Left Termination of the query pattern
qs_in_2(g, a)
w.r.t. the given Prolog program could successfully be proven:
↳ Prolog
↳ PrologToPiTRSProof
Clauses:
qs([], []).
qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys)))).
part(X, .(Y, Xs), .(Y, Ls), Bs) :- ','(less(X, Y), part(X, Xs, Ls, Bs)).
part(X, .(Y, Xs), Ls, .(Y, Bs)) :- part(X, Xs, Ls, Bs).
part(X, [], [], []).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
less(0, s(X)).
less(s(X), s(Y)) :- less(X, Y).
Queries:
qs(g,a).
We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
qs_in: (b,f)
part_in: (b,b,f,f)
less_in: (b,b)
app_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2) = qs_in_ga(x1)
[] = []
qs_out_ga(x1, x2) = qs_out_ga(x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg
U9_gg(x1, x2, x3) = U9_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x2, x6)
part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4) = U4_ga(x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
U8_gga(x1, x2, x3, x4, x5) = U8_gga(x1, x5)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
Pi-finite rewrite system:
The TRS R consists of the following rules:
qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2) = qs_in_ga(x1)
[] = []
qs_out_ga(x1, x2) = qs_out_ga(x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg
U9_gg(x1, x2, x3) = U9_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x2, x6)
part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4) = U4_ga(x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
U8_gga(x1, x2, x3, x4, x5) = U8_gga(x1, x5)
Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:
QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
QS_IN_GA(.(X, Xs), Ys) → PART_IN_GGAA(X, Xs, Littles, Bigs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U9_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → APP_IN_GGA(Ls, .(X, Bs), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U8_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
The TRS R consists of the following rules:
qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2) = qs_in_ga(x1)
[] = []
qs_out_ga(x1, x2) = qs_out_ga(x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg
U9_gg(x1, x2, x3) = U9_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x2, x6)
part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4) = U4_ga(x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
U8_gga(x1, x2, x3, x4, x5) = U8_gga(x1, x5)
APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2)
QS_IN_GA(x1, x2) = QS_IN_GA(x1)
PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x2, x6)
U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6)
U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x2, x6)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4)
U8_GGA(x1, x2, x3, x4, x5) = U8_GGA(x1, x5)
LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x4, x5)
U4_GA(x1, x2, x3, x4) = U4_GA(x4)
U9_GG(x1, x2, x3) = U9_GG(x3)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x4, x5)
We have to consider all (P,R,Pi)-chains
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
Pi DP problem:
The TRS P consists of the following rules:
QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
QS_IN_GA(.(X, Xs), Ys) → PART_IN_GGAA(X, Xs, Littles, Bigs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U9_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GGAA(X, Xs, Ls, Bs)
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → APP_IN_GGA(Ls, .(X, Bs), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U8_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
The TRS R consists of the following rules:
qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2) = qs_in_ga(x1)
[] = []
qs_out_ga(x1, x2) = qs_out_ga(x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg
U9_gg(x1, x2, x3) = U9_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x2, x6)
part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4) = U4_ga(x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
