(0) Obligation:
Clauses:append1([], Ys, Ys).
append1(.(X, Xs), Ys, .(X, Zs)) :- append1(Xs, Ys, Zs).
append2([], Ys, Ys).
append2(.(X, Xs), Ys, .(X, Zs)) :- append2(Xs, Ys, Zs).
sublist(X, Y) :- ','(append1(P, X1, Y), append2(X2, X, P)).
Queries:
sublist(g,g).
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph.
(2) Obligation:
Clauses:append117([], T41, T41).
append117(.(X97, X98), X99, .(X97, T44)) :- append117(X98, X99, T44).
append232([], T81, T81).
append232(.(X158, X159), T86, .(X158, T87)) :- append232(X159, T86, T87).
sublist1([], T15).
sublist1(T5, .(X56, T26)) :- append117(X57, X58, T26).
sublist1(.(T65, T66), .(T65, T26)) :- append117(T66, T35, T26).
sublist1(T73, .(X132, T26)) :- ','(append117(T74, T35, T26), append232(X133, T73, T74)).
Queries:
sublist1(g,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:sublist1_in: (b,b)
append117_in: (f,f,b) (b,f,b)
append232_in: (f,b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
sublist1_in_gg([], T15) → sublist1_out_gg([], T15)
sublist1_in_gg(T5, .(X56, T26)) → U3_gg(T5, X56, T26, append117_in_aag(X57, X58, T26))
append117_in_aag([], T41, T41) → append117_out_aag([], T41, T41)
append117_in_aag(.(X97, X98), X99, .(X97, T44)) → U1_aag(X97, X98, X99, T44, append117_in_aag(X98, X99, T44))
U1_aag(X97, X98, X99, T44, append117_out_aag(X98, X99, T44)) → append117_out_aag(.(X97, X98), X99, .(X97, T44))
U3_gg(T5, X56, T26, append117_out_aag(X57, X58, T26)) → sublist1_out_gg(T5, .(X56, T26))
sublist1_in_gg(.(T65, T66), .(T65, T26)) → U4_gg(T65, T66, T26, append117_in_gag(T66, T35, T26))
append117_in_gag([], T41, T41) → append117_out_gag([], T41, T41)
append117_in_gag(.(X97, X98), X99, .(X97, T44)) → U1_gag(X97, X98, X99, T44, append117_in_gag(X98, X99, T44))
U1_gag(X97, X98, X99, T44, append117_out_gag(X98, X99, T44)) → append117_out_gag(.(X97, X98), X99, .(X97, T44))
U4_gg(T65, T66, T26, append117_out_gag(T66, T35, T26)) → sublist1_out_gg(.(T65, T66), .(T65, T26))
sublist1_in_gg(T73, .(X132, T26)) → U5_gg(T73, X132, T26, append117_in_aag(T74, T35, T26))
U5_gg(T73, X132, T26, append117_out_aag(T74, T35, T26)) → U6_gg(T73, X132, T26, append232_in_agg(X133, T73, T74))
append232_in_agg([], T81, T81) → append232_out_agg([], T81, T81)
append232_in_agg(.(X158, X159), T86, .(X158, T87)) → U2_agg(X158, X159, T86, T87, append232_in_agg(X159, T86, T87))
U2_agg(X158, X159, T86, T87, append232_out_agg(X159, T86, T87)) → append232_out_agg(.(X158, X159), T86, .(X158, T87))
U6_gg(T73, X132, T26, append232_out_agg(X133, T73, T74)) → sublist1_out_gg(T73, .(X132, T26))
The argument filtering Pi contains the following mapping:
sublist1_in_gg(x1, x2) = sublist1_in_gg(x1, x2)
[] = []
sublist1_out_gg(x1, x2) = sublist1_out_gg
.(x1, x2) = .(x1, x2)
U3_gg(x1, x2, x3, x4) = U3_gg(x4)
append117_in_aag(x1, x2, x3) = append117_in_aag(x3)
append117_out_aag(x1, x2, x3) = append117_out_aag(x1, x2)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5)
U4_gg(x1, x2, x3, x4) = U4_gg(x4)
append117_in_gag(x1, x2, x3) = append117_in_gag(x1, x3)
