T91, T97, T90, member27_in_ag(T91, T90))
U3_ag(T91, T97, T90, member27_out_ag(T91, T90)) → U4_ag(T91, T97, T90, members20_in_ag(T97, T90))
members20_in_ag([], T131) → members20_out_ag([], T131)
U4_ag(T91, T97, T90, members20_out_ag(T97, T90)) → members20_out_ag(.(T91, T97), T90)
U8_ag(T27, T31, T39, T40, T30, members20_out_ag(T39, T38)) → color_map1_out_ag(l(region(T27, T31, T39), T40), T30)
color_map1_in_ag(l(region(T27, T31, T39), T72), T30) → U9_ag(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_ag(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_ag(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_ag(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_ag(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
color_map1_in_ag(nil, T137) → color_map1_out_ag(nil, T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(l(region(T27, T31, T39), T72), T30)
The argument filtering Pi contains the following mapping:
color_map1_in_ag(x1, x2) = color_map1_in_ag(x2)
U6_ag(x1, x2, x3, x4, x5, x6) = U6_ag(x6)
select10_in_aga(x1, x2, x3) = select10_in_aga(x2)
.(x1, x2) = .(x1, x2)
select10_out_aga(x1, x2, x3) = select10_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x2, x5)
color_map1_out_ag(x1, x2) = color_map1_out_ag
region(x1, x2, x3) = region(x2, x3)
U7_ag(x1, x2, x3, x4, x5, x6) = U7_ag(x6)
U8_ag(x1, x2, x3, x4, x5, x6) = U8_ag(x6)
members20_in_ag(x1, x2) = members20_in_ag(x2)
U2_ag(x1, x2, x3, x4) = U2_ag(x4)
member27_in_ag(x1, x2) = member27_in_ag(x2)
member27_out_ag(x1, x2) = member27_out_ag(x1)
U5_ag(x1, x2, x3, x4) = U5_ag(x4)
members20_out_ag(x1, x2) = members20_out_ag
U3_ag(x1, x2, x3, x4) = U3_ag(x3, x4)
U4_ag(x1, x2, x3, x4) = U4_ag(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(x6)
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
(35) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(36) Obligation:
Pi DP problem:The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(l(region(T27, T31, T39), T72), T30) → U9_AG(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_AG(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_AG(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_AG(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → COLOR_MAP1_IN_AG(T72, T30)
The TRS R consists of the following rules:
select10_in_aga(T53, .(T53, T54), T54) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(T64, .(T62, T63), .(T62, X73)) → U1_aga(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
members20_in_ag(.(T91, T92), T90) → U2_ag(T91, T92, T90, member27_in_ag(T91, T90))
members20_in_ag(.(T91, T97), T90) → U3_ag(T91, T97, T90, member27_in_ag(T91, T90))
members20_in_ag([], T131) → members20_out_ag([], T131)
U1_aga(T64, T62, T63, X73, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U2_ag(T91, T92, T90, member27_out_ag(T91, T90)) → members20_out_ag(.(T91, T92), T90)
U3_ag(T91, T97, T90, member27_out_ag(T91, T90)) → U4_ag(T91, T97, T90, members20_in_ag(T97, T90))
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U4_ag(T91, T97, T90, members20_out_ag(T97, T90)) → members20_out_ag(.(T91, T97), T90)
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
The argument filtering Pi contains the following mapping:
select10_in_aga(x1, x2, x3) = select10_in_aga(x2)
.(x1, x2) = .(x1, x2)
select10_out_aga(x1, x2, x3) = select10_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x2, x5)
region(x1, x2, x3) = region(x2, x3)
members20_in_ag(x1, x2) = members20_in_ag(x2)
U2_ag(x1, x2, x3, x4) = U2_ag(x4)
member27_in_ag(x1, x2) = member27_in_ag(x2)
member27_out_ag(x1, x2) = member27_out_ag(x1)
U5_ag(x1, x2, x3, x4) = U5_ag(x4)
members20_out_ag(x1, x2) = members20_out_ag
U3_ag(x1, x2, x3, x4) = U3_ag(x3, x4)
U4_ag(x1, x2, x3, x4) = U4_ag(x4)
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
(37) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(38) Obligation:
Q DP problem:The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(T30) → U9_AG(T30, select10_in_aga(T30))
U9_AG(T30, select10_out_aga(T31, T38)) → U10_AG(T30, members20_in_ag(T38))
U10_AG(T30, members20_out_ag) → COLOR_MAP1_IN_AG(T30)
The TRS R consists of the following rules:
select10_in_aga(.(T53, T54)) → select10_out_aga(T53, T54)
select10_in_aga(.(T62, T63)) → U1_aga(T62, select10_in_aga(T63))
members20_in_ag(T90) → U2_ag(member27_in_ag(T90))
members20_in_ag(T90) → U3_ag(T90, member27_in_ag(T90))
members20_in_ag(T131) → members20_out_ag
U1_aga(T62, select10_out_aga(T64, X73)) → select10_out_aga(T64, .(T62, X73))
U2_ag(member27_out_ag(T91)) → members20_out_ag
U3_ag(T90, member27_out_ag(T91)) → U4_ag(members20_in_ag(T90))
member27_in_ag(.(T110, T111)) → member27_out_ag(T110)
member27_in_ag(.(T119, T120)) → U5_ag(member27_in_ag(T120))
U4_ag(members20_out_ag) → members20_out_ag
U5_ag(member27_out_ag(T121)) → member27_out_ag(T121)
The set Q consists of the following terms:
select10_in_aga(x0)
members20_in_ag(x0)
U1_aga(x0, x1)
U2_ag(x0)
U3_ag(x0, x1)
member27_in_ag(x0)
U4_ag(x0)
U5_ag(x0)
We have to consider all (P,Q,R)-chains.
(39) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule COLOR_MAP1_IN_AG(T30) → U9_AG(T30, select10_in_aga(T30)) at position [1] we obtained the following new rules [LPAR04]:
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), select10_out_aga(x0, x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U1_aga(x0, select10_in_aga(x1)))
(40) Obligation:
Q DP problem:The TRS P consists of the following rules:
U9_AG(T30, select10_out_aga(T31, T38)) → U10_AG(T30, members20_in_ag(T38))
U10_AG(T30, members20_out_ag) → COLOR_MAP1_IN_AG(T30)
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), select10_out_aga(x0, x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U1_aga(x0, select10_in_aga(x1)))
The TRS R consists of the following rules:
select10_in_aga(.(T53, T54)) → select10_out_aga(T53, T54)
select10_in_aga(.(T62, T63)) → U1_aga(T62, select10_in_aga(T63))
members20_in_ag(T90) → U2_ag(member27_in_ag(T90))
members20_in_ag(T90) → U3_ag(T90, member27_in_ag(T90))
members20_in_ag(T131) → members20_out_ag
U1_aga(T62, select10_out_aga(T64, X73)) → select10_out_aga(T64, .(T62, X73))
U2_ag(member27_out_ag(T91)) → members20_out_ag
U3_ag(T90, member27_out_ag(T91)) → U4_ag(members20_in_ag(T90))
member27_in_ag(.(T110, T111)) → member27_out_ag(T110)
member27_in_ag(.(T119, T120)) → U5_ag(member27_in_ag(T120))
U4_ag(members20_out_ag) → members20_out_ag
U5_ag(member27_out_ag(T121)) → member27_out_ag(T121)
The set Q consists of the following terms:
select10_in_aga(x0)
members20_in_ag(x0)
U1_aga(x0, x1)
U2_ag(x0)
U3_ag(x0, x1)
member27_in_ag(x0)
U4_ag(x0)
U5_ag(x0)
We have to consider all (P,Q,R)-chains.
