YES Time: 0.217074 TRS: { bsort nil() -> nil(), bsort .(x, y) -> last .(bubble .(x, y), bsort butlast bubble .(x, y)), last nil() -> 0(), last .(x, nil()) -> x, last .(x, .(y, z)) -> last .(y, z), bubble nil() -> nil(), bubble .(x, nil()) -> .(x, nil()), bubble .(x, .(y, z)) -> if(<=(x, y), .(y, bubble .(x, z)), .(x, bubble .(y, z))), butlast nil() -> nil(), butlast .(x, nil()) -> nil(), butlast .(x, .(y, z)) -> .(x, butlast .(y, z))} DP: DP: { bsort# .(x, y) -> bsort# butlast bubble .(x, y), bsort# .(x, y) -> last# .(bubble .(x, y), bsort butlast bubble .(x, y)), bsort# .(x, y) -> bubble# .(x, y), bsort# .(x, y) -> butlast# bubble .(x, y), last# .(x, .(y, z)) -> last# .(y, z), bubble# .(x, .(y, z)) -> bubble# .(x, z), bubble# .(x, .(y, z)) -> bubble# .(y, z), butlast# .(x, .(y, z)) -> butlast# .(y, z)} TRS: { bsort nil() -> nil(), bsort .(x, y) -> last .(bubble .(x, y), bsort butlast bubble .(x, y)), last nil() -> 0(), last .(x, nil()) -> x, last .(x, .(y, z)) -> last .(y, z), bubble nil() -> nil(), bubble .(x, nil()) -> .(x, nil()), bubble .(x, .(y, z)) -> if(<=(x, y), .(y, bubble .(x, z)), .(x, bubble .(y, z))), butlast nil() -> nil(), butlast .(x, nil()) -> nil(), butlast .(x, .(y, z)) -> .(x, butlast .(y, z))} EDG: {(bsort# .(x, y) -> bubble# .(x, y), bubble# .(x, .(y, z)) -> bubble# .(y, z)) (bsort# .(x, y) -> bubble# .(x, y), bubble# .(x, .(y, z)) -> bubble# .(x, z)) (bubble# .(x, .(y, z)) -> bubble# .(x, z), bubble# .(x, .(y, z)) -> bubble# .(y, z)) (bubble# .(x, .(y, z)) -> bubble# .(x, z), bubble# .(x, .(y, z)) -> bubble# .(x, z)) (butlast# .(x, .(y, z)) -> butlast# .(y, z), butlast# .(x, .(y, z)) -> butlast# .(y, z)) (bsort# .(x, y) -> bsort# butlast bubble .(x, y), bsort# .(x, y) -> butlast# bubble .(x, y)) (bsort# .(x, y) -> bsort# butlast bubble .(x, y), bsort# .(x, y) -> bubble# .(x, y)) (bsort# .(x, y) -> bsort# butlast bubble .(x, y), bsort# .(x, y) -> last# .(bubble .(x, y), bsort butlast bubble .(x, y))) (bsort# .(x, y) -> bsort# butlast bubble .(x, y), bsort# .(x, y) -> bsort# butlast bubble .(x, y)) (bsort# .(x, y) -> last# .(bubble .(x, y), bsort butlast bubble .(x, y)), last# .(x, .(y, z)) -> last# .(y, z)) (bubble# .(x, .(y, z)) -> bubble# .(y, z), bubble# .(x, .(y, z)) -> bubble# .(x, z)) (bubble# .(x, .(y, z)) -> bubble# .(y, z), bubble# .(x, .(y, z)) -> bubble# .(y, z)) (last# .(x, .(y, z)) -> last# .(y, z), last# .(x, .(y, z)) -> last# .(y, z)) (bsort# .(x, y) -> butlast# bubble .(x, y), butlast# .(x, .(y, z)) -> butlast# .(y, z))} SCCS (4): Scc: {butlast# .(x, .(y, z)) -> butlast# .(y, z)} Scc: {bubble# .(x, .