U8_gga(x1, x2, x3, x4, x5) = U8_gga(x1, x5)
APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2)
QS_IN_GA(x1, x2) = QS_IN_GA(x1)
PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5, x6) = U6_GGAA(x2, x6)
U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6)
U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x2, x6)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4)
U8_GGA(x1, x2, x3, x4, x5) = U8_GGA(x1, x5)
LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x4, x5)
U4_GA(x1, x2, x3, x4) = U4_GA(x4)
U9_GG(x1, x2, x3) = U9_GG(x3)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x4, x5)
We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 4 SCCs with 9 less nodes.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
The TRS R consists of the following rules:
qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2) = qs_in_ga(x1)
[] = []
qs_out_ga(x1, x2) = qs_out_ga(x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg
U9_gg(x1, x2, x3) = U9_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x2, x6)
part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4) = U4_ga(x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
U8_gga(x1, x2, x3, x4, x5) = U8_gga(x1, x5)
APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2)
We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PiDP
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2)
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
APP_IN_GGA(.(X, Xs), Ys) → APP_IN_GGA(Xs, Ys)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- APP_IN_GGA(.(X, Xs), Ys) → APP_IN_GGA(Xs, Ys)
The graph contains the following edges 1 > 1, 2 >= 2
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
The TRS R consists of the following rules:
qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2) = qs_in_ga(x1)
[] = []
qs_out_ga(x1, x2) = qs_out_ga(x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg
U9_gg(x1, x2, x3) = U9_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x2, x6)
part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4) = U4_ga(x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
U8_gga(x1, x2, x3, x4, x5) = U8_gga(x1, x5)
LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2)
We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GGAA(X, Xs, Ls, Bs)
The TRS R consists of the following rules:
qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2) = qs_in_ga(x1)
[] = []
qs_out_ga(x1, x2) = qs_out_ga(x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg
U9_gg(x1, x2, x3) = U9_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x2, x6)
part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4) = U4_ga(x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
U8_gga(x1, x2, x3, x4, x5) = U8_gga(x1, x5)
PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6)
We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
U5_GGAA(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → PART_IN_GGAA(X, Xs, Ls, Bs)
PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) → U5_GGAA(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN_GGAA(X, Xs, Ls, Bs)
The TRS R consists of the following rules:
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg
U9_gg(x1, x2, x3) = U9_gg(x3)
PART_IN_GGAA(x1, x2, x3, x4) = PART_IN_GGAA(x1, x2)
U5_GGAA(x1, x2, x3, x4, x5, x6) = U5_GGAA(x1, x2, x3, x6)
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
PART_IN_GGAA(X, .(Y, Xs)) → PART_IN_GGAA(X, Xs)
U5_GGAA(X, Y, Xs, less_out_gg) → PART_IN_GGAA(X, Xs)
PART_IN_GGAA(X, .(Y, Xs)) → U5_GGAA(X, Y, Xs, less_in_gg(X, Y))
The TRS R consists of the following rules:
less_in_gg(0, s(X)) → less_out_gg
less_in_gg(s(X), s(Y)) → U9_gg(less_in_gg(X, Y))
U9_gg(less_out_gg) → less_out_gg
The set Q consists of the following terms:
less_in_gg(x0, x1)
U9_gg(x0)
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- PART_IN_GGAA(X, .(Y, Xs)) → U5_GGAA(X, Y, Xs, less_in_gg(X, Y))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3
- PART_IN_GGAA(X, .(Y, Xs)) → PART_IN_GGAA(X, Xs)
