Left Termination of the query pattern color_map_in_2(a, g) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

color_map(.(Region, Regions), Colors) :- ','(color_region(Region, Colors), color_map(Regions, Colors)).
color_map([], Colors).
color_region(region(Name, Color, Neighbors), Colors) :- ','(select(Color, Colors, Colors1), members(Neighbors, Colors1)).
select(X, .(X, Xs), Xs).
select(X, .(Y, Ys), .(Y, Zs)) :- select(X, Ys, Zs).
members(.(X, Xs), Ys) :- ','(member(X, Ys), members(Xs, Ys)).
members([], Ys).
member(X, .(X, X1)).
member(X, .(X2, T)) :- member(X, T).
test_color(Name, Map) :- ','(map(Name, Map), ','(colors(Name, Colors), color_map(Map, Colors))).
map(test, .(region(a, A, .(B, .(C, .(D, [])))), .(region(b, B, .(A, .(C, .(E, [])))), .(region(c, C, .(A, .(B, .(D, .(E, .(F, [])))))), .(region(d, D, .(A, .(C, .(F, [])))), .(region(e, E, .(B, .(C, .(F, [])))), .(region(f, F, .(C, .(D, .(E, [])))), []))))))).
map(west_europe, .(region(portugal, P, .(E, [])), .(region(spain, E, .(F, .(P, []))), .(region(france, F, .(E, .(I, .(S, .(B, .(WG, .(L, []))))))), .(region(belgium, B, .(F, .(H, .(L, .(WG, []))))), .(region(holland, H, .(B, .(WG, []))), .(region(west_germany, WG, .(F, .(A, .(S, .(H, .(B, .(L, []))))))), .(region(luxembourg, L, .(F, .(B, .(WG, [])))), .(region(italy, I, .(F, .(A, .(S, [])))), .(region(switzerland, S, .(F, .(I, .(A, .(WG, []))))), .(region(austria, A, .(I, .(S, .(WG, [])))), []))))))))))).
colors(X, .(red, .(yellow, .(blue, .(white, []))))).

Queries:

color_map(a,g).

(1) 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(.(region(T27, T31, T32), T33), T30) :- select10(T31, T30, X34).
color_map1(.(region(T27, T31, T39), T40), T30) :- ','(select10(T31, T30, T38), members20(T39, T38)).
color_map1(.(region(T27, T31, T39), T72), T30) :- ','(select10(T31, T30, T38), ','(members20(T39, T38), color_map1(T72, T30))).
color_map1([], T137).
color_map1([], 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(.(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(.(region(T27, T31, T32), T33), T30)
color_map1_in_ag(.(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(.(region(T27, T31, T39), T40), T30)
color_map1_in_ag(.(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([], T137) → color_map1_out_ag([], T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(.(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(.(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(.(region(T27, T31, T32), T33), T30)
color_map1_in_ag(.(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(.(region(T27, T31, T39), T40), T30)
color_map1_in_ag(.(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([], T137) → color_map1_out_ag([], T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(.(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(.(region(T27, T31, T32), T33), T30) → U6_AG(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
COLOR_MAP1_IN_AG(.(region(T27, T31, T32), T33), T30) → SELECT10_IN_AGA(T31, T30, X34)
SELECT10_IN_AGA(T64, .(T62, T63), .(T62, X73)) → U1_AGA(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
SELECT10_IN_AGA(T64, .(T62, T63), .(T62, X73)) → SELECT10_IN_AGA(T64, T63, X73)
COLOR_MAP1_IN_AG(.(region(T27, T31, T39), T40), T30) → U7_AG(T27, T31, T39, T40, T30, 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(.(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(.(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(.(region(T27, T31, T32), T33), T30)
color_map1_in_ag(.(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(.(region(T27, T31, T39), T40), T30)
color_map1_in_ag(.(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([], T137) → color_map1_out_ag([], T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(.(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(.(region(T27, T31, T32), T33), T30) → U6_AG(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
COLOR_MAP1_IN_AG(.(region(T27, T31, T32), T33), T30) → SELECT10_IN_AGA(T31, T30, X34)
SELECT10_IN_AGA(T64, .(T62, T63), .(T62, X73)) → U1_AGA(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
SELECT10_IN_AGA(T64, .(T62, T63), .(T62, X73)) → SELECT10_IN_AGA(T64, T63, X73)
COLOR_MAP1_IN_AG(.(region(T27, T31, T39), T40), T30) → U7_AG(T27, T31, T39, T40, T30, 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(.(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(.(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(.(region(T27, T31, T32), T33), T30)
color_map1_in_ag(.(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(.(region(T27, T31, T39), T40), T30)
color_map1_in_ag(.(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([], T137) → color_map1_out_ag([], T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(.(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(.(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(.(region(T27, T31, T32), T33), T30)
color_map1_in_ag(.(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(.(region(T27, T31, T39), T40), T30)
color_map1_in_ag(.(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([], T137) → color_map1_out_ag([], T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(.(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(.(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(.(region(T27, T31, T32), T33), T30)
color_map1_in_ag(.(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(.(region(T27, T31, T39), T40), T30)
color_map1_in_ag(.(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([], T137) → color_map1_out_ag([], T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(.(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(.(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(.(region(T27, T31, T32), T33), T30)
color_map1_in_ag(.(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(.(region(T27, T31, T39), T40), T30)
color_map1_in_ag(.(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([], T137) → color_map1_out_ag([], T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(.(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(.(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(.(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(.(region(T27, T31, T32), T33), T30)
color_map1_in_ag(.(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(.(region(T27, T31, T39), T40), T30)
color_map1_in_ag(.(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([], T137) → color_map1_out_ag([], T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(.(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(.(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:

