(0) Obligation:
Clauses:color_map(.(Region, Regions), Colors) :- ','(color_region(Region, Colors), color_map(Regions, Colors)).
color_map([], Colors).
color_region(region(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, Pairs) :- ','(colors(Name, Colors), ','(color_map(Map, Colors), ','(map(Name, Symbols, Map), symbols(Symbols, Map, Pairs)))).
symbols([], [], []).
symbols(.(S, Ss), .(region(C, N), Rs), .(pair(S, C), Ps)) :- symbols(Ss, Rs, Ps).
map(test, .(a, .(b, .(c, .(d, .(e, .(f, [])))))), .(region(A, .(B, .(C, .(D, [])))), .(region(B, .(A, .(C, .(E, [])))), .(region(C, .(A, .(B, .(D, .(E, .(F, [])))))), .(region(D, .(A, .(C, .(F, [])))), .(region(E, .(B, .(C, .(F, [])))), .(region(F, .(C, .(D, .(E, [])))), []))))))).
map(west_europe, .(portugal, .(spain, .(france, .(belgium, .(holland, .(west_germany, .(luxembourg, .(italy, .(switzerland, .(austria, [])))))))))), .(region(P, .(E, [])), .(region(E, .(F, .(P, []))), .(region(F, .(E, .(I, .(S, .(B, .(WG, .(L, []))))))), .(region(B, .(F, .(H, .(L, .(WG, []))))), .(region(H, .(B, .(WG, []))), .(region(WG, .(F, .(A, .(S, .(H, .(B, .(L, []))))))), .(region(L, .(F, .(B, .(WG, [])))), .(region(I, .(F, .(A, .(S, [])))), .(region(S, .(F, .(I, .(A, .(WG, []))))), .(region(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(T48, .(T48, T49), T49).
select10(T59, .(T57, T58), .(T57, X67)) :- select10(T59, T58, X67).
members20(.(T86, T87), T85) :- member27(T86, T85).
members20(.(T86, T92), T85) :- ','(member27(T86, T85), members20(T92, T85)).
members20([], T126).
member27(T105, .(T105, T106)).
member27(T116, .(T114, T115)) :- member27(T116, T115).
color_map1(.(region(T26, T27), T28), T25) :- select10(T26, T25, X29).
color_map1(.(region(T26, T34), T35), T25) :- ','(select10(T26, T25, T33), members20(T34, T33)).
color_map1(.(region(T26, T34), T67), T25) :- ','(select10(T26, T25, T33), ','(members20(T34, T33), color_map1(T67, T25))).
color_map1([], T132).
color_map1([], T134).
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(T26, T27), T28), T25) → U6_ag(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
select10_in_aga(T48, .(T48, T49), T49) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(T59, .(T57, T58), .(T57, X67)) → U1_aga(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
U1_aga(T59, T57, T58, X67, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U6_ag(T26, T27, T28, T25, select10_out_aga(T26, T25, X29)) → color_map1_out_ag(.(region(T26, T27), T28), T25)
color_map1_in_ag(.(region(T26, T34), T35), T25) → U7_ag(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_ag(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_ag(T26, T34, T35, T25, members20_in_ag(T34, T33))
members20_in_ag(.(T86, T87), T85) → U2_ag(T86, T87, T85, member27_in_ag(T86, T85))
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
U2_ag(T86, T87, T85, member27_out_ag(T86, T85)) → members20_out_ag(.(T86, T87), T85)
members20_in_ag(.(T86, T92), T85) → U3_ag(T86, T92, T85, member27_in_ag(T86, T85))
U3_ag(T86, T92, T85, member27_out_ag(T86, T85)) → U4_ag(T86, T92, T85, members20_in_ag(T92, T85))
members20_in_ag([], T126) → members20_out_ag([], T126)
U4_ag(T86, T92, T85, members20_out_ag(T92, T85)) → members20_out_ag(.(T86, T92), T85)
U8_ag(T26, T34, T35, T25, members20_out_ag(T34, T33)) → color_map1_out_ag(.(region(T26, T34), T35), T25)
color_map1_in_ag(.(region(T26, T34), T67), T25) → U9_ag(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_ag(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_ag(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_ag(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_ag(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
color_map1_in_ag([], T132) → color_map1_out_ag([], T132)
U11_ag(T26, T34, T67, T25, color_map1_out_ag(T67, T25)) → color_map1_out_ag(.(region(T26, T34), T67), T25)
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) = U6_ag(x4, x5)
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)
U7_ag(x1, x2, x3, x4, x5) = U7_ag(x4, x5)
U8_ag(x1, x2, x3, x4, x5) = U8_ag(x4, x5)
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) = U9_ag(x4, x5)
U10_ag(x1, x2, x3, x4, x5) = U10_ag(x4, x5)
U11_ag(x1, x2, x3, x4, x5) = U11_ag(x4, x5)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(4) Obligation:
Pi-finite rewrite system:The TRS R consists of the following rules:
color_map1_in_ag(.(region(T26, T27), T28), T25) → U6_ag(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
select10_in_aga(T48, .(T48, T49), T49) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(T59, .(T57, T58), .(T57, X67)) → U1_aga(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
U1_aga(T59, T57, T58, X67, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U6_ag(T26, T27, T28, T25, select10_out_aga(T26, T25, X29)) → color_map1_out_ag(.(region(T26, T27), T28), T25)
color_map1_in_ag(.(region(T26, T34), T35), T25) → U7_ag(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_ag(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_ag(T26, T34, T35, T25, members20_in_ag(T34, T33))
members20_in_ag(.(T86, T87), T85) → U2_ag(T86, T87, T85, member27_in_ag(T86, T85))
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
U2_ag(T86, T87, T85, member27_out_ag(T86, T85)) → members20_out_ag(.(T86, T87), T85)
members20_in_ag(.(T86, T92), T85) → U3_ag(T86, T92, T85, member27_in_ag(T86, T85))
U3_ag(T86, T92, T85, member27_out_ag(T86, T85)) → U4_ag(T86, T92, T85, members20_in_ag(T92, T85))
members20_in_ag([], T126) → members20_out_ag([], T126)
U4_ag(T86, T92, T85, members20_out_ag(T92, T85)) → members20_out_ag(.(T86, T92), T85)
U8_ag(T26, T34, T35, T25, members20_out_ag(T34, T33)) → color_map1_out_ag(.(region(T26, T34), T35), T25)
color_map1_in_ag(.(region(T26, T34), T67), T25) → U9_ag(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_ag(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_ag(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_ag(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_ag(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
color_map1_in_ag([], T132) → color_map1_out_ag([], T132)
U11_ag(T26, T34, T67, T25, color_map1_out_ag(T67, T25)) → color_map1_out_ag(.(region(T26, T34), T67), T25)
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) = U6_ag(x4, x5)
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)
U7_ag(x1, x2, x3, x4, x5) = U7_ag(x4, x5)
U8_ag(x1, x2, x3, x4, x5) = U8_ag(x4, x5)
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) = U9_ag(x4, x5)
U10_ag(x1, x2, x3, x4, x5) = U10_ag(x4, x5)
U11_ag(x1, x2, x3, x4, x5) = U11_ag(x4, x5)
(5) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:Pi DP problem:
The TRS P consists of the following rules:
COLOR_MAP1_IN_AG(.(region(T26, T27), T28), T25) → U6_AG(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
COLOR_MAP1_IN_AG(.(region(T26, T27), T28), T25) → SELECT10_IN_AGA(T26, T25, X29)
SELECT10_IN_AGA(T59, .(T57, T58), .(T57, X67)) → U1_AGA(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
SELECT10_IN_AGA(T59, .(T57, T58), .(T57, X67)) → SELECT10_IN_AGA(T59, T58, X67)
COLOR_MAP1_IN_AG(.(region(T26, T34), T35), T25) → U7_AG(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_AG(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_AG(T26, T34, T35, T25, members20_in_ag(T34, T33))
U7_AG(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → MEMBERS20_IN_AG(T34, T33)
MEMBERS20_IN_AG(.(T86, T87), T85) → U2_AG(T86, T87, T85, member27_in_ag(T86, T85))
MEMBERS20_IN_AG(.(T86, T87), T85) → MEMBER27_IN_AG(T86, T85)
MEMBER27_IN_AG(T116, .(T114, T115)) → U5_AG(T116, T114, T115, member27_in_ag(T116, T115))
MEMBER27_IN_AG(T116, .(T114, T115)) → MEMBER27_IN_AG(T116, T115)
MEMBERS20_IN_AG(.(T86, T92), T85) → U3_AG(T86, T92, T85, member27_in_ag(T86, T85))
U3_AG(T86, T92, T85, member27_out_ag(T86, T85)) → U4_AG(T86, T92, T85, members20_in_ag(T92, T85))
U3_AG(T86, T92, T85, member27_out_ag(T86, T85)) → MEMBERS20_IN_AG(T92, T85)
COLOR_MAP1_IN_AG(.(region(T26, T34), T67), T25) → U9_AG(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_AG(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_AG(T26, T34, T67, T25, members20_in_ag(T34, T33))
U9_AG(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → MEMBERS20_IN_AG(T34, T33)
U10_AG(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_AG(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
U10_AG(T26, T34, T67, T25, members20_out_ag(T34, T33)) → COLOR_MAP1_IN_AG(T67, T25)
The TRS R consists of the following rules:
color_map1_in_ag(.(region(T26, T27), T28), T25) → U6_ag(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
select10_in_aga(T48, .(T48, T49), T49) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(T59, .(T57, T58), .(T57, X67)) → U1_aga(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
U1_aga(T59, T57, T58, X67, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U6_ag(T26, T27, T28, T25, select10_out_aga(T26, T25, X29)) → color_map1_out_ag(.(region(T26, T27), T28), T25)
color_map1_in_ag(.(region(T26, T34), T35), T25) → U7_ag(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_ag(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_ag(T26, T34, T35, T25, members20_in_ag(T34, T33))
members20_in_ag(.(T86, T87), T85) → U2_ag(T86, T87, T85, member27_in_ag(T86, T85))
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