append117_out_gag(x1, x2, x3) = append117_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5) = U1_gag(x5)
U5_gg(x1, x2, x3, x4) = U5_gg(x1, x4)
U6_gg(x1, x2, x3, x4) = U6_gg(x4)
append232_in_agg(x1, x2, x3) = append232_in_agg(x2, x3)
append232_out_agg(x1, x2, x3) = append232_out_agg(x1)
U2_agg(x1, x2, x3, x4, x5) = U2_agg(x1, 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:
sublist1_in_gg([], T15) → sublist1_out_gg([], T15)
sublist1_in_gg(T5, .(X56, T26)) → U3_gg(T5, X56, T26, append117_in_aag(X57, X58, T26))
append117_in_aag([], T41, T41) → append117_out_aag([], T41, T41)
append117_in_aag(.(X97, X98), X99, .(X97, T44)) → U1_aag(X97, X98, X99, T44, append117_in_aag(X98, X99, T44))
U1_aag(X97, X98, X99, T44, append117_out_aag(X98, X99, T44)) → append117_out_aag(.(X97, X98), X99, .(X97, T44))
U3_gg(T5, X56, T26, append117_out_aag(X57, X58, T26)) → sublist1_out_gg(T5, .(X56, T26))
sublist1_in_gg(.(T65, T66), .(T65, T26)) → U4_gg(T65, T66, T26, append117_in_gag(T66, T35, T26))
append117_in_gag([], T41, T41) → append117_out_gag([], T41, T41)
append117_in_gag(.(X97, X98), X99, .(X97, T44)) → U1_gag(X97, X98, X99, T44, append117_in_gag(X98, X99, T44))
U1_gag(X97, X98, X99, T44, append117_out_gag(X98, X99, T44)) → append117_out_gag(.(X97, X98), X99, .(X97, T44))
U4_gg(T65, T66, T26, append117_out_gag(T66, T35, T26)) → sublist1_out_gg(.(T65, T66), .(T65, T26))
sublist1_in_gg(T73, .(X132, T26)) → U5_gg(T73, X132, T26, append117_in_aag(T74, T35, T26))
U5_gg(T73, X132, T26, append117_out_aag(T74, T35, T26)) → U6_gg(T73, X132, T26, append232_in_agg(X133, T73, T74))
append232_in_agg([], T81, T81) → append232_out_agg([], T81, T81)
append232_in_agg(.(X158, X159), T86, .(X158, T87)) → U2_agg(X158, X159, T86, T87, append232_in_agg(X159, T86, T87))
U2_agg(X158, X159, T86, T87, append232_out_agg(X159, T86, T87)) → append232_out_agg(.(X158, X159), T86, .(X158, T87))
U6_gg(T73, X132, T26, append232_out_agg(X133, T73, T74)) → sublist1_out_gg(T73, .(X132, T26))
The argument filtering Pi contains the following mapping:
sublist1_in_gg(x1, x2) = sublist1_in_gg(x1, x2)
[] = []
sublist1_out_gg(x1, x2) = sublist1_out_gg
.(x1, x2) = .(x1, x2)
U3_gg(x1, x2, x3, x4) = U3_gg(x4)
append117_in_aag(x1, x2, x3) = append117_in_aag(x3)
append117_out_aag(x1, x2, x3) = append117_out_aag(x1, x2)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5)
U4_gg(x1, x2, x3, x4) = U4_gg(x4)
append117_in_gag(x1, x2, x3) = append117_in_gag(x1, x3)
append117_out_gag(x1, x2, x3) = append117_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5) = U1_gag(x5)
U5_gg(x1, x2, x3, x4) = U5_gg(x1, x4)
U6_gg(x1, x2, x3, x4) = U6_gg(x4)
append232_in_agg(x1, x2, x3) = append232_in_agg(x2, x3)
append232_out_agg(x1, x2, x3) = append232_out_agg(x1)
U2_agg(x1, x2, x3, x4, x5) = U2_agg(x1, 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:
SUBLIST1_IN_GG(T5, .(X56, T26)) → U3_GG(T5, X56, T26, append117_in_aag(X57, X58, T26))
SUBLIST1_IN_GG(T5, .(X56, T26)) → APPEND117_IN_AAG(X57, X58, T26)
APPEND117_IN_AAG(.(X97, X98), X99, .(X97, T44)) → U1_AAG(X97, X98, X99, T44, append117_in_aag(X98, X99, T44))