(41) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule U9_AG(T30, select10_out_aga(T31, T38)) → U10_AG(T30, members20_in_ag(T38)) at position [1] we obtained the following new rules [LPAR04]:
U9_AG(y0, select10_out_aga(y1, x0)) → U10_AG(y0, U2_ag(member27_in_ag(x0)))
U9_AG(y0, select10_out_aga(y1, x0)) → U10_AG(y0, U3_ag(x0, member27_in_ag(x0)))
U9_AG(y0, select10_out_aga(y1, x0)) → U10_AG(y0, members20_out_ag)
(42) Obligation:
Q DP problem:The TRS P consists of the following rules:
U10_AG(T30, members20_out_ag) → COLOR_MAP1_IN_AG(T30)
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), select10_out_aga(x0, x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U1_aga(x0, select10_in_aga(x1)))
U9_AG(y0, select10_out_aga(y1, x0)) → U10_AG(y0, U2_ag(member27_in_ag(x0)))
U9_AG(y0, select10_out_aga(y1, x0)) → U10_AG(y0, U3_ag(x0, member27_in_ag(x0)))
U9_AG(y0, select10_out_aga(y1, x0)) → U10_AG(y0, members20_out_ag)
The TRS R consists of the following rules:
select10_in_aga(.(T53, T54)) → select10_out_aga(T53, T54)
select10_in_aga(.(T62, T63)) → U1_aga(T62, select10_in_aga(T63))
members20_in_ag(T90) → U2_ag(member27_in_ag(T90))
members20_in_ag(T90) → U3_ag(T90, member27_in_ag(T90))
members20_in_ag(T131) → members20_out_ag
U1_aga(T62, select10_out_aga(T64, X73)) → select10_out_aga(T64, .(T62, X73))
U2_ag(member27_out_ag(T91)) → members20_out_ag
U3_ag(T90, member27_out_ag(T91)) → U4_ag(members20_in_ag(T90))
member27_in_ag(.(T110, T111)) → member27_out_ag(T110)
member27_in_ag(.(T119, T120)) → U5_ag(member27_in_ag(T120))
U4_ag(members20_out_ag) → members20_out_ag
U5_ag(member27_out_ag(T121)) → member27_out_ag(T121)
The set Q consists of the following terms:
select10_in_aga(x0)
members20_in_ag(x0)
U1_aga(x0, x1)
U2_ag(x0)
U3_ag(x0, x1)
member27_in_ag(x0)
U4_ag(x0)
U5_ag(x0)
We have to consider all (P,Q,R)-chains.
(43) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U9_AG(y0, select10_out_aga(y1, x0)) → U10_AG(y0, U2_ag(member27_in_ag(x0))) we obtained the following new rules [LPAR04]:
U9_AG(.(z0, z1), select10_out_aga(z0, z1)) → U10_AG(.(z0, z1), U2_ag(member27_in_ag(z1)))
U9_AG(.(z0, z1), select10_out_aga(x1, x2)) → U10_AG(.(z0, z1), U2_ag(member27_in_ag(x2)))
(44) Obligation:
Q DP problem:The TRS P consists of the following rules:
U10_AG(T30, members20_out_ag) → COLOR_MAP1_IN_AG(T30)
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), select10_out_aga(x0, x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U1_aga(x0, select10_in_aga(x1)))
U9_AG(y0, select10_out_aga(y1, x0)) → U10_AG(y0, U3_ag(x0, member27_in_ag(x0)))
U9_AG(y0, select10_out_aga(y1, x0)) → U10_AG(y0, members20_out_ag)
U9_AG(.(z0, z1), select10_out_aga(z0, z1)) → U10_AG(.(z0, z1), U2_ag(member27_in_ag(z1)))
U9_AG(.(z0, z1), select10_out_aga(x1, x2)) → U10_AG(.(z0, z1), U2_ag(member27_in_ag(x2)))
The TRS R consists of the following rules:
select10_in_aga(.(T53, T54)) → select10_out_aga(T53, T54)
select10_in_aga(.(T62, T63)) → U1_aga(T62, select10_in_aga(T63))
members20_in_ag(T90) → U2_ag(member27_in_ag(T90))
members20_in_ag(T90) → U3_ag(T90, member27_in_ag(T90))
members20_in_ag(T131) → members20_out_ag
U1_aga(T62, select10_out_aga(T64, X73)) → select10_out_aga(T64, .(T62, X73))
U2_ag(member27_out_ag(T91)) → members20_out_ag
U3_ag(T90, member27_out_ag(T91)) → U4_ag(members20_in_ag(T90))
member27_in_ag(.(T110, T111)) → member27_out_ag(T110)
member27_in_ag(.(T119, T120)) → U5_ag(member27_in_ag(T120))
U4_ag(members20_out_ag) → members20_out_ag
U5_ag(member27_out_ag(T121)) → member27_out_ag(T121)
The set Q consists of the following terms:
select10_in_aga(x0)
members20_in_ag(x0)
U1_aga(x0, x1)
U2_ag(x0)
U3_ag(x0, x1)
member27_in_ag(x0)
U4_ag(x0)
U5_ag(x0)
We have to consider all (P,Q,R)-chains.
(45) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U9_AG(y0, select10_out_aga(y1, x0)) → U10_AG(y0, U3_ag(x0, member27_in_ag(x0))) we obtained the following new rules [LPAR04]:
U9_AG(.(z0, z1), select10_out_aga(z0, z1)) → U10_AG(.(z0, z1), U3_ag(z1, member27_in_ag(z1)))
U9_AG(.(z0, z1), select10_out_aga(x1, x2)) → U10_AG(.(z0, z1), U3_ag(x2, member27_in_ag(x2)))
(46) Obligation:
Q DP problem:The TRS P consists of the following rules:
U10_AG(T30, members20_out_ag) → COLOR_MAP1_IN_AG(T30)
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), select10_out_aga(x0, x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U1_aga(x0, select10_in_aga(x1)))
U9_AG(y0, select10_out_aga(y1, x0)) → U10_AG(y0, members20_out_ag)
U9_AG(.(z0, z1), select10_out_aga(z0, z1)) → U10_AG(.(z0, z1), U2_ag(member27_in_ag(z1)))
U9_AG(.(z0, z1), select10_out_aga(x1, x2)) → U10_AG(.(z0, z1), U2_ag(member27_in_ag(x2)))
U9_AG(.(z0, z1), select10_out_aga(z0, z1)) → U10_AG(.(z0, z1), U3_ag(z1, member27_in_ag(z1)))
U9_AG(.(z0, z1), select10_out_aga(x1, x2)) → U10_AG(.(z0, z1), U3_ag(x2, member27_in_ag(x2)))
The TRS R consists of the following rules:
select10_in_aga(.(T53, T54)) → select10_out_aga(T53, T54)
select10_in_aga(.(T62, T63)) → U1_aga(T62, select10_in_aga(T63))
members20_in_ag(T90) → U2_ag(member27_in_ag(T90))
members20_in_ag(T90) → U3_ag(T90, member27_in_ag(T90))
members20_in_ag(T131) → members20_out_ag
U1_aga(T62, select10_out_aga(T64, X73)) → select10_out_aga(T64, .(T62, X73))
U2_ag(member27_out_ag(T91)) → members20_out_ag
U3_ag(T90, member27_out_ag(T91)) → U4_ag(members20_in_ag(T90))
member27_in_ag(.(T110, T111)) → member27_out_ag(T110)
member27_in_ag(.(T119, T120)) → U5_ag(member27_in_ag(T120))
U4_ag(members20_out_ag) → members20_out_ag
U5_ag(member27_out_ag(T121)) → member27_out_ag(T121)
The set Q consists of the following terms:
select10_in_aga(x0)
members20_in_ag(x0)
U1_aga(x0, x1)
U2_ag(x0)
U3_ag(x0, x1)
member27_in_ag(x0)
U4_ag(x0)
U5_ag(x0)
We have to consider all (P,Q,R)-chains.