U2_ag(T86, T87, T85, member27_out_ag(T86, T85)) → members20_out_ag(.(T86, T87), T85)
members20_in_ag(.(T86, T92), T85) → U3_ag(T86, T92, T85, member27_in_ag(T86, T85))
U3_ag(T86, T92, T85, member27_out_ag(T86, T85)) → U4_ag(T86, T92, T85, members20_in_ag(T92, T85))
members20_in_ag([], T126) → members20_out_ag([], T126)
U4_ag(T86, T92, T85, members20_out_ag(T92, T85)) → members20_out_ag(.(T86, T92), T85)
U8_ag(T26, T34, T35, T25, members20_out_ag(T34, T33)) → color_map1_out_ag(.(region(T26, T34), T35), T25)
color_map1_in_ag(.(region(T26, T34), T67), T25) → U9_ag(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_ag(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_ag(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_ag(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_ag(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
color_map1_in_ag([], T132) → color_map1_out_ag([], T132)
U11_ag(T26, T34, T67, T25, color_map1_out_ag(T67, T25)) → color_map1_out_ag(.(region(T26, T34), T67), T25)
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) = U6_ag(x4, x5)
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)
U7_ag(x1, x2, x3, x4, x5) = U7_ag(x4, x5)
U8_ag(x1, x2, x3, x4, x5) = U8_ag(x4, x5)
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) = U9_ag(x4, x5)
U10_ag(x1, x2, x3, x4, x5) = U10_ag(x4, x5)
U11_ag(x1, x2, x3, x4, x5) = U11_ag(x4, x5)
COLOR_MAP1_IN_AG(x1, x2) = COLOR_MAP1_IN_AG(x2)
U6_AG(x1, x2, x3, x4, x5) = U6_AG(x4, x5)
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) = U7_AG(x4, x5)
U8_AG(x1, x2, x3, x4, x5) = U8_AG(x4, x5)
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) = U9_AG(x4, x5)
U10_AG(x1, x2, x3, x4, x5) = U10_AG(x4, x5)
U11_AG(x1, x2, x3, x4, x5) = U11_AG(x4, x5)
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(T26, T27), T28), T25) → U6_AG(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
COLOR_MAP1_IN_AG(.(region(T26, T27), T28), T25) → SELECT10_IN_AGA(T26, T25, X29)
SELECT10_IN_AGA(T59, .(T57, T58), .(T57, X67)) → U1_AGA(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
SELECT10_IN_AGA(T59, .(T57, T58), .(T57, X67)) → SELECT10_IN_AGA(T59, T58, X67)
COLOR_MAP1_IN_AG(.(region(T26, T34), T35), T25) → U7_AG(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_AG(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_AG(T26, T34, T35, T25, members20_in_ag(T34, T33))
U7_AG(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → MEMBERS20_IN_AG(T34, T33)
MEMBERS20_IN_AG(.(T86, T87), T85) → U2_AG(T86, T87, T85, member27_in_ag(T86, T85))
MEMBERS20_IN_AG(.(T86, T87), T85) → MEMBER27_IN_AG(T86, T85)
MEMBER27_IN_AG(T116, .(T114, T115)) → U5_AG(T116, T114, T115, member27_in_ag(T116, T115))
MEMBER27_IN_AG(T116, .(T114, T115)) → MEMBER27_IN_AG(T116, T115)
MEMBERS20_IN_AG(.(T86, T92), T85) → U3_AG(T86, T92, T85, member27_in_ag(T86, T85))
U3_AG(T86, T92, T85, member27_out_ag(T86, T85)) → U4_AG(T86, T92, T85, members20_in_ag(T92, T85))
U3_AG(T86, T92, T85, member27_out_ag(T86, T85)) → MEMBERS20_IN_AG(T92, T85)
COLOR_MAP1_IN_AG(.(region(T26, T34), T67), T25) → U9_AG(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_AG(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_AG(T26, T34, T67, T25, members20_in_ag(T34, T33))
U9_AG(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → MEMBERS20_IN_AG(T34, T33)
U10_AG(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_AG(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
U10_AG(T26, T34, T67, T25, members20_out_ag(T34, T33)) → COLOR_MAP1_IN_AG(T67, T25)
The TRS R consists of the following rules:
color_map1_in_ag(.(region(T26, T27), T28), T25) → U6_ag(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
select10_in_aga(T48, .(T48, T49), T49) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(T59, .(T57, T58), .(T57, X67)) → U1_aga(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
U1_aga(T59, T57, T58, X67, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U6_ag(T26, T27, T28, T25, select10_out_aga(T26, T25, X29)) → color_map1_out_ag(.(region(T26, T27), T28), T25)
color_map1_in_ag(.(region(T26, T34), T35), T25) → U7_ag(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_ag(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_ag(T26, T34, T35, T25, members20_in_ag(T34, T33))
members20_in_ag(.(T86, T87), T85) → U2_ag(T86, T87, T85, member27_in_ag(T86, T85))
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
U2_ag(T86, T87, T85, member27_out_ag(T86, T85)) → members20_out_ag(.(T86, T87), T85)
members20_in_ag(.(T86, T92), T85) → U3_ag(T86, T92, T85, member27_in_ag(T86, T85))
U3_ag(T86, T92, T85, member27_out_ag(T86, T85)) → U4_ag(T86, T92, T85, members20_in_ag(T92, T85))
members20_in_ag([], T126) → members20_out_ag([], T126)
U4_ag(T86, T92, T85, members20_out_ag(T92, T85)) → members20_out_ag(.(T86, T92), T85)
U8_ag(T26, T34, T35, T25, members20_out_ag(T34, T33)) → color_map1_out_ag(.(region(T26, T34), T35), T25)
color_map1_in_ag(.(region(T26, T34), T67), T25) → U9_ag(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_ag(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_ag(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_ag(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_ag(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
color_map1_in_ag([], T132) → color_map1_out_ag([], T132)
U11_ag(T26, T34, T67, T25, color_map1_out_ag(T67, T25)) → color_map1_out_ag(.(region(T26, T34), T67), T25)
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) = U6_ag(x4, x5)
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)
U7_ag(x1, x2, x3, x4, x5) = U7_ag(x4, x5)
U8_ag(x1, x2, x3, x4, x5) = U8_ag(x4, x5)
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) = U9_ag(x4, x5)
U10_ag(x1, x2, x3, x4, x5) = U10_ag(x4, x5)
U11_ag(x1, x2, x3, x4, x5) = U11_ag(x4, x5)
COLOR_MAP1_IN_AG(x1, x2) = COLOR_MAP1_IN_AG(x2)
U6_AG(x1, x2, x3, x4, x5) = U6_AG(x4, x5)
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) = U7_AG(x4, x5)
U8_AG(x1, x2, x3, x4, x5) = U8_AG(x4, x5)
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) = U9_AG(x4, x5)
U10_AG(x1, x2, x3, x4, x5) = U10_AG(x4, x5)
U11_AG(x1, x2, x3, x4, x5) = U11_AG(x4, x5)
We have to consider all (P,R,Pi)-chains
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 12 less nodes.(8) Complex Obligation (AND)
(9) Obligation:
Pi DP problem:The TRS P consists of the following rules:
MEMBER27_IN_AG(T116, .(T114, T115)) → MEMBER27_IN_AG(T116, T115)
The TRS R consists of the following rules:
color_map1_in_ag(.(region(T26, T27), T28), T25) → U6_ag(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
select10_in_aga(T48, .(T48, T49), T49) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(T59, .(T57, T58), .(T57, X67)) → U1_aga(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
U1_aga(T59, T57, T58, X67, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U6_ag(T26, T27, T28, T25, select10_out_aga(T26, T25, X29)) → color_map1_out_ag(.(region(T26, T27), T28), T25)
color_map1_in_ag(.(region(T26, T34), T35), T25) → U7_ag(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_ag(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_ag(T26, T34, T35, T25, members20_in_ag(T34, T33))
members20_in_ag(.(T86, T87), T85) → U2_ag(T86, T87, T85, member27_in_ag(T86, T85))
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
U2_ag(T86, T87, T85, member27_out_ag(T86, T85)) → members20_out_ag(.(T86, T87), T85)
members20_in_ag(.(T86, T92), T85) → U3_ag(T86, T92, T85, member27_in_ag(T86, T85))
U3_ag(T86, T92, T85, member27_out_ag(T86, T85)) → U4_ag(T86, T92, T85, members20_in_ag(T92, T85))
members20_in_ag([], T126) → members20_out_ag([], T126)
U4_ag(T86, T92, T85, members20_out_ag(T92, T85)) → members20_out_ag(.(T86, T92), T85)
U8_ag(T26, T34, T35, T25, members20_out_ag(T34, T33)) → color_map1_out_ag(.(region(T26, T34), T35), T25)
color_map1_in_ag(.(region(T26, T34), T67), T25) → U9_ag(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_ag(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_ag(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_ag(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_ag(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
color_map1_in_ag([], T132) → color_map1_out_ag([], T132)
U11_ag(T26, T34, T67, T25, color_map1_out_ag(T67, T25)) → color_map1_out_ag(.(region(T26, T34), T67), T25)
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) = U6_ag(x4, x5)
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)
U7_ag(x1, x2, x3, x4, x5) = U7_ag(x4, x5)
U8_ag(x1, x2, x3, x4, x5) = U8_ag(x4, x5)
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) = U9_ag(x4, x5)
U10_ag(x1, x2, x3, x4, x5) = U10_ag(x4, x5)
U11_ag(x1, x2, x3, x4, x5) = U11_ag(x4, x5)
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(T116, .(T114, T115)) → MEMBER27_IN_AG(T116, T115)
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(.(T114, T115)) → MEMBER27_IN_AG(T115)
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(.(T114, T115)) → MEMBER27_IN_AG(T115)
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(.(T86, T92), T85) → U3_AG(T86, T92, T85, member27_in_ag(T86, T85))
U3_AG(T86, T92, T85, member27_out_ag(T86, T85)) → MEMBERS20_IN_AG(T92, T85)
The TRS R consists of the following rules:
color_map1_in_ag(.(region(T26, T27), T28), T25) → U6_ag(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