The graph contains the following edges 1 >= 1, 2 > 2
- U5_GGAA(X, Y, Xs, less_out_gg) → PART_IN_GGAA(X, Xs)
The graph contains the following edges 1 >= 1, 3 >= 2
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
Pi DP problem:
The TRS P consists of the following rules:
QS_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → QS_IN_GA(Littles, Ls)
U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → QS_IN_GA(Bigs, Bs)
The TRS R consists of the following rules:
qs_in_ga([], []) → qs_out_ga([], [])
qs_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) → U5_ggaa(X, Y, Xs, Ls, Bs, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U9_gg(X, Y, less_in_gg(X, Y))
U9_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U5_ggaa(X, Y, Xs, Ls, Bs, less_out_gg(X, Y)) → U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) → U7_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
part_in_ggaa(X, [], [], []) → part_out_ggaa(X, [], [], [])
U7_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) → part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) → U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) → U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) → U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U8_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U8_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) → qs_out_ga(.(X, Xs), Ys)
The argument filtering Pi contains the following mapping:
qs_in_ga(x1, x2) = qs_in_ga(x1)
[] = []
qs_out_ga(x1, x2) = qs_out_ga(x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
part_in_ggaa(x1, x2, x3, x4) = part_in_ggaa(x1, x2)
U5_ggaa(x1, x2, x3, x4, x5, x6) = U5_ggaa(x1, x2, x3, x6)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
s(x1) = s(x1)
less_out_gg(x1, x2) = less_out_gg
U9_gg(x1, x2, x3) = U9_gg(x3)
U6_ggaa(x1, x2, x3, x4, x5, x6) = U6_ggaa(x2, x6)
U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x2, x6)
part_out_ggaa(x1, x2, x3, x4) = part_out_ggaa(x3, x4)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x4, x5)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x4, x5)
U4_ga(x1, x2, x3, x4) = U4_ga(x4)
app_in_gga(x1, x2, x3) = app_in_gga(x1, x2)
app_out_gga(x1, x2, x3) = app_out_gga(x3)
U8_gga(x1, x2, x3, x4, x5) = U8_gga(x1, x5)
QS_IN_GA(x1, x2) = QS_IN_GA(x1)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x4, x5)
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
QS_IN_GA(.(X, Xs)) → U1_GA(X, part_in_ggaa(X, Xs))
U2_GA(X, Bigs, qs_out_ga(Ls)) → QS_IN_GA(Bigs)
U1_GA(X, part_out_ggaa(Littles, Bigs)) → U2_GA(X, Bigs, qs_in_ga(Littles))
U1_GA(X, part_out_ggaa(Littles, Bigs)) → QS_IN_GA(Littles)
The TRS R consists of the following rules:
qs_in_ga([]) → qs_out_ga([])
qs_in_ga(.(X, Xs)) → U1_ga(X, part_in_ggaa(X, Xs))
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg
less_in_gg(s(X), s(Y)) → U9_gg(less_in_gg(X, Y))
U9_gg(less_out_gg) → less_out_gg
U5_ggaa(X, Y, Xs, less_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X, []) → part_out_ggaa([], [])
U7_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
U1_ga(X, part_out_ggaa(Littles, Bigs)) → U2_ga(X, Bigs, qs_in_ga(Littles))
U2_ga(X, Bigs, qs_out_ga(Ls)) → U3_ga(X, Ls, qs_in_ga(Bigs))
U3_ga(X, Ls, qs_out_ga(Bs)) → U4_ga(app_in_gga(Ls, .(X, Bs)))
app_in_gga([], X) → app_out_gga(X)
app_in_gga(.(X, Xs), Ys) → U8_gga(X, app_in_gga(Xs, Ys))
U8_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))
U4_ga(app_out_gga(Ys)) → qs_out_ga(Ys)
The set Q consists of the following terms:
qs_in_ga(x0)
part_in_ggaa(x0, x1)
less_in_gg(x0, x1)
U9_gg(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1)
U6_ggaa(x0, x1)
U1_ga(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
app_in_gga(x0, x1)
U8_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
U2_GA(X, Bigs, qs_out_ga(Ls)) → QS_IN_GA(Bigs)
The remaining pairs can at least be oriented weakly.
QS_IN_GA(.(X, Xs)) → U1_GA(X, part_in_ggaa(X, Xs))
U1_GA(X, part_out_ggaa(Littles, Bigs)) → U2_GA(X, Bigs, qs_in_ga(Littles))