(47) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U9_AG(y0, select10_out_aga(y1, x0)) → U10_AG(y0, members20_out_ag) we obtained the following new rules [LPAR04]:
U9_AG(.(z0, z1), select10_out_aga(z0, z1)) → U10_AG(.(z0, z1), members20_out_ag)
U9_AG(.(z0, z1), select10_out_aga(x1, x2)) → U10_AG(.(z0, z1), members20_out_ag)
(48) Obligation:
Q DP problem:The TRS P consists of the following rules:
U10_AG(T30, members20_out_ag) → COLOR_MAP1_IN_AG(T30)
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), select10_out_aga(x0, x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U1_aga(x0, select10_in_aga(x1)))
U9_AG(.(z0, z1), select10_out_aga(z0, z1)) → U10_AG(.(z0, z1), U2_ag(member27_in_ag(z1)))
U9_AG(.(z0, z1), select10_out_aga(x1, x2)) → U10_AG(.(z0, z1), U2_ag(member27_in_ag(x2)))
U9_AG(.(z0, z1), select10_out_aga(z0, z1)) → U10_AG(.(z0, z1), U3_ag(z1, member27_in_ag(z1)))
U9_AG(.(z0, z1), select10_out_aga(x1, x2)) → U10_AG(.(z0, z1), U3_ag(x2, member27_in_ag(x2)))
U9_AG(.(z0, z1), select10_out_aga(z0, z1)) → U10_AG(.(z0, z1), members20_out_ag)
U9_AG(.(z0, z1), select10_out_aga(x1, x2)) → U10_AG(.(z0, z1), members20_out_ag)
The TRS R consists of the following rules:
select10_in_aga(.(T53, T54)) → select10_out_aga(T53, T54)
select10_in_aga(.(T62, T63)) → U1_aga(T62, select10_in_aga(T63))
members20_in_ag(T90) → U2_ag(member27_in_ag(T90))
members20_in_ag(T90) → U3_ag(T90, member27_in_ag(T90))
members20_in_ag(T131) → members20_out_ag
U1_aga(T62, select10_out_aga(T64, X73)) → select10_out_aga(T64, .(T62, X73))
U2_ag(member27_out_ag(T91)) → members20_out_ag
U3_ag(T90, member27_out_ag(T91)) → U4_ag(members20_in_ag(T90))
member27_in_ag(.(T110, T111)) → member27_out_ag(T110)
member27_in_ag(.(T119, T120)) → U5_ag(member27_in_ag(T120))
U4_ag(members20_out_ag) → members20_out_ag
U5_ag(member27_out_ag(T121)) → member27_out_ag(T121)
The set Q consists of the following terms:
select10_in_aga(x0)
members20_in_ag(x0)
U1_aga(x0, x1)
U2_ag(x0)
U3_ag(x0, x1)
member27_in_ag(x0)
U4_ag(x0)
U5_ag(x0)
We have to consider all (P,Q,R)-chains.
(49) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U10_AG(T30, members20_out_ag) → COLOR_MAP1_IN_AG(T30) we obtained the following new rules [LPAR04]:
U10_AG(.(z0, z1), members20_out_ag) → COLOR_MAP1_IN_AG(.(z0, z1))
(50) Obligation:
Q DP problem:The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), select10_out_aga(x0, x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U1_aga(x0, select10_in_aga(x1)))
U9_AG(.(z0, z1), select10_out_aga(z0, z1)) → U10_AG(.(z0, z1), U2_ag(member27_in_ag(z1)))
U9_AG(.(z0, z1), select10_out_aga(x1, x2)) → U10_AG(.(z0, z1), U2_ag(member27_in_ag(x2)))
U9_AG(.(z0, z1), select10_out_aga(z0, z1)) → U10_AG(.(z0, z1), U3_ag(z1, member27_in_ag(z1)))
U9_AG(.(z0, z1), select10_out_aga(x1, x2)) → U10_AG(.(z0, z1), U3_ag(x2, member27_in_ag(x2)))
U9_AG(.(z0, z1), select10_out_aga(z0, z1)) → U10_AG(.(z0, z1), members20_out_ag)
U9_AG(.(z0, z1), select10_out_aga(x1, x2)) → U10_AG(.(z0, z1), members20_out_ag)
U10_AG(.(z0, z1), members20_out_ag) → COLOR_MAP1_IN_AG(.(z0, z1))
The TRS R consists of the following rules:
select10_in_aga(.(T53, T54)) → select10_out_aga(T53, T54)
select10_in_aga(.(T62, T63)) → U1_aga(T62, select10_in_aga(T63))
members20_in_ag(T90) → U2_ag(member27_in_ag(T90))
members20_in_ag(T90) → U3_ag(T90, member27_in_ag(T90))
members20_in_ag(T131) → members20_out_ag
U1_aga(T62, select10_out_aga(T64, X73)) → select10_out_aga(T64, .(T62, X73))
U2_ag(member27_out_ag(T91)) → members20_out_ag
U3_ag(T90, member27_out_ag(T91)) → U4_ag(members20_in_ag(T90))
member27_in_ag(.(T110, T111)) → member27_out_ag(T110)
member27_in_ag(.(T119, T120)) → U5_ag(member27_in_ag(T120))
U4_ag(members20_out_ag) → members20_out_ag
U5_ag(member27_out_ag(T121)) → member27_out_ag(T121)
The set Q consists of the following terms:
select10_in_aga(x0)
members20_in_ag(x0)
U1_aga(x0, x1)
U2_ag(x0)
U3_ag(x0, x1)
member27_in_ag(x0)
U4_ag(x0)
U5_ag(x0)
We have to consider all (P,Q,R)-chains.
(51) 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'), select10_out_aga(z0', z1')) evaluates to t =U9_AG(.(z0', z1'), select10_out_aga(z0', z1'))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [ ]
- Semiunifier: [ ]
Rewriting sequence
U9_AG(.(z0', z1'), select10_out_aga(z0', z1')) → U10_AG(.(z0', z1'), members20_out_ag)
with rule U9_AG(.(z0'', z1''), select10_out_aga(z0'', z1'')) → U10_AG(.(z0'', z1''), members20_out_ag) at position [] and matcher [z0'' / z0', z1'' / z1']
U10_AG(.(z0', z1'), members20_out_ag) → COLOR_MAP1_IN_AG(.(z0', z1'))
with rule U10_AG(.(z0, z1), members20_out_ag) → COLOR_MAP1_IN_AG(.(z0, z1)) at position [] and matcher [z0 / z0', z1 / z1']
COLOR_MAP1_IN_AG(.(z0', z1')) → U9_AG(.(z0', z1'), select10_out_aga(z0', z1'))
with rule COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), select10_out_aga(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.
(52) NO
(53) PrologToPiTRSProof (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)
members20_in: (f,b)
member27_in: (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
color_map1_in_ag(l(region(T27, T31, T32), T33), T30) → U6_ag(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
select10_in_aga(T53, .(T53, T54), T54) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(T64, .(T62, T63), .(T62, X73)) → U1_aga(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
U1_aga(T64, T62, T63, X73, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U6_ag(T27, T31, T32, T33, T30, select10_out_aga(T31, T30, X34)) → color_map1_out_ag(l(region(T27, T31, T32), T33), T30)
color_map1_in_ag(l(region(T27, T31, T39), T40), T30) → U7_ag(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_ag(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_ag(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
members20_in_ag(.(T91, T92), T90) → U2_ag(T91, T92, T90, member27_in_ag(T91, T90))
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
U2_ag(T91, T92, T90, member27_out_ag(T91, T90)) → members20_out_ag(.(T91, T92), T90)
members20_in_ag(.(T91, T97), T90) → U3_ag(T91, T97, T90, member27_in_ag(T91, T90))
U3_ag(T91, T97, T90, member27_out_ag(T91, T90)) → U4_ag(T91, T97, T90, members20_in_ag(T97, T90))
members20_in_ag([], T131) → members20_out_ag([], T131)
U4_ag(T91, T97, T90, members20_out_ag(T97, T90)) → members20_out_ag(.(T91, T97), T90)
U8_ag(T27, T31, T39, T40, T30, members20_out_ag(T39, T38)) → color_map1_out_ag(l(region(T27, T31, T39), T40), T30)
color_map1_in_ag(l(region(T27, T31, T39), T72), T30) → U9_ag(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_ag(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_ag(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_ag(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_ag(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
color_map1_in_ag(nil, T137) → color_map1_out_ag(nil, T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(l(region(T27, T31, T39), T72), T30)
The argument filtering Pi contains the following mapping:
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)
.(x1, x2) = .(x1, x2)
select10_out_aga(x1, x2, x3) = select10_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x2, x3, x5)
color_map1_out_ag(x1, x2) = color_map1_out_ag(x2)
region(x1, x2, x3) = region(x2, x3)
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)
member27_out_ag(x1, x2) = member27_out_ag(x1, x2)
U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4)
members20_out_ag(x1, x2) = members20_out_ag(x2)
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)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(54) Obligation:
Pi-finite rewrite system:The TRS R consists of the following rules:
color_map1_in_ag(l(region(T27, T31, T32), T33), T30) → U6_ag(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
select10_in_aga(T53, .(T53, T54), T54) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(T64, .(T62, T63), .(T62, X73)) → U1_aga(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
U1_aga(T64, T62, T63, X73, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U6_ag(T27, T31, T32, T33, T30, select10_out_aga(T31, T30, X34)) → color_map1_out_ag(l(region(T27, T31, T32), T33), T30)
color_map1_in_ag(l(region(T27, T31, T39), T40), T30) → U7_ag(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_ag(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_ag(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
members20_in_ag(.(T91, T92), T90) → U2_ag(T91, T92, T90, member27_in_ag(T91, T90))
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
U2_ag(T91, T92, T90, member27_out_ag(T91, T90)) → members20_out_ag(.(T91, T92), T90)
members20_in_ag(.(T91, T97), T90) → U3_ag(T91, T97, T90, member27_in_ag(T91, T90))
U3_ag(T91, T97, T90, member27_out_ag(T91, T90)) → U4_ag(T91, T97, T90, members20_in_ag(T97, T90))
members20_in_ag([], T131) → members20_out_ag([], T131)
U4_ag(T91, T97, T90, members20_out_ag(T97, T90)) → members20_out_ag(.(T91, T97), T90)
U8_ag(T27, T31, T39, T40, T30, members20_out_ag(T39, T38)) → color_map1_out_ag(l(region(T27, T31, T39), T40), T30)