select10_in_aga(T48, .(T48, T49), T49) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(T59, .(T57, T58), .(T57, X67)) → U1_aga(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
U1_aga(T59, T57, T58, X67, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U6_ag(T26, T27, T28, T25, select10_out_aga(T26, T25, X29)) → color_map1_out_ag(.(region(T26, T27), T28), T25)
color_map1_in_ag(.(region(T26, T34), T35), T25) → U7_ag(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_ag(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_ag(T26, T34, T35, T25, members20_in_ag(T34, T33))
members20_in_ag(.(T86, T87), T85) → U2_ag(T86, T87, T85, member27_in_ag(T86, T85))
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
U2_ag(T86, T87, T85, member27_out_ag(T86, T85)) → members20_out_ag(.(T86, T87), T85)
members20_in_ag(.(T86, T92), T85) → U3_ag(T86, T92, T85, member27_in_ag(T86, T85))
U3_ag(T86, T92, T85, member27_out_ag(T86, T85)) → U4_ag(T86, T92, T85, members20_in_ag(T92, T85))
members20_in_ag([], T126) → members20_out_ag([], T126)
U4_ag(T86, T92, T85, members20_out_ag(T92, T85)) → members20_out_ag(.(T86, T92), T85)
U8_ag(T26, T34, T35, T25, members20_out_ag(T34, T33)) → color_map1_out_ag(.(region(T26, T34), T35), T25)
color_map1_in_ag(.(region(T26, T34), T67), T25) → U9_ag(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_ag(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_ag(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_ag(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_ag(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
color_map1_in_ag([], T132) → color_map1_out_ag([], T132)
U11_ag(T26, T34, T67, T25, color_map1_out_ag(T67, T25)) → color_map1_out_ag(.(region(T26, T34), T67), T25)
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) = U6_ag(x4, x5)
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)
U7_ag(x1, x2, x3, x4, x5) = U7_ag(x4, x5)
U8_ag(x1, x2, x3, x4, x5) = U8_ag(x4, x5)
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) = U9_ag(x4, x5)
U10_ag(x1, x2, x3, x4, x5) = U10_ag(x4, x5)
U11_ag(x1, x2, x3, x4, x5) = U11_ag(x4, x5)
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(.(T86, T92), T85) → U3_AG(T86, T92, T85, member27_in_ag(T86, T85))
U3_AG(T86, T92, T85, member27_out_ag(T86, T85)) → MEMBERS20_IN_AG(T92, T85)
The TRS R consists of the following rules:
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
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
(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(T85) → U3_AG(T85, member27_in_ag(T85))
U3_AG(T85, member27_out_ag(T86, T85)) → MEMBERS20_IN_AG(T85)
The TRS R consists of the following rules:
member27_in_ag(.(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(.(T114, T115)) → U5_ag(T114, T115, member27_in_ag(T115))
U5_ag(T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
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.
(21) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule MEMBERS20_IN_AG(T85) → U3_AG(T85, member27_in_ag(T85)) 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)))
(22) Obligation:
Q DP problem:The TRS P consists of the following rules:
U3_AG(T85, member27_out_ag(T86, T85)) → MEMBERS20_IN_AG(T85)
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(.(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(.(T114, T115)) → U5_ag(T114, T115, member27_in_ag(T115))
U5_ag(T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
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.
(23) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U3_AG(T85, member27_out_ag(T86, T85)) → MEMBERS20_IN_AG(T85) 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))
(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, .(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(.(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(.(T114, T115)) → U5_ag(T114, T115, member27_in_ag(T115))
U5_ag(T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
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.
(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, .(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.
(26) NO
(27) Obligation:
Pi DP problem:The TRS P consists of the following rules:
SELECT10_IN_AGA(T59, .(T57, T58), .(T57, X67)) → SELECT10_IN_AGA(T59, T58, X67)
The TRS R consists of the following rules:
color_map1_in_ag(.(region(T26, T27), T28), T25) → U6_ag(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
select10_in_aga(T48, .(T48, T49), T49) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(T59, .(T57, T58), .(T57, X67)) → U1_aga(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
U1_aga(T59, T57, T58, X67, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U6_ag(T26, T27, T28, T25, select10_out_aga(T26, T25, X29)) → color_map1_out_ag(.(region(T26, T27), T28), T25)
color_map1_in_ag(.(region(T26, T34), T35), T25) → U7_ag(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_ag(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_ag(T26, T34, T35, T25, members20_in_ag(T34, T33))
members20_in_ag(.(T86, T87), T85) → U2_ag(T86, T87, T85, member27_in_ag(T86, T85))
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
U2_ag(T86, T87, T85, member27_out_ag(T86, T85)) → members20_out_ag(.(T86, T87), T85)
members20_in_ag(.(T86, T92), T85) → U3_ag(T86, T92, T85, member27_in_ag(T86, T85))
U3_ag(T86, T92, T85, member27_out_ag(T86, T85)) → U4_ag(T86, T92, T85, members20_in_ag(T92, T85))
members20_in_ag([], T126) → members20_out_ag([], T126)
U4_ag(T86, T92, T85, members20_out_ag(T92, T85)) → members20_out_ag(.(T86, T92), T85)
U8_ag(T26, T34, T35, T25, members20_out_ag(T34, T33)) → color_map1_out_ag(.(region(T26, T34), T35), T25)
color_map1_in_ag(.(region(T26, T34), T67), T25) → U9_ag(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_ag(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_ag(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_ag(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_ag(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
color_map1_in_ag([], T132) → color_map1_out_ag([], T132)
U11_ag(T26, T34, T67, T25, color_map1_out_ag(T67, T25)) → color_map1_out_ag(.(region(T26, T34), T67), T25)
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) = U6_ag(x4, x5)
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)
U7_ag(x1, x2, x3, x4, x5) = U7_ag(x4, x5)
U8_ag(x1, x2, x3, x4, x5) = U8_ag(x4, x5)
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) = U9_ag(x4, x5)
U10_ag(x1, x2, x3, x4, x5) = U10_ag(x4, x5)
U11_ag(x1, x2, x3, x4, x5) = U11_ag(x4, x5)
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(T59, .(T57, T58), .(T57, X67)) → SELECT10_IN_AGA(T59, T58, X67)
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(.(T57, T58)) → SELECT10_IN_AGA(T58)
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(.(T57, T58)) → SELECT10_IN_AGA(T58)
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(T26, T34), T67), T25) → U9_AG(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_AG(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_AG(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_AG(T26, T34, T67, T25, members20_out_ag(T34, T33)) → COLOR_MAP1_IN_AG(T67, T25)
The TRS R consists of the following rules:
color_map1_in_ag(.(region(T26, T27), T28), T25) → U6_ag(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
select10_in_aga(T48, .(T48, T49), T49) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(T59, .(T57, T58), .(T57, X67)) → U1_aga(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
U1_aga(T59, T57, T58, X67, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U6_ag(T26, T27, T28, T25, select10_out_aga(T26, T25, X29)) → color_map1_out_ag(.(region(T26, T27), T28), T25)
color_map1_in_ag(.(region(T26, T34), T35), T25) → U7_ag(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_ag(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_ag(T26, T34, T35, T25, members20_in_ag(T34, T33))
members20_in_ag(.(T86, T87), T85) → U2_ag(T86, T87, T85, member27_in_ag(T86, T85))
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
U2_ag(T86, T87, T85, member27_out_ag(T86, T85)) → members20_out_ag(.(T86, T87), T85)
members20_in_ag(.(T86, T92), T85) → U3_ag(T86, T92, T85, member27_in_ag(T86, T85))
U3_ag(T86, T92, T85, member27_out_ag(T86, T85)) → U4_ag(T86, T92, T85, members20_in_ag(T92, T85))
members20_in_ag([], T126) → members20_out_ag([], T126)
U4_ag(T86, T92, T85, members20_out_ag(T92, T85)) → members20_out_ag(.(T86, T92), T85)
U8_ag(T26, T34, T35, T25, members20_out_ag(T34, T33)) → color_map1_out_ag(.(region(T26, T34), T35), T25)
color_map1_in_ag(.(region(T26, T34), T67), T25) → U9_ag(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_ag(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_ag(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_ag(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_ag(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
color_map1_in_ag([], T132) → color_map1_out_ag([], T132)
U11_ag(T26, T34, T67, T25, color_map1_out_ag(T67, T25)) → color_map1_out_ag(.(region(T26, T34), T67), T25)
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) = U6_ag(x4, x5)
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)
U7_ag(x1, x2, x3, x4, x5) = U7_ag(x4, x5)
U8_ag(x1, x2, x3, x4, x5) = U8_ag(x4, x5)
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) = U9_ag(x4, x5)
U10_ag(x1, x2, x3, x4, x5) = U10_ag(x4, x5)