APPEND117_IN_AAG(.(X97, X98), X99, .(X97, T44)) → APPEND117_IN_AAG(X98, X99, T44)
SUBLIST1_IN_GG(.(T65, T66), .(T65, T26)) → U4_GG(T65, T66, T26, append117_in_gag(T66, T35, T26))
SUBLIST1_IN_GG(.(T65, T66), .(T65, T26)) → APPEND117_IN_GAG(T66, T35, T26)
APPEND117_IN_GAG(.(X97, X98), X99, .(X97, T44)) → U1_GAG(X97, X98, X99, T44, append117_in_gag(X98, X99, T44))
APPEND117_IN_GAG(.(X97, X98), X99, .(X97, T44)) → APPEND117_IN_GAG(X98, X99, T44)
SUBLIST1_IN_GG(T73, .(X132, T26)) → U5_GG(T73, X132, T26, append117_in_aag(T74, T35, T26))
U5_GG(T73, X132, T26, append117_out_aag(T74, T35, T26)) → U6_GG(T73, X132, T26, append232_in_agg(X133, T73, T74))
U5_GG(T73, X132, T26, append117_out_aag(T74, T35, T26)) → APPEND232_IN_AGG(X133, T73, T74)
APPEND232_IN_AGG(.(X158, X159), T86, .(X158, T87)) → U2_AGG(X158, X159, T86, T87, append232_in_agg(X159, T86, T87))
APPEND232_IN_AGG(.(X158, X159), T86, .(X158, T87)) → APPEND232_IN_AGG(X159, T86, T87)
The TRS R consists of the following rules:
sublist1_in_gg([], T15) → sublist1_out_gg([], T15)
sublist1_in_gg(T5, .(X56, T26)) → U3_gg(T5, X56, T26, append117_in_aag(X57, X58, T26))
append117_in_aag([], T41, T41) → append117_out_aag([], T41, T41)
append117_in_aag(.(X97, X98), X99, .(X97, T44)) → U1_aag(X97, X98, X99, T44, append117_in_aag(X98, X99, T44))
U1_aag(X97, X98, X99, T44, append117_out_aag(X98, X99, T44)) → append117_out_aag(.(X97, X98), X99, .(X97, T44))
U3_gg(T5, X56, T26, append117_out_aag(X57, X58, T26)) → sublist1_out_gg(T5, .(X56, T26))
sublist1_in_gg(.(T65, T66), .(T65, T26)) → U4_gg(T65, T66, T26, append117_in_gag(T66, T35, T26))
append117_in_gag([], T41, T41) → append117_out_gag([], T41, T41)
append117_in_gag(.(X97, X98), X99, .(X97, T44)) → U1_gag(X97, X98, X99, T44, append117_in_gag(X98, X99, T44))
U1_gag(X97, X98, X99, T44, append117_out_gag(X98, X99, T44)) → append117_out_gag(.(X97, X98), X99, .(X97, T44))
U4_gg(T65, T66, T26, append117_out_gag(T66, T35, T26)) → sublist1_out_gg(.(T65, T66), .(T65, T26))
sublist1_in_gg(T73, .(X132, T26)) → U5_gg(T73, X132, T26, append117_in_aag(T74, T35, T26))
U5_gg(T73, X132, T26, append117_out_aag(T74, T35, T26)) → U6_gg(T73, X132, T26, append232_in_agg(X133, T73, T74))
append232_in_agg([], T81, T81) → append232_out_agg([], T81, T81)
append232_in_agg(.(X158, X159), T86, .(X158, T87)) → U2_agg(X158, X159, T86, T87, append232_in_agg(X159, T86, T87))
U2_agg(X158, X159, T86, T87, append232_out_agg(X159, T86, T87)) → append232_out_agg(.(X158, X159), T86, .(X158, T87))
U6_gg(T73, X132, T26, append232_out_agg(X133, T73, T74)) → sublist1_out_gg(T73, .(X132, T26))
The argument filtering Pi contains the following mapping:
sublist1_in_gg(x1, x2) = sublist1_in_gg(x1, x2)
[] = []
sublist1_out_gg(x1, x2) = sublist1_out_gg
.(x1, x2) = .(x1, x2)
U3_gg(x1, x2, x3, x4) = U3_gg(x4)
append117_in_aag(x1, x2, x3) = append117_in_aag(x3)
append117_out_aag(x1, x2, x3) = append117_out_aag(x1, x2)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5)
U4_gg(x1, x2, x3, x4) = U4_gg(x4)