• Semiunifier: [ ]
• Matcher: [ ]

---

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(.(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(.(region(T27, T31, T32), T33), T30)
color_map1_in_ag(.(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(.(region(T27, T31, T39), T40), T30)
color_map1_in_ag(.(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([], T137) → color_map1_out_ag([], T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(.(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(.(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(.(region(T27, T31, T32), T33), T30)
color_map1_in_ag(.(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(.(region(T27, T31, T39), T40), T30)
color_map1_in_ag(.(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([], T137) → color_map1_out_ag([], T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(.(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(.(region(T27, T31, T32), T33), T30) → U6_AG(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
COLOR_MAP1_IN_AG(.(region(T27, T31, T32), T33), T30) → SELECT10_IN_AGA(T31, T30, X34)
SELECT10_IN_AGA(T64, .(T62, T63), .(T62, X73)) → U1_AGA(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
SELECT10_IN_AGA(T64, .(T62, T63), .(T62, X73)) → SELECT10_IN_AGA(T64, T63, X73)
COLOR_MAP1_IN_AG(.(region(T27, T31, T39), T40), T30) → U7_AG(T27, T31, T39, T40, T30, 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(.(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(.(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(.(region(T27, T31, T32), T33), T30)
color_map1_in_ag(.(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(.(region(T27, T31, T39), T40), T30)
color_map1_in_ag(.(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([], T137) → color_map1_out_ag([], T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(.(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(.(region(T27, T31, T32), T33), T30) → U6_AG(T27, T31, T32, T33, T30, select10_in_aga(T31, T30, X34))
COLOR_MAP1_IN_AG(.(region(T27, T31, T32), T33), T30) → SELECT10_IN_AGA(T31, T30, X34)
SELECT10_IN_AGA(T64, .(T62, T63), .(T62, X73)) → U1_AGA(T64, T62, T63, X73, select10_in_aga(T64, T63, X73))
SELECT10_IN_AGA(T64, .(T62, T63), .(T62, X73)) → SELECT10_IN_AGA(T64, T63, X73)
COLOR_MAP1_IN_AG(.(region(T27, T31, T39), T40), T30) → U7_AG(T27, T31, T39, T40, T30, 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(.(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(.(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(.(region(T27, T31, T32), T33), T30)
color_map1_in_ag(.(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(.(region(T27, T31, T39), T40), T30)
color_map1_in_ag(.(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([], T137) → color_map1_out_ag([], T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(.(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(.(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(.(region(T27, T31, T32), T33), T30)
color_map1_in_ag(.(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(.(region(T27, T31, T39), T40), T30)
color_map1_in_ag(.(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([], T137) → color_map1_out_ag([], T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(.(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(.(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(.(region(T27, T31, T32), T33), T30)
color_map1_in_ag(.(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(.(region(T27, T31, T39), T40), T30)
color_map1_in_ag(.(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([], T137) → color_map1_out_ag([], T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(.(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(.(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(.(region(T27, T31, T32), T33), T30)
color_map1_in_ag(.(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(.(region(T27, T31, T39), T40), T30)
color_map1_in_ag(.(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([], T137) → color_map1_out_ag([], T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(.(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(.(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(.(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(.(region(T27, T31, T32), T33), T30)
color_map1_in_ag(.(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(.(region(T27, T31, T39), T40), T30)
color_map1_in_ag(.(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([], T137) → color_map1_out_ag([], T137)
U11_ag(T27, T31, T39, T72, T30, color_map1_out_ag(T72, T30)) → color_map1_out_ag(.(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(.(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.

(102) NO