U11_ag(x1, x2, x3, x4, x5) = U11_ag(x4, x5)
COLOR_MAP1_IN_AG(x1, x2) = COLOR_MAP1_IN_AG(x2)
U9_AG(x1, x2, x3, x4, x5) = U9_AG(x4, x5)
U10_AG(x1, x2, x3, x4, x5) = U10_AG(x4, x5)
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(T26, T34), T67), T25) → U9_AG(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_AG(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_AG(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_AG(T26, T34, T67, T25, members20_out_ag(T34, T33)) → COLOR_MAP1_IN_AG(T67, T25)
The TRS R consists of the following rules:
select10_in_aga(T48, .(T48, T49), T49) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(T59, .(T57, T58), .(T57, X67)) → U1_aga(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
members20_in_ag(.(T86, T87), T85) → U2_ag(T86, T87, T85, member27_in_ag(T86, T85))
members20_in_ag(.(T86, T92), T85) → U3_ag(T86, T92, T85, member27_in_ag(T86, T85))
members20_in_ag([], T126) → members20_out_ag([], T126)
U1_aga(T59, T57, T58, X67, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U2_ag(T86, T87, T85, member27_out_ag(T86, T85)) → members20_out_ag(.(T86, T87), T85)
U3_ag(T86, T92, T85, member27_out_ag(T86, T85)) → U4_ag(T86, T92, T85, members20_in_ag(T92, T85))
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U4_ag(T86, T92, T85, members20_out_ag(T92, T85)) → members20_out_ag(.(T86, T92), T85)
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
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)
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) = U9_AG(x4, x5)
U10_AG(x1, x2, x3, x4, x5) = U10_AG(x4, x5)
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(T25) → U9_AG(T25, select10_in_aga(T25))
U9_AG(T25, select10_out_aga(T26, T25, T33)) → U10_AG(T25, members20_in_ag(T33))
U10_AG(T25, members20_out_ag(T33)) → COLOR_MAP1_IN_AG(T25)
The TRS R consists of the following rules:
select10_in_aga(.(T48, T49)) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(.(T57, T58)) → U1_aga(T57, T58, select10_in_aga(T58))
members20_in_ag(T85) → U2_ag(T85, member27_in_ag(T85))
members20_in_ag(T85) → U3_ag(T85, member27_in_ag(T85))
members20_in_ag(T126) → members20_out_ag(T126)
U1_aga(T57, T58, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U2_ag(T85, member27_out_ag(T86, T85)) → members20_out_ag(T85)
U3_ag(T85, member27_out_ag(T86, T85)) → U4_ag(T85, members20_in_ag(T85))
member27_in_ag(.(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(.(T114, T115)) → U5_ag(T114, T115, member27_in_ag(T115))
U4_ag(T85, members20_out_ag(T85)) → members20_out_ag(T85)
U5_ag(T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
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.
(39) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule COLOR_MAP1_IN_AG(T25) → U9_AG(T25, select10_in_aga(T25)) 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)))
(40) Obligation:
Q DP problem:The TRS P consists of the following rules:
U9_AG(T25, select10_out_aga(T26, T25, T33)) → U10_AG(T25, members20_in_ag(T33))
U10_AG(T25, members20_out_ag(T33)) → COLOR_MAP1_IN_AG(T25)
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(.(T48, T49)) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(.(T57, T58)) → U1_aga(T57, T58, select10_in_aga(T58))
members20_in_ag(T85) → U2_ag(T85, member27_in_ag(T85))
members20_in_ag(T85) → U3_ag(T85, member27_in_ag(T85))
members20_in_ag(T126) → members20_out_ag(T126)
U1_aga(T57, T58, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U2_ag(T85, member27_out_ag(T86, T85)) → members20_out_ag(T85)
U3_ag(T85, member27_out_ag(T86, T85)) → U4_ag(T85, members20_in_ag(T85))
member27_in_ag(.(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(.(T114, T115)) → U5_ag(T114, T115, member27_in_ag(T115))
U4_ag(T85, members20_out_ag(T85)) → members20_out_ag(T85)
U5_ag(T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
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.
(41) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule U9_AG(T25, select10_out_aga(T26, T25, T33)) → U10_AG(T25, members20_in_ag(T33)) 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))
(42) Obligation:
Q DP problem:The TRS P consists of the following rules:
U10_AG(T25, members20_out_ag(T33)) → COLOR_MAP1_IN_AG(T25)
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(.(T48, T49)) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(.(T57, T58)) → U1_aga(T57, T58, select10_in_aga(T58))
members20_in_ag(T85) → U2_ag(T85, member27_in_ag(T85))
members20_in_ag(T85) → U3_ag(T85, member27_in_ag(T85))
members20_in_ag(T126) → members20_out_ag(T126)
U1_aga(T57, T58, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U2_ag(T85, member27_out_ag(T86, T85)) → members20_out_ag(T85)
U3_ag(T85, member27_out_ag(T86, T85)) → U4_ag(T85, members20_in_ag(T85))
member27_in_ag(.(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(.(T114, T115)) → U5_ag(T114, T115, member27_in_ag(T115))
U4_ag(T85, members20_out_ag(T85)) → members20_out_ag(T85)
U5_ag(T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
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.
(43) 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)))
(44) Obligation:
Q DP problem:The TRS P consists of the following rules:
U10_AG(T25, members20_out_ag(T33)) → COLOR_MAP1_IN_AG(T25)
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(.(T48, T49)) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(.(T57, T58)) → U1_aga(T57, T58, select10_in_aga(T58))
members20_in_ag(T85) → U2_ag(T85, member27_in_ag(T85))
members20_in_ag(T85) → U3_ag(T85, member27_in_ag(T85))
members20_in_ag(T126) → members20_out_ag(T126)
U1_aga(T57, T58, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U2_ag(T85, member27_out_ag(T86, T85)) → members20_out_ag(T85)
U3_ag(T85, member27_out_ag(T86, T85)) → U4_ag(T85, members20_in_ag(T85))
member27_in_ag(.(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(.(T114, T115)) → U5_ag(T114, T115, member27_in_ag(T115))
U4_ag(T85, members20_out_ag(T85)) → members20_out_ag(T85)
U5_ag(T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
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.
(45) 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)))
(46) Obligation:
Q DP problem:The TRS P consists of the following rules:
U10_AG(T25, members20_out_ag(T33)) → COLOR_MAP1_IN_AG(T25)
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(.(T48, T49)) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(.(T57, T58)) → U1_aga(T57, T58, select10_in_aga(T58))
members20_in_ag(T85) → U2_ag(T85, member27_in_ag(T85))
members20_in_ag(T85) → U3_ag(T85, member27_in_ag(T85))
members20_in_ag(T126) → members20_out_ag(T126)
U1_aga(T57, T58, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U2_ag(T85, member27_out_ag(T86, T85)) → members20_out_ag(T85)
U3_ag(T85, member27_out_ag(T86, T85)) → U4_ag(T85, members20_in_ag(T85))
member27_in_ag(.(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(.(T114, T115)) → U5_ag(T114, T115, member27_in_ag(T115))
U4_ag(T85, members20_out_ag(T85)) → members20_out_ag(T85)
U5_ag(T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
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.
(47) 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))
(48) Obligation:
Q DP problem:The TRS P consists of the following rules:
U10_AG(T25, members20_out_ag(T33)) → COLOR_MAP1_IN_AG(T25)
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(.(T48, T49)) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(.(T57, T58)) → U1_aga(T57, T58, select10_in_aga(T58))
members20_in_ag(T85) → U2_ag(T85, member27_in_ag(T85))
members20_in_ag(T85) → U3_ag(T85, member27_in_ag(T85))
members20_in_ag(T126) → members20_out_ag(T126)
U1_aga(T57, T58, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U2_ag(T85, member27_out_ag(T86, T85)) → members20_out_ag(T85)
U3_ag(T85, member27_out_ag(T86, T85)) → U4_ag(T85, members20_in_ag(T85))
member27_in_ag(.(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(.(T114, T115)) → U5_ag(T114, T115, member27_in_ag(T115))
U4_ag(T85, members20_out_ag(T85)) → members20_out_ag(T85)
U5_ag(T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
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.
(49) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U10_AG(T25, members20_out_ag(T33)) → COLOR_MAP1_IN_AG(T25) 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))
(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, .(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(.(T48, T49)) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(.(T57, T58)) → U1_aga(T57, T58, select10_in_aga(T58))
members20_in_ag(T85) → U2_ag(T85, member27_in_ag(T85))
members20_in_ag(T85) → U3_ag(T85, member27_in_ag(T85))
members20_in_ag(T126) → members20_out_ag(T126)
U1_aga(T57, T58, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U2_ag(T85, member27_out_ag(T86, T85)) → members20_out_ag(T85)
U3_ag(T85, member27_out_ag(T86, T85)) → U4_ag(T85, members20_in_ag(T85))
member27_in_ag(.(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(.(T114, T115)) → U5_ag(T114, T115, member27_in_ag(T115))
U4_ag(T85, members20_out_ag(T85)) → members20_out_ag(T85)
U5_ag(T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
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.
(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', .(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.