append117_in_gag(x1, x2, x3) = append117_in_gag(x1, x3)
append117_out_gag(x1, x2, x3) = append117_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5) = U1_gag(x5)
U5_gg(x1, x2, x3, x4) = U5_gg(x1, x4)
U6_gg(x1, x2, x3, x4) = U6_gg(x4)
append232_in_agg(x1, x2, x3) = append232_in_agg(x2, x3)
append232_out_agg(x1, x2, x3) = append232_out_agg(x1)
U2_agg(x1, x2, x3, x4, x5) = U2_agg(x1, x5)
SUBLIST1_IN_GG(x1, x2) = SUBLIST1_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4) = U3_GG(x4)
APPEND117_IN_AAG(x1, x2, x3) = APPEND117_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x1, x5)
U4_GG(x1, x2, x3, x4) = U4_GG(x4)
APPEND117_IN_GAG(x1, x2, x3) = APPEND117_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5) = U1_GAG(x5)
U5_GG(x1, x2, x3, x4) = U5_GG(x1, x4)
U6_GG(x1, x2, x3, x4) = U6_GG(x4)
APPEND232_IN_AGG(x1, x2, x3) = APPEND232_IN_AGG(x2, x3)
U2_AGG(x1, x2, x3, x4, x5) = U2_AGG(x1, x5)
We have to consider all (P,R,Pi)-chains
(6) Obligation:
Pi DP problem:The TRS P consists of the following rules:
SUBLIST1_IN_GG(T5, .(X56, T26)) → U3_GG(T5, X56, T26, append117_in_aag(X57, X58, T26))
SUBLIST1_IN_GG(T5, .(X56, T26)) → APPEND117_IN_AAG(X57, X58, T26)
APPEND117_IN_AAG(.(X97, X98), X99, .(X97, T44)) → U1_AAG(X97, X98, X99, T44, append117_in_aag(X98, X99, T44))
APPEND117_IN_AAG(.(X97, X98), X99, .(X97, T44)) → APPEND117_IN_AAG(X98, X99, T44)
SUBLIST1_IN_GG(.(T65, T66), .(T65, T26)) → U4_GG(T65, T66, T26, append117_in_gag(T66, T35, T26))
SUBLIST1_IN_GG(.(T65, T66), .(T65, T26)) → APPEND117_IN_GAG(T66, T35, T26)
APPEND117_IN_GAG(.(X97, X98), X99, .(X97, T44)) → U1_GAG(X97, X98, X99, T44, append117_in_gag(X98, X99, T44))
APPEND117_IN_GAG(.(X97, X98), X99, .(X97, T44)) → APPEND117_IN_GAG(X98, X99, T44)
SUBLIST1_IN_GG(T73, .(X132, T26)) → U5_GG(T73, X132, T26, append117_in_aag(T74, T35, T26))
U5_GG(T73, X132, T26, append117_out_aag(T74, T35, T26)) → U6_GG(T73, X132, T26, append232_in_agg(X133, T73, T74))
U5_GG(T73, X132, T26, append117_out_aag(T74, T35, T26)) → APPEND232_IN_AGG(X133, T73, T74)
APPEND232_IN_AGG(.(X158, X159), T86, .(X158, T87)) → U2_AGG(X158, X159, T86, T87, append232_in_agg(X159, T86, T87))
APPEND232_IN_AGG(.(X158, X159), T86, .(X158, T87)) → APPEND232_IN_AGG(X159, T86, T87)
The TRS R consists of the following rules:
sublist1_in_gg([], T15) → sublist1_out_gg([], T15)
sublist1_in_gg(T5, .(X56, T26)) → U3_gg(T5, X56, T26, append117_in_aag(X57, X58, T26))
append117_in_aag([], T41, T41) → append117_out_aag([], T41, T41)
append117_in_aag(.(X97, X98), X99, .(X97, T44)) → U1_aag(X97, X98, X99, T44, append117_in_aag(X98, X99, T44))
U1_aag(X97, X98, X99, T44, append117_out_aag(X98, X99, T44)) → append117_out_aag(.(X97, X98), X99, .(X97, T44))
U3_gg(T5, X56, T26, append117_out_aag(X57, X58, T26)) → sublist1_out_gg(T5, .(X56, T26))
sublist1_in_gg(.(T65, T66), .(T65, T26)) → U4_gg(T65, T66, T26, append117_in_gag(T66, T35, T26))
append117_in_gag([], T41, T41) → append117_out_gag([], T41, T41)
append117_in_gag(.(X97, X98), X99, .(X97, T44)) → U1_gag(X97, X98, X99, T44, append117_in_gag(X98, X99, T44))
