↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PrologToPiTRSProof
qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)
QS_IN(.(X, Xs), Ys) → U11(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
QS_IN(.(X, Xs), Ys) → PART_IN(X, Xs, Littles, Bigs)
PART_IN(X, .(Y, Xs), Ls, .(Y, Bs)) → U71(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
PART_IN(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN(X, Xs, Ls, Bs)
PART_IN(X, .(Y, Xs), .(Y, Ls), Bs) → U51(X, Y, Xs, Ls, Bs, less_in(X, Y))
PART_IN(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN(X, Y)
LESS_IN(s(X), s(Y)) → U91(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U51(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U61(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U51(X, Y, Xs, Ls, Bs, less_out(X, Y)) → PART_IN(X, Xs, Ls, Bs)
U11(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U21(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
U11(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → QS_IN(Littles, Ls)
U21(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U31(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U21(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → QS_IN(Bigs, Bs)
U31(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U41(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
U31(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → APP_IN(Ls, .(X, Bs), Ys)
APP_IN(.(X, Xs), Ys, .(X, Zs)) → U81(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)
qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ PrologToPiTRSProof
QS_IN(.(X, Xs), Ys) → U11(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
QS_IN(.(X, Xs), Ys) → PART_IN(X, Xs, Littles, Bigs)
PART_IN(X, .(Y, Xs), Ls, .(Y, Bs)) → U71(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
PART_IN(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN(X, Xs, Ls, Bs)
PART_IN(X, .(Y, Xs), .(Y, Ls), Bs) → U51(X, Y, Xs, Ls, Bs, less_in(X, Y))
PART_IN(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN(X, Y)
LESS_IN(s(X), s(Y)) → U91(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U51(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U61(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U51(X, Y, Xs, Ls, Bs, less_out(X, Y)) → PART_IN(X, Xs, Ls, Bs)
U11(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U21(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
U11(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → QS_IN(Littles, Ls)
U21(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U31(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U21(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → QS_IN(Bigs, Bs)
U31(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U41(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
U31(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → APP_IN(Ls, .(X, Bs), Ys)
APP_IN(.(X, Xs), Ys, .(X, Zs)) → U81(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)
qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)
qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ NonTerminationProof
↳ PiDP
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
APP_IN → APP_IN
APP_IN → APP_IN
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ NonTerminationProof
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
LESS_IN → LESS_IN
LESS_IN → LESS_IN
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PrologToPiTRSProof
PART_IN(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN(X, Xs, Ls, Bs)
U51(X, Y, Xs, Ls, Bs, less_out(X, Y)) → PART_IN(X, Xs, Ls, Bs)
PART_IN(X, .(Y, Xs), .(Y, Ls), Bs) → U51(X, Y, Xs, Ls, Bs, less_in(X, Y))
qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PrologToPiTRSProof
PART_IN(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN(X, Xs, Ls, Bs)
U51(X, Y, Xs, Ls, Bs, less_out(X, Y)) → PART_IN(X, Xs, Ls, Bs)
PART_IN(X, .(Y, Xs), .(Y, Ls), Bs) → U51(X, Y, Xs, Ls, Bs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ PiDP
↳ PrologToPiTRSProof
PART_IN → U51(less_in)
U51(less_out(X)) → PART_IN
PART_IN → PART_IN
less_in → U9(less_in)
less_in → less_out(0)
U9(less_out(X)) → less_out(s(X))
less_in
U9(x0)
PART_IN → U51(less_out(0))
PART_IN → U51(U9(less_in))