(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(T26, T27), T28), T25) → U6_ag(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
select10_in_aga(T48, .(T48, T49), T49) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(T59, .(T57, T58), .(T57, X67)) → U1_aga(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
U1_aga(T59, T57, T58, X67, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U6_ag(T26, T27, T28, T25, select10_out_aga(T26, T25, X29)) → color_map1_out_ag(.(region(T26, T27), T28), T25)
color_map1_in_ag(.(region(T26, T34), T35), T25) → U7_ag(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_ag(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_ag(T26, T34, T35, T25, members20_in_ag(T34, T33))
members20_in_ag(.(T86, T87), T85) → U2_ag(T86, T87, T85, member27_in_ag(T86, T85))
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
U2_ag(T86, T87, T85, member27_out_ag(T86, T85)) → members20_out_ag(.(T86, T87), T85)
members20_in_ag(.(T86, T92), T85) → U3_ag(T86, T92, T85, member27_in_ag(T86, T85))
U3_ag(T86, T92, T85, member27_out_ag(T86, T85)) → U4_ag(T86, T92, T85, members20_in_ag(T92, T85))
members20_in_ag([], T126) → members20_out_ag([], T126)
U4_ag(T86, T92, T85, members20_out_ag(T92, T85)) → members20_out_ag(.(T86, T92), T85)
U8_ag(T26, T34, T35, T25, members20_out_ag(T34, T33)) → color_map1_out_ag(.(region(T26, T34), T35), T25)
color_map1_in_ag(.(region(T26, T34), T67), T25) → U9_ag(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_ag(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_ag(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_ag(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_ag(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
color_map1_in_ag([], T132) → color_map1_out_ag([], T132)
U11_ag(T26, T34, T67, T25, color_map1_out_ag(T67, T25)) → color_map1_out_ag(.(region(T26, T34), T67), T25)
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) = U6_ag(x5)
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
U7_ag(x1, x2, x3, x4, x5) = U7_ag(x5)
U8_ag(x1, x2, x3, x4, x5) = U8_ag(x5)
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) = U9_ag(x4, x5)
U10_ag(x1, x2, x3, x4, x5) = U10_ag(x4, x5)
U11_ag(x1, x2, x3, x4, x5) = U11_ag(x5)
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(T26, T27), T28), T25) → U6_ag(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
select10_in_aga(T48, .(T48, T49), T49) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(T59, .(T57, T58), .(T57, X67)) → U1_aga(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
U1_aga(T59, T57, T58, X67, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U6_ag(T26, T27, T28, T25, select10_out_aga(T26, T25, X29)) → color_map1_out_ag(.(region(T26, T27), T28), T25)
color_map1_in_ag(.(region(T26, T34), T35), T25) → U7_ag(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_ag(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_ag(T26, T34, T35, T25, members20_in_ag(T34, T33))
members20_in_ag(.(T86, T87), T85) → U2_ag(T86, T87, T85, member27_in_ag(T86, T85))
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
U2_ag(T86, T87, T85, member27_out_ag(T86, T85)) → members20_out_ag(.(T86, T87), T85)
members20_in_ag(.(T86, T92), T85) → U3_ag(T86, T92, T85, member27_in_ag(T86, T85))
U3_ag(T86, T92, T85, member27_out_ag(T86, T85)) → U4_ag(T86, T92, T85, members20_in_ag(T92, T85))
members20_in_ag([], T126) → members20_out_ag([], T126)
U4_ag(T86, T92, T85, members20_out_ag(T92, T85)) → members20_out_ag(.(T86, T92), T85)
U8_ag(T26, T34, T35, T25, members20_out_ag(T34, T33)) → color_map1_out_ag(.(region(T26, T34), T35), T25)
color_map1_in_ag(.(region(T26, T34), T67), T25) → U9_ag(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_ag(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_ag(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_ag(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_ag(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
color_map1_in_ag([], T132) → color_map1_out_ag([], T132)
U11_ag(T26, T34, T67, T25, color_map1_out_ag(T67, T25)) → color_map1_out_ag(.(region(T26, T34), T67), T25)
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) = U6_ag(x5)
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
U7_ag(x1, x2, x3, x4, x5) = U7_ag(x5)
U8_ag(x1, x2, x3, x4, x5) = U8_ag(x5)
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) = U9_ag(x4, x5)
U10_ag(x1, x2, x3, x4, x5) = U10_ag(x4, x5)
U11_ag(x1, x2, x3, x4, x5) = U11_ag(x5)
(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(T26, T27), T28), T25) → U6_AG(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
COLOR_MAP1_IN_AG(.(region(T26, T27), T28), T25) → SELECT10_IN_AGA(T26, T25, X29)
SELECT10_IN_AGA(T59, .(T57, T58), .(T57, X67)) → U1_AGA(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
SELECT10_IN_AGA(T59, .(T57, T58), .(T57, X67)) → SELECT10_IN_AGA(T59, T58, X67)
COLOR_MAP1_IN_AG(.(region(T26, T34), T35), T25) → U7_AG(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_AG(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_AG(T26, T34, T35, T25, members20_in_ag(T34, T33))
U7_AG(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → MEMBERS20_IN_AG(T34, T33)
MEMBERS20_IN_AG(.(T86, T87), T85) → U2_AG(T86, T87, T85, member27_in_ag(T86, T85))
MEMBERS20_IN_AG(.(T86, T87), T85) → MEMBER27_IN_AG(T86, T85)
MEMBER27_IN_AG(T116, .(T114, T115)) → U5_AG(T116, T114, T115, member27_in_ag(T116, T115))
MEMBER27_IN_AG(T116, .(T114, T115)) → MEMBER27_IN_AG(T116, T115)
MEMBERS20_IN_AG(.(T86, T92), T85) → U3_AG(T86, T92, T85, member27_in_ag(T86, T85))
U3_AG(T86, T92, T85, member27_out_ag(T86, T85)) → U4_AG(T86, T92, T85, members20_in_ag(T92, T85))
U3_AG(T86, T92, T85, member27_out_ag(T86, T85)) → MEMBERS20_IN_AG(T92, T85)
COLOR_MAP1_IN_AG(.(region(T26, T34), T67), T25) → U9_AG(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_AG(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_AG(T26, T34, T67, T25, members20_in_ag(T34, T33))
U9_AG(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → MEMBERS20_IN_AG(T34, T33)
U10_AG(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_AG(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
U10_AG(T26, T34, T67, T25, members20_out_ag(T34, T33)) → COLOR_MAP1_IN_AG(T67, T25)
The TRS R consists of the following rules:
color_map1_in_ag(.(region(T26, T27), T28), T25) → U6_ag(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
select10_in_aga(T48, .(T48, T49), T49) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(T59, .(T57, T58), .(T57, X67)) → U1_aga(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
U1_aga(T59, T57, T58, X67, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U6_ag(T26, T27, T28, T25, select10_out_aga(T26, T25, X29)) → color_map1_out_ag(.(region(T26, T27), T28), T25)
color_map1_in_ag(.(region(T26, T34), T35), T25) → U7_ag(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_ag(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_ag(T26, T34, T35, T25, members20_in_ag(T34, T33))
members20_in_ag(.(T86, T87), T85) → U2_ag(T86, T87, T85, member27_in_ag(T86, T85))
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
U2_ag(T86, T87, T85, member27_out_ag(T86, T85)) → members20_out_ag(.(T86, T87), T85)
members20_in_ag(.(T86, T92), T85) → U3_ag(T86, T92, T85, member27_in_ag(T86, T85))
U3_ag(T86, T92, T85, member27_out_ag(T86, T85)) → U4_ag(T86, T92, T85, members20_in_ag(T92, T85))
members20_in_ag([], T126) → members20_out_ag([], T126)
U4_ag(T86, T92, T85, members20_out_ag(T92, T85)) → members20_out_ag(.(T86, T92), T85)
U8_ag(T26, T34, T35, T25, members20_out_ag(T34, T33)) → color_map1_out_ag(.(region(T26, T34), T35), T25)
color_map1_in_ag(.(region(T26, T34), T67), T25) → U9_ag(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_ag(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_ag(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_ag(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_ag(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
color_map1_in_ag([], T132) → color_map1_out_ag([], T132)
U11_ag(T26, T34, T67, T25, color_map1_out_ag(T67, T25)) → color_map1_out_ag(.(region(T26, T34), T67), T25)
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) = U6_ag(x5)
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
U7_ag(x1, x2, x3, x4, x5) = U7_ag(x5)
U8_ag(x1, x2, x3, x4, x5) = U8_ag(x5)
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) = U9_ag(x4, x5)
U10_ag(x1, x2, x3, x4, x5) = U10_ag(x4, x5)
U11_ag(x1, x2, x3, x4, x5) = U11_ag(x5)
COLOR_MAP1_IN_AG(x1, x2) = COLOR_MAP1_IN_AG(x2)
U6_AG(x1, x2, x3, x4, x5) = U6_AG(x5)
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) = U7_AG(x5)
U8_AG(x1, x2, x3, x4, x5) = U8_AG(x5)
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) = U9_AG(x4, x5)
U10_AG(x1, x2, x3, x4, x5) = U10_AG(x4, x5)
U11_AG(x1, x2, x3, x4, x5) = U11_AG(x5)
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(T26, T27), T28), T25) → U6_AG(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
COLOR_MAP1_IN_AG(.(region(T26, T27), T28), T25) → SELECT10_IN_AGA(T26, T25, X29)
SELECT10_IN_AGA(T59, .(T57, T58), .(T57, X67)) → U1_AGA(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
SELECT10_IN_AGA(T59, .(T57, T58), .(T57, X67)) → SELECT10_IN_AGA(T59, T58, X67)
COLOR_MAP1_IN_AG(.(region(T26, T34), T35), T25) → U7_AG(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_AG(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_AG(T26, T34, T35, T25, members20_in_ag(T34, T33))
U7_AG(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → MEMBERS20_IN_AG(T34, T33)