color_map1_in_ag(l(region(T27, T31, T39), T72), T30) → U9_ag(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_ag(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_ag(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_ag(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_ag(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
color_map1_in_ag(nil, T137) → color_map1_out_ag(nil, T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(l(region(T27, T31, T39), T72), T30)
The argument filtering Pi contains the following mapping:
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)
.(x1, x2) = .(x1, x2)
select10_out_aga(x1, x2, x3) = select10_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x2, x3, x5)
color_map1_out_ag(x1, x2) = color_map1_out_ag(x2)
region(x1, x2, x3) = region(x2, x3)
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)
member27_out_ag(x1, x2) = member27_out_ag(x1, x2)
U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4)
members20_out_ag(x1, x2) = members20_out_ag(x2)
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)
(55) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:Pi DP problem:
The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(l(region(T27, T31, T32), T33), T30) → U6_AG(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
COLOR_MAP1_IN_AG(l(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(l(region(T27, T31, T39), T40), T30) → U7_AG(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_AG(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_AG(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
U7_AG(T27, T31, T39, T40, T30, select10_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, member27_in_ag(T91, T90))
U3_AG(T91, T97, T90, member27_out_ag(T91, T90)) → U4_AG(T91, T97, T90, members20_in_ag(T97, T90))
U3_AG(T91, T97, T90, member27_out_ag(T91, T90)) → MEMBERS20_IN_AG(T97, T90)
COLOR_MAP1_IN_AG(l(region(T27, T31, T39), T72), T30) → U9_AG(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_AG(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_AG(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U9_AG(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → MEMBERS20_IN_AG(T39, T38)
U10_AG(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_AG(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
U10_AG(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → COLOR_MAP1_IN_AG(T72, T30)
The TRS R consists of the following rules:
color_map1_in_ag(l(region(T27, T31, T32), T33), T30) → U6_ag(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
select10_in_aga(T53, .(T53, T54), T54) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(T64, .(T62, T63), .(T62, X73)) → U1_aga(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
U1_aga(T64, T62, T63, X73, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U6_ag(T27, T31, T32, T33, T30, select10_out_aga(T31, T30, X34)) → color_map1_out_ag(l(region(T27, T31, T32), T33), T30)
color_map1_in_ag(l(region(T27, T31, T39), T40), T30) → U7_ag(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_ag(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_ag(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
members20_in_ag(.(T91, T92), T90) → U2_ag(T91, T92, T90, member27_in_ag(T91, T90))
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
U2_ag(T91, T92, T90, member27_out_ag(T91, T90)) → members20_out_ag(.(T91, T92), T90)
members20_in_ag(.(T91, T97), T90) → U3_ag(T91, T97, T90, member27_in_ag(T91, T90))
U3_ag(T91, T97, T90, member27_out_ag(T91, T90)) → U4_ag(T91, T97, T90, members20_in_ag(T97, T90))
members20_in_ag([], T131) → members20_out_ag([], T131)
U4_ag(T91, T97, T90, members20_out_ag(T97, T90)) → members20_out_ag(.(T91, T97), T90)
U8_ag(T27, T31, T39, T40, T30, members20_out_ag(T39, T38)) → color_map1_out_ag(l(region(T27, T31, T39), T40), T30)
color_map1_in_ag(l(region(T27, T31, T39), T72), T30) → U9_ag(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_ag(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_ag(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_ag(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_ag(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
color_map1_in_ag(nil, T137) → color_map1_out_ag(nil, T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(l(region(T27, T31, T39), T72), T30)
The argument filtering Pi contains the following mapping:
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)
.(x1, x2) = .(x1, x2)
select10_out_aga(x1, x2, x3) = select10_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x2, x3, x5)
color_map1_out_ag(x1, x2) = color_map1_out_ag(x2)
region(x1, x2, x3) = region(x2, x3)
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)
member27_out_ag(x1, x2) = member27_out_ag(x1, x2)
U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4)
members20_out_ag(x1, x2) = members20_out_ag(x2)
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)
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
(56) Obligation:
Pi DP problem:The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(l(region(T27, T31, T32), T33), T30) → U6_AG(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
COLOR_MAP1_IN_AG(l(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(l(region(T27, T31, T39), T40), T30) → U7_AG(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_AG(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_AG(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
U7_AG(T27, T31, T39, T40, T30, select10_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, member27_in_ag(T91, T90))
U3_AG(T91, T97, T90, member27_out_ag(T91, T90)) → U4_AG(T91, T97, T90, members20_in_ag(T97, T90))
U3_AG(T91, T97, T90, member27_out_ag(T91, T90)) → MEMBERS20_IN_AG(T97, T90)
COLOR_MAP1_IN_AG(l(region(T27, T31, T39), T72), T30) → U9_AG(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_AG(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_AG(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U9_AG(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → MEMBERS20_IN_AG(T39, T38)
U10_AG(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_AG(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
U10_AG(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → COLOR_MAP1_IN_AG(T72, T30)
The TRS R consists of the following rules:
color_map1_in_ag(l(region(T27, T31, T32), T33), T30) → U6_ag(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
select10_in_aga(T53, .(T53, T54), T54) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(T64, .(T62, T63), .(T62, X73)) → U1_aga(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
U1_aga(T64, T62, T63, X73, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U6_ag(T27, T31, T32, T33, T30, select10_out_aga(T31, T30, X34)) → color_map1_out_ag(l(region(T27, T31, T32), T33), T30)
color_map1_in_ag(l(region(T27, T31, T39), T40), T30) → U7_ag(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_ag(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_ag(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
members20_in_ag(.(T91, T92), T90) → U2_ag(T91, T92, T90, member27_in_ag(T91, T90))
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
U2_ag(T91, T92, T90, member27_out_ag(T91, T90)) → members20_out_ag(.(T91, T92), T90)
members20_in_ag(.(T91, T97), T90) → U3_ag(T91, T97, T90, member27_in_ag(T91, T90))
U3_ag(T91, T97, T90, member27_out_ag(T91, T90)) → U4_ag(T91, T97, T90, members20_in_ag(T97, T90))
members20_in_ag([], T131) → members20_out_ag([], T131)
U4_ag(T91, T97, T90, members20_out_ag(T97, T90)) → members20_out_ag(.(T91, T97), T90)
U8_ag(T27, T31, T39, T40, T30, members20_out_ag(T39, T38)) → color_map1_out_ag(l(region(T27, T31, T39), T40), T30)
color_map1_in_ag(l(region(T27, T31, T39), T72), T30) → U9_ag(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_ag(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_ag(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_ag(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_ag(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
color_map1_in_ag(nil, T137) → color_map1_out_ag(nil, T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(l(region(T27, T31, T39), T72), T30)
The argument filtering Pi contains the following mapping:
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)
.(x1, x2) = .(x1, x2)
select10_out_aga(x1, x2, x3) = select10_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x2, x3, x5)
color_map1_out_ag(x1, x2) = color_map1_out_ag(x2)
region(x1, x2, x3) = region(x2, x3)
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)
member27_out_ag(x1, x2) = member27_out_ag(x1, x2)
U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4)
members20_out_ag(x1, x2) = members20_out_ag(x2)
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)
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
(57) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 12 less nodes.(58) Complex Obligation (AND)
(59) 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:
color_map1_in_ag(l(region(T27, T31, T32), T33), T30) → U6_ag(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
select10_in_aga(T53, .(T53, T54), T54) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(T64, .(T62, T63), .(T62, X73)) → U1_aga(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