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ NonTerminationProof
↳ PiDP
↳ PrologToPiTRSProof
PART_IN → U51(less_out(0))
PART_IN → U51(U9(less_in))
U51(less_out(X)) → PART_IN
PART_IN → PART_IN
less_in → U9(less_in)
less_in → less_out(0)
U9(less_out(X)) → less_out(s(X))
less_in
U9(x0)
PART_IN → U51(less_out(0))
PART_IN → U51(U9(less_in))
U51(less_out(X)) → PART_IN
PART_IN → PART_IN
less_in → U9(less_in)
less_in → less_out(0)
U9(less_out(X)) → less_out(s(X))
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ PrologToPiTRSProof
U11(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U21(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
U11(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → QS_IN(Littles, Ls)
QS_IN(.(X, Xs), Ys) → U11(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
U21(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → QS_IN(Bigs, Bs)
qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ PrologToPiTRSProof
U21(qs_out) → QS_IN
U11(part_out) → QS_IN
U11(part_out) → U21(qs_in)
QS_IN → U11(part_in)
qs_in → U1(part_in)
part_in → part_out
part_in → U7(part_in)
part_in → U5(less_in)
less_in → U9(less_in)
less_in → less_out(0)
U9(less_out(X)) → less_out(s(X))
U5(less_out(X)) → U6(part_in)
U6(part_out) → part_out
U7(part_out) → part_out
U1(part_out) → U2(qs_in)
qs_in → qs_out
U2(qs_out) → U3(qs_in)
U3(qs_out) → U4(app_in)
app_in → U8(app_in)
app_in → app_out
U8(app_out) → app_out
U4(app_out) → qs_out
qs_in
part_in
less_in
U9(x0)
U5(x0)
U6(x0)
U7(x0)
U1(x0)
U2(x0)
U3(x0)
app_in
U8(x0)
U4(x0)
U11(part_out) → U21(U1(part_in))
U11(part_out) → U21(qs_out)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ PrologToPiTRSProof
U11(part_out) → QS_IN
U21(qs_out) → QS_IN
U11(part_out) → U21(U1(part_in))
QS_IN → U11(part_in)
U11(part_out) → U21(qs_out)
qs_in → U1(part_in)
part_in → part_out
part_in → U7(part_in)
part_in → U5(less_in)
less_in → U9(less_in)
less_in → less_out(0)
U9(less_out(X)) → less_out(s(X))
U5(less_out(X)) → U6(part_in)
U6(part_out) → part_out
U7(part_out) → part_out
U1(part_out) → U2(qs_in)
qs_in → qs_out
U2(qs_out) → U3(qs_in)
U3(qs_out) → U4(app_in)
app_in → U8(app_in)
app_in → app_out
U8(app_out) → app_out
U4(app_out) → qs_out
qs_in
part_in
less_in
U9(x0)
U5(x0)
U6(x0)
U7(x0)
U1(x0)
U2(x0)
U3(x0)
app_in
U8(x0)
U4(x0)
QS_IN → U11(U7(part_in))
QS_IN → U11(U5(less_in))
QS_IN → U11(part_out)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ NonTerminationProof
↳ PrologToPiTRSProof
QS_IN → U11(U7(part_in))
U21(qs_out) → QS_IN
U11(part_out) → QS_IN
QS_IN → U11(part_out)
U11(part_out) → U21(U1(part_in))
QS_IN → U11(U5(less_in))
U11(part_out) → U21(qs_out)
qs_in → U1(part_in)
part_in → part_out
part_in → U7(part_in)
part_in → U5(less_in)
less_in → U9(less_in)
less_in → less_out(0)
U9(less_out(X)) → less_out(s(X))
U5(less_out(X)) → U6(part_in)
U6(part_out) → part_out
U7(part_out) → part_out
U1(part_out) → U2(qs_in)
qs_in → qs_out
U2(qs_out) → U3(qs_in)
U3(qs_out) → U4(app_in)
app_in → U8(app_in)
app_in → app_out
U8(app_out) → app_out
U4(app_out) → qs_out
qs_in
part_in
less_in
U9(x0)
U5(x0)
U6(x0)
U7(x0)
U1(x0)
U2(x0)
U3(x0)
app_in
U8(x0)
U4(x0)
QS_IN → U11(U7(part_in))
U21(qs_out) → QS_IN
U11(part_out) → QS_IN
QS_IN → U11(part_out)
U11(part_out) → U21(U1(part_in))
QS_IN → U11(U5(less_in))
U11(part_out) → U21(qs_out)
qs_in → U1(part_in)
part_in → part_out
part_in → U7(part_in)
part_in → U5(less_in)
less_in → U9(less_in)
less_in → less_out(0)
U9(less_out(X)) → less_out(s(X))
U5(less_out(X)) → U6(part_in)
U6(part_out) → part_out
U7(part_out) → part_out
U1(part_out) → U2(qs_in)
qs_in → qs_out
U2(qs_out) → U3(qs_in)
U3(qs_out) → U4(app_in)
app_in → U8(app_in)
app_in → app_out
U8(app_out) → app_out
U4(app_out) → qs_out
qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)
QS_IN(.(X, Xs), Ys) → U11(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
QS_IN(.(X, Xs), Ys) → PART_IN(X, Xs, Littles, Bigs)
PART_IN(X, .(Y, Xs), Ls, .(Y, Bs)) → U71(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
PART_IN(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN(X, Xs, Ls, Bs)
PART_IN(X, .(Y, Xs), .(Y, Ls), Bs) → U51(X, Y, Xs, Ls, Bs, less_in(X, Y))
PART_IN(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN(X, Y)