MEMBERS20_IN_AG(.(T86, T87), T85) → U2_AG(T86, T87, T85, member27_in_ag(T86, T85))
MEMBERS20_IN_AG(.(T86, T87), T85) → MEMBER27_IN_AG(T86, T85)
MEMBER27_IN_AG(T116, .(T114, T115)) → U5_AG(T116, T114, T115, member27_in_ag(T116, T115))
MEMBER27_IN_AG(T116, .(T114, T115)) → MEMBER27_IN_AG(T116, T115)
MEMBERS20_IN_AG(.(T86, T92), T85) → U3_AG(T86, T92, T85, member27_in_ag(T86, T85))
U3_AG(T86, T92, T85, member27_out_ag(T86, T85)) → U4_AG(T86, T92, T85, members20_in_ag(T92, T85))
U3_AG(T86, T92, T85, member27_out_ag(T86, T85)) → MEMBERS20_IN_AG(T92, T85)
COLOR_MAP1_IN_AG(.(region(T26, T34), T67), T25) → U9_AG(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_AG(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_AG(T26, T34, T67, T25, members20_in_ag(T34, T33))
U9_AG(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → MEMBERS20_IN_AG(T34, T33)
U10_AG(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_AG(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
U10_AG(T26, T34, T67, T25, members20_out_ag(T34, T33)) → COLOR_MAP1_IN_AG(T67, T25)
The TRS R consists of the following rules:
color_map1_in_ag(.(region(T26, T27), T28), T25) → U6_ag(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
select10_in_aga(T48, .(T48, T49), T49) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(T59, .(T57, T58), .(T57, X67)) → U1_aga(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
U1_aga(T59, T57, T58, X67, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U6_ag(T26, T27, T28, T25, select10_out_aga(T26, T25, X29)) → color_map1_out_ag(.(region(T26, T27), T28), T25)
color_map1_in_ag(.(region(T26, T34), T35), T25) → U7_ag(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_ag(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_ag(T26, T34, T35, T25, members20_in_ag(T34, T33))
members20_in_ag(.(T86, T87), T85) → U2_ag(T86, T87, T85, member27_in_ag(T86, T85))
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
U2_ag(T86, T87, T85, member27_out_ag(T86, T85)) → members20_out_ag(.(T86, T87), T85)
members20_in_ag(.(T86, T92), T85) → U3_ag(T86, T92, T85, member27_in_ag(T86, T85))
U3_ag(T86, T92, T85, member27_out_ag(T86, T85)) → U4_ag(T86, T92, T85, members20_in_ag(T92, T85))
members20_in_ag([], T126) → members20_out_ag([], T126)
U4_ag(T86, T92, T85, members20_out_ag(T92, T85)) → members20_out_ag(.(T86, T92), T85)
U8_ag(T26, T34, T35, T25, members20_out_ag(T34, T33)) → color_map1_out_ag(.(region(T26, T34), T35), T25)
color_map1_in_ag(.(region(T26, T34), T67), T25) → U9_ag(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_ag(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_ag(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_ag(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_ag(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
color_map1_in_ag([], T132) → color_map1_out_ag([], T132)
U11_ag(T26, T34, T67, T25, color_map1_out_ag(T67, T25)) → color_map1_out_ag(.(region(T26, T34), T67), T25)
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) = U6_ag(x5)
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
U7_ag(x1, x2, x3, x4, x5) = U7_ag(x5)
U8_ag(x1, x2, x3, x4, x5) = U8_ag(x5)
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) = U9_ag(x4, x5)
U10_ag(x1, x2, x3, x4, x5) = U10_ag(x4, x5)
U11_ag(x1, x2, x3, x4, x5) = U11_ag(x5)
COLOR_MAP1_IN_AG(x1, x2) = COLOR_MAP1_IN_AG(x2)
U6_AG(x1, x2, x3, x4, x5) = U6_AG(x5)
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) = U7_AG(x5)
U8_AG(x1, x2, x3, x4, x5) = U8_AG(x5)
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) = U9_AG(x4, x5)
U10_AG(x1, x2, x3, x4, x5) = U10_AG(x4, x5)
U11_AG(x1, x2, x3, x4, x5) = U11_AG(x5)
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(T116, .(T114, T115)) → MEMBER27_IN_AG(T116, T115)
The TRS R consists of the following rules:
color_map1_in_ag(.(region(T26, T27), T28), T25) → U6_ag(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
select10_in_aga(T48, .(T48, T49), T49) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(T59, .(T57, T58), .(T57, X67)) → U1_aga(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
U1_aga(T59, T57, T58, X67, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U6_ag(T26, T27, T28, T25, select10_out_aga(T26, T25, X29)) → color_map1_out_ag(.(region(T26, T27), T28), T25)
color_map1_in_ag(.(region(T26, T34), T35), T25) → U7_ag(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_ag(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_ag(T26, T34, T35, T25, members20_in_ag(T34, T33))
members20_in_ag(.(T86, T87), T85) → U2_ag(T86, T87, T85, member27_in_ag(T86, T85))
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
U2_ag(T86, T87, T85, member27_out_ag(T86, T85)) → members20_out_ag(.(T86, T87), T85)
members20_in_ag(.(T86, T92), T85) → U3_ag(T86, T92, T85, member27_in_ag(T86, T85))
U3_ag(T86, T92, T85, member27_out_ag(T86, T85)) → U4_ag(T86, T92, T85, members20_in_ag(T92, T85))
members20_in_ag([], T126) → members20_out_ag([], T126)
U4_ag(T86, T92, T85, members20_out_ag(T92, T85)) → members20_out_ag(.(T86, T92), T85)
U8_ag(T26, T34, T35, T25, members20_out_ag(T34, T33)) → color_map1_out_ag(.(region(T26, T34), T35), T25)
color_map1_in_ag(.(region(T26, T34), T67), T25) → U9_ag(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_ag(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_ag(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_ag(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_ag(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
color_map1_in_ag([], T132) → color_map1_out_ag([], T132)
U11_ag(T26, T34, T67, T25, color_map1_out_ag(T67, T25)) → color_map1_out_ag(.(region(T26, T34), T67), T25)
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) = U6_ag(x5)
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
U7_ag(x1, x2, x3, x4, x5) = U7_ag(x5)
U8_ag(x1, x2, x3, x4, x5) = U8_ag(x5)
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) = U9_ag(x4, x5)
U10_ag(x1, x2, x3, x4, x5) = U10_ag(x4, x5)
U11_ag(x1, x2, x3, x4, x5) = U11_ag(x5)
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(T116, .(T114, T115)) → MEMBER27_IN_AG(T116, T115)
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(.(T114, T115)) → MEMBER27_IN_AG(T115)
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(.(T114, T115)) → MEMBER27_IN_AG(T115)
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(.(T86, T92), T85) → U3_AG(T86, T92, T85, member27_in_ag(T86, T85))
U3_AG(T86, T92, T85, member27_out_ag(T86, T85)) → MEMBERS20_IN_AG(T92, T85)
The TRS R consists of the following rules:
color_map1_in_ag(.(region(T26, T27), T28), T25) → U6_ag(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
select10_in_aga(T48, .(T48, T49), T49) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(T59, .(T57, T58), .(T57, X67)) → U1_aga(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
U1_aga(T59, T57, T58, X67, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U6_ag(T26, T27, T28, T25, select10_out_aga(T26, T25, X29)) → color_map1_out_ag(.(region(T26, T27), T28), T25)
color_map1_in_ag(.(region(T26, T34), T35), T25) → U7_ag(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_ag(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_ag(T26, T34, T35, T25, members20_in_ag(T34, T33))
members20_in_ag(.(T86, T87), T85) → U2_ag(T86, T87, T85, member27_in_ag(T86, T85))
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
U2_ag(T86, T87, T85, member27_out_ag(T86, T85)) → members20_out_ag(.(T86, T87), T85)
members20_in_ag(.(T86, T92), T85) → U3_ag(T86, T92, T85, member27_in_ag(T86, T85))
U3_ag(T86, T92, T85, member27_out_ag(T86, T85)) → U4_ag(T86, T92, T85, members20_in_ag(T92, T85))
members20_in_ag([], T126) → members20_out_ag([], T126)
U4_ag(T86, T92, T85, members20_out_ag(T92, T85)) → members20_out_ag(.(T86, T92), T85)
U8_ag(T26, T34, T35, T25, members20_out_ag(T34, T33)) → color_map1_out_ag(.(region(T26, T34), T35), T25)
color_map1_in_ag(.(region(T26, T34), T67), T25) → U9_ag(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_ag(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_ag(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_ag(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_ag(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
color_map1_in_ag([], T132) → color_map1_out_ag([], T132)
U11_ag(T26, T34, T67, T25, color_map1_out_ag(T67, T25)) → color_map1_out_ag(.(region(T26, T34), T67), T25)
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) = U6_ag(x5)
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
U7_ag(x1, x2, x3, x4, x5) = U7_ag(x5)
U8_ag(x1, x2, x3, x4, x5) = U8_ag(x5)
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) = U9_ag(x4, x5)
U10_ag(x1, x2, x3, x4, x5) = U10_ag(x4, x5)
U11_ag(x1, x2, x3, x4, x5) = U11_ag(x5)
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(.(T86, T92), T85) → U3_AG(T86, T92, T85, member27_in_ag(T86, T85))
U3_AG(T86, T92, T85, member27_out_ag(T86, T85)) → MEMBERS20_IN_AG(T92, T85)
The TRS R consists of the following rules:
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
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
(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(T85) → U3_AG(T85, member27_in_ag(T85))
U3_AG(T85, member27_out_ag(T86)) → MEMBERS20_IN_AG(T85)
The TRS R consists of the following rules:
member27_in_ag(.(T105, T106)) → member27_out_ag(T105)
member27_in_ag(.(T114, T115)) → U5_ag(member27_in_ag(T115))
U5_ag(member27_out_ag(T116)) → member27_out_ag(T116)
The set Q consists of the following terms:
member27_in_ag(x0)
U5_ag(x0)
We have to consider all (P,Q,R)-chains.