U1_aga(T64, T62, T63, X73, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U6_ag(T27, T31, T32, T33, T30, select10_out_aga(T31, T30, X34)) → color_map1_out_ag(l(region(T27, T31, T32), T33), T30)
color_map1_in_ag(l(region(T27, T31, T39), T40), T30) → U7_ag(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_ag(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_ag(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
members20_in_ag(.(T91, T92), T90) → U2_ag(T91, T92, T90, member27_in_ag(T91, T90))
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
U2_ag(T91, T92, T90, member27_out_ag(T91, T90)) → members20_out_ag(.(T91, T92), T90)
members20_in_ag(.(T91, T97), T90) → U3_ag(T91, T97, T90, member27_in_ag(T91, T90))
U3_ag(T91, T97, T90, member27_out_ag(T91, T90)) → U4_ag(T91, T97, T90, members20_in_ag(T97, T90))
members20_in_ag([], T131) → members20_out_ag([], T131)
U4_ag(T91, T97, T90, members20_out_ag(T97, T90)) → members20_out_ag(.(T91, T97), T90)
U8_ag(T27, T31, T39, T40, T30, members20_out_ag(T39, T38)) → color_map1_out_ag(l(region(T27, T31, T39), T40), T30)
color_map1_in_ag(l(region(T27, T31, T39), T72), T30) → U9_ag(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_ag(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_ag(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_ag(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_ag(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
color_map1_in_ag(nil, T137) → color_map1_out_ag(nil, T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(l(region(T27, T31, T39), T72), T30)
The argument filtering Pi contains the following mapping:
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)
.(x1, x2) = .(x1, x2)
select10_out_aga(x1, x2, x3) = select10_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x2, x3, x5)
color_map1_out_ag(x1, x2) = color_map1_out_ag(x2)
region(x1, x2, x3) = region(x2, x3)
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)
member27_out_ag(x1, x2) = member27_out_ag(x1, x2)
U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4)
members20_out_ag(x1, x2) = members20_out_ag(x2)
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)
MEMBER27_IN_AG(x1, x2) = MEMBER27_IN_AG(x2)
We have to consider all (P,R,Pi)-chains
(60) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(61) 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
(62) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(63) 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.
(64) 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
(65) YES
(66) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MEMBERS20_IN_AG(.(T91, T97), T90) → U3_AG(T91, T97, T90, member27_in_ag(T91, T90))
U3_AG(T91, T97, T90, member27_out_ag(T91, T90)) → MEMBERS20_IN_AG(T97, T90)
The TRS R consists of the following rules:
color_map1_in_ag(l(region(T27, T31, T32), T33), T30) → U6_ag(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
select10_in_aga(T53, .(T53, T54), T54) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(T64, .(T62, T63), .(T62, X73)) → U1_aga(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
U1_aga(T64, T62, T63, X73, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U6_ag(T27, T31, T32, T33, T30, select10_out_aga(T31, T30, X34)) → color_map1_out_ag(l(region(T27, T31, T32), T33), T30)
color_map1_in_ag(l(region(T27, T31, T39), T40), T30) → U7_ag(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_ag(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_ag(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
members20_in_ag(.(T91, T92), T90) → U2_ag(T91, T92, T90, member27_in_ag(T91, T90))
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
U2_ag(T91, T92, T90, member27_out_ag(T91, T90)) → members20_out_ag(.(T91, T92), T90)
members20_in_ag(.(T91, T97), T90) → U3_ag(T91, T97, T90, member27_in_ag(T91, T90))
U3_ag(T91, T97, T90, member27_out_ag(T91, T90)) → U4_ag(T91, T97, T90, members20_in_ag(T97, T90))
members20_in_ag([], T131) → members20_out_ag([], T131)
U4_ag(T91, T97, T90, members20_out_ag(T97, T90)) → members20_out_ag(.(T91, T97), T90)
U8_ag(T27, T31, T39, T40, T30, members20_out_ag(T39, T38)) → color_map1_out_ag(l(region(T27, T31, T39), T40), T30)
color_map1_in_ag(l(region(T27, T31, T39), T72), T30) → U9_ag(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_ag(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_ag(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_ag(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_ag(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
color_map1_in_ag(nil, T137) → color_map1_out_ag(nil, T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(l(region(T27, T31, T39), T72), T30)
The argument filtering Pi contains the following mapping:
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)
.(x1, x2) = .(x1, x2)
select10_out_aga(x1, x2, x3) = select10_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x2, x3, x5)
color_map1_out_ag(x1, x2) = color_map1_out_ag(x2)
region(x1, x2, x3) = region(x2, x3)
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)
member27_out_ag(x1, x2) = member27_out_ag(x1, x2)
U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4)
members20_out_ag(x1, x2) = members20_out_ag(x2)
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)
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
(67) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(68) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MEMBERS20_IN_AG(.(T91, T97), T90) → U3_AG(T91, T97, T90, member27_in_ag(T91, T90))
U3_AG(T91, T97, T90, member27_out_ag(T91, T90)) → MEMBERS20_IN_AG(T97, T90)
The TRS R consists of the following rules:
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
member27_in_ag(x1, x2) = member27_in_ag(x2)
member27_out_ag(x1, x2) = member27_out_ag(x1, x2)
U5_ag(x1, x2, x3, x4) = U5_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
(69) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(70) Obligation:
Q DP problem:The TRS P consists of the following rules:
MEMBERS20_IN_AG(T90) → U3_AG(T90, member27_in_ag(T90))
U3_AG(T90, member27_out_ag(T91, T90)) → MEMBERS20_IN_AG(T90)
The TRS R consists of the following rules:
member27_in_ag(.(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(.(T119, T120)) → U5_ag(T119, T120, member27_in_ag(T120))
U5_ag(T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
The set Q consists of the following terms:
member27_in_ag(x0)
U5_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(71) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule MEMBERS20_IN_AG(T90) → U3_AG(T90, member27_in_ag(T90)) at position [1] we obtained the following new rules [LPAR04]:
MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), member27_out_ag(x0, .(x0, x1)))
MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), U5_ag(x0, x1, member27_in_ag(x1)))
(72) Obligation:
Q DP problem:The TRS P consists of the following rules:
U3_AG(T90, member27_out_ag(T91, T90)) → MEMBERS20_IN_AG(T90)
MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), member27_out_ag(x0, .(x0, x1)))
MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), U5_ag(x0, x1, member27_in_ag(x1)))
The TRS R consists of the following rules:
member27_in_ag(.(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(.(T119, T120)) → U5_ag(T119, T120, member27_in_ag(T120))
U5_ag(T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
The set Q consists of the following terms:
member27_in_ag(x0)
U5_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(73) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U3_AG(T90, member27_out_ag(T91, T90)) → MEMBERS20_IN_AG(T90) we obtained the following new rules [LPAR04]:
U3_AG(.(z0, z1), member27_out_ag(z0, .(z0, z1))) → MEMBERS20_IN_AG(.(z0, z1))
U3_AG(.(z0, z1), member27_out_ag(x1, .(z0, z1))) → MEMBERS20_IN_AG(.(z0, z1))
(74) Obligation:
Q DP problem:The TRS P consists of the following rules:
MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), member27_out_ag(x0, .(x0, x1)))
MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), U5_ag(x0, x1, member27_in_ag(x1)))
U3_AG(.(z0, z1), member27_out_ag(z0, .(z0, z1))) → MEMBERS20_IN_AG(.(z0, z1))
U3_AG(.(z0, z1), member27_out_ag(x1, .(z0, z1))) → MEMBERS20_IN_AG(.(z0, z1))
The TRS R consists of the following rules:
member27_in_ag(.(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(.(T119, T120)) → U5_ag(T119, T120, member27_in_ag(T120))
U5_ag(T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
The set Q consists of the following terms:
member27_in_ag(x0)
U5_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(75) 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), member27_out_ag(z0, .(z0, z1))) evaluates to t =U3_AG(.(z0, z1), member27_out_ag(z0, .(z0, z1)))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequence
U3_AG(.(z0, z1), member27_out_ag(z0, .(z0, z1))) → MEMBERS20_IN_AG(.(z0, z1))
with rule U3_AG(.(z0', z1'), member27_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), member27_out_ag(z0, .(z0, z1)))
with rule MEMBERS20_IN_AG(.(x0, x1)) → U3_AG(.(x0, x1), member27_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.