LESS_IN(s(X), s(Y)) → U91(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U51(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U61(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U51(X, Y, Xs, Ls, Bs, less_out(X, Y)) → PART_IN(X, Xs, Ls, Bs)
U11(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U21(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
U11(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → QS_IN(Littles, Ls)
U21(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U31(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U21(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → QS_IN(Bigs, Bs)
U31(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U41(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
U31(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → APP_IN(Ls, .(X, Bs), Ys)
APP_IN(.(X, Xs), Ys, .(X, Zs)) → U81(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)
qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
QS_IN(.(X, Xs), Ys) → U11(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
QS_IN(.(X, Xs), Ys) → PART_IN(X, Xs, Littles, Bigs)
PART_IN(X, .(Y, Xs), Ls, .(Y, Bs)) → U71(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
PART_IN(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN(X, Xs, Ls, Bs)
PART_IN(X, .(Y, Xs), .(Y, Ls), Bs) → U51(X, Y, Xs, Ls, Bs, less_in(X, Y))
PART_IN(X, .(Y, Xs), .(Y, Ls), Bs) → LESS_IN(X, Y)
LESS_IN(s(X), s(Y)) → U91(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U51(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U61(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U51(X, Y, Xs, Ls, Bs, less_out(X, Y)) → PART_IN(X, Xs, Ls, Bs)
U11(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U21(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
U11(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → QS_IN(Littles, Ls)
U21(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U31(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U21(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → QS_IN(Bigs, Bs)
U31(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U41(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
U31(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → APP_IN(Ls, .(X, Bs), Ys)
APP_IN(.(X, Xs), Ys, .(X, Zs)) → U81(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)
qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
↳ PiDP
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)
qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PiDP
↳ PiDP
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ NonTerminationProof
↳ PiDP
↳ PiDP
↳ PiDP
APP_IN → APP_IN
APP_IN → APP_IN
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PiDP
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ NonTerminationProof
↳ PiDP
↳ PiDP
LESS_IN → LESS_IN
LESS_IN → LESS_IN
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
PART_IN(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN(X, Xs, Ls, Bs)
U51(X, Y, Xs, Ls, Bs, less_out(X, Y)) → PART_IN(X, Xs, Ls, Bs)
PART_IN(X, .(Y, Xs), .(Y, Ls), Bs) → U51(X, Y, Xs, Ls, Bs, less_in(X, Y))
qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
PART_IN(X, .(Y, Xs), Ls, .(Y, Bs)) → PART_IN(X, Xs, Ls, Bs)
U51(X, Y, Xs, Ls, Bs, less_out(X, Y)) → PART_IN(X, Xs, Ls, Bs)
PART_IN(X, .(Y, Xs), .(Y, Ls), Bs) → U51(X, Y, Xs, Ls, Bs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ PiDP
PART_IN → U51(less_in)
U51(less_out(X)) → PART_IN
PART_IN → PART_IN
less_in → U9(less_in)
less_in → less_out(0)
U9(less_out(X)) → less_out(s(X))
less_in
U9(x0)
PART_IN → U51(less_out(0))
PART_IN → U51(U9(less_in))
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ NonTerminationProof
↳ PiDP
PART_IN → U51(less_out(0))
PART_IN → U51(U9(less_in))
U51(less_out(X)) → PART_IN
PART_IN → PART_IN
less_in → U9(less_in)
less_in → less_out(0)
U9(less_out(X)) → less_out(s(X))
less_in
U9(x0)
PART_IN → U51(less_out(0))
PART_IN → U51(U9(less_in))
U51(less_out(X)) → PART_IN
PART_IN → PART_IN
less_in → U9(less_in)
less_in → less_out(0)
U9(less_out(X)) → less_out(s(X))
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
U11(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U21(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