U1_GA(X, part_out_ggaa(Littles, Bigs)) → QS_IN_GA(Littles)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U8_gga(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
M( part_in_ggaa(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
M( U6_ggaa(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
M( .(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
M( app_in_gga(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
M( less_in_gg(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
M( U1_ga(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
M( U7_ggaa(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
M( part_out_ggaa(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
M( qs_out_ga(x1) ) = | | + | | · | x1 |
M( app_out_gga(x1) ) = | | + | | · | x1 |
M( U3_ga(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
M( U5_ggaa(x1, ..., x4) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
M( U2_ga(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
Tuple symbols:
M( QS_IN_GA(x1) ) = | 0 | + | | · | x1 |
M( U1_GA(x1, x2) ) = | 0 | + | | · | x1 | + | | · | x2 |
M( U2_GA(x1, ..., x3) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 |
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
U7_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
part_in_ggaa(X, []) → part_out_ggaa([], [])
U3_ga(X, Ls, qs_out_ga(Bs)) → U4_ga(app_in_gga(Ls, .(X, Bs)))
U4_ga(app_out_gga(Ys)) → qs_out_ga(Ys)
qs_in_ga([]) → qs_out_ga([])
U2_ga(X, Bigs, qs_out_ga(Ls)) → U3_ga(X, Ls, qs_in_ga(Bigs))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(Y, part_in_ggaa(X, Xs))
qs_in_ga(.(X, Xs)) → U1_ga(X, part_in_ggaa(X, Xs))
U5_ggaa(X, Y, Xs, less_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
U1_ga(X, part_out_ggaa(Littles, Bigs)) → U2_ga(X, Bigs, qs_in_ga(Littles))
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, less_in_gg(X, Y))
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
QS_IN_GA(.(X, Xs)) → U1_GA(X, part_in_ggaa(X, Xs))
U1_GA(X, part_out_ggaa(Littles, Bigs)) → U2_GA(X, Bigs, qs_in_ga(Littles))
U1_GA(X, part_out_ggaa(Littles, Bigs)) → QS_IN_GA(Littles)
The TRS R consists of the following rules:
qs_in_ga([]) → qs_out_ga([])
qs_in_ga(.(X, Xs)) → U1_ga(X, part_in_ggaa(X, Xs))
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg
less_in_gg(s(X), s(Y)) → U9_gg(less_in_gg(X, Y))
U9_gg(less_out_gg) → less_out_gg
U5_ggaa(X, Y, Xs, less_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X, []) → part_out_ggaa([], [])
U7_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
U1_ga(X, part_out_ggaa(Littles, Bigs)) → U2_ga(X, Bigs, qs_in_ga(Littles))
U2_ga(X, Bigs, qs_out_ga(Ls)) → U3_ga(X, Ls, qs_in_ga(Bigs))
U3_ga(X, Ls, qs_out_ga(Bs)) → U4_ga(app_in_gga(Ls, .(X, Bs)))
app_in_gga([], X) → app_out_gga(X)
app_in_gga(.(X, Xs), Ys) → U8_gga(X, app_in_gga(Xs, Ys))
U8_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))
U4_ga(app_out_gga(Ys)) → qs_out_ga(Ys)
The set Q consists of the following terms:
qs_in_ga(x0)
part_in_ggaa(x0, x1)
less_in_gg(x0, x1)
U9_gg(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1)
U6_ggaa(x0, x1)
U1_ga(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
app_in_gga(x0, x1)
U8_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
QS_IN_GA(.(X, Xs)) → U1_GA(X, part_in_ggaa(X, Xs))
U1_GA(X, part_out_ggaa(Littles, Bigs)) → QS_IN_GA(Littles)
The TRS R consists of the following rules:
qs_in_ga([]) → qs_out_ga([])
qs_in_ga(.(X, Xs)) → U1_ga(X, part_in_ggaa(X, Xs))
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg
less_in_gg(s(X), s(Y)) → U9_gg(less_in_gg(X, Y))
U9_gg(less_out_gg) → less_out_gg
U5_ggaa(X, Y, Xs, less_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X, []) → part_out_ggaa([], [])
U7_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
U1_ga(X, part_out_ggaa(Littles, Bigs)) → U2_ga(X, Bigs, qs_in_ga(Littles))
U2_ga(X, Bigs, qs_out_ga(Ls)) → U3_ga(X, Ls, qs_in_ga(Bigs))
U3_ga(X, Ls, qs_out_ga(Bs)) → U4_ga(app_in_gga(Ls, .(X, Bs)))
app_in_gga([], X) → app_out_gga(X)