(76) NO
(77) 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:
color_map1_in_ag(l(region(T27, T31, T32), T33), T30) → U6_ag(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
select10_in_aga(T53, .(T53, T54), T54) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(T64, .(T62, T63), .(T62, X73)) → U1_aga(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
U1_aga(T64, T62, T63, X73, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U6_ag(T27, T31, T32, T33, T30, select10_out_aga(T31, T30, X34)) → color_map1_out_ag(l(region(T27, T31, T32), T33), T30)
color_map1_in_ag(l(region(T27, T31, T39), T40), T30) → U7_ag(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_ag(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_ag(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
members20_in_ag(.(T91, T92), T90) → U2_ag(T91, T92, T90, member27_in_ag(T91, T90))
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
U2_ag(T91, T92, T90, member27_out_ag(T91, T90)) → members20_out_ag(.(T91, T92), T90)
members20_in_ag(.(T91, T97), T90) → U3_ag(T91, T97, T90, member27_in_ag(T91, T90))
U3_ag(T91, T97, T90, member27_out_ag(T91, T90)) → U4_ag(T91, T97, T90, members20_in_ag(T97, T90))
members20_in_ag([], T131) → members20_out_ag([], T131)
U4_ag(T91, T97, T90, members20_out_ag(T97, T90)) → members20_out_ag(.(T91, T97), T90)
U8_ag(T27, T31, T39, T40, T30, members20_out_ag(T39, T38)) → color_map1_out_ag(l(region(T27, T31, T39), T40), T30)
color_map1_in_ag(l(region(T27, T31, T39), T72), T30) → U9_ag(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_ag(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_ag(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_ag(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_ag(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
color_map1_in_ag(nil, T137) → color_map1_out_ag(nil, T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(l(region(T27, T31, T39), T72), T30)
The argument filtering Pi contains the following mapping:
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)
.(x1, x2) = .(x1, x2)
select10_out_aga(x1, x2, x3) = select10_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x2, x3, x5)
color_map1_out_ag(x1, x2) = color_map1_out_ag(x2)
region(x1, x2, x3) = region(x2, x3)
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)
member27_out_ag(x1, x2) = member27_out_ag(x1, x2)
U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4)
members20_out_ag(x1, x2) = members20_out_ag(x2)
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)
SELECT10_IN_AGA(x1, x2, x3) = SELECT10_IN_AGA(x2)
We have to consider all (P,R,Pi)-chains
(78) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(79) 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
(80) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(81) 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.
(82) 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
(83) YES
(84) Obligation:
Pi DP problem:The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(l(region(T27, T31, T39), T72), T30) → U9_AG(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_AG(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_AG(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_AG(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → COLOR_MAP1_IN_AG(T72, T30)
The TRS R consists of the following rules:
color_map1_in_ag(l(region(T27, T31, T32), T33), T30) → U6_ag(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
select10_in_aga(T53, .(T53, T54), T54) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(T64, .(T62, T63), .(T62, X73)) → U1_aga(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
U1_aga(T64, T62, T63, X73, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U6_ag(T27, T31, T32, T33, T30, select10_out_aga(T31, T30, X34)) → color_map1_out_ag(l(region(T27, T31, T32), T33), T30)
color_map1_in_ag(l(region(T27, T31, T39), T40), T30) → U7_ag(T27, T31, T39, T40, T30, select10_in_aga(T31, T30, T38))
U7_ag(T27, T31, T39, T40, T30, select10_out_aga(T31, T30, T38)) → U8_ag(T27, T31, T39, T40, T30, members20_in_ag(T39, T38))
members20_in_ag(.(T91, T92), T90) → U2_ag(T91, T92, T90, member27_in_ag(T91, T90))
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
U2_ag(T91, T92, T90, member27_out_ag(T91, T90)) → members20_out_ag(.(T91, T92), T90)
members20_in_ag(.(T91, T97), T90) → U3_ag(T91, T97, T90, member27_in_ag(T91, T90))
U3_ag(T91, T97, T90, member27_out_ag(T91, T90)) → U4_ag(T91, T97, T90, members20_in_ag(T97, T90))
members20_in_ag([], T131) → members20_out_ag([], T131)
U4_ag(T91, T97, T90, members20_out_ag(T97, T90)) → members20_out_ag(.(T91, T97), T90)
U8_ag(T27, T31, T39, T40, T30, members20_out_ag(T39, T38)) → color_map1_out_ag(l(region(T27, T31, T39), T40), T30)
color_map1_in_ag(l(region(T27, T31, T39), T72), T30) → U9_ag(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_ag(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_ag(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_ag(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → U11_ag(T27, T31, T39, T72, T30, color_map1_in_ag(T72, T30))
color_map1_in_ag(nil, T137) → color_map1_out_ag(nil, T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(l(region(T27, T31, T39), T72), T30)
The argument filtering Pi contains the following mapping:
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)
.(x1, x2) = .(x1, x2)
select10_out_aga(x1, x2, x3) = select10_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x2, x3, x5)
color_map1_out_ag(x1, x2) = color_map1_out_ag(x2)
region(x1, x2, x3) = region(x2, x3)
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)
member27_out_ag(x1, x2) = member27_out_ag(x1, x2)
U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4)
members20_out_ag(x1, x2) = members20_out_ag(x2)
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)
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
(85) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(86) Obligation:
Pi DP problem:The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(l(region(T27, T31, T39), T72), T30) → U9_AG(T27, T31, T39, T72, T30, select10_in_aga(T31, T30, T38))
U9_AG(T27, T31, T39, T72, T30, select10_out_aga(T31, T30, T38)) → U10_AG(T27, T31, T39, T72, T30, members20_in_ag(T39, T38))
U10_AG(T27, T31, T39, T72, T30, members20_out_ag(T39, T38)) → COLOR_MAP1_IN_AG(T72, T30)
The TRS R consists of the following rules:
select10_in_aga(T53, .(T53, T54), T54) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(T64, .(T62, T63), .(T62, X73)) → U1_aga(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
members20_in_ag(.(T91, T92), T90) → U2_ag(T91, T92, T90, member27_in_ag(T91, T90))
members20_in_ag(.(T91, T97), T90) → U3_ag(T91, T97, T90, member27_in_ag(T91, T90))
members20_in_ag([], T131) → members20_out_ag([], T131)
U1_aga(T64, T62, T63, X73, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U2_ag(T91, T92, T90, member27_out_ag(T91, T90)) → members20_out_ag(.(T91, T92), T90)
U3_ag(T91, T97, T90, member27_out_ag(T91, T90)) → U4_ag(T91, T97, T90, members20_in_ag(T97, T90))
member27_in_ag(T110, .(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(T121, .(T119, T120)) → U5_ag(T121, T119, T120, member27_in_ag(T121, T120))
U4_ag(T91, T97, T90, members20_out_ag(T97, T90)) → members20_out_ag(.(T91, T97), T90)
U5_ag(T121, T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
The argument filtering Pi contains the following mapping:
select10_in_aga(x1, x2, x3) = select10_in_aga(x2)
.(x1, x2) = .(x1, x2)
select10_out_aga(x1, x2, x3) = select10_out_aga(x1, x2, x3)
U1_aga(x1, x2, x3, x4, x5) = U1_aga(x2, x3, x5)
region(x1, x2, x3) = region(x2, x3)
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)
member27_out_ag(x1, x2) = member27_out_ag(x1, x2)