U1_gag(X97, X98, X99, T44, append117_out_gag(X98, X99, T44)) → append117_out_gag(.(X97, X98), X99, .(X97, T44))
U4_gg(T65, T66, T26, append117_out_gag(T66, T35, T26)) → sublist1_out_gg(.(T65, T66), .(T65, T26))
sublist1_in_gg(T73, .(X132, T26)) → U5_gg(T73, X132, T26, append117_in_aag(T74, T35, T26))
U5_gg(T73, X132, T26, append117_out_aag(T74, T35, T26)) → U6_gg(T73, X132, T26, append232_in_agg(X133, T73, T74))
append232_in_agg([], T81, T81) → append232_out_agg([], T81, T81)
append232_in_agg(.(X158, X159), T86, .(X158, T87)) → U2_agg(X158, X159, T86, T87, append232_in_agg(X159, T86, T87))
U2_agg(X158, X159, T86, T87, append232_out_agg(X159, T86, T87)) → append232_out_agg(.(X158, X159), T86, .(X158, T87))
U6_gg(T73, X132, T26, append232_out_agg(X133, T73, T74)) → sublist1_out_gg(T73, .(X132, T26))
The argument filtering Pi contains the following mapping:
sublist1_in_gg(x1, x2) = sublist1_in_gg(x1, x2)
[] = []
sublist1_out_gg(x1, x2) = sublist1_out_gg
.(x1, x2) = .(x1, x2)
U3_gg(x1, x2, x3, x4) = U3_gg(x4)
append117_in_aag(x1, x2, x3) = append117_in_aag(x3)
append117_out_aag(x1, x2, x3) = append117_out_aag(x1, x2)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5)
U4_gg(x1, x2, x3, x4) = U4_gg(x4)
append117_in_gag(x1, x2, x3) = append117_in_gag(x1, x3)
append117_out_gag(x1, x2, x3) = append117_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5) = U1_gag(x5)
U5_gg(x1, x2, x3, x4) = U5_gg(x1, x4)
U6_gg(x1, x2, x3, x4) = U6_gg(x4)
append232_in_agg(x1, x2, x3) = append232_in_agg(x2, x3)
append232_out_agg(x1, x2, x3) = append232_out_agg(x1)
U2_agg(x1, x2, x3, x4, x5) = U2_agg(x1, x5)
SUBLIST1_IN_GG(x1, x2) = SUBLIST1_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4) = U3_GG(x4)
APPEND117_IN_AAG(x1, x2, x3) = APPEND117_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x1, x5)
U4_GG(x1, x2, x3, x4) = U4_GG(x4)
APPEND117_IN_GAG(x1, x2, x3) = APPEND117_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5) = U1_GAG(x5)
U5_GG(x1, x2, x3, x4) = U5_GG(x1, x4)
U6_GG(x1, x2, x3, x4) = U6_GG(x4)
APPEND232_IN_AGG(x1, x2, x3) = APPEND232_IN_AGG(x2, x3)
U2_AGG(x1, x2, x3, x4, x5) = U2_AGG(x1, x5)
We have to consider all (P,R,Pi)-chains
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 10 less nodes.(8) Complex Obligation (AND)
(9) Obligation:
Pi DP problem:The TRS P consists of the following rules:
APPEND232_IN_AGG(.(X158, X159), T86, .(X158, T87)) → APPEND232_IN_AGG(X159, T86, T87)
The TRS R consists of the following rules:
sublist1_in_gg([], T15) → sublist1_out_gg([], T15)
sublist1_in_gg(T5, .(X56, T26)) → U3_gg(T5, X56, T26, append117_in_aag(X57, X58, T26))
append117_in_aag([], T41, T41) → append117_out_aag([], T41, T41)
append117_in_aag(.(X97, X98), X99, .(X97, T44)) → U1_aag(X97, X98, X99, T44, append117_in_aag(X98, X99, T44))
U1_aag(X97, X98, X99, T44, append117_out_aag(X98, X99, T44)) → append117_out_aag(.(X97, X98), X99, .(X97, T44))
U3_gg(T5, X56, T26, append117_out_aag(X57, X58, T26)) → sublist1_out_gg(T5, .(X56, T26))
sublist1_in_gg(.(T65, T66), .(T65, T26)) → U4_gg(T65, T66, T26, append117_in_gag(T66, T35, T26))