(71) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule MEMBERS20_IN_AG(T85) → U3_AG(T85, member27_in_ag(T85)) 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)))
(72) Obligation:
Q DP problem:The TRS P consists of the following rules:
U3_AG(T85, member27_out_ag(T86)) → MEMBERS20_IN_AG(T85)
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(.(T105, T106)) → member27_out_ag(T105)
member27_in_ag(.(T114, T115)) → U5_ag(member27_in_ag(T115))
U5_ag(member27_out_ag(T116)) → member27_out_ag(T116)
The set Q consists of the following terms:
member27_in_ag(x0)
U5_ag(x0)
We have to consider all (P,Q,R)-chains.
(73) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U3_AG(T85, member27_out_ag(T86)) → MEMBERS20_IN_AG(T85) 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))
(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))
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(.(T105, T106)) → member27_out_ag(T105)
member27_in_ag(.(T114, T115)) → U5_ag(member27_in_ag(T115))
U5_ag(member27_out_ag(T116)) → member27_out_ag(T116)
The set Q consists of the following terms:
member27_in_ag(x0)
U5_ag(x0)
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)) 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.
(76) NO
(77) Obligation:
Pi DP problem:The TRS P consists of the following rules:
SELECT10_IN_AGA(T59, .(T57, T58), .(T57, X67)) → SELECT10_IN_AGA(T59, T58, X67)
The TRS R consists of the following rules:
color_map1_in_ag(.(region(T26, T27), T28), T25) → U6_ag(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
select10_in_aga(T48, .(T48, T49), T49) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(T59, .(T57, T58), .(T57, X67)) → U1_aga(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
U1_aga(T59, T57, T58, X67, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U6_ag(T26, T27, T28, T25, select10_out_aga(T26, T25, X29)) → color_map1_out_ag(.(region(T26, T27), T28), T25)
color_map1_in_ag(.(region(T26, T34), T35), T25) → U7_ag(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_ag(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_ag(T26, T34, T35, T25, members20_in_ag(T34, T33))
members20_in_ag(.(T86, T87), T85) → U2_ag(T86, T87, T85, member27_in_ag(T86, T85))
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
U2_ag(T86, T87, T85, member27_out_ag(T86, T85)) → members20_out_ag(.(T86, T87), T85)
members20_in_ag(.(T86, T92), T85) → U3_ag(T86, T92, T85, member27_in_ag(T86, T85))
U3_ag(T86, T92, T85, member27_out_ag(T86, T85)) → U4_ag(T86, T92, T85, members20_in_ag(T92, T85))
members20_in_ag([], T126) → members20_out_ag([], T126)
U4_ag(T86, T92, T85, members20_out_ag(T92, T85)) → members20_out_ag(.(T86, T92), T85)
U8_ag(T26, T34, T35, T25, members20_out_ag(T34, T33)) → color_map1_out_ag(.(region(T26, T34), T35), T25)
color_map1_in_ag(.(region(T26, T34), T67), T25) → U9_ag(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_ag(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_ag(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_ag(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_ag(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
color_map1_in_ag([], T132) → color_map1_out_ag([], T132)
U11_ag(T26, T34, T67, T25, color_map1_out_ag(T67, T25)) → color_map1_out_ag(.(region(T26, T34), T67), T25)
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) = U6_ag(x5)
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
U7_ag(x1, x2, x3, x4, x5) = U7_ag(x5)
U8_ag(x1, x2, x3, x4, x5) = U8_ag(x5)
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) = U9_ag(x4, x5)
U10_ag(x1, x2, x3, x4, x5) = U10_ag(x4, x5)
U11_ag(x1, x2, x3, x4, x5) = U11_ag(x5)
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(T59, .(T57, T58), .(T57, X67)) → SELECT10_IN_AGA(T59, T58, X67)
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(.(T57, T58)) → SELECT10_IN_AGA(T58)
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(.(T57, T58)) → SELECT10_IN_AGA(T58)
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(T26, T34), T67), T25) → U9_AG(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_AG(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_AG(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_AG(T26, T34, T67, T25, members20_out_ag(T34, T33)) → COLOR_MAP1_IN_AG(T67, T25)
The TRS R consists of the following rules:
color_map1_in_ag(.(region(T26, T27), T28), T25) → U6_ag(T26, T27, T28, T25, select10_in_aga(T26, T25, X29))
select10_in_aga(T48, .(T48, T49), T49) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(T59, .(T57, T58), .(T57, X67)) → U1_aga(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
U1_aga(T59, T57, T58, X67, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U6_ag(T26, T27, T28, T25, select10_out_aga(T26, T25, X29)) → color_map1_out_ag(.(region(T26, T27), T28), T25)
color_map1_in_ag(.(region(T26, T34), T35), T25) → U7_ag(T26, T34, T35, T25, select10_in_aga(T26, T25, T33))
U7_ag(T26, T34, T35, T25, select10_out_aga(T26, T25, T33)) → U8_ag(T26, T34, T35, T25, members20_in_ag(T34, T33))
members20_in_ag(.(T86, T87), T85) → U2_ag(T86, T87, T85, member27_in_ag(T86, T85))
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
U2_ag(T86, T87, T85, member27_out_ag(T86, T85)) → members20_out_ag(.(T86, T87), T85)
members20_in_ag(.(T86, T92), T85) → U3_ag(T86, T92, T85, member27_in_ag(T86, T85))
U3_ag(T86, T92, T85, member27_out_ag(T86, T85)) → U4_ag(T86, T92, T85, members20_in_ag(T92, T85))
members20_in_ag([], T126) → members20_out_ag([], T126)
U4_ag(T86, T92, T85, members20_out_ag(T92, T85)) → members20_out_ag(.(T86, T92), T85)
U8_ag(T26, T34, T35, T25, members20_out_ag(T34, T33)) → color_map1_out_ag(.(region(T26, T34), T35), T25)
color_map1_in_ag(.(region(T26, T34), T67), T25) → U9_ag(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_ag(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_ag(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_ag(T26, T34, T67, T25, members20_out_ag(T34, T33)) → U11_ag(T26, T34, T67, T25, color_map1_in_ag(T67, T25))
color_map1_in_ag([], T132) → color_map1_out_ag([], T132)
U11_ag(T26, T34, T67, T25, color_map1_out_ag(T67, T25)) → color_map1_out_ag(.(region(T26, T34), T67), T25)
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) = U6_ag(x5)
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
U7_ag(x1, x2, x3, x4, x5) = U7_ag(x5)
U8_ag(x1, x2, x3, x4, x5) = U8_ag(x5)
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) = U9_ag(x4, x5)
U10_ag(x1, x2, x3, x4, x5) = U10_ag(x4, x5)
U11_ag(x1, x2, x3, x4, x5) = U11_ag(x5)
COLOR_MAP1_IN_AG(x1, x2) = COLOR_MAP1_IN_AG(x2)
U9_AG(x1, x2, x3, x4, x5) = U9_AG(x4, x5)
U10_AG(x1, x2, x3, x4, x5) = U10_AG(x4, x5)
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(T26, T34), T67), T25) → U9_AG(T26, T34, T67, T25, select10_in_aga(T26, T25, T33))
U9_AG(T26, T34, T67, T25, select10_out_aga(T26, T25, T33)) → U10_AG(T26, T34, T67, T25, members20_in_ag(T34, T33))
U10_AG(T26, T34, T67, T25, members20_out_ag(T34, T33)) → COLOR_MAP1_IN_AG(T67, T25)
The TRS R consists of the following rules:
select10_in_aga(T48, .(T48, T49), T49) → select10_out_aga(T48, .(T48, T49), T49)
select10_in_aga(T59, .(T57, T58), .(T57, X67)) → U1_aga(T59, T57, T58, X67, select10_in_aga(T59, T58, X67))
members20_in_ag(.(T86, T87), T85) → U2_ag(T86, T87, T85, member27_in_ag(T86, T85))
members20_in_ag(.(T86, T92), T85) → U3_ag(T86, T92, T85, member27_in_ag(T86, T85))
members20_in_ag([], T126) → members20_out_ag([], T126)
U1_aga(T59, T57, T58, X67, select10_out_aga(T59, T58, X67)) → select10_out_aga(T59, .(T57, T58), .(T57, X67))
U2_ag(T86, T87, T85, member27_out_ag(T86, T85)) → members20_out_ag(.(T86, T87), T85)
U3_ag(T86, T92, T85, member27_out_ag(T86, T85)) → U4_ag(T86, T92, T85, members20_in_ag(T92, T85))
member27_in_ag(T105, .(T105, T106)) → member27_out_ag(T105, .(T105, T106))
member27_in_ag(T116, .(T114, T115)) → U5_ag(T116, T114, T115, member27_in_ag(T116, T115))
U4_ag(T86, T92, T85, members20_out_ag(T92, T85)) → members20_out_ag(.(T86, T92), T85)
U5_ag(T116, T114, T115, member27_out_ag(T116, T115)) → member27_out_ag(T116, .(T114, T115))
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)
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) = U9_AG(x4, x5)
U10_AG(x1, x2, x3, x4, x5) = U10_AG(x4, x5)
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(T25) → U9_AG(T25, select10_in_aga(T25))
U9_AG(T25, select10_out_aga(T26, T33)) → U10_AG(T25, members20_in_ag(T33))
U10_AG(T25, members20_out_ag) → COLOR_MAP1_IN_AG(T25)
The TRS R consists of the following rules:
select10_in_aga(.(T48, T49)) → select10_out_aga(T48, T49)
select10_in_aga(.(T57, T58)) → U1_aga(T57, select10_in_aga(T58))
members20_in_ag(T85) → U2_ag(member27_in_ag(T85))
members20_in_ag(T85) → U3_ag(T85, member27_in_ag(T85))
members20_in_ag(T126) → members20_out_ag
U1_aga(T57, select10_out_aga(T59, X67)) → select10_out_aga(T59, .(T57, X67))
U2_ag(member27_out_ag(T86)) → members20_out_ag
U3_ag(T85, member27_out_ag(T86)) → U4_ag(members20_in_ag(T85))
member27_in_ag(.(T105, T106)) → member27_out_ag(T105)
member27_in_ag(.(T114, T115)) → U5_ag(member27_in_ag(T115))
U4_ag(members20_out_ag) → members20_out_ag
U5_ag(member27_out_ag(T116)) → member27_out_ag(T116)
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.