U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4)
members20_out_ag(x1, x2) = members20_out_ag(x2)
U3_ag(x1, x2, x3, x4) = U3_ag(x3, x4)
U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4)
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
(87) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(88) Obligation:
Q DP problem:The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(T30) → U9_AG(T30, select10_in_aga(T30))
U9_AG(T30, select10_out_aga(T31, T30, T38)) → U10_AG(T30, members20_in_ag(T38))
U10_AG(T30, members20_out_ag(T38)) → COLOR_MAP1_IN_AG(T30)
The TRS R consists of the following rules:
select10_in_aga(.(T53, T54)) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(.(T62, T63)) → U1_aga(T62, T63, select10_in_aga(T63))
members20_in_ag(T90) → U2_ag(T90, member27_in_ag(T90))
members20_in_ag(T90) → U3_ag(T90, member27_in_ag(T90))
members20_in_ag(T131) → members20_out_ag(T131)
U1_aga(T62, T63, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U2_ag(T90, member27_out_ag(T91, T90)) → members20_out_ag(T90)
U3_ag(T90, member27_out_ag(T91, T90)) → U4_ag(T90, members20_in_ag(T90))
member27_in_ag(.(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(.(T119, T120)) → U5_ag(T119, T120, member27_in_ag(T120))
U4_ag(T90, members20_out_ag(T90)) → members20_out_ag(T90)
U5_ag(T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
The set Q consists of the following terms:
select10_in_aga(x0)
members20_in_ag(x0)
U1_aga(x0, x1, x2)
U2_ag(x0, x1)
U3_ag(x0, x1)
member27_in_ag(x0)
U4_ag(x0, x1)
U5_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(89) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule COLOR_MAP1_IN_AG(T30) → U9_AG(T30, select10_in_aga(T30)) at position [1] we obtained the following new rules [LPAR04]:
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), select10_out_aga(x0, .(x0, x1), x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U1_aga(x0, x1, select10_in_aga(x1)))
(90) Obligation:
Q DP problem:The TRS P consists of the following rules:
U9_AG(T30, select10_out_aga(T31, T30, T38)) → U10_AG(T30, members20_in_ag(T38))
U10_AG(T30, members20_out_ag(T38)) → COLOR_MAP1_IN_AG(T30)
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), select10_out_aga(x0, .(x0, x1), x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U1_aga(x0, x1, select10_in_aga(x1)))
The TRS R consists of the following rules:
select10_in_aga(.(T53, T54)) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(.(T62, T63)) → U1_aga(T62, T63, select10_in_aga(T63))
members20_in_ag(T90) → U2_ag(T90, member27_in_ag(T90))
members20_in_ag(T90) → U3_ag(T90, member27_in_ag(T90))
members20_in_ag(T131) → members20_out_ag(T131)
U1_aga(T62, T63, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U2_ag(T90, member27_out_ag(T91, T90)) → members20_out_ag(T90)
U3_ag(T90, member27_out_ag(T91, T90)) → U4_ag(T90, members20_in_ag(T90))
member27_in_ag(.(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(.(T119, T120)) → U5_ag(T119, T120, member27_in_ag(T120))
U4_ag(T90, members20_out_ag(T90)) → members20_out_ag(T90)
U5_ag(T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
The set Q consists of the following terms:
select10_in_aga(x0)
members20_in_ag(x0)
U1_aga(x0, x1, x2)
U2_ag(x0, x1)
U3_ag(x0, x1)
member27_in_ag(x0)
U4_ag(x0, x1)
U5_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(91) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule U9_AG(T30, select10_out_aga(T31, T30, T38)) → U10_AG(T30, members20_in_ag(T38)) at position [1] we obtained the following new rules [LPAR04]:
U9_AG(y0, select10_out_aga(y1, y0, x0)) → U10_AG(y0, U2_ag(x0, member27_in_ag(x0)))
U9_AG(y0, select10_out_aga(y1, y0, x0)) → U10_AG(y0, U3_ag(x0, member27_in_ag(x0)))
U9_AG(y0, select10_out_aga(y1, y0, x0)) → U10_AG(y0, members20_out_ag(x0))
(92) Obligation:
Q DP problem:The TRS P consists of the following rules:
U10_AG(T30, members20_out_ag(T38)) → COLOR_MAP1_IN_AG(T30)
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), select10_out_aga(x0, .(x0, x1), x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U1_aga(x0, x1, select10_in_aga(x1)))
U9_AG(y0, select10_out_aga(y1, y0, x0)) → U10_AG(y0, U2_ag(x0, member27_in_ag(x0)))
U9_AG(y0, select10_out_aga(y1, y0, x0)) → U10_AG(y0, U3_ag(x0, member27_in_ag(x0)))
U9_AG(y0, select10_out_aga(y1, y0, x0)) → U10_AG(y0, members20_out_ag(x0))
The TRS R consists of the following rules:
select10_in_aga(.(T53, T54)) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(.(T62, T63)) → U1_aga(T62, T63, select10_in_aga(T63))
members20_in_ag(T90) → U2_ag(T90, member27_in_ag(T90))
members20_in_ag(T90) → U3_ag(T90, member27_in_ag(T90))
members20_in_ag(T131) → members20_out_ag(T131)
U1_aga(T62, T63, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U2_ag(T90, member27_out_ag(T91, T90)) → members20_out_ag(T90)
U3_ag(T90, member27_out_ag(T91, T90)) → U4_ag(T90, members20_in_ag(T90))
member27_in_ag(.(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(.(T119, T120)) → U5_ag(T119, T120, member27_in_ag(T120))
U4_ag(T90, members20_out_ag(T90)) → members20_out_ag(T90)
U5_ag(T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
The set Q consists of the following terms:
select10_in_aga(x0)
members20_in_ag(x0)
U1_aga(x0, x1, x2)
U2_ag(x0, x1)
U3_ag(x0, x1)
member27_in_ag(x0)
U4_ag(x0, x1)
U5_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(93) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U9_AG(y0, select10_out_aga(y1, y0, x0)) → U10_AG(y0, U2_ag(x0, member27_in_ag(x0))) we obtained the following new rules [LPAR04]:
U9_AG(.(z0, z1), select10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), U2_ag(z1, member27_in_ag(z1)))
U9_AG(.(z0, z1), select10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), U2_ag(x2, member27_in_ag(x2)))
(94) Obligation:
Q DP problem:The TRS P consists of the following rules:
U10_AG(T30, members20_out_ag(T38)) → COLOR_MAP1_IN_AG(T30)
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), select10_out_aga(x0, .(x0, x1), x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U1_aga(x0, x1, select10_in_aga(x1)))
U9_AG(y0, select10_out_aga(y1, y0, x0)) → U10_AG(y0, U3_ag(x0, member27_in_ag(x0)))
U9_AG(y0, select10_out_aga(y1, y0, x0)) → U10_AG(y0, members20_out_ag(x0))
U9_AG(.(z0, z1), select10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), U2_ag(z1, member27_in_ag(z1)))
U9_AG(.(z0, z1), select10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), U2_ag(x2, member27_in_ag(x2)))
The TRS R consists of the following rules:
select10_in_aga(.(T53, T54)) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(.(T62, T63)) → U1_aga(T62, T63, select10_in_aga(T63))
members20_in_ag(T90) → U2_ag(T90, member27_in_ag(T90))
members20_in_ag(T90) → U3_ag(T90, member27_in_ag(T90))
members20_in_ag(T131) → members20_out_ag(T131)
U1_aga(T62, T63, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U2_ag(T90, member27_out_ag(T91, T90)) → members20_out_ag(T90)
U3_ag(T90, member27_out_ag(T91, T90)) → U4_ag(T90, members20_in_ag(T90))
member27_in_ag(.(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(.(T119, T120)) → U5_ag(T119, T120, member27_in_ag(T120))
U4_ag(T90, members20_out_ag(T90)) → members20_out_ag(T90)
U5_ag(T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
The set Q consists of the following terms:
select10_in_aga(x0)
members20_in_ag(x0)
U1_aga(x0, x1, x2)
U2_ag(x0, x1)
U3_ag(x0, x1)
member27_in_ag(x0)
U4_ag(x0, x1)
U5_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(95) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U9_AG(y0, select10_out_aga(y1, y0, x0)) → U10_AG(y0, U3_ag(x0, member27_in_ag(x0))) we obtained the following new rules [LPAR04]:
U9_AG(.(z0, z1), select10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), U3_ag(z1, member27_in_ag(z1)))