app_in_gga(.(X, Xs), Ys) → U8_gga(X, app_in_gga(Xs, Ys))
U8_gga(X, app_out_gga(Zs)) → app_out_gga(.(X, Zs))
U4_ga(app_out_gga(Ys)) → qs_out_ga(Ys)
The set Q consists of the following terms:
qs_in_ga(x0)
part_in_ggaa(x0, x1)
less_in_gg(x0, x1)
U9_gg(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1)
U6_ggaa(x0, x1)
U1_ga(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
app_in_gga(x0, x1)
U8_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
QS_IN_GA(.(X, Xs)) → U1_GA(X, part_in_ggaa(X, Xs))
U1_GA(X, part_out_ggaa(Littles, Bigs)) → QS_IN_GA(Littles)
The TRS R consists of the following rules:
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, less_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X, []) → part_out_ggaa([], [])
U7_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
less_in_gg(0, s(X)) → less_out_gg
less_in_gg(s(X), s(Y)) → U9_gg(less_in_gg(X, Y))
U5_ggaa(X, Y, Xs, less_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
U9_gg(less_out_gg) → less_out_gg
The set Q consists of the following terms:
qs_in_ga(x0)
part_in_ggaa(x0, x1)
less_in_gg(x0, x1)
U9_gg(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1)
U6_ggaa(x0, x1)
U1_ga(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
app_in_gga(x0, x1)
U8_gga(x0, x1)
U4_ga(x0)
We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
qs_in_ga(x0)
U1_ga(x0, x1)
U2_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
app_in_gga(x0, x1)
U8_gga(x0, x1)
U4_ga(x0)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
QS_IN_GA(.(X, Xs)) → U1_GA(X, part_in_ggaa(X, Xs))
U1_GA(X, part_out_ggaa(Littles, Bigs)) → QS_IN_GA(Littles)
The TRS R consists of the following rules:
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, less_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X, []) → part_out_ggaa([], [])
U7_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
less_in_gg(0, s(X)) → less_out_gg
less_in_gg(s(X), s(Y)) → U9_gg(less_in_gg(X, Y))
U5_ggaa(X, Y, Xs, less_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
U9_gg(less_out_gg) → less_out_gg
The set Q consists of the following terms:
part_in_ggaa(x0, x1)
less_in_gg(x0, x1)
U9_gg(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1)
U6_ggaa(x0, x1)
We have to consider all (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
QS_IN_GA(.(X, Xs)) → U1_GA(X, part_in_ggaa(X, Xs))
The remaining pairs can at least be oriented weakly.
U1_GA(X, part_out_ggaa(Littles, Bigs)) → QS_IN_GA(Littles)
Used ordering: Polynomial interpretation [25]:
POL(.(x1, x2)) = 1 + x2
POL(0) = 0
POL(QS_IN_GA(x1)) = x1
POL(U1_GA(x1, x2)) = x2
POL(U5_ggaa(x1, x2, x3, x4)) = 1 + x3
POL(U6_ggaa(x1, x2)) = 1 + x2
POL(U7_ggaa(x1, x2)) = 1 + x2
POL(U9_gg(x1)) = 0
POL([]) = 0
POL(less_in_gg(x1, x2)) = 0
POL(less_out_gg) = 0
POL(part_in_ggaa(x1, x2)) = x2
POL(part_out_ggaa(x1, x2)) = x1
POL(s(x1)) = 0
The following usable rules [17] were oriented:
U5_ggaa(X, Y, Xs, less_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(Y, part_in_ggaa(X, Xs))
U7_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, less_in_gg(X, Y))
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
part_in_ggaa(X, []) → part_out_ggaa([], [])
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
U1_GA(X, part_out_ggaa(Littles, Bigs)) → QS_IN_GA(Littles)
The TRS R consists of the following rules:
part_in_ggaa(X, .(Y, Xs)) → U5_ggaa(X, Y, Xs, less_in_gg(X, Y))
part_in_ggaa(X, .(Y, Xs)) → U7_ggaa(Y, part_in_ggaa(X, Xs))
part_in_ggaa(X, []) → part_out_ggaa([], [])
U7_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(Ls, .(Y, Bs))
less_in_gg(0, s(X)) → less_out_gg
less_in_gg(s(X), s(Y)) → U9_gg(less_in_gg(X, Y))
U5_ggaa(X, Y, Xs, less_out_gg) → U6_ggaa(Y, part_in_ggaa(X, Xs))
U6_ggaa(Y, part_out_ggaa(Ls, Bs)) → part_out_ggaa(.(Y, Ls), Bs)
U9_gg(less_out_gg) → less_out_gg
The set Q consists of the following terms:
part_in_ggaa(x0, x1)
less_in_gg(x0, x1)
U9_gg(x0)
U5_ggaa(x0, x1, x2, x3)
U7_ggaa(x0, x1)
U6_ggaa(x0, x1)
We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.