append117_in_gag([], T41, T41) → append117_out_gag([], T41, T41)
append117_in_gag(.(X97, X98), X99, .(X97, T44)) → U1_gag(X97, X98, X99, T44, append117_in_gag(X98, X99, T44))
U1_gag(X97, X98, X99, T44, append117_out_gag(X98, X99, T44)) → append117_out_gag(.(X97, X98), X99, .(X97, T44))
U4_gg(T65, T66, T26, append117_out_gag(T66, T35, T26)) → sublist1_out_gg(.(T65, T66), .(T65, T26))
sublist1_in_gg(T73, .(X132, T26)) → U5_gg(T73, X132, T26, append117_in_aag(T74, T35, T26))
U5_gg(T73, X132, T26, append117_out_aag(T74, T35, T26)) → U6_gg(T73, X132, T26, append232_in_agg(X133, T73, T74))
append232_in_agg([], T81, T81) → append232_out_agg([], T81, T81)
append232_in_agg(.(X158, X159), T86, .(X158, T87)) → U2_agg(X158, X159, T86, T87, append232_in_agg(X159, T86, T87))
U2_agg(X158, X159, T86, T87, append232_out_agg(X159, T86, T87)) → append232_out_agg(.(X158, X159), T86, .(X158, T87))
U6_gg(T73, X132, T26, append232_out_agg(X133, T73, T74)) → sublist1_out_gg(T73, .(X132, T26))
The argument filtering Pi contains the following mapping:
sublist1_in_gg(x1, x2) = sublist1_in_gg(x1, x2)
[] = []
sublist1_out_gg(x1, x2) = sublist1_out_gg
.(x1, x2) = .(x1, x2)
U3_gg(x1, x2, x3, x4) = U3_gg(x4)
append117_in_aag(x1, x2, x3) = append117_in_aag(x3)
append117_out_aag(x1, x2, x3) = append117_out_aag(x1, x2)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5)
U4_gg(x1, x2, x3, x4) = U4_gg(x4)
append117_in_gag(x1, x2, x3) = append117_in_gag(x1, x3)
append117_out_gag(x1, x2, x3) = append117_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5) = U1_gag(x5)
U5_gg(x1, x2, x3, x4) = U5_gg(x1, x4)
U6_gg(x1, x2, x3, x4) = U6_gg(x4)
append232_in_agg(x1, x2, x3) = append232_in_agg(x2, x3)
append232_out_agg(x1, x2, x3) = append232_out_agg(x1)
U2_agg(x1, x2, x3, x4, x5) = U2_agg(x1, x5)
APPEND232_IN_AGG(x1, x2, x3) = APPEND232_IN_AGG(x2, x3)
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:
APPEND232_IN_AGG(.(X158, X159), T86, .(X158, T87)) → APPEND232_IN_AGG(X159, T86, T87)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APPEND232_IN_AGG(x1, x2, x3) = APPEND232_IN_AGG(x2, x3)
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:
APPEND232_IN_AGG(T86, .(X158, T87)) → APPEND232_IN_AGG(T86, T87)
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:
- APPEND232_IN_AGG(T86, .(X158, T87)) → APPEND232_IN_AGG(T86, T87)
The graph contains the following edges 1 >= 1, 2 > 2
(15) YES
(16) Obligation:
Pi DP problem:The TRS P consists of the following rules:
APPEND117_IN_GAG(.(X97, X98), X99, .(X97, T44)) → APPEND117_IN_GAG(X98, X99, T44)
The TRS R consists of the following rules:
sublist1_in_gg([], T15) → sublist1_out_gg([], T15)
sublist1_in_gg(T5, .(X56, T26)) → U3_gg(T5, X56, T26, append117_in_aag(X57, X58, T26))
append117_in_aag([], T41, T41) → append117_out_aag([], T41, T41)
append117_in_aag(.(X97, X98), X99, .(X97, T44)) → U1_aag(X97, X98, X99, T44, append117_in_aag(X98, X99, T44))
U1_aag(X97, X98, X99, T44, append117_out_aag(X98, X99, T44)) → append117_out_aag(.(X97, X98), X99, .(X97, T44))