U11(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → QS_IN(Littles, Ls)
QS_IN(.(X, Xs), Ys) → U11(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
U21(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → QS_IN(Bigs, Bs)
qs_in(.(X, Xs), Ys) → U1(X, Xs, Ys, part_in(X, Xs, Littles, Bigs))
part_in(X, [], [], []) → part_out(X, [], [], [])
part_in(X, .(Y, Xs), Ls, .(Y, Bs)) → U7(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
part_in(X, .(Y, Xs), .(Y, Ls), Bs) → U5(X, Y, Xs, Ls, Bs, less_in(X, Y))
less_in(s(X), s(Y)) → U9(X, Y, less_in(X, Y))
less_in(0, s(X)) → less_out(0, s(X))
U9(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U5(X, Y, Xs, Ls, Bs, less_out(X, Y)) → U6(X, Y, Xs, Ls, Bs, part_in(X, Xs, Ls, Bs))
U6(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), .(Y, Ls), Bs)
U7(X, Y, Xs, Ls, Bs, part_out(X, Xs, Ls, Bs)) → part_out(X, .(Y, Xs), Ls, .(Y, Bs))
U1(X, Xs, Ys, part_out(X, Xs, Littles, Bigs)) → U2(X, Xs, Ys, Bigs, qs_in(Littles, Ls))
qs_in([], []) → qs_out([], [])
U2(X, Xs, Ys, Bigs, qs_out(Littles, Ls)) → U3(X, Xs, Ys, Ls, qs_in(Bigs, Bs))
U3(X, Xs, Ys, Ls, qs_out(Bigs, Bs)) → U4(X, Xs, Ys, app_in(Ls, .(X, Bs), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U8(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U8(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U4(X, Xs, Ys, app_out(Ls, .(X, Bs), Ys)) → qs_out(.(X, Xs), Ys)
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
U21(qs_out) → QS_IN
U11(part_out) → QS_IN
U11(part_out) → U21(qs_in)
QS_IN → U11(part_in)
qs_in → U1(part_in)
part_in → part_out
part_in → U7(part_in)
part_in → U5(less_in)
less_in → U9(less_in)
less_in → less_out(0)
U9(less_out(X)) → less_out(s(X))
U5(less_out(X)) → U6(part_in)
U6(part_out) → part_out
U7(part_out) → part_out
U1(part_out) → U2(qs_in)
qs_in → qs_out
U2(qs_out) → U3(qs_in)
U3(qs_out) → U4(app_in)
app_in → U8(app_in)
app_in → app_out
U8(app_out) → app_out
U4(app_out) → qs_out
qs_in
part_in
less_in
U9(x0)
U5(x0)
U6(x0)
U7(x0)
U1(x0)
U2(x0)
U3(x0)
app_in
U8(x0)
U4(x0)
U11(part_out) → U21(U1(part_in))
U11(part_out) → U21(qs_out)
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
U11(part_out) → QS_IN
U21(qs_out) → QS_IN
U11(part_out) → U21(U1(part_in))
QS_IN → U11(part_in)
U11(part_out) → U21(qs_out)
qs_in → U1(part_in)
part_in → part_out
part_in → U7(part_in)
part_in → U5(less_in)
less_in → U9(less_in)
less_in → less_out(0)
U9(less_out(X)) → less_out(s(X))
U5(less_out(X)) → U6(part_in)
U6(part_out) → part_out
U7(part_out) → part_out
U1(part_out) → U2(qs_in)
qs_in → qs_out
U2(qs_out) → U3(qs_in)
U3(qs_out) → U4(app_in)
app_in → U8(app_in)
app_in → app_out
U8(app_out) → app_out
U4(app_out) → qs_out
qs_in
part_in
less_in
U9(x0)
U5(x0)
U6(x0)
U7(x0)
U1(x0)
U2(x0)
U3(x0)
app_in
U8(x0)
U4(x0)
QS_IN → U11(U7(part_in))
QS_IN → U11(U5(less_in))
QS_IN → U11(part_out)
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ NonTerminationProof
QS_IN → U11(U7(part_in))
U21(qs_out) → QS_IN
U11(part_out) → QS_IN
QS_IN → U11(part_out)
U11(part_out) → U21(U1(part_in))
QS_IN → U11(U5(less_in))
U11(part_out) → U21(qs_out)
qs_in → U1(part_in)
part_in → part_out
part_in → U7(part_in)
part_in → U5(less_in)
less_in → U9(less_in)
less_in → less_out(0)
U9(less_out(X)) → less_out(s(X))
U5(less_out(X)) → U6(part_in)
U6(part_out) → part_out
U7(part_out) → part_out
U1(part_out) → U2(qs_in)
qs_in → qs_out
U2(qs_out) → U3(qs_in)
U3(qs_out) → U4(app_in)
app_in → U8(app_in)
app_in → app_out
U8(app_out) → app_out
U4(app_out) → qs_out
qs_in
part_in
less_in
U9(x0)
U5(x0)
U6(x0)
U7(x0)
U1(x0)
U2(x0)
U3(x0)
app_in
U8(x0)
U4(x0)
QS_IN → U11(U7(part_in))
U21(qs_out) → QS_IN
U11(part_out) → QS_IN
QS_IN → U11(part_out)
U11(part_out) → U21(U1(part_in))
QS_IN → U11(U5(less_in))
U11(part_out) → U21(qs_out)
qs_in → U1(part_in)
part_in → part_out
part_in → U7(part_in)
part_in → U5(less_in)
less_in → U9(less_in)
less_in → less_out(0)
U9(less_out(X)) → less_out(s(X))
U5(less_out(X)) → U6(part_in)
U6(part_out) → part_out
U7(part_out) → part_out
U1(part_out) → U2(qs_in)
qs_in → qs_out
U2(qs_out) → U3(qs_in)
U3(qs_out) → U4(app_in)
app_in → U8(app_in)
app_in → app_out
U8(app_out) → app_out
U4(app_out) → qs_out