(89) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule COLOR_MAP1_IN_AG(T25) → U9_AG(T25, select10_in_aga(T25)) 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)))
(90) Obligation:
Q DP problem:The TRS P consists of the following rules:
U9_AG(T25, select10_out_aga(T26, T33)) → U10_AG(T25, members20_in_ag(T33))
U10_AG(T25, members20_out_ag) → COLOR_MAP1_IN_AG(T25)
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(.(T48, T49)) → select10_out_aga(T48, T49)
select10_in_aga(.(T57, T58)) → U1_aga(T57, select10_in_aga(T58))
members20_in_ag(T85) → U2_ag(member27_in_ag(T85))
members20_in_ag(T85) → U3_ag(T85, member27_in_ag(T85))
members20_in_ag(T126) → members20_out_ag
U1_aga(T57, select10_out_aga(T59, X67)) → select10_out_aga(T59, .(T57, X67))
U2_ag(member27_out_ag(T86)) → members20_out_ag
U3_ag(T85, member27_out_ag(T86)) → U4_ag(members20_in_ag(T85))
member27_in_ag(.(T105, T106)) → member27_out_ag(T105)
member27_in_ag(.(T114, T115)) → U5_ag(member27_in_ag(T115))
U4_ag(members20_out_ag) → members20_out_ag
U5_ag(member27_out_ag(T116)) → member27_out_ag(T116)
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.
(91) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule U9_AG(T25, select10_out_aga(T26, T33)) → U10_AG(T25, members20_in_ag(T33)) 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)
(92) Obligation:
Q DP problem:The TRS P consists of the following rules:
U10_AG(T25, members20_out_ag) → COLOR_MAP1_IN_AG(T25)
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(.(T48, T49)) → select10_out_aga(T48, T49)
select10_in_aga(.(T57, T58)) → U1_aga(T57, select10_in_aga(T58))
members20_in_ag(T85) → U2_ag(member27_in_ag(T85))
members20_in_ag(T85) → U3_ag(T85, member27_in_ag(T85))
members20_in_ag(T126) → members20_out_ag
U1_aga(T57, select10_out_aga(T59, X67)) → select10_out_aga(T59, .(T57, X67))
U2_ag(member27_out_ag(T86)) → members20_out_ag
U3_ag(T85, member27_out_ag(T86)) → U4_ag(members20_in_ag(T85))
member27_in_ag(.(T105, T106)) → member27_out_ag(T105)
member27_in_ag(.(T114, T115)) → U5_ag(member27_in_ag(T115))
U4_ag(members20_out_ag) → members20_out_ag
U5_ag(member27_out_ag(T116)) → member27_out_ag(T116)
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.
(93) 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)))
(94) Obligation:
Q DP problem:The TRS P consists of the following rules:
U10_AG(T25, members20_out_ag) → COLOR_MAP1_IN_AG(T25)
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(.(T48, T49)) → select10_out_aga(T48, T49)
select10_in_aga(.(T57, T58)) → U1_aga(T57, select10_in_aga(T58))
members20_in_ag(T85) → U2_ag(member27_in_ag(T85))
members20_in_ag(T85) → U3_ag(T85, member27_in_ag(T85))
members20_in_ag(T126) → members20_out_ag
U1_aga(T57, select10_out_aga(T59, X67)) → select10_out_aga(T59, .(T57, X67))
U2_ag(member27_out_ag(T86)) → members20_out_ag
U3_ag(T85, member27_out_ag(T86)) → U4_ag(members20_in_ag(T85))
member27_in_ag(.(T105, T106)) → member27_out_ag(T105)
member27_in_ag(.(T114, T115)) → U5_ag(member27_in_ag(T115))
U4_ag(members20_out_ag) → members20_out_ag
U5_ag(member27_out_ag(T116)) → member27_out_ag(T116)
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.
(95) 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)))
(96) Obligation:
Q DP problem:The TRS P consists of the following rules:
U10_AG(T25, members20_out_ag) → COLOR_MAP1_IN_AG(T25)
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(.(T48, T49)) → select10_out_aga(T48, T49)
select10_in_aga(.(T57, T58)) → U1_aga(T57, select10_in_aga(T58))
members20_in_ag(T85) → U2_ag(member27_in_ag(T85))
members20_in_ag(T85) → U3_ag(T85, member27_in_ag(T85))
members20_in_ag(T126) → members20_out_ag
U1_aga(T57, select10_out_aga(T59, X67)) → select10_out_aga(T59, .(T57, X67))
U2_ag(member27_out_ag(T86)) → members20_out_ag
U3_ag(T85, member27_out_ag(T86)) → U4_ag(members20_in_ag(T85))
member27_in_ag(.(T105, T106)) → member27_out_ag(T105)
member27_in_ag(.(T114, T115)) → U5_ag(member27_in_ag(T115))
U4_ag(members20_out_ag) → members20_out_ag
U5_ag(member27_out_ag(T116)) → member27_out_ag(T116)
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.
(97) 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)
(98) Obligation:
Q DP problem:The TRS P consists of the following rules:
U10_AG(T25, members20_out_ag) → COLOR_MAP1_IN_AG(T25)
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(.(T48, T49)) → select10_out_aga(T48, T49)
select10_in_aga(.(T57, T58)) → U1_aga(T57, select10_in_aga(T58))
members20_in_ag(T85) → U2_ag(member27_in_ag(T85))
members20_in_ag(T85) → U3_ag(T85, member27_in_ag(T85))
members20_in_ag(T126) → members20_out_ag
U1_aga(T57, select10_out_aga(T59, X67)) → select10_out_aga(T59, .(T57, X67))
U2_ag(member27_out_ag(T86)) → members20_out_ag
U3_ag(T85, member27_out_ag(T86)) → U4_ag(members20_in_ag(T85))
member27_in_ag(.(T105, T106)) → member27_out_ag(T105)
member27_in_ag(.(T114, T115)) → U5_ag(member27_in_ag(T115))
U4_ag(members20_out_ag) → members20_out_ag
U5_ag(member27_out_ag(T116)) → member27_out_ag(T116)
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.
(99) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule U10_AG(T25, members20_out_ag) → COLOR_MAP1_IN_AG(T25) we obtained the following new rules [LPAR04]:
U10_AG(.(z0, z1), members20_out_ag) → 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, 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(.(T48, T49)) → select10_out_aga(T48, T49)
select10_in_aga(.(T57, T58)) → U1_aga(T57, select10_in_aga(T58))
members20_in_ag(T85) → U2_ag(member27_in_ag(T85))
members20_in_ag(T85) → U3_ag(T85, member27_in_ag(T85))
members20_in_ag(T126) → members20_out_ag
U1_aga(T57, select10_out_aga(T59, X67)) → select10_out_aga(T59, .(T57, X67))
U2_ag(member27_out_ag(T86)) → members20_out_ag
U3_ag(T85, member27_out_ag(T86)) → U4_ag(members20_in_ag(T85))
member27_in_ag(.(T105, T106)) → member27_out_ag(T105)
member27_in_ag(.(T114, T115)) → U5_ag(member27_in_ag(T115))
U4_ag(members20_out_ag) → members20_out_ag
U5_ag(member27_out_ag(T116)) → member27_out_ag(T116)
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.
(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', z1')) evaluates to t =U9_AG(.(z0', z1'), select10_out_aga(z0', z1'))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [ ]
- Semiunifier: [ ]
Rewriting sequence
U9_AG(.(z0', z1'), select10_out_aga(z0', z1')) → U10_AG(.(z0', z1'), members20_out_ag)
with rule U9_AG(.(z0'', z1''), select10_out_aga(z0'', z1'')) → U10_AG(.(z0'', z1''), members20_out_ag) at position [] and matcher [z0'' / z0', z1'' / z1']
U10_AG(.(z0', z1'), members20_out_ag) → COLOR_MAP1_IN_AG(.(z0', z1'))
with rule U10_AG(.(z0, z1), members20_out_ag) → COLOR_MAP1_IN_AG(.(z0, z1)) at position [] and matcher [z0 / z0', z1 / z1']
COLOR_MAP1_IN_AG(.(z0', z1')) → U9_AG(.(z0', z1'), select10_out_aga(z0', z1'))
with rule COLOR_MAP1_IN_AG(.(x0, x1)) → U9_AG(.(x0, x1), select10_out_aga(x0, x1))
Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
All these steps are and every following step will be a correct step w.r.t to Q.