U3_gg(T5, X56, T26, append117_out_aag(X57, X58, T26)) → sublist1_out_gg(T5, .(X56, T26))
sublist1_in_gg(.(T65, T66), .(T65, T26)) → U4_gg(T65, T66, T26, append117_in_gag(T66, T35, T26))
append117_in_gag([], T41, T41) → append117_out_gag([], T41, T41)
append117_in_gag(.(X97, X98), X99, .(X97, T44)) → U1_gag(X97, X98, X99, T44, append117_in_gag(X98, X99, T44))
U1_gag(X97, X98, X99, T44, append117_out_gag(X98, X99, T44)) → append117_out_gag(.(X97, X98), X99, .(X97, T44))
U4_gg(T65, T66, T26, append117_out_gag(T66, T35, T26)) → sublist1_out_gg(.(T65, T66), .(T65, T26))
sublist1_in_gg(T73, .(X132, T26)) → U5_gg(T73, X132, T26, append117_in_aag(T74, T35, T26))
U5_gg(T73, X132, T26, append117_out_aag(T74, T35, T26)) → U6_gg(T73, X132, T26, append232_in_agg(X133, T73, T74))
append232_in_agg([], T81, T81) → append232_out_agg([], T81, T81)
append232_in_agg(.(X158, X159), T86, .(X158, T87)) → U2_agg(X158, X159, T86, T87, append232_in_agg(X159, T86, T87))
U2_agg(X158, X159, T86, T87, append232_out_agg(X159, T86, T87)) → append232_out_agg(.(X158, X159), T86, .(X158, T87))
U6_gg(T73, X132, T26, append232_out_agg(X133, T73, T74)) → sublist1_out_gg(T73, .(X132, T26))
The argument filtering Pi contains the following mapping:
sublist1_in_gg(x1, x2) = sublist1_in_gg(x1, x2)
[] = []
sublist1_out_gg(x1, x2) = sublist1_out_gg
.(x1, x2) = .(x1, x2)
U3_gg(x1, x2, x3, x4) = U3_gg(x4)
append117_in_aag(x1, x2, x3) = append117_in_aag(x3)
append117_out_aag(x1, x2, x3) = append117_out_aag(x1, x2)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5)
U4_gg(x1, x2, x3, x4) = U4_gg(x4)
append117_in_gag(x1, x2, x3) = append117_in_gag(x1, x3)
append117_out_gag(x1, x2, x3) = append117_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5) = U1_gag(x5)
U5_gg(x1, x2, x3, x4) = U5_gg(x1, x4)
U6_gg(x1, x2, x3, x4) = U6_gg(x4)
append232_in_agg(x1, x2, x3) = append232_in_agg(x2, x3)
append232_out_agg(x1, x2, x3) = append232_out_agg(x1)
U2_agg(x1, x2, x3, x4, x5) = U2_agg(x1, x5)
APPEND117_IN_GAG(x1, x2, x3) = APPEND117_IN_GAG(x1, x3)
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:
APPEND117_IN_GAG(.(X97, X98), X99, .(X97, T44)) → APPEND117_IN_GAG(X98, X99, T44)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APPEND117_IN_GAG(x1, x2, x3) = APPEND117_IN_GAG(x1, x3)
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:
APPEND117_IN_GAG(.(X97, X98), .(X97, T44)) → APPEND117_IN_GAG(X98, T44)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(21) 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:
- APPEND117_IN_GAG(.(X97, X98), .(X97, T44)) → APPEND117_IN_GAG(X98, T44)
The graph contains the following edges 1 > 1, 2 > 2
(22) YES
(23) Obligation:
Pi DP problem:The TRS P consists of the following rules:
APPEND117_IN_AAG(.(X97, X98), X99, .(X97, T44)) → APPEND117_IN_AAG(X98, X99, T44)
The TRS R consists of the following rules:
sublist1_in_gg([], T15) → sublist1_out_gg([], T15)
sublist1_in_gg(T5, .(X56, T26)) → U3_gg(T5, X56, T26, append117_in_aag(X57, X58, T26))
append117_in_aag([], T41, T41) → append117_out_aag([], T41, T41)