U9_AG(.(z0, z1), select10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), U3_ag(x2, member27_in_ag(x2)))
(96) Obligation:
Q DP problem:The TRS P consists of the following rules:
U10_AG(T30, members20_out_ag(T38)) → COLOR_MAP1_IN_AG(T30)
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), select10_out_aga(x0, .(x0, x1), x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U1_aga(x0, x1, select10_in_aga(x1)))
U9_AG(y0, select10_out_aga(y1, y0, x0)) → U10_AG(y0, members20_out_ag(x0))
U9_AG(.(z0, z1), select10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), U2_ag(z1, member27_in_ag(z1)))
U9_AG(.(z0, z1), select10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), U2_ag(x2, member27_in_ag(x2)))
U9_AG(.(z0, z1), select10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), U3_ag(z1, member27_in_ag(z1)))
U9_AG(.(z0, z1), select10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), U3_ag(x2, member27_in_ag(x2)))
The TRS R consists of the following rules:
select10_in_aga(.(T53, T54)) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(.(T62, T63)) → U1_aga(T62, T63, select10_in_aga(T63))
members20_in_ag(T90) → U2_ag(T90, member27_in_ag(T90))
members20_in_ag(T90) → U3_ag(T90, member27_in_ag(T90))
members20_in_ag(T131) → members20_out_ag(T131)
U1_aga(T62, T63, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U2_ag(T90, member27_out_ag(T91, T90)) → members20_out_ag(T90)
U3_ag(T90, member27_out_ag(T91, T90)) → U4_ag(T90, members20_in_ag(T90))
member27_in_ag(.(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(.(T119, T120)) → U5_ag(T119, T120, member27_in_ag(T120))
U4_ag(T90, members20_out_ag(T90)) → members20_out_ag(T90)
U5_ag(T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
The set Q consists of the following terms:
select10_in_aga(x0)
members20_in_ag(x0)
U1_aga(x0, x1, x2)
U2_ag(x0, x1)
U3_ag(x0, x1)
member27_in_ag(x0)
U4_ag(x0, x1)
U5_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(97) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U9_AG(y0, select10_out_aga(y1, y0, x0)) → U10_AG(y0, members20_out_ag(x0)) we obtained the following new rules [LPAR04]:
U9_AG(.(z0, z1), select10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), members20_out_ag(z1))
U9_AG(.(z0, z1), select10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), members20_out_ag(x2))
(98) Obligation:
Q DP problem:The TRS P consists of the following rules:
U10_AG(T30, members20_out_ag(T38)) → COLOR_MAP1_IN_AG(T30)
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), select10_out_aga(x0, .(x0, x1), x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U1_aga(x0, x1, select10_in_aga(x1)))
U9_AG(.(z0, z1), select10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), U2_ag(z1, member27_in_ag(z1)))
U9_AG(.(z0, z1), select10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), U2_ag(x2, member27_in_ag(x2)))
U9_AG(.(z0, z1), select10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), U3_ag(z1, member27_in_ag(z1)))
U9_AG(.(z0, z1), select10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), U3_ag(x2, member27_in_ag(x2)))
U9_AG(.(z0, z1), select10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), members20_out_ag(z1))
U9_AG(.(z0, z1), select10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), members20_out_ag(x2))
The TRS R consists of the following rules:
select10_in_aga(.(T53, T54)) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(.(T62, T63)) → U1_aga(T62, T63, select10_in_aga(T63))
members20_in_ag(T90) → U2_ag(T90, member27_in_ag(T90))
members20_in_ag(T90) → U3_ag(T90, member27_in_ag(T90))
members20_in_ag(T131) → members20_out_ag(T131)
U1_aga(T62, T63, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U2_ag(T90, member27_out_ag(T91, T90)) → members20_out_ag(T90)
U3_ag(T90, member27_out_ag(T91, T90)) → U4_ag(T90, members20_in_ag(T90))
member27_in_ag(.(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(.(T119, T120)) → U5_ag(T119, T120, member27_in_ag(T120))
U4_ag(T90, members20_out_ag(T90)) → members20_out_ag(T90)
U5_ag(T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
The set Q consists of the following terms:
select10_in_aga(x0)
members20_in_ag(x0)
U1_aga(x0, x1, x2)
U2_ag(x0, x1)
U3_ag(x0, x1)
member27_in_ag(x0)
U4_ag(x0, x1)
U5_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(99) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U10_AG(T30, members20_out_ag(T38)) → COLOR_MAP1_IN_AG(T30) we obtained the following new rules [LPAR04]:
U10_AG(.(z0, z1), members20_out_ag(x1)) → COLOR_MAP1_IN_AG(.(z0, z1))
U10_AG(.(z0, z1), members20_out_ag(z1)) → COLOR_MAP1_IN_AG(.(z0, z1))
(100) Obligation:
Q DP problem:The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), select10_out_aga(x0, .(x0, x1), x1))
COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), U1_aga(x0, x1, select10_in_aga(x1)))
U9_AG(.(z0, z1), select10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), U2_ag(z1, member27_in_ag(z1)))
U9_AG(.(z0, z1), select10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), U2_ag(x2, member27_in_ag(x2)))
U9_AG(.(z0, z1), select10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), U3_ag(z1, member27_in_ag(z1)))
U9_AG(.(z0, z1), select10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), U3_ag(x2, member27_in_ag(x2)))
U9_AG(.(z0, z1), select10_out_aga(z0, .(z0, z1), z1)) → U10_AG(.(z0, z1), members20_out_ag(z1))
U9_AG(.(z0, z1), select10_out_aga(x1, .(z0, z1), x2)) → U10_AG(.(z0, z1), members20_out_ag(x2))
U10_AG(.(z0, z1), members20_out_ag(x1)) → COLOR_MAP1_IN_AG(.(z0, z1))
U10_AG(.(z0, z1), members20_out_ag(z1)) → COLOR_MAP1_IN_AG(.(z0, z1))
The TRS R consists of the following rules:
select10_in_aga(.(T53, T54)) → select10_out_aga(T53, .(T53, T54), T54)
select10_in_aga(.(T62, T63)) → U1_aga(T62, T63, select10_in_aga(T63))
members20_in_ag(T90) → U2_ag(T90, member27_in_ag(T90))
members20_in_ag(T90) → U3_ag(T90, member27_in_ag(T90))
members20_in_ag(T131) → members20_out_ag(T131)
U1_aga(T62, T63, select10_out_aga(T64, T63, X73)) → select10_out_aga(T64, .(T62, T63), .(T62, X73))
U2_ag(T90, member27_out_ag(T91, T90)) → members20_out_ag(T90)
U3_ag(T90, member27_out_ag(T91, T90)) → U4_ag(T90, members20_in_ag(T90))
member27_in_ag(.(T110, T111)) → member27_out_ag(T110, .(T110, T111))
member27_in_ag(.(T119, T120)) → U5_ag(T119, T120, member27_in_ag(T120))
U4_ag(T90, members20_out_ag(T90)) → members20_out_ag(T90)
U5_ag(T119, T120, member27_out_ag(T121, T120)) → member27_out_ag(T121, .(T119, T120))
The set Q consists of the following terms:
select10_in_aga(x0)
members20_in_ag(x0)
U1_aga(x0, x1, x2)
U2_ag(x0, x1)
U3_ag(x0, x1)
member27_in_ag(x0)
U4_ag(x0, x1)
U5_ag(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(101) 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'), select10_out_aga(z0', .(z0', z1'), z1')) evaluates to t =U9_AG(.(z0', z1'), select10_out_aga(z0', .(z0', z1'), z1'))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [ ]
- Semiunifier: [ ]
Rewriting sequence
U9_AG(.(z0', z1'), select10_out_aga(z0', .(z0', z1'), z1')) → U10_AG(.(z0', z1'), members20_out_ag(z1'))
with rule U9_AG(.(z0'', z1''), select10_out_aga(z0'', .(z0'', z1''), z1'')) → U10_AG(.(z0'', z1''), members20_out_ag(z1'')) at position [] and matcher [z0'' / z0', z1'' / z1']
U10_AG(.(z0', z1'), members20_out_ag(z1')) → COLOR_MAP1_IN_AG(.(z0', z1'))
with rule U10_AG(.(z0, z1), members20_out_ag(x1')) → COLOR_MAP1_IN_AG(.(z0, z1)) at position [] and matcher [z0 / z0', z1 / z1', x1' / z1']
COLOR_MAP1_IN_AG(.(z0', z1')) → U9_AG(.(z0', z1'), select10_out_aga(z0', .(z0', z1'), z1'))
with rule COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), select10_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.