(y, z)) -> bubble# .(x, z), bubble# .(x, .(y, z)) -> bubble# .(y, z)} Scc: {last# .(x, .(y, z)) -> last# .(y, z)} Scc: {bsort# .(x, y) -> bsort# butlast bubble .(x, y)} SCC (1): Strict: {butlast# .(x, .(y, z)) -> butlast# .(y, z)} Weak: { bsort nil() -> nil(), bsort .(x, y) -> last .(bubble .(x, y), bsort butlast bubble .(x, y)), last nil() -> 0(), last .(x, nil()) -> x, last .(x, .(y, z)) -> last .(y, z), bubble nil() -> nil(), bubble .(x, nil()) -> .(x, nil()), bubble .(x, .(y, z)) -> if(<=(x, y), .(y, bubble .(x, z)), .(x, bubble .(y, z))), butlast nil() -> nil(), butlast .(x, nil()) -> nil(), butlast .(x, .(y, z)) -> .(x, butlast .(y, z))} SPSC: Simple Projection: pi(butlast#) = 0 Strict: {} Qed SCC (2): Strict: {bubble# .(x, .(y, z)) -> bubble# .(x, z), bubble# .(x, .(y, z)) -> bubble# .(y, z)} Weak: { bsort nil() -> nil(), bsort .(x, y) -> last .(bubble .(x, y), bsort butlast bubble .(x, y)), last nil() -> 0(), last .(x, nil()) -> x, last .(x, .(y, z)) -> last .(y, z), bubble nil() -> nil(), bubble .(x, nil()) -> .(x, nil()), bubble .(x, .(y, z)) -> if(<=(x, y), .(y, bubble .(x, z)), .(x, bubble .(y, z))), butlast nil() -> nil(), butlast .(x, nil()) -> nil(), butlast .(x, .(y, z)) -> .(x, butlast .(y, z))} POLY: Mode: weak, max_in=1, output_bits=-1, dnum=1, ur=true Interpretation: [if](x0, x1, x2) = x0 + 1, [.](x0, x1) = x0 + 1, [<=](x0, x1) = 0, [bsort](x0) = x0 + 1, [last](x0) = x0, [bubble](x0) = x0 + 1, [butlast](x0) = 0, [nil] = 1, [0] = 0, [bubble#](x0) = x0 Strict: bubble# .(x, .(y, z)) -> bubble# .(y, z) 2 + 0x + 0y + 1z >= 1 + 0y + 1z bubble# .(x, .(y, z)) -> bubble# .(x, z) 2 + 0x + 0y + 1z >= 1 + 0x + 1z Weak: Qed SCC (1): Strict: {last# .(x, .(y, z)) -> last# .(y, z)} Weak: { bsort nil() -> nil(), bsort .(x, y) -> last .(bubble .(x, y), bsort butlast bubble .(x, y)), last nil() -> 0(), last .(x, nil()) -> x, last .(x, .(y, z)) -> last .(y, z), bubble nil() -> nil(), bubble .(x, nil()) -> .(x, nil()), bubble .(x, .(y, z)) -> if(<=(x, y), .(y, bubble .(x, z)), .(x, bubble .(y, z))), butlast nil() -> nil(), butlast .(x, nil()) -> nil(), butlast .(x, .(y, z)) -> .(x, butlast .(y, z))} SPSC: Simple Projection: pi(last#) = 0 Strict: {} Qed SCC (1): Strict: {bsort# .(x, y) -> bsort# butlast bubble .(x, y)} Weak: { bsort nil() -> nil(), bsort .(x, y) -> last .(bubble .(x, y), bsort butlast bubble .(x, y)), last nil() -> 0(), last .(x, nil()) -> x, last .(x, .(y, z)) -> last .(y, z), bubble nil() -> nil(), bubble .(x, nil()) -> .(x, nil()), bubble .(x, .(y, z)) -> if(<=(x, y), .(y, bubble .(x, z)), .(x, bubble .