append117_in_aag(.(X97, X98), X99, .(X97, T44)) → U1_aag(X97, X98, X99, T44, append117_in_aag(X98, X99, T44))
U1_aag(X97, X98, X99, T44, append117_out_aag(X98, X99, T44)) → append117_out_aag(.(X97, X98), X99, .(X97, T44))
U3_gg(T5, X56, T26, append117_out_aag(X57, X58, T26)) → sublist1_out_gg(T5, .(X56, T26))
sublist1_in_gg(.(T65, T66), .(T65, T26)) → U4_gg(T65, T66, T26, append117_in_gag(T66, T35, T26))
append117_in_gag([], T41, T41) → append117_out_gag([], T41, T41)
append117_in_gag(.(X97, X98), X99, .(X97, T44)) → U1_gag(X97, X98, X99, T44, append117_in_gag(X98, X99, T44))
U1_gag(X97, X98, X99, T44, append117_out_gag(X98, X99, T44)) → append117_out_gag(.(X97, X98), X99, .(X97, T44))
U4_gg(T65, T66, T26, append117_out_gag(T66, T35, T26)) → sublist1_out_gg(.(T65, T66), .(T65, T26))
sublist1_in_gg(T73, .(X132, T26)) → U5_gg(T73, X132, T26, append117_in_aag(T74, T35, T26))
U5_gg(T73, X132, T26, append117_out_aag(T74, T35, T26)) → U6_gg(T73, X132, T26, append232_in_agg(X133, T73, T74))
append232_in_agg([], T81, T81) → append232_out_agg([], T81, T81)
append232_in_agg(.(X158, X159), T86, .(X158, T87)) → U2_agg(X158, X159, T86, T87, append232_in_agg(X159, T86, T87))
U2_agg(X158, X159, T86, T87, append232_out_agg(X159, T86, T87)) → append232_out_agg(.(X158, X159), T86, .(X158, T87))
U6_gg(T73, X132, T26, append232_out_agg(X133, T73, T74)) → sublist1_out_gg(T73, .(X132, T26))
The argument filtering Pi contains the following mapping:
sublist1_in_gg(x1, x2) = sublist1_in_gg(x1, x2)
[] = []
sublist1_out_gg(x1, x2) = sublist1_out_gg
.(x1, x2) = .(x1, x2)
U3_gg(x1, x2, x3, x4) = U3_gg(x4)
append117_in_aag(x1, x2, x3) = append117_in_aag(x3)
append117_out_aag(x1, x2, x3) = append117_out_aag(x1, x2)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5)
U4_gg(x1, x2, x3, x4) = U4_gg(x4)
append117_in_gag(x1, x2, x3) = append117_in_gag(x1, x3)
append117_out_gag(x1, x2, x3) = append117_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5) = U1_gag(x5)
U5_gg(x1, x2, x3, x4) = U5_gg(x1, x4)
U6_gg(x1, x2, x3, x4) = U6_gg(x4)
append232_in_agg(x1, x2, x3) = append232_in_agg(x2, x3)
append232_out_agg(x1, x2, x3) = append232_out_agg(x1)
U2_agg(x1, x2, x3, x4, x5) = U2_agg(x1, x5)
APPEND117_IN_AAG(x1, x2, x3) = APPEND117_IN_AAG(x3)
We have to consider all (P,R,Pi)-chains
(24) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(25) Obligation:
Pi DP problem:The TRS P consists of the following rules:
APPEND117_IN_AAG(.(X97, X98), X99, .(X97, T44)) → APPEND117_IN_AAG(X98, X99, T44)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APPEND117_IN_AAG(x1, x2, x3) = APPEND117_IN_AAG(x3)
We have to consider all (P,R,Pi)-chains
(26) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(27) Obligation:
Q DP problem:The TRS P consists of the following rules:
APPEND117_IN_AAG(.(X97, T44)) → APPEND117_IN_AAG(T44)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(28) 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:
- APPEND117_IN_AAG(.(X97, T44)) → APPEND117_IN_AAG(T44)
The graph contains the following edges 1 > 1