(y, z))), butlast nil() -> nil(), butlast .(x, nil()) -> nil(), butlast .(x, .(y, z)) -> .(x, butlast .(y, z))} BOUND: Bound: top(-raise)-DP-bounded by 1 Automaton: { 0_0() -> 7* <=_1(7, 7) -> 15* <=_0(10, 10) -> 7* <=_0(10, 9) -> 7* <=_0(10, 8) -> 7* <=_0(10, 7) -> 7* <=_0(9, 10) -> 7* <=_0(9, 9) -> 7* <=_0(9, 8) -> 7* <=_0(9, 7) -> 7* <=_0(8, 10) -> 7* <=_0(8, 9) -> 7* <=_0(8, 8) -> 7* <=_0(8, 7) -> 7* <=_0(7, 10) -> 7* <=_0(7, 9) -> 7* <=_0(7, 8) -> 7* <=_0(7, 7) -> 7* if_1(15, 16, 17) -> 12* if_0(10, 10, 10) -> 7* if_0(10, 10, 9) -> 7* if_0(10, 10, 8) -> 7* if_0(10, 10, 7) -> 7* if_0(10, 9, 10) -> 7* if_0(10, 9, 9) -> 7* if_0(10, 9, 8) -> 7* if_0(10, 9, 7) -> 7* if_0(10, 8, 10) -> 7* if_0(10, 8, 9) -> 7* if_0(10, 8, 8) -> 7* if_0(10, 8, 7) -> 7* if_0(10, 7, 10) -> 7* if_0(10, 7, 9) -> 7* if_0(10, 7, 8) -> 7* if_0(10, 7, 7) -> 7* if_0(9, 10, 10) -> 7* if_0(9, 10, 9) -> 7* if_0(9, 10, 8) -> 7* if_0(9, 10, 7) -> 7* if_0(9, 9, 10) -> 7* if_0(9, 9, 9) -> 7* if_0(9, 9, 8) -> 7* if_0(9, 9, 7) -> 7* if_0(9, 8, 10) -> 7* if_0(9, 8, 9) -> 7* if_0(9, 8, 8) -> 7* if_0(9, 8, 7) -> 7* if_0(9, 7, 10) -> 7* if_0(9, 7, 9) -> 7* if_0(9, 7, 8) -> 7* if_0(9, 7, 7) -> 7* if_0(8, 10, 10) -> 7* if_0(8, 10, 9) -> 7* if_0(8, 10, 8) -> 7* if_0(8, 10, 7) -> 7* if_0(8, 9, 10) -> 7* if_0(8, 9, 9) -> 7* if_0(8, 9, 8) -> 7* if_0(8, 9, 7) -> 7* if_0(8, 8, 10) -> 7* if_0(8, 8, 9) -> 7* if_0(8, 8, 8) -> 7* if_0(8, 8, 7) -> 7* if_0(8, 7, 10) -> 7* if_0(8, 7, 9) -> 7* if_0(8, 7, 8) -> 7* if_0(8, 7, 7) -> 7* if_0(7, 10, 10) -> 7* if_0(7, 10, 9) -> 7* if_0(7, 10, 8) -> 7* if_0(7, 10, 7) -> 7* if_0(7, 9, 10) -> 7* if_0(7, 9, 9) -> 7* if_0(7, 9, 8) -> 7* if_0(7, 9, 7) -> 7* if_0(7, 8, 10) -> 7* if_0(7, 8, 9) -> 7* if_0(7, 8, 8) -> 9 | 7 if_0(7, 8, 7) -> 7* if_0(7, 7, 10) -> 7* if_0(7, 7, 9) -> 7* if_0(7, 7, 8) -> 7* if_0(7, 7, 7) -> 7* butlast_1(12) -> 13* butlast_0(10) -> 7* butlast_0(9) -> 10* butlast_0(8) -> 7* butlast_0(7) -> 7* bubble_1(11) -> 12* bubble_0(10) -> 7* bubble_0(9) -> 7* bubble_0(8) -> 9* bubble_0(7) -> 7* ._1(7, 14) -> 12* ._1(7, 12) -> 17 | 16 ._1(7, 7) -> 11* ._0(10, 10) -> 8* ._0(10, 9) -> 8* ._0(10, 8) -> 8* ._0(10, 7) -> 8* ._0(9, 10) -> 8* ._0(9, 9) -> 8* ._0(9, 8) -> 8* ._0(9, 7) -> 8* ._0(8, 10) -> 8* ._0(8, 9) -> 8* ._0(8, 8) -> 8* ._0(8, 7) -> 8* ._0(7, 10) -> 8* ._0(7, 9) -> 8* ._0(7, 8) -> 8* ._0(7, 7) -> 10 | 9 | 8 last_0(10) -> 7* last_0(9) -> 7* last_0(8) -> 7* last_0(7) -> 7* bsort#_1(13) -> 3* bsort#_0(10) -> 3* bsort_0(10) -> 7* bsort_0(9) -> 7* bsort_0(8) -> 7* bsort_0(7) -> 7* nil_1() -> 14 | 13 nil_0() -> 10 | 7 10 -> 7* 9 -> 7* 8 -> 7*} Strict: {} Qed